A study of spatio-temporal spread of infectious disease: SARS€¦ · mental epidemic models to...

A Study of Spatio-Temporal Spread of

Infectious Disease:SARS

Afia Naheed

A thesis submitted in fulfilment of requirements for the Degree of

Doctor of Philosophy

Mathematics Department

Faculty of Science Engineering and Technology

Swinburne University of Technology

Australia

2015

Dedicated

To my loving parents and husband, my source of strength and inspiration

i

Abstract

This thesis is based on using three different types of deterministic compart-

mental epidemic models to investigate the transmission dynamics of Severe Acute

Respiratory Syndrome (SARS). These models are represented by ordinary and

partial differential equations, with the inclusion of a reaction-diffusion system. The

first model assumes Susceptible-Exposed-Infected-Diagnosed-Recovered (SEIJR)

populations, whereas the second and third models are extensions of the first model,

with treatment and quarantine compartments added. For different initial popula-

tion distributions the system of differential equations, representing different com-

partments, has been solved in the presence of diffusion in the first three compart-

ments i.e susceptible, exposed and infected. In this study the effects of diffusion on

SARS transmission are investigated as are the effects of some intervention strate-

gies. It is shown that diffusion and initial population distribution play crucial roles

in disease transmission. Then, the same system is solved numerically with diffusion

in the susceptible, exposed and infected compartments and with cross-diffusion in

the susceptible and exposed compartments for different cases. Using clinical and

demographic information for SARS, the SEIJR model is further extended to

a Susceptible-Exposed-Infected-Diagnosed-Treated-Recovered (SEIJTR) model.

The SEIJTR model’s parameters are estimated, again using available field data

on the 2003 SARS epidemic in Hong Kong. After that, model parameters are

analysed for sensitivity and uncertainty. Three different techniques are used to per-

form the sensitivity analysis. The effect of the treatment compartment on SARS

transmission is then numerically studied. Stability analyses of steady state and

treatment-reduced basic reproduction number are performed. Studies show that

availability of treatment can reduce infection significantly. Finally, a quarantine

compartment is added to the SEIJTR model in order to study the effects of quar-

antine on SARS transmission. The results for this extended SEQIJTR model are

compared with those for the SEIJTR model in which only isolation and treatment

but no quarantine measures, are used as intervention. The investigations show that

the presence of quarantine measures effectively reduces disease transmission.

ii

Acknowledgements

I would like to express the deepest appreciation to my coordinating supervisor

Dr. Manmohan Singh for his extreme kindness, extraordinary guidance, intellec-

tual and moral support through out all stages of my doctorate. I am very grateful

to him for introducing me to exciting subject of Mathematical Biology. Without

his thoughtful attention, vast knowledge and persistent help this thesis would not

have been possible.

I’m also grateful to my associate supervisors Dr. David Lucy and Mr. David

Richards for their thoughtful attention, guidance and invaluable suggestions. Their

constructive criticism, valuable comments and constant encouragement have re-

markably improved the quality of my thesis. I have become a stronger writer with

their kind help in editing my written pieces. My sincere thanks goes to Dr. Md

Samsuzzoha for his help and fruitful discussions during the completion of this work.

My special gratitude to Professor Geoffrey Brooks, Professor Billy Todd, and other

staff from Mathematics Department, Faculty of Science, Engineering and Technol-

ogy, for their kind help and friendly attitude during the preparation and completion

of the thesis.

I would like to thank Swinburne University of Technology for providing me Swin-

burne University Postgraduate Research Award (SUPRA) to carry out PhD. It

has been challenging, enjoyable and worthwhile time. I am deeply indebted to the

Faculty of Science Engineering and Technology for their technological support.

My sincere thanks to my amazing friends and coleagues for their love, care, advice,

and support through out my doctorate. I would like to share a special thanks to

Afsana Ahmed, Fatemeh Mekanic, Hou Wen, Qudsia Arooj, Rashida Bashir, Shab-

nam Sabah and Wajeehah Aayeshah for always being available for me and offering

the positive words and advice to keep me afloat .

Finally and most importantly, I would like to extend my gratitude to my wonder-

ful family (regular and new). Thank you mama and papa for your love, unlimited

support and inspiration of all my goals in life. You taught me to love in silence

and instilled in me the strength and confidence to continue in the way of success

to make my dreams come true. A big thanks to my siblings who have always been

iii

the best friends to me and my inlaws who gave me a home away from home. I have

no words to thanks my dear husband for his love, support and encouragement in

the journey of making this work possible.

iv

Declaration

The candidate hereby declares that the work in this thesis, presented for the Degree

of Doctorate in Mathematics submitted in the faculty of Science, Engineering and

Technology, Swinburne University of Technology:

1. is that the candidate alone and has not been submitted previously, in whole

or in part, in respect of any other academic award and has not been published

in any form by other person except where due references are given, and

2. has been carried out during the period from March 2011 to February 2015

under the supervision of Dr. Manmohan Singh, Dr. David Lucy and Mr.

David Richards.

—————————

Afia Naheed Date: February, 2015

v

Contents

1 Summary of the thesis 1

1.1 Research overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Literature Survey 6

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Mathematical epidemiology a brief look at history . . . . . . . . . . 9

2.3 The case of severe acute respiratory syndrome . . . . . . . . . . . . 17

2.3.1 The outbreak . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.2 SARS virus . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.3 Symptoms of SARS . . . . . . . . . . . . . . . . . . . . . . 22

2.3.4 Transmission route . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.5 Mathematical modeling of SARS . . . . . . . . . . . . . . . 24

3 Numerical Study of SARS Model with the Inclusion of Diffusion

in the System 28

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 The SEIJR epidemic model . . . . . . . . . . . . . . . . . . . . . . 29

3.2.1 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.2 Initial and boundary conditions . . . . . . . . . . . . . . . . 30

3.3 Numerical scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4.1 Reproduction number and disease-free equilibrium (DFE) . 34

3.4.2 Disease-free equilibrium and stability analysis . . . . . . . . 36

3.4.3 Stability of endemic equilibrium without diffusion . . . . . . 37

vi

3.4.4 Stability of endemic equilibrium with diffusion . . . . . . . . 38

3.4.5 Excited mode and bifurcation value . . . . . . . . . . . . . . 40

3.5 Numerical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.5.1 Solutions of SEIJR model without diffusion (Case 1) . . . . 42

3.5.2 Solutions of SEIJR model with diffusion (Case 1) . . . . . 47

3.5.3 Other cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4 Numerical Simulation of Cross Diffusion on Transmission Dynam-

ics of SARS 57

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2 The SEIJR epidemic model . . . . . . . . . . . . . . . . . . . . . . 59

4.2.1 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59



4.3.1 Reproduction number and disease-free equilibrium (DFE) . 62

4.3.2 Stability of endemic equilibrium with cross-diffusion . . . . . 63

4.3.3 Reproduction number with diffusion . . . . . . . . . . . . . 65




4.5.1 Numerical solution for initial condition (i) . . . . . . . . . . 69

4.5.2 Numerical solution for initial condition (ii) . . . . . . . . . . 72

4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5 Parameter Estimation with Uncertainty and Sensitivity Analysis

for the SARS Outbreak in Hong Kong 86

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.2 Model formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.2.1 Reproduction number RIT for SARS . . . . . . . . . . . . . 92

5.2.2 Epidemiological data . . . . . . . . . . . . . . . . . . . . . . 92

5.2.3 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . 92

vii

5.2.4 Validation statistics . . . . . . . . . . . . . . . . . . . . . . . 94

5.3 Uncertainty and sensitivity analysis . . . . . . . . . . . . . . . . . . 97

5.3.1 Sensitivity indices of RIT . . . . . . . . . . . . . . . . . . . 99

5.3.2 Partial rank correlation coefficient (PRCC) . . . . . . . . . 101

5.3.3 Factor prioritization by reduction of variance . . . . . . . . . 107

5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6 Numerical Study of SARS Model with Treatment (SEIJTR) and

Diffusion in the System 112

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.2 SEIJTR epidemic model . . . . . . . . . . . . . . . . . . . . . . . 113

6.2.1 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113




6.4.1 Disease-free equilibrium (DFE) . . . . . . . . . . . . . . . . 121

6.4.2 Endemic equilibrium without diffusion . . . . . . . . . . . . 122

6.4.3 Endemic equilibrium with diffusion . . . . . . . . . . . . . . 124



6.5.1 Solutions of SEIJTR model in the absence of diffusion (Case

1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.5.2 Solutions of SEIJTR model in the presence of diffusion

(Case 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.5.3 Other cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

7 Simulating the Effect of Quarantine on Isolation Treatment Model

for SARS Epidemic 144

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

7.2 The SEQIJTR epidemic model . . . . . . . . . . . . . . . . . . . . 147

7.2.1 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

viii


7.2.3 Numerical scheme . . . . . . . . . . . . . . . . . . . . . . . . 152

7.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

7.3.1 Reproduction number without diffusion . . . . . . . . . . . . 154

7.3.2 Disease-free equilibrium (DFE) . . . . . . . . . . . . . . . 155

7.3.3 Endemic equilibrium without diffusion . . . . . . . . . . . . 157

7.3.4 Endemic equilibrium with diffusion . . . . . . . . . . . . . . 159

7.3.5 Reproduction number with diffusion . . . . . . . . . . . . . 161



7.4.1 Numerical solution without diffusion . . . . . . . . . . . . . 166

7.4.2 Numerical solution with diffusion . . . . . . . . . . . . . . . 170

7.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

8 Conclusions 177

Appendix i

List of Publications and Conference Presentations xxxiv

ix

List of Figures

2.1 Emerging and re-emerging infectious diseases in the world. Red de-

notes newly emerging diseases, blue, re-emerging/resurging diseases,

black, a deliberately emerging disease [69]. . . . . . . . . . . . . . . 7

2.2 Annual deaths worldwide due to infectious diseases [236]. . . . . . 8

2.3 Sever acute respiratory syndrome (SARS) a deadly threat [124]. . . 19

2.4 Recent patient of SARS-like disease [123]. . . . . . . . . . . . . . . 20

2.5 SARS corona-virus under the microscope [122]. . . . . . . . . . . . 21

2.6 Symptoms of SARS [117]. . . . . . . . . . . . . . . . . . . . . . . . 22

2.7 Transmission route of SARS virus [119]. . . . . . . . . . . . . . . . 23

3.1 Initial conditions (i)− (iv). . . . . . . . . . . . . . . . . . . . . . . 32

3.2 Determination of first excited mode with β as an unknown parameter. 41

3.3 Solutions with initial condition (i) and without diffusion. . . . . . . 43

3.4 Solutions with initial condition (ii) and without diffusion. . . . . . . 44

3.5 Solutions with initial condition (iii) and without diffusion. . . . . . 45

3.6 Solutions with initial condition (iv) and without diffusion. . . . . . 46

3.7 Solutions with initial condition (i) and with diffusion. . . . . . . . . 47

3.8 Solutions with initial condition (ii) and with diffusion. . . . . . . . 48

3.9 Solutions with initial condition (iii) and with diffusion. . . . . . . . 49

3.10 Solutions with initial condition (iv) and with diffusion. . . . . . . . 50

4.1 Initial Conditions (i) and (ii). . . . . . . . . . . . . . . . . . . . . . 62

4.2 Determination of first excited mode with β as an unknown parameter. 67

4.3 Solutions with initial condition (i) for Case (a). . . . . . . . . . . . 72

4.4 Solutions with initial condition (i) for Case (b). . . . . . . . . . . . 73

x

4.5 Solutions with initial condition (i) for Case (c). . . . . . . . . . . . 74

4.6 Solutions with initial condition (i) for Case (d). . . . . . . . . . . . 75

4.7 Solutions with initial condition (ii) for Case (a). . . . . . . . . . . . 78

4.8 Solutions with initial condition (ii) for Case (b). . . . . . . . . . . . 79

4.9 Solutions with initial condition (ii) for Case (c). . . . . . . . . . . . 80

4.10 Solutions with initial condition (ii) for Case (d). . . . . . . . . . . . 81

5.1 SARS infected incidence data, Hong Kong 2003. . . . . . . . . . . . 93

5.2 Predicted model of SARS. . . . . . . . . . . . . . . . . . . . . . . . 96

5.3 The Model fitted to the data for the infected individuals : Initial (a)

and final fit (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.4 Residual (a) and residual correlation (b). . . . . . . . . . . . . . . . 97

5.5 Plots of probability distributions for all parameters generated with

10, 000 sample size. . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.6 Scatter plots for the basic reproduction number and eight sampled

input parameters values with 10, 000 random samples. . . . . . . . . 103

5.7 Scatter plots for the basic reproduction number and eight sampled

input parameters values with 10, 000 LHS samples. . . . . . . . . . 104

5.8 PRCCs for the full range of parameters from Table 5.6 for LHSb1 =

10, 000 (a) and RSa1 = 10, 000 (b). . . . . . . . . . . . . . . . . . . 105

5.9 PRCCs for the full range of parameters from Table 5.6 LHSb2 =

20, 000 (a) and RSa2 = 20, 000 (b). . . . . . . . . . . . . . . . . . . 106

5.10 Pie chart of factor prioritization sensitivity indices LHSb1 and RSa1. 108

5.11 Pie chart of factor prioritization sensitivity indices LHSb2 and RSa2. 109

6.1 Initial conditions (i) and (ii). . . . . . . . . . . . . . . . . . . . . . 118

6.2 Initial condition (iii). . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.3 Determination of first excited mode with β as an unknown parameter.127

6.4 Solutions for initial condition (i) without diffusion. . . . . . . . . . 130

6.5 Solutions for initial condition (ii) without diffusion. . . . . . . . . . 132

6.6 Solutions for initial condition (iii) without diffusion. . . . . . . . . . 133

6.7 Solutions for initial condition (i) with diffusion. . . . . . . . . . . . 135

xi

6.8 Solutions for initial condition (ii) with diffusion. . . . . . . . . . . . 136

6.9 Solutions for initial condition (iii) with diffusion. . . . . . . . . . . 138

7.1 Initial conditions (i) and (ii). . . . . . . . . . . . . . . . . . . . . . 152

7.2 Determination of first excited mode with β as an unknown parameter.165

7.3 Solutions for initial condition (i) without diffusion. . . . . . . . . . 168

7.4 Solutions for initial condition (ii) without diffusion. . . . . . . . . . 169

7.5 Solutions for initial condition (i) with diffusion. . . . . . . . . . . . 172

7.6 Solutions for initial condition (ii) with diffusion. . . . . . . . . . . . 173

1 Bifurcation diagram for β without diffusion for Case(1)-(4) . . . . . x

2 Bifurcation diagram for β with diffusion Case(1)-(4) . . . . . . . . . xi

3 Bifurcation diagram for γ1 without diffusion Case(1)-(4) . . . . . . xi

4 Bifurcation diagram for γ1 with diffusion Case(1)-(4) . . . . . . . . xii

5 Bifurcation diagram for γ2 without diffusion Case(1)-(4) . . . . . . xii

6 Bifurcation diagram for γ2 with diffusion Case(1)-(4) . . . . . . . . xiii

7 Bifurcation diagram for β for Case(a)-(d) . . . . . . . . . . . . . . . xv

8 Bifurcation diagram for γ1 for Case(a)-(d) . . . . . . . . . . . . . . xvi

9 Bifurcation diagram for γ2 for Case(a)-(d) . . . . . . . . . . . . . . xvi

10 Bifurcation diagrams for without diffusion . . . . . . . . . . . . . . xxvi

11 Bifurcation diagrams with diffusion . . . . . . . . . . . . . . . . . . xxvii

12 Bifurcation diagrams without diffusion . . . . . . . . . . . . . . . . xxxii

13 Bifurcation diagrams with diffusion . . . . . . . . . . . . . . . . . . xxxiii

xii

List of Tables

3.1 Interpretation of parameters (per day) . . . . . . . . . . . . . . . . 30

3.2 Values of LHS Routh-Hurwitz criterion of equilibrium without dif-

fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 Values of LHS Routh-Hurwitz criterion of equilibrium with diffusion 40

3.4 Bifurcation value of β . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.5 Bifurcation values of γ1 andγ2 . . . . . . . . . . . . . . . . . . . . . 41

3.6 Four cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.7 Peak values of susceptible (S) and exposed (E) (without diffusion) 52

3.8 Peak values of infected (I) and recovered (R) (without diffusion) . . 52

3.9 Peak values of susceptible (S) and exposed (E) (with diffusion) . . 53

3.10 Peak values of infected (I) and recovered (R) (with diffusion) . . . 53

3.11 Peak values of infected at t = 20 . . . . . . . . . . . . . . . . . . . 54

4.1 Interpretation of parameters (per day) . . . . . . . . . . . . . . . . 60

4.2 Cases for cross-diffusion . . . . . . . . . . . . . . . . . . . . . . . . 62

4.3 Routh-Hurwitz criteria with and without cross-diffusion . . . . . . . 65

4.4 Reproduction number . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.5 Bifurcation value of β, γ1 and γ2 with cross-diffusion . . . . . . . . 67

5.1 Biological definition of parameters and state variables . . . . . . . . 91

5.2 Estimated parameters value for model . . . . . . . . . . . . . . . . 95

5.3 Covariance relations among parameters of SEIJTR model . . . . . 99

5.4 Parameters’ sensitivity analysis . . . . . . . . . . . . . . . . . . . . 100

5.5 Probability distribution functions (PDF ) for parameters . . . . . . 102

5.6 Estimates of partial rank correlation coefficients . . . . . . . . . . . 105

xiii

5.7 Percentage values of sensitivity index based on reduction of variance 108

6.1 Biological definition of parameters . . . . . . . . . . . . . . . . . . . 115

6.2 Parameters’ value for model . . . . . . . . . . . . . . . . . . . . . . 116

6.3 Values of LHS of Routh-Hurwitz criteria of equilibrium without dif-

fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.4 Values of LHS of Routh-hurwitz criteria of equilibrium with diffusion126

6.5 Bifurcation values of β and α. . . . . . . . . . . . . . . . . . . . . . 127

6.6 Peak values of susceptible(S), exposed(E), infective(I) and recovered(R)

(without diffusion) . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.7 Peak values of susceptible(S), exposed(E), infective(I) and recovered(R)

(with diffusion) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.8 Basic reproduction number RIT . . . . . . . . . . . . . . . . . . . . 142

7.1 Biological definition of parameters . . . . . . . . . . . . . . . . . . 149

7.2 Parameters values for model . . . . . . . . . . . . . . . . . . . . . . 150

7.3 Values for Routh-Hurwitz criteria of equilibrium . . . . . . . . . . . 159

7.4 Value of reproduction number (without diffusion) . . . . . . . . . . 164

7.5 Bifurcation value of influential parameters . . . . . . . . . . . . . . 165

7.6 Peak values for initial condition (ii) without diffusion . . . . . . . . 167

7.7 Peak values for initial condition (ii) without diffusion . . . . . . . . 171

xiv

Chapter 1

Summary of the thesis

1.1 Research overview

The main aim of this thesis is to understand the spatio-temporal spread of Severe

Acute Respiratory Syndrome (SARS) through the numerical study of the factors

related to the transmission dynamics of the disease in order to develop effective

control measures and strategies for its better control. SARS is an infectious disease

caused by a corona virus, called SARS associated corona virus (SARS − Cov).

At first it was thought to be transmitted by close person-to-person contact most

readily by droplet and airborne spread [210], but later many studies showed that

it can also spread through faecal-oral transmission [191, 226]. It was first reported

in Asia in February 2003 and with-in the next few months, the illness spread to

more than two dozen countries in North America, South America, Europe and

Asia. According to the World Health Organization (WHO) [238], a total of 8, 450

people worldwide became sick with SARS during the 2003 outbreak and of these,

774 died. In 2004 and 2005 many separate outbreaks emerged in Taiwan, China

and Singapore due to the accidental release of virus from laboratories or infected

animals. Recently, in 2012, a SARS−like disease named afterwards as MERS or

Middle East Respiratory Syndrome was reported [248]. All this shows the danger

of appearance of such deadly disease anytime in future. There is no specific treat-

ment or vaccine available for SARS till now. So to control any future outbreak

of such infectious disease, it is crucial to understand its transmission dynamics

1

and factors involved in aggravating its transmission. Mathematical modelling of

SARS has played a crucial role in understanding and predicting the transmission

of SARS for different parts of world. This has also helped in applying medical and

non medical intervention in order to control not only SARS, but also any deadly

new emerging infectious disease [31, 39, 40, 67, 86, 102, 153, 191, 101, 212, 233,

231, 241, 243, 249, 239, 233, 240, 245, 241, 239, 240, 245].

The focus of this thesis is to study the transmission dynamics of SARS epidemic

as well as the efficiency of different control strategies. Different initial conditions

are used to study the disease transmission with different population distributions.

The operator splitting method has been used to solve the differential equations

governing the system under investigation. The bifurcation effects for the most sen-

sitive parameters in all models have been investigated. In order to investigate the

effects of cross and diffusion on transmission dynamics, the differential equations

have been solved with and without diffusion in the system. Different cases have

been constructed in order to investigate the medical (treatment) and non medical

(quarantine, diagnoses and isolation) interventions for the disease. The nature of

the periodic solutions in case of cross diffusion have been studied. In order to study

the effect of quarantine, isolation and treatment, three different models with the

addition of three different classes or compartment J (diagnosed and isolated), T

(treatment) and Q (quarantined) in the basic SEIR model have been formulated.

The main threshold parameter, the reproduction number, R0 is calculated for all

models. R0 is calculated both with self and cross diffusion for SEIJR model. In

order to re-estimate the parameters for SEIJTR model, field data from the Hong

Kong SARS epidemic 2003, has been used.

Main aims of this thesis are to:

• investigate the effect of diagnosis and isolation on the spread of SARS,

• study the effects of self and cross diffusion on the transmission dynamics of

the SARS outbreak 2003,

• estimate the parameters for the SARS epidemic in Hong Kong based on field

data,

2

• determine the sensitivity and uncertainty in the model parameters in order

to examine the influence of parameters on the spread of disease,

• study the effects of treatment in controlling SARS disease,

• investigate the effects of quarantine in controlling SARS disease.

MATHEMATICA, MATLAB and Sampling and Sensitivity Analysis Tools (SaSAT )

are the softwares that have been used for the numerical calculations. LATEX and

WinEdt are used for the preparation of this thesis.

1.2 Thesis outline

This thesis is divided into eight chapters and an appendix.

Main aims of this thesis are:

• Chapter 1. Summary of the Thesis

This chapter contains an introduction to the work done in this thesis.

• Chapter 2. Literature Survey

This chapter reviews the history of mathematical biology with mathemati-

cal modelling of infectious diseases. A review of Severe Acute Respiratory

Syndrome (SARS) epidemic and mathematical models that represent the

transmission dynamics of SARS is given here.

• Chapter 3. Numerical Study of SARS Epidemic Model with the

Inclusion of Diffusion in the System

This chapter discusses the numerical study of population model based on

the epidemics of Severe Acute Respiratory Syndrome (SARS). The SEIJR

(Susceptible, Exposed, Infected, Diagnosed, Recovered) model of SARS epi-

demic is considered with net inflow of individuals into a region. Transmission

of the disease is analyzed by solving the system of differential equations using

numerical methods with different initial population distributions. The effect

of diffusion on the spread of disease is examined. Stability is established for

3

the numerical solutions. The reproduction numbers for all possible cases with

and without diffusion are estimated and compared. Effects of interventions

(medical and non medical) are also analyzed.

• Chapter 4. Numerical Simulation of Cross-Diffusion on Transmission

Dynamics of SARS Epidemic

This chapter describes the numerical study of spatial distribution dynam-

ics of SARS epidemic under the influence of self and cross-diffusion using

an SEIJR (Susceptible, Exposed, Infected, Diagnosed, Recovered) compart-

mental model. For both self-diffusion and cross-diffusion, the nonlinear par-

tial differential system is solved using the operator splitting and forward

difference technique. Criteria for local stability are obtained and the effect of

cross-diffusion on stability is also analysed.

• Chapter 5. Parameter Estimation with Uncertainty and Sensitivity

Analysis for the SARS Outbreak in Hong Kong

Parameter estimation is a key issue in systems biology, as it represents the

crucial step to obtain predictions from computational models of biological

systems. Based on the SARS epidemic of 2003 the parameters are esti-

mated using Runge-Kutta (Dormand-Prince pairs) and Least squares meth-

ods. Graphical and numerical techniques are used to validate the estimates.

The effect of the model parameters on the dynamics of the disease is examined

using sensitivity and uncertainty analysis.

• Chapter 6. Numerical Study of SARS Epidemic Model with Treat-

ment (SEIJTR) and Diffusion in the System

A numerical study of Severe Acute Respiratory Syndrome (SARS) using an

open population model consisting of six compartments i.e SEIJTR (Sus-

ceptible, Exposed, Infected, Diagnosed, Treated, Recovered) has been done

in this chapter. Different population distributions are used to generate the

numerical simulations for the disease in different situations. Stability of the

system is analysed. In order to observe the effect on disease transmission,

different cases are studied in the absence and presence of diffusion in the

4

system. The impact of treatment on the transmission of the disease is also

analysed

• Chapter 7. Simulating the Effect of Quarantine on Isolation Treat-

ment Model for SARS Epidemic

This chapter describes a compartmental model to predict the geographic

spread of infectious diseases in the presence of quarantine measures. The

model described in Chapter 6 with the inclusion of quarantine compartment is

numerically studied, in this chapter. Stability of the disease free and endemic

equilibria are analysed with the help of the Routh-Hurwitz criterion. The

reproduction number in the presence and absence of diffusion and quarantine

is estimated and analysed. Finally, the numerical simulations of the present

model are compared with the SEIJTR model discussed in Chapter 6.

• Chapter 8. Conclusions

This chapter concludes with the crucial findings and suggests opportunities

for future work.

5

Chapter 2

Literature Survey

2.1 Introduction

Through out human history, infectious diseases have been a serious cause of mor-

bidity and mortality. According to historians with-in 50 years of 1500, more than

half of the population of the American region reduced as a result of the spread

of infectious diseases [209]. Although the presence of infectious diseases is not ig-

norable at all times in human population but the epidemics are more noticeable

and striking. One of such major epidemic in history was Black Death, that caused

25 million causalities in Europe in 14th century [44]. According to Meltzer [168]

“It is often said that in the centuries after Columbus landed in the New World on

12 October 1492, more native Americans died each year from infectious diseases

brought by the European settlers than were born.” Small pox was the first disease

to appear in America in 1518 in Hispaniola, from where the disease spread rapidly

to Mexico in 1520 and then to the whole world. In the years of 1525 − 26, this

disease killed half of the population in Aztecs, Guatemala and the territories of

Incas. A few years later another infectious disease, influenza attracted the world’s

attention through a massive epidemic in 1558 − 59. With-in one year in 1919, 20

million people died due to influenza world wide [8]. Other than these epidemics,

typhus caused half a million deaths from 1918 to 1921 in Russia.

Although in the beginning of 21st century with the appearance of different med-

ical and non-medical interventions like vaccines, antibiotics, early quarantine and

6

Figure 2.1: Emerging and re-emerging infectious diseases in the world. Red denotes

newly emerging diseases, blue, re-emerging/resurging diseases, black, a deliberately

emerging disease [69].

isolation of infected, reduced the danger of infectious diseases and the world’s atten-

tion diverted to chronic diseases like diabetes, cancer, arthritis and cardiovascular

disease, in developed countries. On the other hand, infectious diseases not only con-

tinued to cause sufferings in developing countries but also their agents developed

and re developed, as a result many infectious diseases emerged, re-emerged and de-

liberately emerged as shown in Fig. 2.1). In a short period of nineteen years from

1975 to 1993, the world came across many deadly new infectious disease like Lyme

(1975), Legionnaires (1976) toxic-shock syndrome (1978), the sexually transmitted

disease Acquired Immunodeficiency Syndrome (AIDS) (1981), hepatitis C (1989),

hepatitis E (1990) and hantavirus (1993). Many diseases such as gonorrhea (sexu-

ally transmitted diseases), tuberculosis, pneumonia, malaria, dengue, yellow fever

reappeared while the epidemics of cholera, hemorrhagic fevers (Bolivian, Ebola,

Lassa, Marburg, etc.) [108] and SARS emerged as new infections. After bacte-

ria, helminths (worms), viruses and protozoa, a new agent called prions has joined

the group. Prions are considered to be the cause of spongiform encephalopathies

such as bovine Creutzfeldt-Jakob Disease (CJD), scabies in sheep, bovine spongi-

7

Figure 2.2: Annual deaths worldwide due to infectious diseases [236].

form encephalopathy (BSE, mad cow disease) and kuru [182]. The pandemics of

influenza in the years of 1918, 1957, 1968, 1977 and 2009 and Middle East respira-

tory syndrome (MERS) are a few of the infectious diseases that still threaten the

world.

The rate of mortality due to infectious diseases is higher in developing countries

[94], specifically as a major cause of child death in developing areas of the world. A

rough estimate of a study shows that three million children die every year because of

diarrhoeal diseases and malaria alone [236]. According to Morens [172] “ About 15

million (> 25%) of 57 million annual deaths worldwide are estimated to be related

directly to infectious diseases; this figure does not include the additional millions

of deaths that occur as a consequence of past infections (for example, streptococcal

rheumatic heart disease), or because of complications associated with chronic in-

fections, such as liver failure and hepatocellular carcinoma in people infected with

hepatitis B or C viruses.” Fig 2.2 gives a brief overview of worldwide causes of

deaths where infectious disease contribute to a major number of deaths. In this

situation, in addition to medicines epidemiological models are the most helpful

tools to improve the efficiency of control strategies by understanding the under

lying mechanism of infectious diseases and finally, can wipe out the infection from

population. The present chapter outlines the history of infectious diseases and his-

tory of mathematical modelling for disease dynamics. It also describes the story of

Sever Acute Respiratory Syndrome (SARS), the fatal epidemic which appeared in

2003 in the context of mathematical modelling.

8

2.2 Mathematical epidemiology a brief look at

history

MacMahon and Pugh [157] defined epidemiology as “ The study of the distribution

and determinants of disease frequency in man.” Epidemiology is a blend of four

objectives given as follows [109]:

• First objective is to define dispersion of disease in real world.

• Second objective is to determine the causes, risk factors and complexities

that effect the disease dynamics.

• Third objective is to develop and test theories.

• Fourth objective is to develop, maintain and apply control measures.

An infection in a community can be studied in the best possible way with the

help of its epidemiological information using mathematical models. According to

Nokes [179] “A major goal of theoretical or mathematical study in epidemiology

is to develop understanding of the interplay between the variables that determine

the course of infection within an individual, and the variables that control the pat-

tern of infections within communities of people. In view of the successes achieved

by combining empirical and theoretical work in the physical sciences, it is sur-

prising that many people still question the potential usefulness of mathematical

models in epidemiology.” If we look at the literature mathematical epidemiology

can be traced back more than two hundred years. the Great Plague of London

(1665− 1666) is said to be the first epidemic, that was investigated through mod-

elling techniques while in the 18th century a small pox model to measure the effect

of variolation for common public health was constructed and estimated by Daniel

Bernoulli [22] in 1760. This was followed by a long gap till the middle of 19th

century when the popular dynamical systems techniques were used and the deter-

ministic epidemiology era started. In 1840, Dr. William Farr [68] fitted a normal

curve model to small pox induced mortality data (1837-1839) for Wales and Eng-

land. The work done by Enko [55] between 1873 and 1894 is considered as the first

9

step to develop modern mathematical epidemiology. Dr John Brownlee continued

Farr’s work and developed theories for epidemic systems. He published excellent

work on epidemics, where in the beginning he relied on normal curves but after

analysing many epidemics he found that Pearson Frequency Distribution fits bet-

ter for epidemics. According to Brownlee “An epidemic is an organic phenomenon,

the course of which seems to depend on the acquisition by an organism of a high

grade of infectivity at the point where the epidemic starts, this infectivity being

lost from that period till the end of the epidemic at a rate approaching to the terms

of a geometrical progression.” His work is considered as an important contribution

to the statistical approach in mathematical modelling [28, 29, 30]. Mathematical

epidemiology is used to study the mechanism of transmission of infectious diseases.

There are several ways of transfer of infections for these diseases. Some of them

are transferred by viral agents such as influenza, ebola, measles, severe acute res-

piratory syndrome, rubella (German measles), rabies and chicken pox. Some are

transmitted through bacteria, for example tuberculosis, pneumonia, meningitis,

and gonorrhea while some are transferred through vectors such as West Nile virus

malaria, dengue, chagas disease, lyme disease and yellow fever etc.

According to O’Neill [181] “The models for infectious disease transmission essen-

tially subdivide into two categories, namely deterministic models and stochastic

models. The former are frequently defined via a system of ordinary or partial dif-

ferential equations, which describe how the numbers or proportions of individuals

in different states (susceptible, infective, etc.) evolve through time. An attractive

feature of deterministic models is that it is usually fairly straight forward to obtain

numerical solutions for a given set of parameter values. They are generally most

effective as descriptions of reality in large populations, where, roughly speaking,

laws of large numbers act to reduce the order of stochastic effects. Stochastic mod-

els are usually thought of as more realistic, although their mathematical analysis

is often much harder. They can capture the stochasticity seen in real-life disease

outbreaks, for example, the phenomenon of fade-out in endemic diseases. They

are generally defined at the level of individuals, for instance, specifying probability

distributions that describe the latent or infectious periods.” The most used branch

10

of deterministic modelling is compartmental models, where a population is divided

in number of compartments based on their epidemiological characteristics. Accord-

ing to Sattenspiel [206] “The compartmental models are mathematical models for

the spread of disease. The process of building a mathematical model begins with

a series of assumptions about how the disease process works and the development

of a simplified model to describe the process. As a consequence of the analysis

of the initial model, comparison with actual disease data, and evaluation of the

assumptions, the model is reformulated in a more realistic manner. In general, the

process of mathematical modeling proceeds from simple to complex.”

Deterministic epidemiological modelling based on compartmental models flourished

immensely between 1900 and 1935 with the research work of Sir Ross, Hamer, McK-

endrick and Kermack. Hamer [100] published his work on diseases transmission.

He formulated a simple mathematical model to study the transmission dynamics

of measles. He introduced the idea of “mass action principal” that is considered

as basic concept for epidemic models nowadays. Inspired by Hamer, Dr Ronald

Ross became interested in the prevalence and control of malaria and he started

working, in 1899, on (bird) malaria where he extended Hammer’s discrete time-

model to continuous-time model in order to study transmission of disease. In 1904,

Ross [194] published his paper about the irregular movements of mosquitoes. Ross

published many papers describing malaria transmission dynamics [195, 196]. In

these papers he developed his “theory of happenings” that is considered as the

basis of modern epidemic theory. Numerous epidemiological models were then de-

veloped by Ross and Hudson, Martini and Lotka [12, 53]. Based on the work of

Hamer and Ross, Kermack and McKendrick developed a theory for the spread of

infection through population and published three articles in 1927, 1932, and 1933

respectively [134, 135, 136] based on simple a compartmental model to study the

progress of an epidemic in homogenous close and open population and developed

the threshold theory. They explained that threshold value of a particular epidemic

depends on infectivity, recovery and death rates for that epidemic and also the

population density should exceed a certain threshold value for a diseases to spread

and become epidemic. This theory gave rise to the vaccination with an impact on

11

immunity, proving that in order to eliminate an infection, it is not necessary to

vaccinate the whole population. This was proved in the case of small pox in 1970,

where eradication of the virus was achieved with only 80% vaccine coverage world-

wide. They also explained that small change in the rate of infection might cause

a huge epidemic. Following the work of Hamer, Soper [215] published his work in

1929 on the periodicity of epidemics. He explained that the reemergence of measles

in any certain region depends on two major factors. First, the dreadful virus of the

disease and secondly, the number of susceptible population for that disease. All

these mathematical models were based on Hamer’s “mass action principal”.

Models on an alternative technique of this principal was developed by Lowell Reed

and Wade Hampton Frost [2, 159] in 1920 and was published in 1950. According

to Sattenspiel [206] “In this model, the transmission of infection is defined in terms

of a probability of effective contact rather than a proportion of contacts that result

in transmission. Effective contact is the type of contact necessary for transmission

of infection, not necessarily just casual contact.” Starting with Reed Frost [2, 159]

the importance of stochastic models for the transmission dynamics of infectious

diseases has been analysed in many ways while Bailey [12] and Becker [18] pro-

vided many examples of stochastic versions of SIR models.

Epidemiological modelling grew significantly after 20th century tremendously. The

first edition of Bailey’s book in 1957 [12] is considered as a significant milestone in

this regard. His scholarly review “The Mathematical Theory of Infectious Diseases”

documented 539 research papers on mathematical epidemiology in its bibliography

written between 1900 and 1973 and 336 of these articles were published in 1964-

1973. Anderson and May [8] made major contributions to the understanding of

population biology in infectious diseases. They investigated the relations between

transmission parameters of their models. They explained the direct and indirect

transmission of microparasites (viruses, bacteria and protozoa) and macroparasites

(helminths and arthropods) through intermediate hosts. A huge variety of mathe-

matical models for infectious diseases were developed in the 21st century. Most of

these models explained the concept of steady loss of vaccination, disease vectors,

mixing of population, age structure, spacial spread, self and cross-diffusion, pas-

12

sive immunity, disease acquired immunity, quarantine, isolation, stage of infection

chemotherapy and vector transmission [17, 32]. According to Hethcote [109] “An

advantage of mathematical modeling of infectious diseases is the economy, clarity

and precision of a mathematical formulation. A model using difference, differential,

integral or functional differential equations is not ambiguous or vague. Of course,

the parameters must be defined precisely and each term in the equations must be

explained in terms of mechanisms, but the resulting model is a definitive statement

of the basic principles involved. Once the mathematical formulation is complete,

there are many mathematical techniques available for determining the threshold,

equilibrium, periodic solutions, and their local and global stability. Thus the full

power of mathematics is available for the analysis of the equations. Moreover,

information about the model can also be obtained by numerical simulation on dig-

ital computers of the equations describing the model. The mathematical analyses

and computer simulations can identify important combinations of parameters and

essential aspects or variables in the model. In order to choose and use epidemio-

logical modeling effectively on specific diseases, one must understand the behavior

of the available formulations and the implications of choosing a particular formula-

tion. Thus mathematical epidemiology provides a foundation for the applications

[106, 107].”

Since 1983 HIV and AIDS are the two main sexually transmitted diseases. Ac-

cording to estimates in 2002, 45 million population got infected from AIDS and

HIV epidemics. Each year, there are almost five million infected and three mil-

lion deaths are recorded in Africa. In the United Kingdom syphilis, chlamydia

and gonorrhea have been reported in increasing rates in the last ten years [133].

This alarming situation drew the attention of epidemiologist to model the trans-

mission dynamics of these sexually transmitted diseases in order to find effective

control measures. On the other hand due to the lack of efficient computers in

early times mathematical methods developed were based on the idea of random

mixing of population that make the models simple enough to generate meaningful

results. However in reality, populations do not mix up randomly. The development

of non-random or heterogenous mixed population models started in mid 1970 and

13

refinement of compartmental models came into light with the paper by Cooke and

Yorke [49]. The heterogenous behavior was initially included in the risk structure

models of sexually transmitted diseases.

Structured models were used to study the transmission dynamics of various in-

fectious diseases like measles, influenza, smallpox and hepatitis. Haggett [98, 99]

developed structured model to investigate the spatial patterns in the spread of

measles outbreak and determined the importance of different spatial processes at

different stages for the development of the epidemic. Cliff et al. [47], developed

the static aspects of spatial structure, regional dynamics, spatial autocorrelation

and spatial forecasting for the mathematical models. Some structured models were

developed for Influenza by Bayroyan et al. [16], Rvachev and Longini [193], Longini

[156]. Sattenspiel [203], Sattenspiel and Simon [205] formulated structured models

for the transmission dynamics of hepatitis. Travis and Lenhart [225] established

structured model for infectious diseases in heterogenous populations for small pox.

They developed the essential conditions to eradicate the diseases with the help of

vaccination. Furthermore Andreasen and Christiansen [9] analysed effects of popu-

lation structure on infectious diseases. All these models were developed on the idea

that, human population is structured essentially in some manner. Thus random

mixing of population in a model is inappropriate for any disease. Another kind of

demographic model also appeared as age-structured models either with continuous

age models or models with age groups. These models were developed on the con-

cept of more interaction of people from the same age group as compared to different

age groups. So the difference between risk-structure (sexually transmitted disease)

and age-structure models is the inclusion of individuals age. The work done by

Anderson and May [6], Castillo-Chavez et al. [32], Dietz, [5], Dietz and Schenzle

[54] and Hethcote [107] contributed as foundation of age structured epidemiological

modeling. According to Hethcote “Indeed, some of the early epidemiology models

incorporated continuous age structure [22, 134]. Modern mathematical analysis

of age-structured models appears to have started with Hoppensteadt [115], who

formulated epidemiological models with both continuous chronological age and in-

fection class age (time since infection), showed that they were well posed, and found

14

threshold conditions for endemicity.”

The new term “Community epidemiology” has recently appeared in mathematical

epidemiology. It is based on the theoretical expansion of simple one-pathogene

one-host model where one pathogene is transferred to multiple host species. There

are two kind of multiple host models. Vector borne diseases such as malaria,

dengue fever and leishmania are among a few of the most dangerous and challeng-

ing multiple host diseases. Also more and more models are required to optimize the

control for macro-parasitic infectious diseases as avian influenza, hantavirus, Lyme

disease, bubonic plague, Q-fever, SARS, rabies, West Nile virus, toxoplasmosis,

trypanosomiasis etc. Ferguson et al. [73], in 1999, developed multiple host model

to investigate effect of antibody-dependent improvement in the multi pathogene

transmission dynamics. They predicted the chaotic behaviour of the epidemic with

the frequent antibody-dependent improvement. In 2002 Gog and Grenfell [93] de-

veloped a model to answer the fundamental question in strain dynamics where they

used specific example of Influenza while in 2003 Ferguson et al. [74], studied dif-

ferent strains of Influenza virus to investigate the factors effecting the variational

epidemiological dynamics patterns with the help of a multi-strain mathematical

model. According to Keeling [133] “ Models incorporating multiple pathogenes

allow us to investigate questions of disease evolution, from theoretical questions

such as understanding current disease behavior in terms of an optimal strategy for

transmission to more applied issues such as predicting the influenza strains for the

coming year or understanding the effects of strain specific control. Finally multi-

strain models offer insights into the increasing prevalence of drug resistant bacteria

and how to limit their control.”

Nowadays contact network models are also very popular not only in epidemiology

but also in various other branches of science. Networks have been used in modelling

to study the spread of infectious disease for the last fifty years [12, 53]. With the

availability of huge and accurate data, computational efficiency and improvement

in methodologies in the last twenty years, the theory of network modelling has been

widely used for the investigation of human diseases. Eubank et al. [65], Meyers et

al. [170], Bansal et al. [14], have successfully developed epidemiological models to

15

study several infectious diseases . According to Welch et al. [232], “The network-

based studies to date have largely focused on the impact of network structure on

disease dynamics and the effect of control strategies. Network structure has largely

been determined by collecting host data to inform probabilistic models of host in-

teractions, which are then used to generate simulated networks over which disease

spread can be studied. Network-based models provide an elegant alternative to

homogeneous-mixing models by intuitively capturing diversity in the underlying

patterns of interaction in a population.”

The effect of diffusion is extensively studied in epidemic of infectious diseases.

Spatial diffusion for measles was first studied by Soper [215] in 1929. His work

contributed to the foundation of the study of spatial and temporal aspects of epi-

demics. In 1975, Bailey [12], summarized some of such spatial diffusion complex

but highly idealized models. The book written by Cliff et al. [48], in the year 1981,

is considered as the most important addition in the study of spatial diffusion of

infectious diseases. Liu et al. [154], states that as “we do not know what kind of

epidemic outbreaks, when it outbreaks, and how it diffuses. Generally, after an epi-

demic outbreaks, public officials are faced with many critical and complex issues,

the most important of which is to make certain how the epidemic diffuses so that

the rescue operation efficiency is maximized.” There is a large number of diffusive

compartmental models [171, 218, 151, 250, 255]. Most recently Kim et al. [138],

examined a diffusive model to study the transmission of avian influenza among

human and birds. Their study suggests that by eradicating the infected birds and

by reducing the contact rate of susceptible and infected humans the danger of pan-

demic of influenza can be reduced. Samsuzzoha et al. [198], formulated an SEIR

diffusive compartmental model and numerically studied the transmission dynamics

of the 1918 Influenza pandemic with different initial population distributions. Also

Samsuzzoha et al. [199], developed SV EIR model in the presence of diffusion to

study the effect of vaccination on transmission dynamics of influenza. Very recently

Lui and Xiao [154] developed a susceptible-infected-susceptible epidemic diffusive

model to study the effect of population migration between two cities on the spread

of an epidemic. They showed that to control the diffusion of an epidemic in both

16

cities the migration of individuals in both cities needs to be controlled in same

manner.

According to Kramer et al. [142], “Preventing and reducing the spread of infectious

disease among humans is an essential function of public health. Epidemiology is

often called the core science of public health, which studies the distribution and de-

terminants of disease risk in human populations. Starting in the middle of the 19th

century, infectious disease epidemiology applies the fundamentals of epidemiology

to study infectious diseases and deals with questions about conditions for disease

emergence, spread and persistence. It describes the prevalence and incidence of

infectious diseases through which the epidemiological trends can be characterized

for different world regions.”

2.3 The case of severe acute respiratory syndrome

Severe acute respiratory syndrome also called SARS is a viral respiratory illness

caused by a coronavirus termed as SARS-associated coronavirus (SARS − CoV )

[131]. This epidemic is different from the conventional atypical pneumonia. The

SARS outbreak which happened in 2003, had no prior history like bird flu, human

enterovirus and Nipah virus. The person infected with SARS experience a severe

lack of oxygen and need medical aid for breathing. In addition the infection is

communicable and can infect massive number of individuals in some situations.

That is why world health organization (WHO) named it as severe acute respira-

tory syndrome. It is refereed as most unrivaled disease in the history of WHO

records. The SARS epidemic was hard to handle due to three main reasons [149]:

• Firstly, The symptoms of SARS are similar to common flu. So its usually

not possible to distinguish between SARS and flu patient. That’s why many

SARS patient were treated with antibiotics for common flu and sent back

home, spreading the disease more.

• Secondly, SARS virus can survive outside the human body for a few hours

which causes rapid transfer of the infection with close contact. That is why at

17

the beginning of SARS outbreak mostly family members, friends and health

workers of the infected individual were also infected.

• Thirdly, the incubation period of SARS is less than ten days, meaning it

not only spreads fast, it also kills faster. Also it is more dangerous for older

people as they have weaker immunity and also they suffer already with health

problems like high blood pressure, diabetes, and heart diseases.

2.3.1 The outbreak

It was in November 2002 when first case of SARS was actually diagnosed in

Guangzhou, the capital of a Chinese province Guangdong. Afterwards, many new

cases of SARS were diagnosed in Heyuan, Jianmen, Shengzhen, Zhaoqing and

Zhongshan between November 2002 and January 2003; the locations of these cities

following a major cluster in Guangzhou [255]. But Chinese government kept it

secret to maintain public confidence, and informed the World Health Organization

(WHO) about the SARS outbreak only in late February 2003, when SARS epi-

demic was beyond their control. Till 9 February 2003, 792 probable cases of SARS

were diagnosed with 31 deaths in China. At the end of the SARS outbreak in

China alone the total number of SARS probable cases reached the figure of 5, 327

with 343 deaths. China is one of the countries where the SARS outbreak hit the

hardest [254]. According to the assessment of a WHO report [237] “If SARS is not

brought under control in China, there will be no chance of controlling the global

threat of SARS. Achieving control of SARS is a major challenge especially in a

country as large and diverse as China.” According to Leung [149] “The disease

soon found its way to Hong Kong, and subsequently, to more than 20 countries

in the rest of the world. According to investigations carried out by Hong Kong

authorities, the territory’s first “index case” was a 64 year old doctor from China

who had treated SARS patients in Guangzhou. The doctor, who stayed at the

Metropole Hotel in Monk Kok on 21st February 2003, was admitted to hospital

with SARS symptoms on the next day, where he died on 4, March 2003. It was

later discovered that five other guests of the hotel who had stayed on the same floor

as the doctor (the ninth floor) also contracted the disease. Three of these guests

18

Figure 2.3: Sever acute respiratory syndrome (SARS) a deadly threat [124].

were female tourists from Singapore while the other two were Canadians. Experts

think that it is highly probable that five of the guests from the hotel contracted

the disease from the doctor, who brought the virus into Hong Kong from mainland

China.”

There is no doubt that SARS took real advantage of international air travel.

Modern fast transport system made the spread of SARS virus very easy and rapid.

Hitoshi Oshitani, [237] the WHO coordinator for SARS, calls this “the most sig-

nificant outbreak that has been spread through air travel in history”. According

to Leung [149] “In Hanoi, Vietnam, a 48 year old businessman from the US fell ill

and was admitted to the French Hospital on 26 February 2003, He had traveled to

Hong Kong and China before arriving in Vietnam, shortly after that, an outbreak

occurred in Vietnam. The outbreak in Singapore is believed to have been caused

by the three women who were infected in the Metropole Hotel in Hong Kong. After

returning to their home country, all three of them fell ill, were hospitalized and were

found to have contracted SARS. The SARS cases in Canada are linked to the

two Canadian tourists who stayed at the Metropole Hotel in Hong Kong. Similar

to what happened in Singapore, the disease was brought back to their home coun-

tries when they returned. Ontario was most badly affected, while British Columbia

saw a couple of cases. By the end of March 2003, the former had more than 40

cases of infection while the latter witnessed 2 cases. Both of the source patients

later succumbed to the illness and died. By mid-March 2003, numerous cases of

19

Figure 2.4: Recent patient of SARS-like disease [123].

SARS were reported in many other countries such as US, Taiwan, Thailand and

several European countries including France, Germany, Italy, Republic of Ireland,

Spain, Switzerland and the United Kingdom. Initial investigations revealed that

all the outbreaks had origins in Asia. The viral carrier, when that could be traced,

was almost always someone who had made a recent trip to China or Hong Kong.”

According to the estimates of WHO [238] between November 2002 and June 2003,

8, 450 people were infected. Among those infected, 21% were health care workers.

There were 774 deaths in 33 countries over the five continents. Outside Asia the

largest outbreak happened in Canada. After June 2003 and again in the start of

2004 many cases of SARS were reported either by the individuals who were in con-

tact with the SARS viruses in laboratories or with the animals carring SARS-like

virus.

Recently a SARS-like virus was diagnosed in humans. According to the director-

general of the United Nations this virus is “a threat to the entire world”. At

first the virus was named as novel coronavirus showing the same symptoms as

SARS virus [123]. According to Chris et al. [43], “The emergence in 2012 of a

new disease-causing coronavirus has generated substantial concern. As of June 26,

2013, middle east respiratory syndrome coronavirus (MERS-CoV ) had caused 77

laboratory-confirmed cases and 40 deaths. The virus is related to the Severe Acute

Respiratory Syndrome coronavirus (SARS-CoV ) that emerged in 2002-03 and as

SARS-CoV had during its prepandemic stage, MERS-CoV has probably been

20

Figure 2.5: SARS corona-virus under the microscope [122].

transmitted from an unknown animal host to human beings repeatedly in the past

year. Cases of human-to-human transmission have also been documented in several

countries.”

2.3.2 SARS virus

Corona-virus belongs to a large class of viruses that cause common cold or minor

respiratory problems. In some severe cases they can also cause pneumonia or acute

respiratory distress syndrome (ARDS) [1]. The exact origin of corona-virus is still

not known, but it is believed that transmission of corona-virus to humans may

happened through a cat-like mammal called civets. SARS first appeared in the

Chinese province of Guangdong and this region is rich in its number of civets. A

similar SARS-like virus is found in Horseshoe bats. There is a possibility that these

bats transferred the virus to civets that infected the humans with SARS disease.

As a result of this discovery civets were banned and slaughtered in Guangdong

province. According to Leung [149] “Coronaviruses, so called because of their

spiky crown of protein globules, are generally not mortally harmful. They are a

pest to livestock, and in humans are responsible for more than one-third of common

cold cases. But in this case, researchers believe that the bugs have mutated into

something far deadlier as a rogue virus that triggers a killer pneumonia, now widely

known as SARS. The new coronavirus was isolated in Vero E6 cells from nasal and

throat swab specimens of two patients in Thailand and Hong Kong with suspected

21

Figure 2.6: Symptoms of SARS [117].

SARS. The isolate was identified initially as a coronavirus by electron microscopy

(EM). The little hooks sticking out of the viral body are the telltale characteristics

that help.”

2.3.3 Symptoms of SARS

Prompt identification of the features of SARS − CoV , is not yet available, so the

diagnosis of SARS disease is based on the existence of clinical symptoms and the

evidence of exposure to an infected SARS patient. If a person meets these two cri-

teria then he/she have been termed as ’probable’ case of SARS. Initially flu, cough,

fever, chills, loss of appetite, muscle aches and sore throat can be the symptoms of

SARS. According to the report of Center for Disease Control (CDC) [210] “In gen-

eral, SARS begins with a high fever (temperature greater than 100.4F [> 38.0C]).

Other symptoms may include headache, an overall feeling of discomfort, and body

aches. Some people also have mild respiratory symptoms at the outset. About 10

percent to 20 percent of patients have diarrhea. After 2 to 7 days, SARS patients

may develop a dry cough. Most patients develop pneumonia.” The usual changes

in chest X-ray were irregular fortification leading to bilateral bronchopneumonia

for five to ten days. In most cases symptoms of the disease appear after 3 to 17

22

Figure 2.7: Transmission route of SARS virus [119].

days of latent period, and most of the patients recover within two weeks. The rate

of mortality for SARS was over all 15% where as among the elderly this rate was

higher 50% [95].

2.3.4 Transmission route

The transmission route of severe acute respiratory syndrome (SARS) is quite un-

certain. In some mathematical models it is discovered to be transmitted by close

person-to-person contact similar to Influenza usually where droplets of respiratory

secretions transfer the virus from one person to another. When an infected person

coughs or sneezes these droplets can travel over 3 feet through air and fall on the

mucous membrane of nose, eyes or mouth of surrounding individuals. This trans-

mission is also possible if a person touches his nose or mouth or eye after touching

a contaminated surface. The report of the Center for Disease Control (CDC) [210]

defines the close person-to-person contact as “In the context of SARS, close con-

tact means having cared for or lived with someone with SARS or having direct

contact with respiratory secretions or body fluids of a patient with SARS. Exam-

ples of close contact include kissing or hugging, sharing eating or drinking utensils,

talking to someone within 3 feet and touching someone directly. Close contact does

not include activities like walking by a person or briefly sitting across a waiting

room or office.” On the other hand in some cases the airborne spread of SARS

virus is also identified [126]. Some studies even suspect the fecal-oral transmission

23

for the SARS virus [191, 226].

2.3.5 Mathematical modeling of SARS

When the SARS outbreak happened in 2003, there were no vaccines or specific

treatment available, this situation still holds true today. Quarantine and isolation

were the only means to restrict or control and predicting the disease transmission.

Also many basic questions regarding the number of new infections, its peak, the

time of peak arrival ,and the intensity and length of its peak needed answers to

develop the control policies [191]. In this situation mathematical models facili-

tated the researchers with the tools to provide best answers to these important

questions. Various mathematical models were formulated and solved to study the

SARS transmission dynamics in order to predict its behavior and to plan control

policies. The mathematical models provide the precise quantitative details of the

effect of control strategies. As quarantine and isolation were the first non-medial

interventions used for the prevention of SARS and thus they appeared quite early

in the mathematical modelling of SARS. SARS models of quarantine measures

showed many surprising results. Lipsitch et al. [152], established deterministic

and stochastic mixed homogenous compartmental models to study the effect of

isolation and quarantine on the transmission dynamics of SARS in Hong Kong

and Singapore. They showed that more than one control measure is needed to

control a SARS outbreak that also effect the reproduction number and reduce the

chances of developing outbreak. They also found that the average time of quar-

antine can be reduced by an quarantining a large number of individuals around

infected person under a certain threshold. Lloyd-Smith et al. [155], formulated

a stochastic compartmental model to study the effect of different control mea-

sures. Their studies shows that the increase in the reproduction number of SARS

strongly depends on efficient quarantine and the process is nonlinear. They also

found that applying quarantine strategy to people in hospital is much more effec-

tive than applying it to diagnosed people. In 2003 Riley et al. [191], formulated

a stochastic matapopulation compartmental model to estimate the reproduction

number for SARS for Hong Kong without superspreading events (SSEs). They

24

concluded that implementation of control measures and restricting the contact and

movement of individuals in a region can lead to a slow down of the transmission of

the disease.

Several models of SARS studies show that if the duration between onset of symp-

toms and isolation of diagnosed is reduced some how, it can greatly effect the

transmission of SARS and can be helpful in controlling the outbreak [191, 152, 45].

Many models were used to study the nosocomial outbreaks of SARS disease. One

of such studies was done by Hsieh et al. [116], with the help of a deterministic com-

partmental modal. The results of studies shows that time during the admission of

suspected SARS cases and their classification as probable SARS cases is very im-

portant to increase or decrease the nosocomial transmission of disease. According

to Chris et al [43], “A recurring theme in many SARS models is the importance of

timely application of control measures [95, 152, 155, 177, 230, 231]. These models

show that quarantine and isolation have a disproportionate impact on epidemic

control if applied early in the outbreak. Conversely, delays in imposing control

can lead to large case burdens or even failure of potentially successful containment

measures. The dependence of outbreak size on the time of application is nonlin-

ear: there are crucial periods early in the outbreak beyond which the effectiveness

of control measures is severely degraded.” In the beginning the SARS epidemic

was studied based on homogenous populations but soon different mathematical

models illustrated heterogeneity for SARS transmission in space. Several math-

ematical studies significantly demonstrated the difference in parameter estimates

and prediction about epidemics [45, 191]. Although much difference of infectious-

ness is observed in different age groups of population for SARS [45, 158, 192]. The

concepts of possible treatment, spatial and social structures and susceptibility vari-

ations were also incorporated to study and understand the transmission of SARS

epidemic [152, 155, 163, 170].

Occurrence of super spreading events (SSEs) was one of the captivating features

of the 2003 epidemic of severe acute respiratory syndrome (SARS). They have a

large influence on the early spread of SARS [152, 191, 57]. The major Hong Kong

and Singapore SARS epidemics were the result of two different SSEs in Amoy

25

Gardens estate and hospitals [51, 223, 235] and five sperate SSEs in Singapore

outbreak [148]. So to prevent such SARS epidemics in future, it is essential to un-

derstand the factors of SSEs. According to Fang et al. [66], “SSEs are those rare

events where, in a particular setting, an individual may generate many more than

the average number of secondary cases.” He also states that “The understanding

of SSEs is critical to the containment of SARS. From the mathematical modeling

it is found that SSEs can happen even when the virulences are equal for all the

infective individuals. The long latent periods play a critical role in the appearance

of SSEs. Early awareness of the epidemic, which is also effected by the long latent

periods, is vital for the reduction of the possibility of SSEs and the containment

of SARS.”

China was the place where SARS epidemic hit the hardest. After the appear-

ance of SARS, Chinese mathematical modelling in epidemiology experienced a

new age of modelling. Mathematical models were formulated to study the trans-

mission dynamics of SARS disease with real−time data. From 1994 − 2006 the

total number of mathematical modelling papers published on infectious disease in

Chinese journals were 375. Among those 64 papers were only on SARS model-

ing. In 2003 there was further increase in the number [102]. According to Han et

al. [102], “Clearly, the SARS epidemic has resulted in an enormous boost to this

scientific discipline, leading to disproportional large numbers of papers on SARS

models equalling the annual production of modelling papers in the years before

the outbreak. Perhaps surprisingly, this overall extension of the research field

has not lead to a decrease in the number of modelling papers on other infectious

diseases.” He also states that “For the SARS modelling, relatively often more

complex techniques were used, e.g. individual-based modelling, spatio-temporal

models and other types (Autoregressive Integrated Moving Average (ARIMA)

modelling and small world network models). Spatio-temporal models were largely

confined to application on SARS.” Out of the 64 published papers on the SARS

epidemic in Chinese journals six consist of deterministic compartmental models,

eight consist of stochastic compartmental models, one based on a small-world net-

work model, and two focused on spatiotemporal models, while the remaining 43

26

models were based on simple to complex curve fitting techniques like ARIMA

[56] that described SARS outbreak patterns and explained the intervention ap-

plied [102]. The SARS outbreaks in specific regions were focused and modeled

successfully. Numerous mathematical models were formulated for the epidemic in

Beijing [31, 39, 67, 86, 101, 212, 233, 231, 241, 243, 249, 239], Guangdong province

[40, 233, 240, 245, 241], Inner Mongolia [241], Hong Kong [239, 240, 245, 153]. Some

of the Chinese modelers used data from outside China due to the unavailability of

data [153, 211]. These models also provided the estimates of reproduction number

R0 in the range 3.5 − 4.5. [233, 40, 153]. The value of reproduction number was

verified by other researchers Lipsitch et al. [152] and Riley et al. [191]. According

to Han et al. [102], “It is clear that the Chinese modelling initiatives have had lim-

ited implications for policy advice about the SARS epidemic itself, simply because

the reports were only available in the scientific literature after the epidemic. Also,

as far as we can conclude from the papers, no modelling initiative had a direct

link with decision makers before publication. However, from these papers policy

makers may have learned how models can support their decision making, so that

there may be more interaction in case of re-emergence of SARS or an outbreak of

another infectious disease. In fact, these and other modelling initiatives (including

most of the curve-fitting studies) purely focused on mathematical issues and did

not really consider practical implications with public health relevance. Perhaps the

most important benefit of the Chinese SARS modelling efforts is not preparation

for possible re-emergence of SARS, but rather preparation any future outbreak of

infectious diseases.”

27

Chapter 3

Numerical Study of SARS Model

with the Inclusion of Diffusion in

the System

3.1 Introduction

Epidemiological models are considered as one of the most powerful tools to analyze

and understand the spread and control of infectious diseases. Analysis of trans-

mission dynamics of infectious diseases can lead to the better methodologies to

slow their transmission. These models ranges from simple curve fitting models to

standard compartmental models [8] (MSEIR, MSEIRS, SEIJR, SIR, SIRS,

SEIR, SEIS, SI, and SIS etc.) to complex stochastic models. Fast computer

systems and the availability of huge data bases has made it possible to use complex

mathematical models to analyze the data.

In the beginning, epidemiologist, tried to find out the reasons for the cause and

spread of SARS. They tried to find measures to control it, but the main emphasis

was on research work concerned with the biological properties of the corona virus

[202]. Some work was done to investigate the transmission dynamics and the effect

of various control measures. G. Chowell et al. [45] fitted an SEIJR model for

SARS epidemic for the data from Toronto, Hong Kong and Singapore. Chowell

et al. predicted the behavior of the disease and the role of diagnosis and isola-

28

tion as a control mechanism in these regions showing the difference between the

epidemic dynamics occurred in these three cities. Yang et al. [245] established a

compartmental model to describe the SARS epidemic in spatial-temporal dimen-

sions determining whether people traveling in buses and trains infect one another

or not. They concluded that SARS can spread through people traveling in buses

and trains. In their SEIR models based on data from Beijing and Hong Kong, Wu

et al. [233] and Chen et al. [40] estimated the source of super-spreading events of

SARS with the calculation of the reproductive rate of the disease based on data

from Beijing and Hong Kong.

In this chapter, a SARS model (Chowell et al. [45]) is considered with the in-

clusion of diffusion in the system. Diffusion is introduced in the system to study

the spatial spread of disease. Different initial population distributions are chosen

to investigate the effect of diffusion on the spread of SARS. Also intervention

strategies have been proposed to investigate the effect on spread of disease.

3.2 The SEIJR epidemic model

3.2.1 Equations

This model is based on the SEIJR model (G Chowell et al. [45]) with the inclusion

of diffusion in the equations governing the system. Total population is supposed

to be N where N = S + E + I + J +R.

∂S

∂t= −β

(I + qE + lJ)

NS − µS +Π+ d1

∂2S

∂x2(3.1)

∂E

∂t= β

(I + qE + lJ)

NS − (µ+ κ)E + d2

∂2E

∂x2(3.2)

∂I

∂t= κE − (µ+ α + γ1 + δ)I + d3

∂2I

∂x2(3.3)

∂J

∂t= αI − (µ+ γ2 + δ)J + d4

∂2J

∂x2(3.4)

∂R

∂t= γ1I + γ2J − µR + d5

∂2R

∂x2(3.5)

where the variables S, E, I, J and R denote the proportion of susceptible, exposed,

infected, diagnosed and recovered individuals respectively. d1, d2, d3, d4 and d5 are

29

Table 3.1: Interpretation of parameters (per day)

Parameter Description Values

Π Rate of inflow of susceptible individuals into region 3.3× 10−5b

β Transmission rate 0.75a

µ Rate of natural mortality 3.4× 10−5b

l Relative measure of reduced risk among diagnosed 0.38a

κ Rate of progression from exposed to the infected 0.33a

q Relative measure of infectiousness for exposed individuals 0.1a

α Rate of progression from infected to diagnosed 0.33a

γ1 recovery rate of infected individuals 0.125a

γ2 recovery rate of diagnosed individuals 0.2a

δ SARS induced mortality rate .006a

a(Chowell G. et. al. [45]), b (Gummel A. B. et.al. [95] )

the diffusivity constants. Table 3.1 provides the description and the values of the

parameters.

3.2.2 Initial and boundary conditions

The domain of all the calculations is considered as [−2, 2]. Boundary and initial

conditions are chosen as follows:

∂S(−2, t)

∂x=

∂E(−2, t)

∂x=

∂I(−2, t)

∂x=

∂J(−2, t)

∂x=

∂R(−2, t)

∂x= 0 (3.6)

∂S(2, t)

∂x=

∂E(2, t)

∂x=

∂I(2, t)

∂x=

∂J(2, t)

∂x=

∂R(2, t)

∂x= 0 (3.7)

30

(i)

S0 = 0.98Sech(5x− 1), − 2 ≤ x ≤ 2.

E0 = 0, − 2 ≤ x ≤ 2.

I0 = 0.02Sech(5x− 1), − 2 ≤ x ≤ 2.

J0 = 0, − 2 ≤ x ≤ 2.

R0 = 0, − 2 ≤ x ≤ 2.

(ii)

S0 = 0.98 exp(−5x2)3, − 2 ≤ x ≤ 2.

E0 = 0, − 2 ≤ x ≤ 2.

I0 =

0, − 2 ≤ x < −0.4,

0.02, − 0.4 ≤ x ≤ 0.4,

0, 0.4 < x ≤ 2.

J0 = 0, − 2 ≤ x ≤ 2.

R0 = 0, − 2 ≤ x ≤ 2.

(iii)

S0 = 0.97 exp(−5(x− 1)2), − 2 ≤ x ≤ 2.

E0 = 0, − 2 ≤ x ≤ 2.

I0 = 0.03 exp, (−5(x+ 1)2), − 2 ≤ x ≤ 2.

J0 = 0, − 2 ≤ x ≤ 2.

R0 = 0, − 2 ≤ x ≤ 2.

31

t = 0

t = 0(i)

-2 -1 0 1 2x

0.2

0.4

0.6

0.8

1.0S,I

t = 0

(ii)

-2 -1 0 1 2x

0.2

0.4

0.6

0.8

1.0S,I

t = 0

(iii)

-2 -1 0 1 2x

0.2

0.4

0.6

0.8

1.0S,I

t = 0

(iv)

-2 -1 0 1 2x

0.2

0.4

0.6

0.8

1.0

S,I

Figure 3.1: Initial conditions (i)− (iv).

(iv)

S0 ={

0.96Sech(15x), − 2 ≤ x ≤ 2.

E0 = 0, − 2 ≤ x ≤ 2.

I0 =

0, − 2 ≤ x < −.6,

0.04, − .6 ≤ x ≤ .6,

0, .6 < x ≤ 2.

J0 = 0, − 2 ≤ x ≤ 2.

R0 = 0, − 2 ≤ x ≤ 2.

The initial conditions are shown in Fig. 3.1. In initial condition (i) a large pro-

portion of susceptible and infected populations is concentrated towards the right

half of the main domain. Initial condition (ii) shows both S and I concentrated

around the middle of the main domain. In initial condition (iii), I has high con-

centration in the left half of the domain [−2, 2] and population S has concentration

on the right half of the domain [−2, 2]. Initial condition (iv) shows susceptible S

around the middle of domain [−2, 2] and infectious individuals around the middle

but beyond the domain of S.

32

3.3 Numerical scheme

In this section the operator splitting technique [244] has been used to solve the

SEIJRmodel equations. The equations are divided in two groups of sub equations.

The first group comprises the nonlinear reaction equations to be used for the first

half-time step as given:

1

2

∂S

∂t= −β

(I + qE + lJ)

NS − µS +Π (3.8)

1

2

∂E

∂t= β

(I + qE + lJ)

NS − (µ+ κ)E (3.9)

1

2

∂I

∂t= κE − (µ+ α + γ1 + δ)I (3.10)

1

2

∂J

∂t= αI − (µ+ γ2 + δ)J (3.11)

1

2

∂R

∂t= γ1I + γ2J − µR (3.12)

The second group consists of the linear diffusion equations, to be used for the

second half-time step as follows:

1

2

∂S

∂t= d1

∂2S

∂x2(3.13)

1

2

∂E

∂t= d2

∂2E

∂x2(3.14)

1

2

∂I

∂t= d3

∂2I

∂x2(3.15)

1

2

∂J

∂t= d4

∂2J

∂x2(3.16)

1

2

∂R

∂t= d5

∂2R

∂x2(3.17)

By the forward Euler scheme the above equations transform to

Sj+ 1

2i = Sj

i +∆t(−β(Iji + qEj

i + lJ ji )

N ji

Sji − µSj

i +Π) (3.18)

Ej+ 1

2i = Ej

i +∆t(β(Iji + qEj

i + lJ ji )

N ji

Sji − (µ+ κ)Ej

i ) (3.19)

Ij+ 1

2i = Iji +∆t(κEj

i − (µ+ α + γ1 + δ)Iji ) (3.20)

Jj+ 1

2i = J j

i +∆t(αIji − (µ+ γ2 + δ)J ji ) (3.21)

33

Rj+ 1

2i = Rj

i +∆t(γ1Iji + γ2J

ji − µRj

i ) (3.22)

where Sji , E

ji , I

ji , J

ji and Rj

i are the approximated values of S, E, I, J and R at

position −2 + i∆x, for i = 0, 1, . . . and time j∆t, j = 0, 1, . . . and Sj+ 1

2i , E

j+ 12

i ,

Ij+ 1

2i , J

j+ 12

i and Rj+ 1

2i denote their values at the first half-time step. Similarly, for

the second half-time step,

Sj+1i = S

j+ 12

i + d1∆t

(∆x)2(S

j+ 12

i−1 − 2Sj+ 1

2i + S

j+ 12

i+1 ) (3.23)

Ej+1i = E

j+ 12

i + d2∆t

(∆x)2(E

j+ 12

i−1 − 2Ej+ 1

2i + E

j+ 12

i+1 ) (3.24)

Ij+1i = I

j+ 12

i + d3∆t

(∆x)2(I

j+ 12

i−1 − 2Ij+ 1

2i + I

j+ 12

i+1 ) (3.25)

J j+1i = J

j+ 12

i + d4∆t

(∆x)2(J

j+ 12

i−1 − 2Jj+ 1

2i + J

j+ 12

i+1 ) (3.26)

Rj+1i = R

j+ 12

i + d5∆t

(∆x)2(R

j+ 12

i−1 − 2Rj+ 1

2i +R

j+ 12

i+1 ) (3.27)

The stability condition satisfied by the above described numerical method is given

as:dn∆t

(∆x)2≤ 0.5, n = 1, 2, 3, 4, 5. (3.28)

In each case, ∆x = 0.1, d1 = 0.025, d2 = 0.01, d3 = 0.001, d4 = 0.0, d5 = 0.0 and

∆t = 0.03 are used.

3.4 Stability analysis

3.4.1 Reproduction number and disease-free equilibrium

(DFE)

The threshold parameter for any DFE is R0, referred to as the basic reproduction

number. It is defined as the expected number of secondary cases produced, in a

completely susceptible population, by a typical infected individual [52]. To deter-

mine the expression for the basic reproduction number represented as RI for the

system (3.1) - (3.5), the matrices F and W [60] are chosen as follows:

34

F =

β (I+qE+lJ)N

S

0

0

0

0

and W =

(µ+ κ)E

−κE + (µ+ α + γ1 + δ)I

−αI + (µ+ γ2 + δ)J

−γ1I − γ2J + µR

β (I+qE+lJ)N

S + µS − Π

where F describe the transmission route for infection and W denotes the remaining

dynamics of compartments E, I, J , R and S. For the model represented by the

system of Equations (3.1) - (3.5), E, I and J represent the infected compartments.

Therefore the following matrices F represents the paths to infection and W repre-

sents the remaining dynamics corresponding to the compartments E, I and J . Thus

F =

Coefficient of ES

NCoefficient of IS

NCoefficient of JS

N

Coefficient of ESN

Coefficient of ISN

Coefficient of JSN

Coefficient of ESN

Coefficient of ISN

Coefficient of JSN

=

qβ β lβ

0 0 0

0 0 0

and

W =

Coefficient of E Coefficient of I Coefficient of J



=

(µ+ κ) 0 0

−κ (µ+ α + γ1 + δ) 0

0 −α (µ+ γ2 + δ)

This gives

W−1 =

[1

µ+κ0 0

κ(µ+α+γ1+δ)(µ+κ)

1µ+α+γ1+δ

0

ακ(µ+α+γ1+δ)(µ+γ2+δ)(µ+κ)

ακ+µα(µ+α+γ1+δ)(µ+γ2+δ)(µ+κ)

1µ+γ2+δ

]and

F .W−1 =

[qβ

(µ+κ)+ βκ

(µ+α+γ1+δ)(µ+κ)+ lβκα

(µ+α+γ1+δ)(µ+γ2+δ)(µ+κ)β

(µ+α+γ1+δ)+

lβ(µα+κα)(µ+α+γ1+δ)(µ+γ2+δ)(µ+κ)

lβµ+γ2+δ

0 0 0

0 0 0

]

The eigenvalues of F .W−1 are

λ1 = 0,

λ2 = 0,

35

λ3 =qβµ+κ

+ βκ(µ+α+γ1+δ)(µ+κ)

+ lβκα(µ+γ1+α+δ)(µ+γ2+δ)(µ+κ)

.

The dominant eigenvalue i.e, basic reproduction number is given as:

RI =qβ(µ+α+γ1+δ)(µ+γ2+δ)+κ(µ+γ2+δ)+lακ

(µ+α+γ1+δ)(µ+γ2+δ)(κ+µ).

3.4.2 Disease-free equilibrium and stability analysis

The variational matrix of the (3.1) - (3.5) at the disease-free equilibrium P0 =

(1, 0, 0, 0, 0), can be given as:

V0 =

A11 A12

A21 A22

where

A11 =

−µ −qβ

0 (qβ − (κ+ µ)

, A12 =

−β −lβ 0

β lβ 0

,

A21 =

0 κ

0 0

0 0

and A22 =

−(α + γ1 + δ + µ) 0 0

α −(µ+ γ2 + δ) 0

γ1 γ2 −µ

.

The stability of the point of equilibrium, P0(1, 0, 0, 0, 0) depends on the char-

acteristic of the eigenvalues of the matrices A11 and A22. The eigenvalues of the

matrix A11 and A22 are qβ−(κ+µ), −µ and −(α+γ1+δ+µ), −(γ2+δ+µ) and −µ

respectively. All eigenvalues of A22 are clearly negative and real. One eigenvalue

of A11 is negative for other eigenvalue it is shown that:

If RI < 1 then βq(µ+ α+ γ1 + δ)(µ+ γ2 + δ) + βκ(µ+ γ2 + δ) + lβακ

< (µ+ α + γ1 + δ)(µ+ γ2 + δ)(κ+ µ)

⇒ βq < (κ+ µ)

⇒ βq − (κ+ µ) < 0

This implies that P0(1, 0, 0, 0, 0) is stable for ℜ0 < 1.

36

3.4.3 Stability of endemic equilibrium without diffusion

The variational matrix of the system of equations (3.1) - (3.5) at P ∗(S∗, E∗, I∗, J∗R∗),

is given by

V ∗ =

a11 a12 a13 a14 a15

a21 a22 a23 a24 a25

a31 a32 a33 a34 a35

a41 a42 a43 a44 a45

a51 a52 a53 a54 a55

where

a11 = β (I∗+qE∗+lJ∗)S∗

(S∗+E∗+I∗+J∗+R∗)2− β (I∗+qE∗+lJ∗)

(S∗+E∗+I∗+J∗+R∗)− µ,

a12 = β (I∗+qE∗+lJ∗)S∗

(S∗+E∗+I∗+J∗+R∗)2− β qS∗

(S∗+E∗+I∗+J∗+R∗),

a13 = β (I∗+qE∗+lJ∗)S∗

(S∗+E∗+I∗+J∗+R∗)2− β S∗

(S∗+E∗+I∗+J∗+R∗),

a14 = β (I∗+qE∗+lJ∗)S∗

(S∗+E∗+I∗+J∗+R∗)2− β lS∗

(S∗+E∗+I∗+J∗+R∗),

a21 = β 1(S∗+E∗+I∗+J∗+R∗)

− β (I∗+qE∗+lJ∗)S∗

(S∗+E∗+I∗+J∗+R∗)2,

a22 = β qS∗

(S∗+E∗+I∗+J∗+R∗)− β (I∗+qE∗+lJ∗)S∗

(S∗+E∗+I∗+J∗+R∗)2− κ− µ,

a23 = β S∗

(S∗+E∗+I∗+J∗+R∗)− β (I∗+qE∗+lJ∗)S∗

(S∗+E∗+I∗+J∗+R∗)2,

a24 = β lS∗

(S∗+E∗+I∗+J∗+R∗)− β (I∗+qE∗+lJ∗)S∗

(S∗+E∗+I∗+J∗+R∗)2,

a15 = β (I∗+qE∗+lJ∗)S∗

(S∗+E∗+I∗+J∗+R∗)2, a25 = −β (I∗+qE∗+lJ∗)S∗

(S∗+E∗+I∗+J∗+R∗)2,

a32 = κ, a33 = −α− γ1 − δ − µ, a43 = α, a44 = −γ2 − δ − µ, a53 = γ1,

a54 = γ2, a55 = −µ, and a31 = a34 = a35 = a41 = a42 = a45 = a51 = a52 = 0. The

characteristic equation for P ∗(S∗, E∗, I∗, J∗, R∗) can be written as

λ5 + p1λ4 + p2λ

3 + p3λ2 + p4λ+ p5 = 0 (3.29)

Where p1, p2, p3, p4 and p5 are calculated as in [198] and are given in appendix

A.3. The Routh-Hurwitz criterion for the stability is given as in [200]:

C1 : p1 > 0, C2 : p5 > 0,

C3 : p1p2 − p3 > 0,

C4 : p1p2p3 + p1p5 − (p23 + p21p4) > 0,

C5 : (p1p4 − p5)(p1p2p3 − p23 − p21p4) + p21p4p5 − (p5(p1p2 − p3)2 + p1p

25) > 0.

and P1, P2, P3 and P4 are points of equilibrium.

P1 = (.331581, 0.000064, 0.000046, 0.000074, .607887),

P2 = (.357187, .000062, .000044, .000058, .585632),

37

Table 3.2: Values of LHS Routh-Hurwitz criterion of equilibrium without diffusion

Case Equil. Pt. C1 C2 C3 C4 C5 Stability

1 P1 0.97677 6.902× 10−7 .20805 5.343× 10−6 7.613× 10−12 Stable

2 P2 1.02482 7.641× 10−11 .24969 6.903× 10−6 1.114× 10−11 Stable

3 P3 1.02416 6.560× 10−11 .23359 6.059× 10−6 8.399× 10−12 Stable

4 P4 0.97676 6.199× 10−11 .20804 5.109× 10−6 6.421× 10−12 Stable

P3 = (.365393, .000061, .000039, .000064, .577199),

P4 = (.355572, .000062, .000044, .000072, .584715).

3.4.4 Stability of endemic equilibrium with diffusion

To calculate the small perturbations S1(x, t), E1(x, t), I1(x, t),J1(x, t) and R1(x, t),

the equations are linearized about the point of equilibrium P ∗(S∗, E∗, I∗, J∗, R∗)

as described in [35, 201].

∂S1

∂t= a11S1 + a12E1 + a13I1 + a14J1 + a15R1 + d1

∂2S1

∂x2(3.30)

∂E1

∂t= a21S1 + a22E1 + a23I1 + a24J1 + a25R1 + d2

∂2E1

∂x2(3.31)

∂I1∂t

= a31S1 + a32E1 + a33I1 + a34J1 + a35R1 + d3∂2I1∂x2

(3.32)

∂J1∂t

= a41S1 + a42E1 + a43I1 + a44J1 + a45R1 + d4∂2J1∂x2

(3.33)

∂R1

∂t= a51S1 + a52E1 + a53I1 + a54J1 + a55R1 + d5

∂2R1

∂x2(3.34)

where a11, a12, a13 etc are the elements of the variational matrix V ∗ calculated using

the same method as described in [198]. Assume a Fourier series solution exists of

equations (3.30) - (3.34) of the form:

S1(x, t) =∑k

Skeλt cos(kx) (3.35)

E1(x, t) =∑k

Ekeλt cos(kx) (3.36)

I1(x, t) =∑k

Ikeλt cos(kx) (3.37)

38

J1(x, t) =∑k

Jkeλt cos(kx) (3.38)

R1(x, t) =∑k

Rkeλt cos(kx) (3.39)

where k = nπ2, (n = 1, 2, 3, · · · · · · ) is the wave number for the node n. Substituting

the value of S1, E1, I1, R1 in the equations (3.30) - (3.34), the equations are

transformed into∑k

(a11 − d1k2 − λ)Sk +

∑k

a12Ek +∑k

a13Ik +∑k

a14Jk +∑k

a15Rk = 0 (3.40)

∑k

a21Sk +∑k

(a22 − d2k2 − λ)Ek +

∑k

a23Ik +∑k

a24Jk +∑k

a25Rk = 0 (3.41)

∑k

a32Ek +∑k

(a33 − d3k2 − λ)Ik = 0 (3.42)

∑k

a43Ik +∑k

(a44 − d4k2 − λ)Jk = 0 (3.43)

∑k

a53Ik +∑k

a54Jk +∑k

(a55 − d5k2 − λ)Rk = 0 (3.44)

The Variational matrix V for the equations (3.40) - (3.44)

V =

a11 − d1k2 a12 a13 a14 a15

a21 a22 − d2k2 a23 a24 a25

0 a32 a33 − d3k2 0 0

0 0 a43 a44 − d4k2 0

0 0 a53 a54 a55 − d5k2

The characteristic equation for the variational matrix V is given as

λ5 + q1λ4 + q2λ

3 + q3λ2 + q4λ+ q5 = 0 (3.45)

where q1, q2, q3, q4 and q5 are calculated with the same technique as used in [198]

and are given in appendix A.3.

Routh-Hurwitz Conditions are given as:

C1 : q1 > 0, C2 : q5 > 0,

C3 : q1q2 − p3 > 0,

C4 : q1q2q3 + q1q5 − (q23 + q21q4) > 0,

C5 : (q1q4 − q5)(q1q2q3 − q23 − q21q4) + q21q4q5 − (q5(q1q2 − q3)2q1q

25) > 0.

39

Table 3.3: Values of LHS Routh-Hurwitz criterion of equilibrium with diffusion

Case Equil. Pt. C1 C2 C3 C4 C5 Stability

1 P1 1.0656 4.859× 10−8 .29522 .00478 8.061× 10−7 Stable

2 P2 1.11365 5.820× 10−8 .345648 .006458 1.411× 10−6 Stable

3 P3 1.1109 4.696× 10−8 .322143 .004815 9.623× 10−8 Stable

4 P4 1.06559 4.70× 10−8 .29521 .004779 8.049× 10−7 Stable

3.4.5 Excited mode and bifurcation value

The first excited mode of the oscillation n is calculated by the same technique as

used in [35]. According to the definition of mode of excitation the curve

f(β) = (q1q4 − q5)(q1q2q3 − q23 − q21q4) + q21q4q5 − (q5(q1q2 − q3)2q1q

25). (3.46)

for n = 1 represents the first mode of excitation as being closest to the β-axis

as shown in Fig 2. Similarly, n = 1 is first mode of excitation for Cases 2 − 4.

Bifurcation values of the transmission coefficient β are given in Table 3.4. It is

observed that the bifurcation value of transmission coefficient with diffusion is

greater than the value of transmission coefficient without diffusion. Bifurcation

values of the recovery coefficients γ1 and γ2 for which the point of equilibrium

remains stable [36] are given in Table 3.5. Here the bifurcation value of recovery

coefficients with diffusion are smaller than without diffusion. The corresponding

bifurcation diagrams and calculation of bifurcation values of β, γ1 and γ2 are given

in appendix A.3.

3.5 Numerical solutions

Four cases with the variation of β, the transmission coefficient, γ1, the recovery

coefficient in the infectious class and γ2, the recovery coefficient in the diagnosed

class have been chosen as given in Table 3.6. Numerical solutions are obtained

both with and without diffusion for all cases specified in Table 3.6.

40

n=1

0.5 1.0 1.5 2.0 2.5 3.0Β

0.00005

0.00010

0.00015

0.00020

0.00025

f HΒL

n=2

0.5 1.0 1.5 2.0 2.5 3.0Β

0.005

0.010

0.015

0.020

f HΒL

n=3

0.5 1.0 1.5 2.0 2.5 3.0Β

0.1

0.2

0.3

0.4

f HΒL

Figure 3.2: Determination of first excited mode with β as an unknown parameter.

Table 3.4: Bifurcation value of β

Cases Value of β Considered Bifurcation Value

Without Diffusion With Diffusion

1 0.75 0.750435 0.810383

2 0.75 0.750367 0.810239

3 0.75 0.750387 0.809841

4 0.70 0.700381 0.756276

Table 3.5: Bifurcation values of γ1 andγ2

Cases γ1 Without Diffusion With Diffusion γ2 Without Diffusion With Diffusion

1 0.125 0.124708 0.087493 0.2 0.199660 0.161489

2 0.125 0.124752 0.0873489 0.25 0.249586 0.196281

3 0.175 0.17471 0.133485 0.2 0.199694 0.161499

4 0.125 0.124726 0.0875375 0.2 0.199680 0.161530

41

Table 3.6: Four cases

Case Transmission Coefficient(β) Recovery Coefficient(γ1) Recovery Coefficient (γ2)

1 0.75 0.125 0.2

2 0.75 0.125 0.25

3 0.75 0.175 0.2

4 0.7 0.125 0.2

3.5.1 Solutions of SEIJR model without diffusion (Case 1)

Fig. 3.3, shows the output with initial condition (i) and without diffusion. Here

the susceptible population decreases slowly in the first five days of disease but after

that there is a rapid decrease till t = 20 days. There is a rapid increase in popula-

tion exposed to disease in the first five days and this increase continues till t = 10.

It slows down however after five days. After ten days there is a quick decrease

in exposed individuals. Only a few individuals are in the exposed compartment

after fifteen days. There is a slow increase in infected individuals till the fifth day.

But a sudden increase in infected is observed in the next five days. After reaching

maximum level, a decline in infected is observed till t = 15. With the increase of

infected individuals, the number of diagnosed has also increased rapidly but after

attaining the maximum in the first ten days of disease, there is a slow decrease till

t = 15. After that a quick decline is observed at t = 20. Recovered individuals

increase slowly in first ten days of disease but after that a rapid increase in recovery

is observed.

Fig. 3.4, shows the output with initial condition (ii) without diffusion. Here

the proportion of the susceptible population decreases rapidly between 5-10 days

and after that there is very low level of susceptible population. The population

becomes exposed very quickly during first five days. After ten days, there is a

sudden decrease which continues till t = 20 days. Infected individuals increase for

the first ten days with rapid increase between 5-10 days. After that there is a rapid

decrease till t = 15. Then a decrease occurs slowly till t = 20. The proportion of

diagnosed shows an increase till t = 10 and after that diagnosed individuals reduce

42

t = 0

t = 5

t = 10 t = 15t = 20

t=20

-2 -1 0 1 2x

0.2

0.4

0.6

0.8

1.0

S

t = 5

t = 10

t = 15

t = 20t=0

-2 -1 0 1 2x

0.05

0.10

0.15

0.20

0.25E

t = 0

t=10

t=5

t=15t=0

t=20

-2 -1 0 1 2x

0.05

0.10

0.15

0.20

0.25I

t = 20

t=10

t=15

t=0

t=5

-2 -1 0 1 2x

0.05

0.10

0.15

0.20

0.25J

t = 0t = 5

t = 10

t = 15

t = 20

-2 -1 0 1 2x

0.2

0.4

0.6

0.8

1.0R

Figure 3.3: Solutions with initial condition (i) and without diffusion.

43

t = 0

t = 5

t = 10t = 15 t = 20-2 -1 0 1 2

x

0.2

0.4

0.6

0.8

1.0

S

t = 5

* * * * * * * * * * * * * * * *

*

*

*

*

*

*

*

*

*

* * * * * * * * * * * * * * * *

t = 10

t = 15

t=0t=20

-2 -1 0 1 2x

0.05

0.10

0.15

0.20

0.25E

t = 0

* * * * * * * * * * * * * * * *

*

*

*

*

*

*

*

*

*

* * * * * * * * * * * * * * * *

t = 20

t=10

t=5

t=15

-2 -1 0 1 2x

0.05

0.10

0.15

0.20

I

t = 0

t = 15

t = 20

t=10

t=5

-2 -1 0 1 2x

0.05

0.10

0.15

0.20

0.25J

t = 0t = 5

t = 10

t = 15

t = 20

-2 -1 0 1 2x

0.2

0.4

0.6

0.8

1.0R

Figure 3.4: Solutions with initial condition (ii) and without diffusion.

44

t = 0t=5

t=10t=15

t=20

-2 -1 0 1 2x

0.2

0.4

0.6

0.8

1.0

S

t = 10

t = 15

t=20

t=5t=0-2 -1 0 1 2

x

0.05

0.10

0.15

0.20

0.25

0.30E

t = 0

t=5

t=15

t=20

t=10

-2 -1 0 1 2x

0.02

0.04

0.06

0.08

0.10

0.12I

t = 0

t=20

t=5t=10

t=15

-2 -1 0 1 2x

0.01

0.02

0.03

0.04

0.05

0.06

0.07

J

t = 0

t = 10

t=5

t=20

t=15

-2 -1 0 1 2x

0.02

0.04

0.06

0.08

R

Figure 3.5: Solutions with initial condition (iii) and without diffusion.

with a quick fall between t = 15 and 20 days. Recovery is slow initially but after

t = 10, it is fairly quick.

Fig. 3.5, shows the output with initial condition (iii) and without diffusion. The

behavior of the susceptible is quite different as compared to the initial conditions

(i) and (ii). Susceptible move to the right of the initial domain of concentration

slowly and slowly without much change in proportion of susceptible population.

Initially the main concentration of the susceptible population is in the interval

[0, 2] but at t = 20 this shifts to [0.6, 2]. More and more of the population become

exposed during t=10 to 20 days of onset of the disease. During the first five days

of the onset of SARS, the proportion of infected people goes down in its domain

[−2, 0] and after that starts moving to domain [0, 2] with gradual increase. A sharp

increase is observed between t = 10 and 20 days [0, 1]. The number of diagnosed

45

t = 0

t = 5

t = 10t = 15 t = 20-2 -1 0 1 2

x

0.2

0.4

0.6

0.8

1.0S

t = 5

t = 10

t = 15 t = 20

-2 -1 0 1 2x

0.05

0.10

0.15

0.20

0.25

0.30E

t = 0

t=10

t=5

t=15t=20

-2 -1 0 1 2x

0.05

0.10

0.15

0.20I

t = 0

t = 15

t=10

t=20t=5

-2 -1 0 1 2x

0.05

0.10

0.15

0.20

0.25

J

t = 0t = 5

t = 10

t = 15

t=20

-2 -1 0 1 2x

0.2

0.4

0.6

0.8

1.0R

Figure 3.6: Solutions with initial condition (iv) and without diffusion.

individuals increases during the first five days in the domain [−2, 0]. After that

diagnosed individuals decrease with a slow pace. Also the concentration of the

diagnosed moves to the domain [0, 2]. A rapid increase of diagnosed can be seen

between t = 15 and 20 days. Till t = 10 recovery increases in the domain [−2, 0]

and slowly moves to domain [0, 1]. From t = 15 to t = 20 recovery attains maxi-

mum values in the domain [0, 1].

Fig 3.6, shows the output with the initial condition (iv) and without diffusion.

Susceptible individuals decrease rapidly after t = 5. At t = 15, the susceptible re-

duce to a very low level of concentration. Exposed individuals reach the maximum

level in first five days and after that start reducing. Infected individuals increase

till t = 10 and after that there is a sudden fall till t = 20. Diagnosed individuals

increase in the first ten days and after that there is a gradual decrease. There is

46

t = 0

t = 5

t = 10 t = 15 t = 20

-2 -1 0 1 2x

0.2

0.4

0.6

0.8

1.0

S

t = 5

t = 10

t = 20t=15

-2 -1 0 1 2x

0.02

0.04

0.06

0.08

0.10

0.12E

t = 0t = 20

t=15

t=5

t=10

-2 -1 0 1 2x

0.02

0.04

0.06

0.08

I

t = 0

t = 20

t=5

t=15

t=10

-2 -1 0 1 2x

0.02

0.04

0.06

0.08

0.10

0.12J

t = 0

t = 5

t = 10

t = 15t = 20

t=0-2 -1 0 1 2

x

0.1

0.2

0.3

0.4R

Figure 3.7: Solutions with initial condition (i) and with diffusion.

gradual increase in recovered individuals as shown in Fig 3.6.

3.5.2 Solutions of SEIJR model with diffusion (Case 1)

Fig 3.7, shows the output with initial condition (i) and with diffusion. With

the inclusion of diffusion in the system, susceptible spread in the entire region

at t = 5 with peak value 0.2228567. Initially exposed are mainly confined in do-

main [−1, 1.5]. At t = 5 exposed spread in the domain [−1, 1.5] with peak value

0.118055. At t = 10, exposed spread in the whole domain with peak value 0.068133.

After this a rapid decrease in the exposed individuals occurs. Infected individuals

spread in the whole domain [−2, 2] with the passage of time and at t = 10, infected

individual attains its maximum with peak value 0.077894. After this there is a

47

t = 0

t = 5

t = 10t = 15 t = 20-2 -1 0 1 2

x

0.2

0.4

0.6

0.8

1.0

S

t = 5

t = 10

t = 20t=15 t=0

-2 -1 0 1 2x

0.02

0.04

0.06

0.08

0.10

0.12

0.14

E

t = 0

t = 20

t=5

t=10

t=15

-2 -1 0 1 2x

0.01

0.02

0.03

0.04

0.05

0.06I

t = 0

t = 15

t = 20

t=10

t=5

-2 -1 0 1 2x

0.02

0.04

0.06

0.08

0.10J

t = 0

t = 5

t = 10

t = 20

t=10

t=15

-2 -1 0 1 2x

0.1

0.2

0.3

0.4R

Figure 3.8: Solutions with initial condition (ii) and with diffusion.

fall in the infected individuals and at t = 20, there is very low level of infected

individuals. A steady spread of diagnosed is observed in the main domain [−2, 2].

At t = 10 diagnosed are observed in the entire domain with peak value 0.105364.

After t = 10, there starts a decrease in the diagnosed population. Recovery starts

spreading in the domain and at t = 20 days, it completely spreads over the whole

domain with peak value 0.383669.

Fig 3.8, represents the results with the initial condition (ii) and with diffu-

sion. Susceptible quickly spreads in the whole domain [−2, 2] with low peak value

0.129482 at t = 5. In the first five days exposed spread in the domain [−1, 1]

with peak value 0.117691. In first ten days, exposed spread to whole domain with

peak value 0.040644. After ten days the maximum proportion of population gets

infected and spread in the main domain [−2, 2]. Though the population in the

48

t = 0

t = 5

t = 10

t = 15

t=20

-2 -1 0 1 2x

0.2

0.4

0.6

0.8

1.0

S

t = 10

t=5

t=20

t=15

t=0

-2 -1 0 1 2x

0.02

0.04

0.06

0.08

0.10

0.12

0.14

E

t = 0

t=10t=20

t=15

t=5

-2 -1 0 1 2x

0.02

0.04

0.06

0.08I

t = 0

t=20

t=15

t=5

t=10

-2 -1 0 1 2x

0.02

0.04

0.06

0.08J

t = 0t = 5

t = 10

t = 15

t = 20

-2 -1 0 1 2x

0.05

0.10

0.15

0.20

0.25R

Figure 3.9: Solutions with initial condition (iii) and with diffusion.

whole domain [−2, 2] is infected but main concentration of infected lies in domain

[−1, 1]. At time t = 10, diagnosed spread over the whole domain with peak value

0.085846. Diagnosed thereafter start reducing and at time t = 15 days reduce

to peak value 0.051532. Recovery starts slowly and spread to domain [−.6, .6] at

t = 5. At t = 20, recovered are spread over the whole domain with peak value

0.311524.

Fig 3.9, shows the results with the initial condition (iii) and with diffusion. Suscep-

tible spread from the initial domain of concentration [0, 2] to the domain [−1.5, 2]

at t = 5. After 5 days, susceptible start moving back. Susceptible are confined to

the domain [0.5, 2] at t = 15. Exposed individuals also shifts from domain [−2, 0]

to [0, 2] with the passage of time. Initially infected are confined to domain [−2, 0].

49

t = 0

t = 5 t = 10 t = 15 t = 20

-2 -1 0 1 2x

0.2

0.4

0.6

0.8

1.0

S

t = 20t=15

t=10

t=5

-2 -1 0 1 2x

0.01

0.02

0.03

0.04

0.05E

t = 20

t=0

t=5

t=10

t=15

-2 -1 0 1 2x

0.01

0.02

0.03

0.04

I

t = 0

t = 20

t=5

t=10

t=15

-2 -1 0 1 2x

0.005

0.010

0.015

0.020

0.025

0.030

0.035J

t = 0t = 10

t=5

t=10

t=20

t=15

-2 -1 0 1 2x

0.05

0.10

0.15

R

Figure 3.10: Solutions with initial condition (iv) and with diffusion.

Infected spread at very slow pace and a small pulse with peak value 0.004476 can

be observed at t = 5. After ten days, the domain of concentration of infected

people moves to [−1, 1]. At t = 15, infection spread in the whole domain [−2, 2].

At t = 5, a small proportion of diagnosed remain in the domain [−2, 0]. There

is, however, a rapid increase in the number of diagnosed with peak value 0.074575

at t = 20. At t = 5, recovery is restricted to the domain [−2, 0]. After 20 days,

recovered spread in the domain [−2, 2], with peak value 0.206879.

Fig 3.10, shows the results with initial condition (iv) and with diffusion. A sudden

fall in the susceptible is observed at t = 5 with peak value 0.005466. During the

same time, exposed spread to domain [−1.5, 1.5] with peak value 0.043322. At

t = 10, infected spread in the domain [−1.8, 1.8] with peak value 0.010391. At

50

t = 5, diagnosed spread to domain[−1, 1] with peak value 0.033966. Diagnosed

spread further in the domain [−1.8, 1.8] at t = 15 with peak value 0.01385. Recov-

ery initially occurs in the domain [−1, 1] at t = 5 with peak value 0.045636. There

is a further increase of recovered at t = 10 with peak value 0.106593. At t = 20

there is maximum recovery with peak value 0.147773.

3.5.3 Other cases

Graphs of numerical solutions of Cases 2-4, obtained both with and without diffu-

sion, for all cases specified in Table 3.6 are quite similar to Case 1. Thus graphs for

Cases 2-4 are not reproduced here. Summarized results for Cases 2-4 are shown in

Tables 3.7, 3.8, 3.9 and 3.10. Here Sj, Ej, Ij and Rj for j = (i), (ii), (iii) and (iv)

represent the proportion of susceptible, exposed, infected and recovered population

at critical points in the domain [−2, 2] without and with diffusion, for the initial

condition (i), (ii), (iii) and (iv) respectively. The following description is based on

the information provided in Tables 3.7 and 3.10.

In Case 2, there is an increase in the recovery coefficient of diagnosed individual,

γ2 = 0.25 while keeping values of transmission coefficient, β and recovery rate of

infected, γ1 the same as in Case 1. There is a slow decrease in susceptible popu-

lation as compared to Case 1, for all initial conditions, with and without diffusion

in the system. Fewer individuals seem to be exposed and infected to disease in the

first five days with conditions (i), (ii) and (iv) and after five days their proportion

is greater in comparison to Case 1. But with condition (iii), there is a smaller

proportion of exposed all the time. There is higher proportion of recovered in Case

2 as compared to Case 1, both with and without diffusion.

In Case 3, there is increase in recovery rate of infected individuals, γ1 = 0.175 as

compared to Case 1 and Case 2. There is slow decrease in susceptible population

during the first five days for condition (i) − (iv), with and without diffusion as

compared to Cases 1 and 2. Initially exposed are small in proportion as compared

to Case 1 but after five days the proportion of exposed is more than in Case 1

with initial condition (i), (ii) and (iv). On the other hand with initial condition

(iii), there is a decrease in exposed individuals. Population of infected reduces

51

Table 3.7: Peak values of susceptible (S) and exposed (E) (without diffusion)

Case t S(i) S(ii) S(iii) S(iv) E(i) E(ii) E(iii) E(iv)

1 00 98.0 98.0 97.0 96.0 0.00 0.00 0.00 0.00

05 66.4 66.4 96.9 46.3 19.5 19.5 .171 29.0

10 .073 2.34 96.8 .039 21.3 21.3 1.25 .039

15 .001 2.36 96.7 .059 2.94 2.94 7.87 2.05

20 .002 2.38 96.6 .078 .410 .410 30.1 0.288

2 00 98.0 98.0 97.0 96.0 0.00 0.00 0.00 0.00

05 68.2 68.2 96.9 48.6 18.3 18.3 .161 27.57

10 .156 2.34 96.9 .039 22.6 22.6 1.12 15.7

15 .001 2.36 96.8 .059 3.12 3.12 7.38 2.16

20 .003 2.38 96.7 .078 .434 .434 26.3 .303

3 00 98.0 98.0 97.0 96.0 0.00 0.00 0.00 0.00

05 68.7 68.7 96.9 49.3 18.0 18.0 .159 27.3

10 .151 2.34 96.8 .039 22.8 22.8 1.11 15.9

15 .001 2.36 96.8 .059 3.15 3.15 7.36 2.18

20 .002 2.38 96.7 .078 .438 .438 26.2 .306

4 00 98.0 98.0 97.0 96.0 0.00 0.00 0.00 0.00

05 69.7 68.2 96.9 50.5 17.4 18.3 .155 26.6

10 .181 2.34 96.8 .039 23.4 22.6 1.04 16.3

15 .001 2.36 96.8 .059 3.24 3.12 6.92 2.23

20 .002 2.38 96.7 .078 .451 .434 22.7 .313

Table 3.8: Peak values of infected (I) and recovered (R) (without diffusion)

Case t I(i) I(ii) I(iii) I(iv) R(i) R(ii) R(iii) R(iv)

1 002.00 2.00 3.00 4.00 0.00 0.00 0.00 0.00

05 6.91 6.91 .189 11.4 3.53 3.53 1.86 6.61

10 21.4 21.4 .416 17.7 32.4 32.4 2.59 41.1

15 5.37 5.37 2.98 3.91 69.9 69.9 2.833 74.8

20 .891 .891 11.9 .632 87.6 87.6 8.47 89.4

2 00 2.00 2.00 3.00 4.00 0.00 0.00 0.00 0.00

05 6.55 6.55 .189 10.9 3.71 3.71 2.01 6.98

10 21.7 21.7 .403 18.1 33.7 33.7 2.71 43.2

15 5.64 5.64 2.62 4.09 73.4 73.4 2.89 78.3

20 .942 .942 11.6 .665 90.3 90.3 8.39 91.9

3 00 2.00 2.00 3.00 4.00 0.00 0.00 0.00 0.00

05 6.08 6.08 .141 10.1 3.92 3.92 2.01 7.35

10 19.8 19.8 .377 16.2 34.8 34.8 2.66 43.9

15 4.72 4.72 2.48 3.38 72.5 72.5 2.86 77.0

20 .742 .742 10.9 .521 88.9 88.9 8.82 90.5

4 00 2.00 2.00 3.00 4.00 0.00 0.00 0.00 0.00

05 6.20 6.55 .189 10.4 3.32 3.71 1.86 6.27

10 21.9 21.7 .394 18.4 30.5 33.7 2.60 39.4

15 5.82 5.64 2.35 4.23 68.7 73.4 2.83 73.8

20 .975 .942 11.2 .687 87.1 90.3 7.26 89.0

52

Table 3.9: Peak values of susceptible (S) and exposed (E) (with diffusion)

Case t S(i) S(ii) S(iii) S(iv) E(i) E(ii) E(iii) E(iv)

1 00 98.0 98.0 97.0 96.0 0.00 0.00 0.00 0.00

05 22.9 12.9 48.4 .547 11.8 11.8 1.18 4.33

10 .595 .859 34.9 .021 6.81 4.06 8.39 .584

15 .001 .002 15.1 .002 .886 .531 13.2 .077

20 .002 .003 .004 .005 .123 .077 3.05 .017

2 00 98.0 98.0 97.0 96.0 0.00 0.00 0.00 0.00

05 23.8 13.7 48.5 .669 .111 11.2 1.09 4.30

10 .728 .917 34.9 .024 .072 4.25 7.99 .603

15 .002 .002 15.7 .003 .009 .556 13.1 .079

20 .003 .004 .005 .007 .001 .079 3.24 .016

3 00 98.0 98.0 97.0 96.0 0.00 0.00 0.00 0.00

05 23.9 13.9 48.5 .683 11.0 11.1 1.10 4.33

10 .760 .949 35.0 .026 7.21 4.29 7.97 .609

15 .001 .002 15.9 .003 .942 .561 13.2 .079

20 .003 .003 .005 .005 .130 .081 3.28 .017

4 00 98.0 98.0 97.0 96.0 0.00 0.00 0.00 0.00

05 24.4 13.7 48.5 .700 10.7 11.2 1.10 4.35

10 .876 .917 7.78 .029 7.39 4.25 7.78 .615

15 .001 .002 16.7 .003 .966 .556 13.0 .080

20 .002 .004 .005 .005 .134 .079 3.51 .017

Table 3.10: Peak values of infected (I) and recovered (R) (with diffusion)

Case t I(i) I(ii) I(iii) I(iv) R(i) R(ii) R(iii) R(iv)

1 00 2.00 2.00 3.00 4.00 0.00 0.00 0.00 0.00

05 4.63 5.03 .448 3.30 2.94 3.21 1.84 4.56

10 7.79 5.27 3.61 1.04 17.4 16.0 2.78 10.7

15 1.72 1.07 7.73 .166 32.0 26.7 10.5 13.7

20 .272 .167 3.98 .028 38.4 31.2 20.7 14.8

2 00 2.00 2.00 3.00 4.00 0.00 0.00 0.00 0.00

05 3.10 3.40 1.99 4.89 3.10 3.40 1.99 4.89

10 18.1 16.8 2.91 11.2 18.1 16.8 2.91 11.2

15 33.3 27.8 10.9 14.1 33.3 27.8 10.9 14.1

20 39.2 31.7 21.6 14.9 39.2 31.7 21.6 14.9

3 00 2.00 2.00 3.00 4.00 0.00 0.00 0.00 0.00

05 4.08 4.46 .394 2.95 3.25 3.57 1.99 5.03

10 7.11 4.77 3.20 .895 18.4 16.9 2.89 11.1

15 1.48 .905 7.05 .135 32.8 27.3 11.1 13.9

20 .223 .135 3.65 .022 38.6 31.3 21.3 14.8

4 00 2.00 2.00 3.00 4.00 0.00 0.00 0.00 0.00

05 4.21 4.82 .422 3.21 2.79 3.39 35.1 4.45

10 7.99 5.36 3.32 1.07 16.5 16.8 2.78 10.5

15 1.84 1.11 7.53 .174 31.3 27.8 9.98 13.5

20 .294 .174 4.27 .029 37.8 31.7 19.9 14.6

53

Table 3.11: Peak values of infected at t = 20

Cases Peak values for initial conditions Reproductive Number

(i) (ii) (iii) (iv)

1 0.891 0.891 11.90 0.632 2.83

2 0.942 0.942 11.60 0.665 2.64

3 0.742 0.742 10.90 0.521 2.58

4 0.975 0.942 11.20 0.687 2.64

remarkably in Case 3. Diagnosed class also has a decrease in individuals. More

individuals recover in Case 3 as compared to Case 1. But the recovered population

in Case 3 is less than that in Case 2.

In Case 4, there is a reduced value of the transmission coefficient, β = 0.7 as com-

pared to Cases 1, 2, 3. This causes a slow decrease in the susceptible population as

compared to Cases 1, 2 and 3 till t = 10. Exposed behave similarly as in Cases 2

and 3 with the number of individuals first decreasing and then increasing as com-

pared to Case 1. Population of infected is less than Cases 1 at t=5. After that, till

t=20 days , more infected individuals are observed in Case 4 as compared to Cases

1. Recovered individuals slow down here and the number of recovered population

is less in this case as compared to other cases.

3.6 Discussion

An SEIJR Model for SARS (G. Chowell et al. [45]) is considered with the inclu-

sion of diffusion in the system. Four different initial conditions are taken for the

population distribution. The equation governing the system are solved numerically

using operator splitting method. The reproduction number RI is calculated for the

disease. It is shown that disease dies out for RI < 1, in disease-free equilibrium.

It however prevails for endemic equilibrium, where RI > 1 as shown in Table 3.11.

Stability of solutions with and without diffusion is established using Routh-Hurwitz

conditions. The value of the reproduction number RI depends on ten parameters.

54

The parameters transmission coefficient β, recovery rate in infectious class γ1 and

recovery coefficient in diagnosed γ2 have been varied to observe the effects on the

spread of disease. Hence four cases are produced to see the effect on the spread of

disease. Bifurcation values of transmission coefficient β and recovery coefficients

γ1 and γ2 are calculated. It is observed that diffusion causes an increase in the

bifurcation value of β and a decrease in the value of recovery coefficients. This

shows that the system can be stable for larger value of β and smaller values of

recovery rates γ1 and γ2 in the presence of diffusion.

Numerical solution with initial condition (i), as shown in Fig 3.3 and Fig 3.7, that

in the absence of diffusion, only the population in the domain [−1, 1.5] becomes sus-

ceptible, but when diffusion is introduced susceptible spread over the whole domain

in first five days. Similarly the exposed remain confined in the interval [−1, 1.5] in

the absence of diffusion. But with diffusion population outside the domain [−1, 1.5]

also become exposed to infection and after ten days exposed population spread to

whole domain. With and without diffusion in the system, infection reached its peak

in the first ten days. With the inclusion of diffusion, however, infection spreads

over the whole domain [−2, 2]. Recovered also follows the same pattern. Numerical

solution of initial condition (ii), as shown in Fig 3.4 and Fig 3.8, follows the same

pattern as in condition (i) with only difference in concentration of population in

different domain. In the absence of diffusion infected population fluctuate inside

the domain [−.5, .5], but with diffusion in the system fluctuations follows with the

spread in the whole domain after ten days.

Numerical solution with initial condition (iii), as shown in Fig. 3.5 and Fig. 3.9,

shows the main concentration of susceptible shifts slightly to right of domain [0, 2]

in 20 days. Infected move to domain [0, 2] from domain [−2, 0] and with that di-

agnosed and recovered also follow the same pattern. With diffusion in the system

Susceptible start spreading to the left of the domain [−2, 2] but are mainly con-

fined in [−0.8, 2]. Exposed grows in the smaller domain [−1, 0.5] first and then

with passage of time spread in the domain [−1.5, 2]. Infected shifts their domain

from [−2, 0] to [−1, 2] followed with decrease, increase and then again decrease in

proportion. Thus with diffusion more individuals get infected within a short time

55

and infection spreads quickly and reaches its maximum in 15 days covering almost

the whole domain. In the absence of diffusion the maximum number of infected

are observed after twenty days in domain [0, 1], reflecting the intensity of infection

more than that with diffusion during the same time. A large proportion of popu-

lation is recovered with diffusion in the system during the same time as compared

to without diffusion.

Numerical solution with initial condition (iv), as shown in Fig. 3.6 and Fig. 3.10,

demonstrate that diffusion causes the infection to spread out from domain [−.7, .7]

to [−2, 2]. The intensity of the infection also becomes less than the initial intensity

as it spreads to [−1.5, 1.5]. It has been observed in Tables 3.7 and 3.10 that when

recovery is improved in diagnosed class with an increased value of γ2 as in Case

2, a smaller proportion of the population becomes infected and the proportion of

the recovered increases. Even better result is obtained with the greater recovery of

infected with an increased value of γ1 as shown in Case 3, where the proportion of

recovered is higher than previous Case 2 even with a lower value of diagnosed re-

covery, γ2. With recovery coefficients γ1 the same and decreasing the transmission

coefficient β as in Case 4, recovery is observed to be slower than the original Case 1.

In the next chapter the SEIJR model is further examined for the impact of

cross diffusion on the transmission dynamics of SARS. Different cases of positive

and negative cross diffusion are considered to study the effects of cross-diffusion.

56

Chapter 4

Numerical Simulation of Cross

Diffusion on Transmission

Dynamics of SARS

4.1 Introduction

Recently, ecological and epidemiological modelling are increasingly focused on spa-

tially structured models and emergent heterogeneity. The wave of infection is often

caused by the diffusion within the populations in a given spatial region, that gener-

ates periodic infection [219]. Spatial epidemiology can be helpful in this situation

to develop strategies to control the transmission of disease. As a result, spatial

epidemiology with self-diffusion and cross-diffusion has arisen as the principal sci-

entific discipline devoted to understand the causes and consequences of spatial

heterogeneity in infectious diseases, particularly in zoonoses diseases such as in-

fluenza, SARS and MERS [219] that are transmitted to humans from non-human

vertebrate reservoirs. The most important and difficult aims are now to incorpo-

rate spatial effects and specify the dispersion of individuals. Reaction−diffusion

systems have gained much importance in recent years in epidemiological modelling

to target these spacial effects. The investigation of dynamical systems with diffu-

sion in living organism or species has become a fascinating phenomenon in system

biology. Various terms such as self-diffusion, tracer diffusion, mutual diffusion, up-

57

hill diffusion, inter diffusion and cross-diffusion are used to describe diffusion of

objects/species [62, 224]. The term diffusion was first analysed experimentally for

binary liquid mixtures in 1850 [87] and later, in 1855, Fick [75] developed theories

for it. However it was not till 1955 when the existence of cross-diffusion was ex-

perimentally verified by Baldwin et al. [13].

Self-diffusion is termed as passive diffusion where the diffusing object moves along

its intensification. On the other hand, cross-diffusion involves reverse-mobility. So

in terms of population compartments the phenomenon of cross-diffusion can be

defined as the tendency of the susceptible to keep away from the infected as the

susceptible individual has the ability to recognize the infected and move away from

them [213]. This fact has been generally overlooked despite of its potential eco-

logical reality and intrinsic theoretical interest. The value of the cross-diffusion

coefficient can possibly be positive, negative or zero. Positive cross-diffusion refers

to the movement of these susceptible towards lower density of infected while nega-

tive cross-diffusion means that susceptible tend to diffuse towards regions of higher

densities of the infected [219].

Cross-diffusion with reaction diffusion systems occur in living, social and physico-

chemical contexts etc [229]. A number of ecologists and mathematicians have made

contributions to investigate the stability behavior of a system of interacting popu-

lations by taking into account the effect of self-diffusion as well as cross-diffusion.

Kerner [137] found that cross-diffusion can induce pattern-forming instability in an

ecological situation. Gurtin [97] developed mathematical models to investigate the

effects of cross-diffusion and self diffusion on the population dynamic and showed

that cross-diffusion could give rise to the segregation of two species. In the litera-

ture, Jorne [128, 129], Freedman and Shukla [79], Gatto and Rinaldi [82], Hastings

[103], Okubo [180] and Chattopadhyay et al. [38], developed models for interactive

populations to study the effect of cross-diffusion. Shukla and Verma [214] showed

that the cross-diffusion of species may lead to stability, depending upon the na-

ture and the magnitudes of the self and cross-diffusion coefficients. Kuznetsov

et al. [145] developed a mathematical model of cross-diffusion with two interacting

components qualitatively describing the spatial-temporal dynamics of a mixed-age

58

monospecies forest. Zhang and Fu [253] analysed the stability of the nonnegative

constant steady states for a realistic and mathematically complex cross-diffusive

prey-predator system. They showed that in the presence of cross-diffusion, the

unique positive constant steady state is asymptotically stable, also non-constant

positive steady solutions can exist. Chattopadhyay and Tapaswi [37] showed the

crucial role of negative cross-diffusion in the structure of tumour growth. Sun et al.

[219] investigated the spatial model with cross-diffusion in the susceptible. In re-

cent years several researchers investigated the dynamics of interacting populations

with self and cross-diffusion [21, 176, 221]. However untill now, little work has been

done to study the importance of cross-diffusion phenomena for the spatial spread

and transmission of infectious diseases.

The present study is a numerical investigation of the effects of cross-diffusion on

the transmission dynamics of Severe Acute Respiratory Syndrome (SARS). Here

the self diffusion is defined as the effect of the population pressure on the diffusion

of its own compartmental population and cross-diffusion as the density pressure

on compartmental population because of movement in another compartment. So

here, each individual is affected by pressures based on its own population and the

population of the other compartments, due to mixing between different compart-

ments. The spatial epidemic model with both self-and cross-diffusion along with

the parameters of the model, are defined in Sec. 2. Stability of point equilibrium

and bifurcation analysis of the model are derived in Sec. 3. Numerical scheme

and solutions are given in Sec. 4 and 5. In Sec. 6 the results are summarised and

concluded. In order to differentiate between two types of diffusion simple diffusion

has been referred as self diffusion in this chapter.

4.2 The SEIJR epidemic model

4.2.1 Equations

This model is based on the SEIJR model discussed in Chapter 3 with the inclusion

of self and cross-diffusion in the equations governing the system. Cross-diffusion

is introduced in susceptible (S) and exposed to disease (E) populations only. The

59

Table 4.1: Interpretation of parameters (per day)

Parameter Description Values

Π Rate of inflow of susceptible individuals into region 3.3× 10−5b

β Transmission rate 0.75a

µ Rate of natural mortality 3.4× 10−5b

l Relative measure of reduced risk among diagnosed 0.38a

κ Rate of progression from exposed to infected 0.33a

q Relative measure of infectiousness for exposed individuals 0.1a

α Rate of progression from infective to diagnosed 0.33a

γ1 Recovery rate of infected individuals 0.125a

γ2 Recovery rate of diagnosed individuals 0.2a

δ SARS induced mortality rate .006a

a(Chowell G. et al. [45]), b (Gummel A. B. et al. [95] )

total population proportion is to be N where N = S + E + I + J +R.

∂S

∂t= −β

(I + qE + lJ)

NS − µS +Π+ d1

∂2S

∂x2+ dse

∂2E

∂x2(4.1)

∂E

∂t= β

(I + qE + lJ)

NS − (µ+ κ)E + d2

∂2E

∂x2+ des

∂2S

∂x2(4.2)

∂I

∂t= κE − (µ+ α + γ1 + δ)I + d3

∂2I

∂x2(4.3)

∂J

∂t= αI − (µ+ γ2 + δ)J + d4

∂2J

∂x2(4.4)

∂R

∂t= γ1I + γ2J − µR + d5

∂2R

∂x2(4.5)

where the variables S, E, I, J and R denote the proportions of susceptible, exposed,

infected, diagnosed and recovered individuals respectively. d1, d2, d3, d4 and d5 are

self diffusion constants while dse and des are cross-diffusion constants. Table 4.1

provides description and the values of the parameters.

60


The domain of all the calculations is taken to be [−2, 2]. Boundary and initial


∂S(−2, t)

∂x=

∂E(−2, t)

∂x=

∂I(−2, t)

∂x=

∂J(−2, t)

∂x=

∂R(−2, t)

∂x= 0 (4.6)

∂S(2, t)

∂x=

∂E(2, t)

∂x=

∂I(2, t)

∂x=

∂J(2, t)

∂x=

∂R(2, t)

∂x= 0 (4.7)

(i)

S0 = 0.98Sech(5x− 1), − 2 ≤ x ≤ 2.

E0 = 0, − 2 ≤ x ≤ 2.

I0 = 0.02Sech(5x− 1), − 2 ≤ x ≤ 2.

J0 = 0, − 2 ≤ x ≤ 2.

R0 = 0, − 2 ≤ x ≤ 2.

(ii)

S0 ={

0.96Sech(15x), − 2 ≤ x ≤ 2.

E0 = 0, − 2 ≤ x ≤ 2.

I0 =

0, − 2 ≤ x < −.6,

0.04, − .6 ≤ x ≤ .6,

0, .6 < x ≤ 2.

J0 = 0, − 2 ≤ x ≤ 2.

R0 = 0, − 2 ≤ x ≤ 2.

The initial conditions are graphed in Fig. 4.1. Under initial condition (i) large

proportions of the susceptible and infected populations are concentrated towards

the right half of the main domain. For initial condition (ii) most susceptible S

are near the middle of the domain [−2, 2] and infectious individuals widely spread

around the middle of the domain.

Four cases with different pairs of values for the cross-diffusion coefficient des and

61

t = 0

t = 0

(i)-2 -1 0 1 2

x

0.2

0.4

0.6

0.8

1.0

1.2S,I

t = 0

t = 0

(ii)

-2 -1 0 1 2x

0.2

0.4

0.6

0.8

1.0S,I

Figure 4.1: Initial Conditions (i) and (ii).

dse have been chosen, as given in Table 4.2. Numerical solutions are given for all

these cases.

Table 4.2: Cases for cross-diffusion

Case dse des

(a) 0.00000 0.00000

(b) 0.01250 0.00000

(c) 0.01250 0.00350

(d) −0.0125 0.00350


4.3.1 Reproduction number and disease-free equilibrium

(DFE)

The variation matrix for the system of equations (4.1) - (4.5) and the conditions

for disease-free equilibrium are chosen to be the same as in the previous chapter.

Thus the expression for the reproduction number, without diffusion is as follows:

RI=q(µ+α+γ1+δ)(µ+γ2+δ)+κ(µ+γ2+δ)+lακ

(µ+α+γ1+δ)(µ+γ2+δ)(κ+µ).

62

4.3.2 Stability of endemic equilibrium with cross-diffusion

To calculate the small perturbations S1(x, t), E1(x, t), I1(x, t),J1(x, t) and R1(x, t),

the equations are linearized about the point of equilibrium P ∗(S∗, E∗, I∗, J∗, R∗)

as described in [35, 201].

∂S1

∂t= a11S1 + a12E1 + a13I1 + a14J1 + a15R1 + d1

∂2S1

∂x2+ dse

∂2E1

∂x2(4.8)

∂E1

∂t= a21S1 + a22E1 + a23I1 + a24J1 + a25R1 + d2

∂2E1

∂x2+ des

∂2S1

∂x2(4.9)

∂I1∂t

= a31S1 + a32E1 + a33I1 + a34J1 + a35R1 + d3∂2I1∂x2

(4.10)

∂J1∂t

= a41S1 + a42E1 + a43I1 + a44J1 + a45R1 + d4∂2J1∂x2

(4.11)

∂R1

∂t= a51S1 + a52E1 + a53I1 + a54J1 + a55R1 + d5

∂2R1

∂x2(4.12)

where a11, a12, a13 etc are the elements of the variational matrix V ∗ calculated using

the same method as described in [198] and are same as given in Chapter 3. Assume

a Fourier series solution exists for equations (4.8) - (4.12) of the form:

S1(x, t) =∑k


E1(x, t) =∑k


I1(x, t) =∑k


J1(x, t) =∑k


R1(x, t) =∑k


where k = nπ2, (n = 1, 2, 3, · · · · · · ) is the wave number for the node n. Substituting

the values of S1, E1, I1, J1, R1 in the equations (4.8) - (4.12), the equations are

transformed into∑k

(a11 − d1k2 − λ)Sk +

∑k

(a12 − dsek2)Ek +

∑k

a13Ik +∑k

a14Jk +∑k

a15Rk = 0

(4.18)∑k

(a21 − desk2)Sk +

∑k

(a22 − d2k2 − λ)Ek +

∑k

a23Ik +∑k

a24Jk +∑k

a25Rk = 0

(4.19)

63

∑k

a32Ek +∑k

(a33 − d3k2 − λ)Ik = 0 (4.20)

∑k

a43Ik +∑k

(a44 − d4k2 − λ)Jk = 0 (4.21)

∑k

a53Ik +∑k

a54Jk +∑k

(a55 − d5k2 − λ)Rk = 0 (4.22)

The Variational matrix V d for the equations (4.18) - (4.22) is

V d =

a11 − d1k2 a12 − dsek

2 a13 a14 a15

a21 − desk2 a22 − d2k

2 a23 a24 a25

0 a32 a33 − d3k2 0 0

0 0 a43 a44 − d4k2 0

0 0 a53 a54 a55 − d5k2

Where a11, a12, a13,... are same as given in Chapter 3. The characteristic equation

for the variational matrix V d is given as

λ5 + q1λ4 + q2λ

3 + q3λ2 + q4λ+ q5 = 0 (4.23)

where q1, q2, q3, q4 and q5 are calculated with the same technique as used in [198].

Routh-Hurwitz Conditions for stability are given as:

C1 : q1 > 0,

C2 : q5 > 0,

C3 : q1q2 − q3 > 0,

C4 : q1q2q3 + q1q5 − (q23 + q21q4) > 0,

C5 : (q1q4 − q5)(q1q2q3 − q23 − q21q4) + q21q4q5 − (q5(q1q2 − q3)2q1q

25) > 0.

The numerical results of L.H.S of Routh-Hurwitz Conditions on the point of

equilibrium, P1 = (0.331581, 0.000064, 0.000046, 0.000074, 0.607887) are given in

Table 4.3 for all cases taken in Table 4.2.

64

Table 4.3: Routh-Hurwitz criteria with and without cross-diffusion

Case dse des C1 C2 C3 C4 C5 Stability

(a) 0.0000 0.0000 1.066 4.9 × 10−8 0.295 0.0048 8.1 × 10−7 Stable

(b) 0.0125 0.0000 1.013 1.8 × 10−8 0.242 0.0016 4.1 × 10−8 Stable

(c) 0.0125 0.0035 1.066 4.4 × 10−8 0.296 0.0048 −7.2 × 10−7 Unstable

(d) −0.0125 0.0035 1.013 1.7 × 10−8 0.242 0.0016 −1.2 × 10−7 Unstable

4.3.3 Reproduction number with diffusion

Variational matrix method is used to calculate reproduction number with diffusionRd

I . The variational matrix with diffusion for P0 = (1, 0, 0, 0, 0) is given as followes:

Vd

=

−(d1k2 + µ) −(dsek

2 + qβ) −β −lβ 0

−desk2 qβ − (d2k

2 + κ + µ) β lβ 0

0 κ −(d3k2 + α + γ1 + δ + µ) 0 0

0 0 α −(d4k2 + γ2 + δ + µ) 0

0 0 γ1 γ2 −(d5k2 + µ)

It is observed that last eigenvalue −(d5k2 + µ) of variational matrix V d is negative

and all entries above it are zero. This allows to eliminate the last row and column.

So, the reduced matrix is given as follows:

V d =

−(d1k

2 + µ) −(dsek2 + qβ) −β −lβ

−desk2 qβ − (d2k

2 + κ+ µ) β lβ

0 κ −(d3k2 + α + γ1 + δ + µ) 0

0 0 α −(d4k2 + γ2 + δ + µ)

The characteristic equation for the above matrix is given as :

λ4 + p1λ3 + p2λ

2 + p3λ1 + p4 = 0 (4.24)

Where p1, p2, p3 and p4 are calculated as in [198] and are given in appendix A.4.

The Routh-Hurwitz criterion for the stability is given as in [200]. The Routh Hur-

witz condition p4 > 0 gives the following expression for reproduction number, RdI

with diffusion:

RdI =

k2(A+B + C) +D

(d1k2 + µ)(d3k2 + α + γ1 + δ + µ)(d4k2 + γ2 + δ + µ)(d2k2 + κ+ µ)(4.25)

where

A = (desdsek2 + d1qβ + desqβ)(α + γ1 + δ + µ)(γ2 + δ + µ)

65

B = (d3(d4k2 + γ2 + δ + µ)(des(dsek

4 + k2qβ) + qβ(d1k2 + µ)) − d4(β(d1k

2 +

µ)(κ+ q(α+ γ1+ δ+µ))+ desk2(dsek

2(α+ γ1+ δ+µ)+β(κ+ q(α+ γ1+ δ+µ)))))

C = (d1 + des)βκ(lα + γ2 + δ + µ)

D = βµ(q(α + γ1 + δ + µ)(γ2 + δ + µ) + κ(lα + γ2 + δ + µ))

The values ofRdI for the cases (a) - (d) as defined in Table 4.2 are given in Table4.4.

Table 4.4: Reproduction number

Cases Value of RdI

(a) 2.6

(b) 3.95

(c) 3.96

(d) 1.31


The first excited mode of the oscillation n is calculated by the same technique as

used in [35]. In Case (a), n = 1 represents the first mode of excitation as being

closest to the β-axis of the curve given by equation (4.26). Similarly, n = 1 is the

first mode of excitation for Cases (b)− (d).

f(β) = (q1q4 − q5)(q1q2q3 − q23 − q21q4) + q21q4q5 − (q5(q1q2 − q3)2q1q

25). (4.26)

Bifurcation values of the transmission coefficient β, recovery rate of infected indi-

viduals γ1 and recovery rate of diagnosed individuals γ2 are given in Table 4.5. It

is observed that the bifurcation value of the transmission coefficient, β increases

in Cases (a) and (b) and decreases in Cases (c) and (d). So the system remains

stable for higher values in Cases (a) and (b) while in Cases (c) and (d), the system

becomes stable at smaller bifurcation values. For the infected and diagnosed recov-

ery coefficients, γ1 and γ2, the system becomes stable at smaller bifurcation values

66

n=1

0.5 1.0 1.5 2.0Β

2.´ 10-6

4.´ 10-6

6.´ 10-6

8.´ 10-6

0.00001

fHΒL

n=2

0.5 1.0 1.5 2.0Β

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

fHΒL

n=3

0.5 1.0 1.5 2.0Β

-0.005

0.005

0.010

0.015

0.020

0.025

fHΒL


Table 4.5: Bifurcation value of β, γ1 and γ2 with cross-diffusion

Cases Value Considered Bifurcation Value

β γ1 γ2 β γ1 γ2

(a) 0.75 0.125 0.20 0.776 0.109 0.182

(b) 0.75 0.125 0.20 0.869 0.108 0.181

(c) 0.75 0.125 0.20 0.727 0.180 0.279

(d) 0.75 0.125 0.20 0.734 0.174 0.269

in Case (a) and (b). In Cases (c) and (d), the system becomes stable at higher

bifurcation values. The corresponding bifurcation diagrams are given in appendix

A.4.


The operator splitting technique [244] has been used to solve the SEIJR model

equations. The equations are divided into two groups of sub equations. The first

group comprises the nonlinear reaction equations to be used for the first half-time

67

step:1

2

∂S

∂t= −β

(I + qE + lJ)

NS − µS +Π (4.27)

1

2

∂E

∂t= β

(I + qE + lJ)

NS − (µ+ κ)E (4.28)

1

2

∂I

∂t= κE − (µ+ α + γ1 + δ)I (4.29)

1

2

∂J

∂t= αI − (µ+ γ2 + δ)J (4.30)

1

2

∂R

∂t= γ1I + γ2J − µR (4.31)



1

2

∂S

∂t= d1

∂2S

∂x2+ dse

∂2E

∂x2(4.32)

1

2

∂E

∂t= d2

∂2E

∂x2+ des

∂2S

∂x2(4.33)

1

2

∂I

∂t= d3

∂2I

∂x2(4.34)

1

2

∂J

∂t= d4

∂2J

∂x2(4.35)

1

2

∂R

∂t= d5

∂2R

∂x2(4.36)

The solution of the first half-time step is given in Chapter 3. For the second half-

time step,

Sj+1i = S

j+ 12

i +∆t

(∆x)2(d1(S

j+ 12

i−1 −2Sj+ 1

2i +S

j+ 12

i+1 )+dse(Ej+ 1

2i−1 −2E

j+ 12

i +Ej+ 1

2i+1 )) (4.37)

Ej+1i = E

j+ 12

i +∆t

(∆x)2(d2(E

j+ 12

i−1 −2Ej+ 1

2i +E

j+ 12

i+1 )+des(Sj+ 1

2i−1 −2S

j+ 12

i +Sj+ 1

2i+1 )) (4.38)

Ij+1i = I

j+ 12

i + d3∆t

(∆x)2(I

j+ 12

i−1 − 2Ij+ 1

2i + I

j+ 12

i+1 ) (4.39)

J j+1i = J

j+ 12

i + d4∆t

(∆x)2(J

j+ 12

i−1 − 2Jj+ 1

2i + J

j+ 12

i+1 ) (4.40)

Rj+1i = R

j+ 12

i + d5∆t

(∆x)2(R

j+ 12

i−1 − 2Rj+ 1

2i +R

j+ 12

i+1 ) (4.41)

The stability condition satisfied by the numerical method described above is given

as:dn∆t

(∆x)2≤ 0.5, n = 1, 2, 3, 4, 5. (4.42)

In each case, ∆x = 0.1, ∆t = 0.03, d1 = 0.025, d2 = 0.01, d3 = 0.001, d4 = 0.0 and

d5 = 0.0 are used.

68


Four cases, with different pairs of values of the cross-diffusion coefficients des and

dse, have been chosen, as given in Table 4.2. Numerical solutions are given for all

these cases.

4.5.1 Numerical solution for initial condition (i)

Fig. 4.3, shows the numerical solution with initial condition (i) and self diffusion

for Case (a). In the first five days a significant proportion of the susceptible pop-

ulation moves to the exposed class and the effective domain of susceptible alters

from [−1, 1.5] to [−2, 2], producing a pulse with peak value 0.222857. At t = 10

days almost the whole proportion of susceptible have been exposed to the disease.

After that the proportion of susceptible is not apparent in Fig. 4.3, but it has

pulses with peak values 0.005952, 0.000013, 0.000023 at t = 10, t = 15 and t = 20

days respectively. Initially the exposed are mainly confined to domain [−1, 1.5]

producing a pulse with peak value 0.118055 at t = 5 days. At t = 10 days, the

exposed population is spread accross domain [−2, 2] producing a pulse with peak

value 0.068133. This is followed by a rapid decrease in the exposed proportion and

thus at t = 15 days, a very small proportion of exposed is observed, producing a

pulse with peak value 0.008858. Infection grows with the passage of time. In the

first five days of disease infected are concentrated in domain [−1, 1.5] producing a

pulse with peak value 0.046294. The proportion of the infected spreads across the

domain [−2, 2] with the passage of time and at t = 10, produces a pulse with peak

value 0.077894. After this there is a fall in the infected population proportion and

at t = 20, there is a very low level of infected, producing a pulse with peak value

0.002717 confined to domain [−2, 2]. The diagnosed proportion is fairly small in

the first five days of the disease producing a pulse with peak value 0.026952 in the

domain [−1, 1.5]. After t = 5 days, there is a sudden increase in the diagnosed

proportion of population across the entire domain [−2, 2]. The proportion of the

diagnosed population produces a pulse with peak value 0.105364 at t = 10. After

that, there is a gradual decrease in the diagnosed population proportion producing

69

a pulse with peak value 0.072124 at t = 15 days. In the last five days of study of

disease a rapid decrease is observed in diagnosed population proportion, producing

a pulse with peak of only 0.028397 at t = 20 days. There is a small recovery in the

first five days of the disease but in the next fifteen days recovery rapidly increases

in proportion. At t = 10 days, the peak value of the pulse of proportion of recov-

ered population reaches 0.173848. At t = 15 days, peak value of pulse produced by

recovered population reaches 0.320015. At t = 20 days, the pulse attains a peak

value 0.383669 with the domain extending to [−2, 2].

Fig. 4.4, shows the output for initial condition (i) for Case (b). Here cross-diffusion,

dse, is introduced among the susceptible population. In this case, due to the den-

sity pressure of the exposed population, susceptible move in the direction of low

concentration of the exposed population because of cross-diffusion in the system.

This situation causes a change in the transmission of the disease such that at

t = 5 days, the peak value of the pulse of the susceptible population proportion is

0.217763 and this peak value remains steady in the domain [−0.4, 0.7]. The peak

value of pulses produced for the exposed proportion are 0.116768, 0.060374 and

0.007727 at t = 5, t = 10 and t = 15 days, respectively. At t = 5 days, the peak

value of the infected population pulse is 0.046061 in the domain [−1, 1.5]. The

pulse produced by infected proportion keeps on increasing and reaches peak value

0.071505 at t = 10 days in domain [−2, 2]. At t = 15 days the proportion of popu-

lation infected decreases with the peak value 0.015325. At t = 20, there is a further

decrease in infected proportion producing pulse with peak value 0.002382. Here,

the intensities of the proportions of diagnosed and infected are slightly lower for

Case (a). This indicates a slow transmission pattern of the disease. The recovered

population follows the same pattern as in Case (a) except that the pulses produced

here have lower peak values.

Fig. 4.5, shows the output with initial condition (i) for Case (c). In this case the

susceptible and exposed populations cause motility to the lower concentration of

each other due to cross-diffusion in the system. This generates a different pattern

of disease transmission by giving rise to oscillations. There is a sudden fall in the

population proportion of susceptible, producing a pulse with peak value 0.329880 at

70

t = 5 days in the domain [−1.5, 2]. The susceptible population move to the exposed

compartment quite quickly and after t = 10 days, there is a negligible proportion

of susceptible left. The proportion of population exposed accordingly increases in

the first five days of the disease with the production of oscillations. The first pulse

appears in the domain [−2, 0.2] with value 0.068716. The lowest amplitude of this

pulse is observed to be 0.018522. The next pulse appears in the domain [0.2, 2]

with same peak value 0.068716. At t = 10 days, exposed population spread across

the domain [−2, 2] producing a single pulse with peak value 0.074724. In next five

days the proportion of the exposed population suddenly reduces producing a pulse

with peak value 0.008375 at t = 15 days. Infection slowly grows in the first five

days, producing oscillations with the first pulse having peak value 0.026041 in the

domain [−2, 0.2]. This pulse reaches the lowest amplitude of 0.005 and the second

pulse appear in the domain [0.2, 2] with peak value 0.026041. There is a sudden

increase in the infected population at t = 10 days, with pulses having peak value

0.067428 in the domain [−1, 1]. There is a sudden increase in the proportion of

diagnosed population between t = 5 and t = 10 days. Oscillations are generated

here too, producing pulses with peak value 0.074977 in the domain [−1, 1] at t = 10

days. Proportion of recovered population also shows oscillations, producing pulses

having highest amplitude of 0.276379 at t = 20 days.

Fig. 4.6, shows the output for initial condition (i) for Case (d). In this case the

negative value of the diffusivity coefficient, dse, shows that density pressure of the

exposed causes the mobility of the susceptible towards higher concentration of the

exposed population. This also has significant impact on the transmission of disease.

This case also gives rise to oscillations as in Case (c). There are however slight

differences in the behavior of the oscillations as compared to Case (c). For example

the lowest amplitude of the first pulse is higher in this case than with Case (c) in

almost all compartments. Here at t = 10 exposed population produce pulses in the

interval [−1, 1] with little difference in their peak values. Also the domain of the

exposed population narrows down slightly at t = 10 days, as compared to Case (c).

At t = 15 days the exposed and infected remain steady in the domain [−0.4, 0.8]

unlike for Case (c). Exposed and recovered populations behave in the same way as

71

t = 0

t = 5

t = 10 t = 15 t = 20

-2 -1 0 1 2x

0.2

0.4

0.6

0.8

1.0

S

t = 5

t = 10

t = 20t=15

-2 -1 0 1 2x

0.02

0.04

0.06

0.08

0.10

0.12E

t = 0t = 20

t=15

t=5

t=10

-2 -1 0 1 2x

0.02

0.04

0.06

0.08

I

t = 0

t = 20

t=5

t=15

t=10

-2 -1 0 1 2x

0.02

0.04

0.06

0.08

0.10

0.12J

t = 0

t = 5

t = 10

t = 15t = 20

t=0-2 -1 0 1 2

x

0.1

0.2

0.3

0.4R

Figure 4.3: Solutions with initial condition (i) for Case (a).

for Case (c), apart from differences peak values of the pulses along the domain of

activity. Moreover, the effective domain of all compartments reduce as compared

to Case (c).

4.5.2 Numerical solution for initial condition (ii)

Fig. 4.7, shows the results with initial condition (ii) for Case (a). A sudden fall

in the proportion of susceptible is observed at t = 5 producing pulses with peak

value 0.005466. Susceptible proportions produce pulse with peak values 0.000213,

0.000025 and 0.000048 at t = 10, t = 15 and t = 20 days respectively. At t = 5

days, the exposed population spreads across domain [−1.2, 1.2] producing a pulse

with peak value 0.043322. The exposed population proportion shows rapid de-

crease and produces a pulse with peak value 0.005844 at t = 10 days with the

72

t = 0

t = 5

t = 10

t = 20-2 -1 1 2

x

0.2

0.4

0.6

0.8

1.0

S

t = 5

t = 10

t=15

-2 -1 0 1 2x

0.02

0.04

0.06

0.08

0.10

0.12E

t = 0

t = 20

t=10

t=5

t=15

-2 -1 0 1 2x

0.02

0.04

0.06

0.08I

t = 15

t = 20

t=10

t=5

-2 -1 0 1 2x

0.02

0.04

0.06

0.08

0.10J

t = 0

t = 5

t = 15

t = 20

t=10

-2 -1 0 1 2x

0.1

0.2

0.3

0.4R

Figure 4.4: Solutions with initial condition (i) for Case (b).

73

t = 0

t = 5

t = 20-2 -1 1 2

x

0.2

0.4

0.6

0.8

1.0

S

t = 5t = 10

t = 20t=15

-2 -1 0 1 2x

0.02

0.04

0.06

0.08E

t = 0

t = 20

t=5

t=10

-2 -1 0 1 2x

0.02

0.04

0.06

0.08I

t = 20

t=10

t=15

t=5

-2 -1 0 1 2x

0.02

0.04

0.06

0.08J

t = 5

t = 15

t = 20

t=10

-2 -1 0 1 2x

0.05

0.10

0.15

0.20

0.25

0.30R

Figure 4.5: Solutions with initial condition (i) for Case (c).

74

t = 0

t = 5

t = 20-2 -1 1 2

x

0.2

0.4

0.6

0.8

1.0

S

t = 5

t = 10

t = 20

t=15

-2 -1 0 1 2x

0.02

0.04

0.06

0.08E

t = 0

t = 20

t=10

t=5t=15

-2 -1 0 1 2x

0.02

0.04

0.06

0.08I

t = 20

t=10

t=15

t=20

t=5

-2 -1 0 1 2x

0.02

0.04

0.06

0.08J

t = 0

t = 5

t = 15

t = 20

t=10

-2 -1 0 1 2x

0.05

0.10

0.15

0.20

0.25

0.30R

Figure 4.6: Solutions with initial condition (i) for Case (d).

75

main concentration of the population in the domain [−1.8, 1.8]. Infection reduces

in first five days of the disease, producing a pulse with peak value 0.032972 in the

domain [−1, 1]. At t = 10, infected population spreads in the domain [−1.5, 1.5]

producing a pulse with peak value 0.010391. At t = 5 days, proportion of the

population diagnosed with SARS in the domain [−1, 1] is producing a pulse with

peak value 0.033966. This diagnosed proportion of the population spreads to do-

main [−1.4, 1.4] at t = 10 days, producing a pulse with peak value 0.032229. After

that there is a quick decrease in the proportion of diagnosed, which spread across

domain [−1.5, 1.5] producing a pulse with peak value 0.013849 at t = 15 days.

This proportion reduce to 0.004733 at t = 20 days. Recovery initially occurs in

the domain [−1, 1] at t = 5, producing a pulse with peak value 0.045636. There is

a further increase in the recovered at t = 10 producing a pulse with a peak value

0.106593 in the domain [−1.2, 1.2]. At t = 20 there is maximum recovery producing

a pulse with peak value 0.147773. Here, the population of recovered concentrates

in domain [−1.5, 1.5].

Fig. 4.8, shows the output for the initial condition (ii) and Case (b). Values of

the susceptible proportion over time are very low and thus difficult to observe in

the graph. The proportion of susceptible is, however, higher in magnitude than for

Case (a) in the first ten days and lower in the next ten days of the disease. The

peak value of the pulse produced by the exposed proportion of the population at

t = 5 is 0.039203 in the domain [−1.2, 1.2]. It undergoes a decline over the next five

days, producing a pulse with peak value 0.004513 in the domain [−2, 2] . There is

a reduction in infected population between t = 5 and t = 10 days, producing pulses

with peak values 0.031539 and 0.008701 respectively with an increase in length of

effective domain compared to Case (a). At t = 15 and t = 20 days the propor-

tion of population infected is small, producing pulses with peak value 0.001352

and 0.002333 respectively. At t = 5 the proportion of the population diagnosed

has a pulse with peak value 0.033472. At t = 10 days, the peak value of pulse

for the diagnosed is 0.029586 which decreases further at t = 15 days, to a pulse

with peak value 0.012348. At this stage the diagnosed population spread over the

whole domain [−2, 2]. At t = 5 days, the proportion of the population recovered

76

has a pulse with peak value 0.053113 in the domain [−1, 1]. At t = 10 days, the

recovered population spread over the domain [−1.4, 1.4], and the pulse has a higher

peak value 0.102424. At t = 15 and t = 20 days, the proportion of the population

recovered spreads to the domain [−1.8, 1.8] and has pulses with increased peak

values 0.129854 and 0.139435. The improvement in recovery is greatest between

t = 5 and t = 10 days, after which the improvement slows.

Fig. 4.9, shows the output for initial condition (ii) and Case (c). The introduction

of cross-diffusion in the susceptible and exposed populations causes a change in

the transmission of the disease. The susceptible are found to be greater at t = 5

days, than for Case (a) and in the last fifteen days of the disease, the proportion

is less than for Case (a). At t = 5 days, the effective domain of the exposed pop-

ulation proportion is [−1.5, 1.5] and the pulse has peak value 0.037877. At t = 10

days, there is an increase in the domain from [−1.5, 1.5] to [−2, 2] with a decline

in the exposed population giving a pulse with peak value 0.004453. At t = 5,

the infected population has two pulses with peak value 0.027584 in the interval

[−1.2, 1.2]. At t = 10 there is a single pulse with peak value 0.008502, which is

quite low compared to Case (a). In the first five days, the proportion of population

diagnosed has two pulses in the domain [−1.2, 0] and [0, 1.2] with common peak

value 0.030418. At t = 10 and t = 15 days the peak values of the pulses reduce to

0.027121 and 0.011435 with domain extended to [−1.6, 1.6]. These peak values of

pulses are lower than for Case (a), with significant difference at t = 15 days. The

recovered population also oscillates with increase in the peak values of its pulses

with the passage of time. The effective domain of the recovered population is wider

compared to Case (a).

Fig. 4.10, shows the output for the initial condition (ii) and Case (d). Here again

the negative value of the diffusivity coefficient, dse, from susceptible to exposed,

population density pressure of the exposed causes the motility of the susceptible

towards higher concentration of the exposed population. This case also gives rise

to oscillations (as in the previous Case (c)), with significant differences in the peaks

of pulses observed at various time steps. In this case the diagnosed population has

pulses with higher peak values at t = 10 days, compared to Case (c). At t = 15

77

t = 0

t = 5 t = 10 t = 15 t = 20

-2 -1 0 1 2x

0.2

0.4

0.6

0.8

1.0

S

t = 20t=15

t=10

t=5

-2 -1 0 1 2x

0.01

0.02

0.03

0.04

0.05E

t = 20

t=0

t=5

t=10

t=15

-2 -1 0 1 2x

0.01

0.02

0.03

0.04

I

t = 0

t = 20

t=5

t=10

t=15

-2 -1 0 1 2x

0.005

0.010

0.015

0.020

0.025

0.030

0.035J

t = 0t = 10

t=5

t=10

t=20

t=15

-2 -1 0 1 2x

0.05

0.10

0.15

R

Figure 4.7: Solutions with initial condition (ii) for Case (a).

there is a smooth curve for diagnosed in the interval [−1.2, 1.2]. The effective do-

main of population in all compartments is narrower than in Case (c). Recovery

in this case is comparatively slower than in Case (a). There are oscillations pro-

duced in this case too. At t=5 days, two pulses are produced with shared peak

value 0.042747 in effective domain [−0.8, 0.8]. At t = 10 days, two pulses are also

produced with peak value 0.099205 in the domain [−1.2, 1.2]. At t = 20 days, the

proportion of population recovered concentrates inside the domain [−1.4, 1.4] with

smaller peak values of pulses than for Case (a).

78

t = 0

-2 -1 1 2x

0.2

0.4

0.6

0.8

1.0

S

t = 10

t=5

t=15

-2 -1 0 1 2x

0.01

0.02

0.03

0.04

0.05E

t = 0

t=5

t=10t=15

-2 -1 0 1 2x

0.01

0.02

0.03

0.04I

t = 20

t=5

t=10

t=15

-2 -1 0 1 2x

0.005

0.010

0.015

0.020

0.025

0.030

0.035J

t = 10

t=20

t=15

t=10

t=5

-2 -1 0 1 2x

0.02

0.04

0.06

0.08

0.10

0.12

0.14

R

Figure 4.8: Solutions with initial condition (ii) for Case (b).

79

t = 0

t = 20-2 -1 1 2

x

0.2

0.4

0.6

0.8

1.0

S

t = 10

t=5

t=15

-2 -1 0 1 2x

0.01

0.02

0.03

0.04

0.05E

t = 0

t=5

t=10

t=15

-2 -1 0 1 2x

0.01

0.02

0.03

0.04

I

t = 20

t=5

t=10

t=15

-2 -1 0 1 2x

0.005

0.010

0.015

0.020

0.025

0.030

0.035J

t = 10

t=20

t=15

t=10

t=5

-2 -1 0 1 2x

0.05

0.10

0.15

R

Figure 4.9: Solutions with initial condition (ii) for Case (c).

80

t = 0

-2 -1 1 2x

0.2

0.4

0.6

0.8

1.0

S

t = 10

t=5

t=15

-2 -1 0 1 2x

0.01

0.02

0.03

0.04

0.05E

t = 0

t=5

t=10

t=15

-2 -1 0 1 2x

0.01

0.02

0.03

0.04

I

t = 20

t=5

t=10

t=15

-2 -1 0 1 2x

0.005

0.010

0.015

0.020

0.025

0.030

0.035J

t=20

t=15

t=10

t=5

-2 -1 0 1 2x

0.05

0.10

0.15

R

Figure 4.10: Solutions with initial condition (ii) for Case (d).

81

4.6 Discussion

In this paper SEIJR model for SARS described in Chapter 1 is considered, with

self and cross-diffusion included in the system. Two different initial conditions

are taken for the population distribution. Differential equations governing the sys-

tem are solved numerically using the operator splitting technique with forward

and central difference schemes. The stability of solutions with and without cross-

diffusion is established using Routh-Hurwitz conditions. Four different cases with

cross-diffusion coefficients in the susceptible and exposed compartments are chosen,

with a view to see the effect on the spread of disease.

Bifurcation values of the transmission coefficient β and recovery coefficients γ1 and

γ2 are obtained. It is observed that in Case (b), the system remains stable for a

higher value of β as than in the other cases. Case (a) has bifurcation value of β

higher than Cases (c) and (d). Case (c) has the lowest bifurcation value. The bi-

furcation values for cases (c) and (d) is less than the considered value of β = 0.75.

This shows that when the susceptible and exposed population cross-diffuse, the

system is destabilised for a smaller value of β. Case (c) gives the highest values

of bifurcation for the infected and diagnosed recovery coefficients γ1 and γ2. This

shows that with positive cross-diffusion in the susceptible and exposed compart-

ments, the system stabilises for higher values of the recovery coefficients γ1 and γ2.

With negative cross-diffusion in the susceptible compartment, the system gets sta-

bilised slightly earlier than in Case (c) for the recovery coefficients γ1 and γ2. With

no cross-diffusion in the exposed compartment in Case (b), the system stabilises

much earlier than in the other cases. This implies that with positive cross-diffusion

in the susceptible, a proportion of susceptible move towards the lower concentra-

tions of exposed, thus stabilises the system earlier compared to systems without

cross-diffusion, for recovery coefficients γ1 and γ2.

Graphs of numerical solutions for the initial condition (i) are shown in Figs. (4.3)-

(4.6). For initial condition (i) with cross-diffusion in the susceptible proportion of

population, Case (b), the transmission of disease to the exposed population pro-

duces pulse with lower peak value than for Case (a), without cross-diffusion, at

t = 10 days, of disease as shown in Fig. 4.4. Thus cross-diffusion slows down

82

the transmission of disease to reach its peak value. With cross-diffusion in both

susceptible and exposed populations, oscillations appear in the exposed population

at t = 5 days. Here, two loops are generated in the given domain [−2, 2]. At t = 10

days, the exposed population reduces to one pulse with peak value higher than at

t = 5 days. This resulted in two pulses in the infected at t = 5 days with the

prominent peak values. However at t = 10 days there are two pulses in the infected

proportion of population with no prominent peak values. As a result of oscillations

produced in the exposed population, oscillations are generated in infected, diag-

nosed and recovered proportions of the population. In the case of cross-diffusion

in susceptible and exposed populations, the intensity of disease reduces right from

the stage of exposed to diagnosed. Thus, in this case the recovery process slows.

With cross-diffusion in the exposed proportion of the population, oscillations with

small pulses are produced in all compartments of the population. There is a dip

in proportion of the population exposed near the middle of the domain at t = 5

days because of cross-diffusion in the system. Due to cross-diffusion in the system,

the exposed proportion of the population starts moving away from the middle of

domain to areas of low density, thus creating a dip in this proportion around the

middle of the domain at t = 5 days. With negative cross-diffusion in the susceptible

compartment, the peak values of the pulses of the exposed proportion are lower

than for Case (c) in the first five days, but higher after day ten of the disease. As

an effect of negative diffusion in the susceptible population, the proportion of pop-

ulation infected is higher after day five. Recovery is a little higher with negative

cross-diffusion in the system. The introduction of cross-diffusion in the susceptible

proportion of the population has not helped to contain the disease remarkably.

With cross-diffusion in both the susceptible and exposed proportions of the popu-

lation, small pockets of population have been generated in each compartment. In

the first five days the proportion of the population in the exposed compartment is

lower in Case (d) than in Case (c). After day five of the disease, transmission from

the infected to the diagnosed compartment is slightly higher in Case (d) than in

Case (c). This shows that in Case (d), there is comparatively fast transmission of

disease.

83

For Initial condition (ii), numerical solutions are shown in Figs. (4.7)- (4.10), for

Cases (a) − (d). In the absence of cross-diffusion, the largest spread of infection

goes to the domain [−1.8, 1.8]. When positive cross-diffusion is introduced in the

susceptible compartment the transmission of disease slows and the proportion of

susceptible in the first ten days of disease increases. The intensity of the exposed

and infected populations also goes down, especially in the first ten days, whereas the

domain of proportion expands. When positive cross-diffusion is included in both

the exposed and the susceptible compartments, oscillations are produced where

exposed at t = 5 days doesn’t show any oscillation as in condition (i). The pro-

portion of susceptible is higher in the first five days. Infection spreads slower with

lower peaks of the proportion than for Case (a), and the domain of the proportion

does not change as much as in condition (i). When negative cross-diffusion occurs

in the susceptible while the exposed have positive cross-diffusion, domains of the

population reduces in after t = 5 days. Here the infected population attains higher

peaks than in Case (a). The proportion of population exposed is higher in Case (d)

for all days of the disease under study i.e t = 20 days. The domain of the exposed

population is significantly smaller in Case (d) at t = 5 and t = 10 days than for

Case (c). The proportion of infected shows higher peaks in the first fifteen days

in Case (d), compared with Case (c). The main domain of the infected proportion

is shortened to [−1, 1] in Case (d). The proportions of diagnosed follow the same

transmission pattern as the infected and have the same differences in Cases (d)

and (c). The proportion of the population recovered is higher in Case (d) than in

Case (c) after day five of the disease. The domain of concentration of recovered

proportion is smaller in Case (d) in the last fifteen days of study of the disease than

for Case (c).

The numerical values of reproduction number with and without diffusion are given

in Table 4.4. The numerical value for the reproduction number is 2.83 without

diffusion as given in Chapter 3. Table 4.4 shows that with the inclusion of self-

diffusion, the value of the reproduction number decreases as in Case (a). The

inclusion of positive cross-diffusion in the susceptible compartment, as in Case (b),

and in the susceptible and exposed compartments as in Case (c), cause significant

84

increase in the reproduction number. This leads to an acceleration of the spread of

disease. In Case (d), where cross-diffusion in the exposed compartment is negative,

diffusion of the susceptible towards high concentrations of the exposed has reduced

the reproduction number. Thus the spread of the disease is slowed.

It is now proposed to include treatment compartment in the SEIJR model. In

the view of that all the parameters governing the system of equations have been

re-estimated using SARS data for the 2003 outbreak in Hong Kong, in next chap-

ter.

85

Chapter 5

Parameter Estimation with

Uncertainty and Sensitivity

Analysis for the SARS Outbreak

in Hong Kong

5.1 Introduction

Mathematical modeling of real-life processes often requires the estimation of un-

known parameters. The problem of parameter estimation belongs to the class of

inverse problems in which the knowledge of the dynamical system is derived from

the input as well as output data values of the system. This process is empirical

in nature and often ill-posed because, in many instances, it is possible that some

different model can be fitted to the same response. This process of determining

the unknown parameters of a mathematical model from noisy data based on input

and output values is termed as parameter estimation. Parameter estimation is an

important step in the development of systems biology, as it helps to obtain predic-

tions from computational models of biological systems.

Parameter estimation in dynamic systems is a wide area involving many different

aspects of mathematical as well as statistical analysis. Parameters are estimated

from a data fit depending on their type such as epidemiological or demographic,

86

and, with the resulting model, predictions are made that can be tested with further

experiments. There are a number of difficulties and complexities involved in the

estimation of parameters of non-linear systems. On the basis of prior theoretical

knowledge the structure of a model is suggested defining the state variables and pa-

rameters. The parameter estimation criteria (hardly ever a single criterion) reflect

the desired properties of the estimates. A great deal of effort in recent decades has

gone into developing parameter estimation with the help of algorithms having good

theoretical properties (Norton, 1986; Soderstrom and Stoica, 1989; Ljung, 1999). A

Few of them are Maximum Likelihood Estimator (MLE), Bayes Estimators (BE),

Principal Differential Analysis (PDA), Methods of Moments Estimators (MME),

Minimum Variance Unbiased Estimator (MV UE), Particle Filter, Maximum a

Posteriori (MAP ), Minimum Mean Squared Error (MMSE), Best Linear Unbi-

ased Estimator (BLUE), Markov Chain Monte Carlo (MCMC) method, Kalman

Filter and Ensemble Kalman Filter (EnKF ) [144, 174]. The most used algorithms

to solve the non-linear inverse problems are based on deterministic and stochastic

methods.

Once the parameter estimates have been computed by the means of optimization,

it is very important to know how reliable they are and what is the quality of the

parameter estimates especially if the parameter values are used to draw biological

conclusions from the model. According to Jakeman et al. [127], “Uncertainty in

models stems from incomplete system understanding (which processes to include,

which processes to interact); from imprecise, finite and often sparse data and mea-

surements; and from uncertainty in the baseline inputs and conditions for model

runs, including predicted inputs.” Whereas Marino et al. states that [161], “Un-

certainty analysis is performed to investigate the uncertainty in the model output

generated from uncertainty in parameter inputs. Sensitivity analysis naturally fol-

lows uncertainty analysis as it assesses how variations in model outputs can be

apportioned, qualitatively or quantitatively, to different input sources.” This type

of uncertainty is termed epistemic or subjective or reducible type B uncertainty.

Epistemic uncertainty derives from a lack of knowledge about the adequate value

for a parameter/input/quantity that is assumed to be constant throughout model

87

analysis [160]. Numerous approaches to uncertainty and sensitivity analysis are

available, such as [105]

• Differential analysis, which involves approximating a model with a

Taylor series and then using variance propagation formulas to obtain

uncertainty and sensitivity analysis results.

• Response surface methodology, which is based on using classical

experimental designs to select points for use in developing a response

surface replacement for a model and then using this replacement model

in subsequent uncertainty and sensitivity analyses based on Monte

Carlo simulation and variance propagation.

• The Fourier amplitude sensitivity test (FAST ) and other variance

decomposition procedures, which involve the determination of

uncertainty and sensitivity analysis results on the basis of the variance

of model predictions and the contributions of individual variables to

this variance.

• Fast probability integration, which is primarily an uncertainty analysis

procedure used to estimate the tails of uncertainty distributions for

model predictions.

• Sampling-based (i.e. Monte Carlo) methods, that are based on the

formation and exploration of a probabilistically based mapping from

analysis inputs to analysis results.

Sampling based methods using random and Latin Hypercube Sampling (LHS)

are focused in this study for sensitivity and uncertainty analysis. The calculation

of uncertainty values with large number of parameters is based on formulation of

the multi-dimensional parameter space. It is done with an aim to find an

appropriate and computationally effectual way such as Latin Hypercube

Sampling. It was McKay et al. [113] who extended Latin Square sampling and

developed Latin Hypercube Sampling (LHS). Whereas Iman et al. [113],

improved it. Latin Hypercube Sampling (LHS) belongs to the group of Monte

Carlo sampling.

88

Donnelly et al. [57], was one of the leading groups who assessed the epidemiology

of SARS in Hong Kong. They estimated the key epidemiological parametric

distributions using the integrated data bases constructed from several sources

containing information about epidemiological, demographic and clinical variables

which provided the base line for the parameters of SARS models by Chowell et

al. [45, 46], Gummel et al. [95], Riley et al. [191], Yan et al. [242] and Lipsitich et

al. [152]. Chowell et al. [45, 46], fitted an SEIJR model to the data from

Toronto, Hong Kong and Singapore outbreaks, calculated the reproductive

number and used uncertainty and sensitivity analysis of reproductive number to

assess the role that model parameters play in outbreak control. Riley et al. [191]

and Lipsitch et al. [152], developed comparatively complex dynamical models for

the SARS transmission in Hong Kong and Singapore. But these models provided

the researchers with enough information and understanding to estimate many

influential parameters that can access the danger of disease spread in future.

Wallinga and Teunis [230] developed a likelihood-based estimation procedure that

infers the temporal pattern of effective reproduction numbers from an observed

epidemic curve. Zhou et al. [254], formulated a discrete mathematical model to

investigate the transmission of SARS and estimated the parameters of the model

on the basis of statistical data. Numerical simulations have been carried out to

describe the transmission process for SARS in China. Wanga and Ruanb [231]

proposed a mathematical model to simulate the SARS outbreak in Beijing by

estimating the reproduction number and estimated certain important

epidemiological parameters using the available data after simplifying the model in

suspect-probable and single-compartment model.

The model considered here is an extension of the SEIJR open population SARS

model presented in Chapter 3. It consists of six sub-populations compartments,

namely susceptible(S), exposed(E), infected(I), diagnosed(J) and recovered(R)

along with treatment class(T ) named as SEIJTR model. The aim of this chapter

is to estimate the finite set of parameters for the new SEIJTR. To calculate the

parameters for the new model SEIJTR data from the SARS outbreak in Hong

Kong [152] is used. Uncertainty and sensitivity analysis of the estimated

89

parameters are also investigated on the basis of local derivative and sampling

techniques in the context of a deterministic dynamical system.

5.2 Model formulation

The SEIJTR model is considered to study the SARS epidemic which occurred

in Hong Kong in 2003. This model of the SARS epidemic consists of a system of

non-linear ordinary differential equations:

dS

dt= πΛ− β

(I + qE + lJ)

NS − µS, (5.1)

dE

dt= (1− π)Λ + β

(I + qE + lJ)

NS − (µ+ κ)E, (5.2)

dI

dt= κE − (µ+ α + δ)I, (5.3)

dJ

dt= αI − (µ+ γ1 + δ + ζ)J, (5.4)

dT

dt= ζJ − (γ2 + µ+ δ(1− θ))T, (5.5)

dR

dt= γ1J + γ2T − µR. (5.6)

With initial conditions

S(0) = S0, E(0) = E0, I(0) = I0, J(0) = J0, T (0) = T0, R(0) = R0 and

N = S + E + I + J + T +R. The parameter estimation problem given by the

system (5.1)- (5.6) is basically set up using the objective function and is given as

as follows:

Φ = g(t,Φ,Ψ), Φ(t0) = Φ0, (5.7)

where t denotes time (independent variable), Φ denotes the vector of dependent

variable (S,E,I,J,T,R) and Ψ represents vector of unknown parameters (Λ, β, l, κ,

α, γ1, γ2, ζ and δ). Then according to [217] if the solution of Eq. (5.7) for ith

component at the time ti is denoted by Φi(ti,Ψ), then the ith residual can be

given as:

ri = Φi − g(ti,Ψ) (5.8)

90

Table 5.1: Biological definition of parameters and state variables

Parameter Description

Λ Rate at which the new recruits enter the population

π Proportion of new recruits into the population that are susceptible

1− π Proportion of new recruits into the population that are are exposed

β Transmission Rate

µ Rate of natural mortality

l Relative measure of reduced risk among diagnosed

κ Rate of progression from exposed to the infectives

q Relative measure of infectiousness for exposed individuals

α Rate of progression from infective to diagnosed

γ1 Natural recovery rate

γ2 Recovery rate in treatment class

ζ Treatment Rate

δ SARS induced mortality rate

θ Effectiveness of the drugs as a reduction factor in disease-induced death of

infectious individuals(0 ≤ θ ≤ 1)

S0 Initial Susceptible Population

E0 Initial Exposed Population

I0 Initial Infected Population

J0 Initial Diagnosed Population

T0 Initial Treated Population

R0 Initial Recovered Population

91

5.2.1 Reproduction number RIT for SARS

In epidemiology, the basic reproduction number RIT is the number of individuals

infected by a single infected individual during his or her entire infectious period,

in a population which is entirely susceptible. If RIT < 1, then disease dies out,

but if RIT > 1 infection spreads and causes epidemic. The reproduction number

of SEIJTR model that represents the effect of isolation after diagnosis and

treatment into standard SEIR model is calculated. The detailed calculation for

reproduction number RIT is given in appendix A.5 and the formula is given as

follows:

RIT =Λβ(q(µ+ δ + α)(µ+ δ + γ1 + ζ) + κ(µ+ δ + γ1 + ζ + lα))

µ(µ+ κ)(µ+ δ + α)(µ+ δ + γ1 + ζ)(5.9)

5.2.2 Epidemiological data

Parameters estimation leads to a comparison with experimental data. Often a

model contains the parameters that need to be adjusted to obtain a best fit to the

data. The accuracy of the parameter estimate depends on the nature of the data,

the noise in the data and the structure of the model. In some circumstances a

small error in the data will cause a vastly magnified error in the parameters [220].

In 2003, a SARS (Severe Acute Respiratory Syndrome) epidemic spread globally.

The data considered here [121] is taken from one of the pandemic waves of SARS

in Hong Kong. It consists of daily reported infected and recovered. The SARS

epidemic in Hong Kong went through three phases appearing in a teaching

hospital then in a community followed by eight hospitals and 170 housing states

[152]. The data on infection is of dates from 17 March to 12 July 2003. Fig. 5.1

shows the infection data per day.

5.2.3 Parameter estimation

Estimation of parameters is a non-linear problem as the system consists of

non-linear ODES with respect to parameters and system state variables. The

solution of non-linear problems require a numerical solver and an optimizer to

92

0 10 20 30 40 50 60 70 80 900

20

40

60

80

100

120

140

160

Time (Days)

Infe

cted

Indi

vidu

las

Figure 5.1: SARS infected incidence data, Hong Kong 2003.

estimate the parameters of system. The series of tests for the selection of better

solver and optimizer in [144] suggests that Runge-Kutta methods provides quite

accurate solutions with better convergence for the solution of initial value

problem as compared to all avialable numerical methods and

Levenberg-Marquardt works as best optimizer [144]. Hence Dormand-Prince

Pairs (Runge-Kutta methods) method has been selected as solver for the system

of non-linear ordinary differential and the least square optimization problem is

solved using Levenberg-Marquardt method. MATLAB software has been used for

calculation throughout. Various steps of the calculations are as follows:

• System of ordinary differential equations is solved using the random

choosen initial values of parameters and system state variables with

MATLAB ode45 routine.

• The model output is then compared with field data. Levenberg-Marquardt

optimization algorithm is then used to estimate a new set of parameter

values that better fits the field data.

• The system of ordinary differential equations is again solved numerically

then using the new parameters values estimated with the help of

93

Levenberg-Marquardt optimization algorithm. The model output is again

compared with the field data.

• This comparison of the updated parameter values and field data is continued

till convergence criterion is obtained. About 1010 and 1015 simulations for

models have been run with more than a thousand values chosen randomly.

There are 13 unknown parameters and 6 state variables in the SEIJTR model.

Among these, the value of natural death rate µ, is obtained from the available

literature. The proportion of new recruits π and effectiveness of drug θ are

assumed. The remaining parameters are estimated from field data. The estimated

parameters are given in Table 5.2.

5.2.4 Validation statistics

With a view to examine possible errors in the estimated values of parameters as

well as to check their reliability, model validation has been performed. Many

statistical tools for model validation are available and among them the primary

and most applicable tool is graphical residual analysis that compares the model

and the field data. Fig. 5.2, shows the predicted model for infected population

and Fig. 5.3, shows the initial (the basic solution) and the final fit obtained after

optimization to the available SARS data. The residual plot for the prediction

model and field data as well as correlogram of residuals for the output infected

population of the corresponding estimates are shown in Fig. 5.4. In the initial

days of epidemic, a partial pattern of the residuals can be seen as a result of

difference of field data values from the model values. In the last phase of disease,

values in field data are quite small. This leads to negative values of residual

indicating reliability of estimated values of the parameters.

The concept of randomness is critically important for all measurement

procedures. Many of the statistical results cannot be assured without checking

the randomness of the model. Autocorrelation plot or correlogram is the best way

to check the randomness of the residuals. From Fig. 5.4, it is clear that

autocorrelation of residuals of corresponding estimates are inside the 99%

94

Table 5.2: Estimated parameters value for model

Parameter Value(per days) Source

Λ 0.00002 Estimated

π 0.85 Assumed

β 0.24 Estimated

µ .000035 [114]

l 0.65 Estimated

κ 0.195 Estimated

q 0.1 [57]

α 0.238 Estimated

γ1 0.046 Estimated

γ2 0.05 Estimated

ζ 0.2 Estimated

δ 0.024 Estimated

θ 0.25 Assumed

S0 18440 Estimated

E0 19 Estimated

I0 18 Estimated

J0 0 Estimated

T0 0 Estimated

R0 0 Estimated

95

0 10 20 30 40 50 60 70 80

5

10

15

20

25

30

35

40Predicted output SEIJTR Model

SA

RS

Inci

denc

e

Time (day)

Figure 5.2: Predicted model of SARS.

confidence interval. This gives evidence of randomness. Since adjacent

observations do not co-relate and so there is no significant autocorrelations. The

model fits the data well.

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

Time (day)

Infe

cted

Pop

ulat

ion

DATAModel Fit

(a)

0 10 20 30 40 50 60 70 800

10

20

30

40

50

60

Infected Population. (1−step pred)

Time (day)

Infe

cted

Pop

ulat

ion

DATAModel Fit

(b)

Figure 5.3: The Model fitted to the data for the infected individuals : Initial (a)

and final fit (b).

96

0 10 20 30 40 50 60 70 80

−15

−10

−5

0

5

10

15

20

25

Prediction error

SA

RS

Inci

denc

e

Time (day)

(a)

0 10 20 30 40 50 60 70 80 90−0.4

−0.2

0

0.2

0.4

0.6

0.8

1Correlation function of residuals. Output Infected Population

lag

(b)

Figure 5.4: Residual (a) and residual correlation (b).

5.3 Uncertainty and sensitivity analysis

Uncertainty and sensitivity analysis is a fundamental component of

epidemiological modeling. It is based on the uncertainty in output derived from

the uncertainty in the input and the relationships between the uncertainty of

output and the uncertainty in the individual input. An effort to understand the

nature of the real field data and the uncertainty associated with it, is the key to

evaluating the parameters with the greater accuracy. The uncertainty of the

model parameters and the inconsistency in the model output can be determined

by the covariance matrix. All covariance matrices are symmetrical. The absolute

values in the leading diagonal of the covariance matrix provides the information

about accuracy. The inter-connections of the parameters can be studied having a

glance at the other values in the matrix. This matrix provides the variability in

the parameter estimation that interpret relations in uncertainties of

measurements. The matrix∑

shows the covariance matrix for the SEIJTR

97

model.

∑=

Λ β l κ α γ1 γ2 ζ δ

Λ 1.227 0.6277 6.830 −0.5926 13.316 −28.31 9.834 21.93 −1.237

β 0.6277 0.4485 4.181 −0.3782 8.577 −21.12 6.266 20.15 −0.7925

l 6.830 4.181 23.57 −4.025 25.08 −27.53 24.36 26.78 −8.386

κ −0.5926 −0.3782 −4.025 0.3468 −7.869 20.94 −5.646 −19.78 0.7241

α 13.32 8.577 25.08 −7.869 25.88 −28.85 26.18 28.53 −16.43

γ1 −28.31 −21.12 −27.53 27.28 −28.85 30.57 −28.03 −29.55 24.01

γ2 9.833 6.266 24.36 −5.646 26.18 −28.03 21.56 27.98 −11.80

ζ 21.93 20.15 26.78 −19.79 28.53 −29.55 27.98 29.02 −28.03

δ −1.237 −0.7925 −8.386 0.7241 −16.43 24.01 −11.80 −28.03 1.512

(5.10)

The leading diagonal elements of the matrix∑

represent the variance of each

parameter. The rate of recovery in the diagnosed class γ1, has the largest variance

(30.57) followed by the treatment rate, ζ (29.01), recovery rate in treatment class,

γ2 (21.56), rate of progression from infective to diagnosed, α (25.88), relative

measure of reduced risk among diagnosed, l (23.57), SARS induced mortality

rate, δ (1.512), Net inflow of individuals, Λ (1.227), transmission rate, β (0.4485)

and rate of progression from exposed to infectivs, κ (0.3468). If the covariance

between two parameters is positive then it implies that the two parameters move

in the same direction but if the covariance between two parameters is negative it

shows they move in the opposite direction. Calculated relations of covariance for

all parameters of the model are given in Table 5.3. An uncertainty analysis is

performed to determine how the uncertainty in the selection of input factor

(parameter) causes variability in the model output(s).The uncertainty in the

output results of prediction models due to the variability in the model inputs is

investigated through sensitivity analysis. Sensitivity analysis identifies the

importance of parameters based on the variability in their uncertainty

contributing to the variability in the outcome. In fact sensitivity analysis

determines the robustness of the model predictions to parameter values because

of errors in data collection and assumed values of parameters.

SARS is considered as one of the short term and fast spreading infectious

diseases. Its control basically depends on the study of those factors or parameters

which directly effect the reproduction number RIT . So with a precise knowledge

of the sensitivity of the factors effecting reproduction number, transmission of a

disease can be controlled. In order to identify key parameters in the transmission

of SARS, we have conducted the sensitivity analysis based on two methods. The

98

Table 5.3: Covariance relations among parameters of SEIJTR model

Covariance Direction Covariance Direction Covariance Direction

CΛ,β +ve Cβ,γ2 +ve Cκ,ζ −ve

CΛ,l +ve Cβ,ζ +ve Cκ,δ +ve

CΛ,κ −ve Cβ,δ −ve Cα,γ1 −ve

CΛ,α +ve Cl,κ −ve Cα,γ2 +ve

CΛ,γ1 −ve Cl,α +ve Cα,ζ +ve

CΛ,γ2 +ve Cl,γ1 −ve Cα,δ −ve

CΛ,ζ +ve Cl,γ2 +ve Cγ1,γ2 −ve

CΛ,δ −ve Cl,ζ +ve Cγ1,ζ −ve

Cβ,l +ve Cl,δ −ve Cγ1,δ +ve

Cβ,κ −ve Cκ,α −ve Cγ2,ζ +ve

Cβ,α +ve Cκ,γ1 +ve Cγ2,δ −ve

Cβ,γ1 −ve Cκ,γ2 −ve Cζ,δ −ve

first method is based on fixed point estimates of model parameters and the

second method is based on uncertainty in the model parameter estimation.

5.3.1 Sensitivity indices of RIT

Sensitivity analysis using the fixed point estimation has been applied to

determine the relative importance of different parameters responsible for the

SARS transmission related to the reproductive number RIT [42]. The normalized

forward sensitivity index of a variable p, that depends differentially on a

parameter x, is defined as [186]:

Υpx =

∂p

∂x× x

p(5.11)

The sensitivity indices have been calculated for the reproduction numbers RIT for

all the parameters using the definition given in Eq. (5.11). Expressions for

normalized sensitivity indices for all parameters are given in appendix A.5. The

calculated numerical values for each parameters are as given in Table 5.4. The

sensitivity index of RIT with respect to parameter Λ and β is 1, being not

99

Table 5.4: Parameters’ sensitivity analysis

Parameters Sensitivity Index for RIT Percentage Change

Λ +1.00000 −1.00000

π −−−−− −−−−−

β +1.00000 −1.00000

µ −1.00032 +0.99968

l +0.27682 −3.61242

κ −0.085780 +11.6577

q +0.08595 −11.6341

α −0.55023 +1.81742

γ1 −0.04273 +23.4049

γ2 −−−−− −−−−−

ζ −0.210844 +4.74284

δ −0.11009 +9.08299

θ −−−−− −−−−−

100

dependent on any parameter value. Table 5.4 shows that the sensitivity indices of

Λ, β, l, q are positive and µ, κ, α, γ1, δ and ζ are negative. Table 5.4 shows

corresponding percentage changes in the parameters with change of 1% in the

value of RIT . The value of RIT decreases 1% with the decrease in the value of Λ,

β, l, q, by 1%, 1%, 3.61242% and 11.6341%, respectively. While in order to

decrease RIT by 1% we need to increase µ, κ, α, γ1 and δ and ζ by 0.99968%,

11.6577%, 1.81742%, 23.4049%, 9.0829% and 4.74284% respectively.

5.3.2 Partial rank correlation coefficient (PRCC)

In this section, sensitivity analysis is performed to find out the most important

parameters or factors in contributing the variability in the output of the

reproduction number based on the uncertainty in their estimation. PRCC

technique has been used here for the parameter ranking. It is a powerful

sensitivity measure for non-linear but monotonic relationships between input

parameters and output of model, as long as small or no correlation exists between

the inputs. PRCCs are appropriate for determining the most effective

parameters but not for evaluating how much change appears in the outcome by

changing the value of the input parameter. However, the signs (negative or

positive) of PRCCs can suggest the direction of change in the outcome variable

due to the change in the input parameter [166].

There are eight parameters Λ, β, α, δ, l, ζ, κ, γ1, involved in Eq. (5.9), whose

estimated values are given in Table 5.2. In order to examine the sensitivity of the

estimated parameters, all parameters are assumed to be random variables with a

corresponding density probability functions. The assumption of the probability

distribution functions is a critical decision which is based on the biological

understanding, and information of the natural history of the concerned disease.

The assumed probability distributions for SARS using available information

[57, 85] are described in Table 5.5 and their plots are shown in Fig. 5.5. Random

Sampling (RS) and Latin Hypercube Sampling (LHS) have been used to

generate the samples for the parameters. Random Sampling is generally

preferable to univariate approaches for uncertainty and sensitivity analysis. In

101

Table 5.5: Probability distribution functions (PDF ) for parameters

Parameters Type of Distribution PDF Parameter Values Source

Λ Exponential Mean= .000021 Assumed

β Exponential Mean=.24 Assumed

l Beta a= 1,b= 2 Assumed

1/κ Gamma a= 2.4, b= 2.8 [46]

1/α Gamma a= 8.3, b= 2.7 [46]

1/γ1 Gamma a= 1.8, b= 2.6 [46]

ζ Beta a= 2, b= 1 Assumed

1/δ Gamma a= 2.25, b=15.5 [46]

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

I

Pro

babi

lity

Dis

trib

utio

n F

unct

ion

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Zeta

Pro

babi

lity

Dis

trib

utio

n F

unct

ion

Figure 5.5: Plots of probability distributions for all parameters generated with

10, 000 sample size.

102

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14

I

RIT

Figure 5.6: Scatter plots for the basic reproduction number and eight sampled

input parameters values with 10, 000 random samples.

Random Sampling each sample element is generated independently of all other

sample elements. The more efficient technique is Latin Hypercube Sampling

(LHS) which was introduced to the field of disease modeling by Blower [24].

Both sampling techniques are used to select 10, 000 and 20, 000 samples with each

probability distribution. The scatter plots comparing the basic reproduction

number RIT and eight parameters with 10,000 and 20,000 samples are shown in

the Fig. 5.6 and Fig. 5.7, for Random and Latin Hypercube Sampling. Some

parameters show clear and some partial linear monotonic relationship between

the reproduction number RIT and the parameters. The partial rank correlation

coefficients (PRCCs) are calculated using technique given in [113] between the

values of each of the eight parameters and the values of RIT in order to rank the

103

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

I

RIT

Figure 5.7: Scatter plots for the basic reproduction number and eight sampled

input parameters values with 10, 000 LHS samples.

104

Table 5.6: Estimates of partial rank correlation coefficients

Parameters Λ β l κ α γ1 ζ δ

RSa1 +0.9168 +0.9153 +0.2579 +0.1088 +0.5569 +0.0427 −0.1438 +0.3104

LHSb1 +0.9149 +0.9144 +0.25383 +0.1420 +0.5594 +0.0464 −0.1536 +0.2944

RSa2 +0.9158 +0.9156 +0.2530 +0.1121 +0.5523 +0.0515 −0.1429 +0.3095

LHSb2 +0.9156 +0.9152 +0.2564 +0.1257 +0.56322 +0.0329 −.1455 +0.2927

Importance Λ β α δ l ζ κ γ1

RSa1 = 10, 000 Random Samples, LHSb1 = 10, 000 Latin Hypercube Samples


parameters according to the magnitude of their effect on RIT . The higher the

value of the PRCCs, the greater effect of input parameter is on the magnitude of

RIT . The order of these PRCCs directly aggregate to the level of statistical

influence, the corresponding input parameter has on the variability of the

reproduction number RIT . It is based on its own estimation uncertainty. Table

5.6, represents the estimated partial rank correlation coefficients for all considered

eight parameters. All results satisfy the condition of significance i.e (p < 0.05).

(a) (b)

Figure 5.8: PRCCs for the full range of parameters from Table 5.6 for LHSb1 =

10, 000 (a) and RSa1 = 10, 000 (b).

Random Sampling shows a strong correlation between RIT and Λ and β with

+0.9168 and +0.9153 using 10, 000 samples and +0.9158, +0.9156 using 20,000

samples respectively as shown in Table 5.6. Moderate correlation is observed

between α and δ with RIT with the corresponding values as +0.5569 and +0.3104

with 10, 000 and +0.5523, +0.3095 with 20, 000 samples. Weak correlation is

105

(a) (b)

Figure 5.9: PRCCs for the full range of parameters from Table 5.6 LHSb2 = 20, 000

(a) and RSa2 = 20, 000 (b).

observed between l, ζ, κ and γ1 with RIT with corresponding values +0.2579,

−0.1438, 0.1088 and +0.0427 using 10, 000 samples and +0.2530, −0.1429,

+0.1121 and +0.0515 with 20, 000 samples.

With Latin Hypercube Sampling, Λ and β correlate with RIT strongly with the

highest correlation coefficient with values +0.9149 and +0.9144 using 10, 000

samples and +0.9156, +0.9152 using 20, 000 samples respectively. Moderate

correlation of α and δ with RIT and the corresponding values are +0.5594 and

+0.2944 with 10, 000 and +0.5632, +0.2927 with 20, 000 samples, similarly again

weak correlation is observed between l, ζ, κ and γ1 with RIT with corresponding

values +0.2538, −0.1536, +0.1420 and 0.0464 using 10, 000 samples and +0.2564,

−0.1455, +0.1257 and +0.0329 with 20, 000 samples as shown in Table 5.6.

Tornado plots of partial rank correlation coefficients, indicating the importance of

each parameter’s uncertainty in contributing to the variability in the time to

eradicate infection are shown in Fig. 5.8 and 5.9. Values of PRCCs and the

corresponding parameters under consideration are shown on x-axis and y-axis

respectively. Here treatment rate ζ moves in the negative direction while Λ, β, α,

δ, l, κ and γ1 move in the positive direction showing that with the increase in

treatment rate ζ, RIT will decrease in magnitude and with the increase in Λ, β,

α, δ, l, κ and γ1, RIT will increase. It is shown in Fig. 5.8 and 5.9, that the net

106

inflow rate of individuals Λ and the transmission rate β plays the most important

role in the transmission of disease. Other parameters in order of decreasing

importance follow as α, δ, l, ζ, κ, γ1.

5.3.3 Factor prioritization by reduction of variance

Factor prioritization is a vast concept representing a group of statistical

approaches for classifying the importance of variables in contributing to

particular outcomes. After being popular in various disciplines variance based

measures for factor prioritization are becoming famous in computational

modeling. The main target of this method is to determine factor which can lead

to the maximum reduction in the variance of outcome. After identifying first

factor the second most important factor in reducing the variance is determined

this procedure continues till all independent input factors are ranked. The

concept of importance is thus particularly associated to a reduction of the

variance of the outcome. The sensitivity index attain values between 0 and 1.

The higher the value of the sensitivity index the more important is the random

variable. Variance based measures, such as the sensitivity index just defined, are

compact, and easy to understand and convey. This is an appropriate measure of

sensitivity to use to rank the input factors in order of importance even if the

input factors are correlated [113]. Furthermore, this method is completely

’model-free’. The sensitivity index is interpreted as being the proportion of the

total variance attributable to the random variable [113].

Factor prioritization by reduction of variance method has been used to calculate

the sensitivity [113] and the results obtained are shown in Table 5.7. As shown in

Table 5.7, when Random Sampling is selected Λ and β shows the maximum

variability in the output RIT with value 46.6268% and 45.7693% with 10, 000

samples and 46.12%, 45.9968% with 20, 000 samples. The next parameters in

variability are α, δ, l, ζ, k, γ1 with values 5.4738%, 0.9135%, 0.6753%, 0.3466%,

0.1714%, 0.0233% with 10, 000 samples and 5.6176%, 0.9968%, 0.7357%, 0.3479%,

0.1545%, 0.0308% with 20, 000 samples.

When Latin hyper cube sampling is considered, the parameter Λ shows again

107

Table 5.7: Percentage values of sensitivity index based on reduction of variance

Outcome for RT Λ β l κ α γ1 ζ δ

RSa1 46.6268% 45.7693% 0.6753% 0.1714% 5.4738% 0.0233% 0.3466% 0.9135%

LHSb1 46.351% 46.0087% 0.6799% 0.1803% 5.2876% 0.0311% 0.3031% 1.1583%

RSa2 46.12% 45.9968% 0.7357% 0.1545% 5.6176% 0.0308% 0.3479% 0.9968%

LHSb2 46.9666% 45.2418% 0.7461% 0.1591% 5.7453% 0.0331% 0.2929% 0.8152%



Figure 5.10: Pie chart of factor prioritization sensitivity indices LHSb1 and RSa1.

high variability with value 46.351% followed by β causing variability with

46.0087%, α with 5.2876%, δ with 1.1583%, l with 0.6799%, ζ with 0.3031% κ

with 0.1803% and γ1 for 0.0311% with 10, 000 samples. Similarly for the 20, 000

samples calculation shows Λ causing the highest variability 46.9666% and γ1

accounts for the least variability 0.0331% in the reproduction number RIT as

shown in the Table 5.7. Pie charts of factor prioritization sensitivity indices are

shown in Fig. 5.10 and Fig 5.11, which clearly shows the dominance of the net

inflow rate of the individuals Λ and transmission rate of disease β for the model.

In Fig. 5.10 and Fig. 5.11, δ, l, ζ, κ, γ1 are put together under title “Other” as

the sensitivity indices of these parameters are small compared to α, β and Λ.

108

Figure 5.11: Pie chart of factor prioritization sensitivity indices LHSb2 and RSa2.

5.4 Discussion

In order to investigate the effects of transmission of dynamics of SARS along

with treatment rate and the net flow of the individuals to the region of the

disease, a deterministic model SEIJTR has been constructed. First of all,

parameters involved in the model have been estimated based on the best fit to the

field data [121], published as daily reports on SARS epidemic in Hong Kong in

2003 by world health organization. A numerical method called Dormand-Prince

pairs method has been used as system solver for the non-linear differential model

SEIJTR and Levenberg-Marquardt technique has been used as the least square

optimizer in order to determine the best fit to the field data. MATLAB has been

used for all the calculations. A large number of simulations have been run to

estimate the parameters. The estimation of some parameters is obtained from the

demographic information on the city of Hong Kong. Different graphical and

numerical methods have been used to verify the estimation. Autocorrelation of

the residuals are within the confidence interval as shown in Fig. 5.4. This shows

that there is no correlation among residuals hence the estimates are reliable and

the model fits the data well. Uncertainty in the estimation of parameters is

investigated, followed by sensitivity analysis. The covariance matrix∑

provides

information on uncertainties in parameter estimation based on the covariance

between them. The information about interrelationship of the parameter given by

109

covariance matrix helps to determine the uncertainties in their measurements. It

is shown in Table 5.3 that the net inflow rate of individuals Λ moves in the same

direction as β, l, α, γ2 and ζ and moves in the opposite direction of κ, γ1 and δ.

Transmission rate of disease β have positive relation with l, α, γ2 and ζ but

negative relation with κ, γ1, and δ. The third most influential parameter α moves

in the same direction as γ2, l and ζ while in the opposite direction of κ, γ1, and δ.

For sensitivity analysis all the newly estimated parameters appearing in the

equation for reproductive number RIT are considered. Methods of Perturbation

of fixed point, Partial rank correlation and Factor prioritization by reduction of

variance have been applied for the calculation of sensitivity analysis.

• Method of finding the sensitivity indices based on the perturbation of fixed

point shows that the most important parameter for RIT are the rate at

which the new recruits enter the population (net inflow rate of individuals)

Λ and transmission rate, β as shown in Table 5.4. In order to decrease RIT

by 1% it is necessary to decrease Λ and β by 1%. Rate of progression from

infective to diagnosed class α is the third influential parameter. In order to

obtain 1% decrease in the value of RIT it is necessary to increase α by

1.81742% other important parameters are µ, l, ζ, δ q, κ and γ1 respectively.

• Partial Rank Correlation Coefficients of the parameters are calculated using

Random and Latin hyper cube sampling technique for two independently

generated samples of size 10, 000, 20, 000 each and the results are compared

as shown in Table 5.6. For both approaches, the rate at which the new

recruits enter the population, Λ and transmission rate, β have the highest

correlation with RIT , showing that these are the most important

parameters for RIT . The order of the importance of the remaining

parameters is α, δ, l, ζ, κ, γ1 respectively. The positive sign of PRCC

indicates that if we increase a parameter with positive sign, the

reproduction number RIT also increases. The negative sign with the value

of parameter suggests that if we increase it, reproduction number RIT will

decrease. Only the treatment rate, ζ shows the negative correlation with

RIT . All other parameters show positive correlation. Hence with the

110

increase in ζ, the reproduction number will decrease and thus slow down

the disease transmission. Also β and Λ have the greatest potential to make

the epidemic worse on increasing. However ζ is the parameter with the

maximum potential to decrease the intensity of epidemic when maximized.

• The results obtained in Table 5.7, using the factor prioritization variance

based technique, with the inclusion of both sampling techniques, determine

that the parameter Λ, the rate of new recruits entering the population, is

the cause of maximum variability in the reproduction number RIT .

Parameter β, transmission rate, is the next for accounting the variability in

RIT . Where as parameter γ1, natural recovery rate, causes the least

variability in the reproduction number RIT .

On the basis of sensitivity analysis using all the techniques, rate of new recruits

entering the population, Λ, transmission rate, β and rate of progression from

infective to diagnosed, α have been shown in Tables 5.4, 5.6 and 5.7, to be the

most influential parameters for the reproduction number. In order to control the

initial growth of the disease it is very important to control the net inflow of the

individuals to the concerned region and the transmission rate to control the

transmission of the disease. After the transmission rate, β diagnosis rate, α is the

most important factor to develop the control strategies. Faster movement of the

infective to diagnosed will reduce the transmission of disease. Although there was

not any specific treatment available at the time of SARS epidemic, but

sensitivity analysis comes up with the result that treatment rate, ζ is the

parameter with the potential to reduce the epidemic when it is maximized as

shown in Figs. 5.8 and 5.9.

In the next chapter SARS model with treatment (SEIJTR) has been

investigated numerically to study the impact of treatment on the transmission

dynamics of disease.

111

Chapter 6

Numerical Study of SARS Model

with Treatment (SEIJTR) and

Diffusion in the System

6.1 Introduction

Recent years have seen an increasing trend of utilizing mathematical models for

the prediction of and insight in infectious diseases. These models are considered

as conceptual tools to explain the behavior of disease at different scales, and allow

us to understand the spread of infection in the real world and the impact of

various factors on disease dynamics. The key concepts associated with

mathematical modeling, such as basic and effective reproduction number,

generation time, epidemic growth rates, mortality rates, transmission rates,

incubation periods, heterogeneities, disease transmission routes, risk factors for

diseases spread and pre-clinical infectiousness play significant roles in the

epidemiological analysis and control of diseases. The process of modeling in

epidemiology has, at its heart, the same underlying philosophy and goals as

ecological modelling. Both endeavors share the ultimate aim of attempting to

understand the prevalence and distribution of a species, together with the factors

that determine incidence, spread, and persistence [4, 64].

The spatio-temporal spread of infectious diseases is the most significant area of

112

epidemic modelling. Accurate and precise mathematical models enable scientists

to understand the risk factors of disease transmission and to develop workable

control strategies for possible future outbreaks. There are obvious public health

and/or economic benefits in understanding the infectious dynamics of diseases in

humans, animals, and plants. Furthermore, it is well understood how important

the spatial aspect of these dynamics is to understand disease spread [147]. In the

case of emerging and re-emerging outbreaks of an infectious disease, it is crucial

to quantify the characteristics of a disease in order to estimate the potential

threat. Accurate estimation of these characteristics relies on modified

epidemiological information.

This chapter is based on the numerical study of a mathematical model to

investigate the transmission of SARS. This model includes exposed, infected,

diagnosed, treatment and recovered classes (compartment). Diffusion has been

included in the system to examine its role in transmission of the disease. The

compartmental model for SARS transmission is given in Section 2. The numerical

scheme to solve the model is described in Section 3. The stability of numerical

solutions with and without diffusion is analyzed in Section 4. Section 5 shows

numerical simulations. Further discussion and conclusion are given in Section 6.

6.2 SEIJTR epidemic model

6.2.1 Equations

This model of SARS consists of the following system of non-linear partial

differential equations.

∂S

∂t= πΛ− β

(I + qE + lJ)

NS − µS + d1

∂2S

∂x2(6.1)

∂E

∂t= (1− π)Λ + β

(I + qE + lJ)

NS − (µ+ κ)E + d2

∂2E

∂x2(6.2)

∂I

∂t= κE − (µ+ α + δ)I + d3

∂2I

∂x2(6.3)

∂J

∂t= αI − (µ+ γ1 + δ + ζ)J + d4

∂2J

∂x2(6.4)

113

∂T

∂t= ζJ − (γ2 + µ+ δ(1− θ))T + d5

∂2T

∂x2(6.5)

∂R

∂t= γ1J + γ2T − µR + d6

∂2R

∂x2(6.6)

with initial conditions

S(0) = S0, E(0) = E0, I(0) = I0, J(0) = J0, T (0) = T0 and R(0) = R0 where

S,E, I, J, T and R represent susceptible, exposed, infected, diagnosed, treated

and recovered classes respectively and N denotes the total population,

N = S +E + I + J + T +R. d1, d2, d3, d4, d5 and d6 are the diffusivity constants.

Table 6.1 and Table 6.2, provide, respectively the description and the values of

the parameters.

To scale the population size in each compartment by the total population sizes by

substituting s = S/N , e = E/N , i = I/N , j = J/N , t1 = T/N , r = R/N

Π = Λ/N in the system of equations (6.1) - (6.6). After simplification replacing s

by S, e by E, i by I, j by J , t1 by T and r by R, the following dimensionless

system of equations is obtained:

∂S

∂t= −β(I+qE+lJ)S+(π−S)Π+γ1IS+δ(I+J+(1−θ)T )S+d1

∂2S

∂x2(6.7)

∂E

∂t= (1−π)Π+β(I+qE+ lJ)S−(Π+κ)E+δ(I+J+(1−θ)T )E+d2

∂2E

∂x2(6.8)

∂I

∂t= κE − (α + δ +Π)I + δ(I + J + (1− θ)T )I + d3

∂2I

∂x2(6.9)

∂J

∂t= αI − (Π + γ1 + ζ)J + δ(I + J + (1− θ)T )J + d4

∂2J

∂x2(6.10)

∂T

∂t= ζJ − (Π+γ2)T + δ(I+J +(θ− 1)+ (1− θ)T )T +d5

∂2T

∂x2(6.11)

∂R

∂t= γ1J + γ2T −ΠR+ δ(I + J + (1− θ)T )R+ d6

∂2R

∂x2(6.12)

where S + E + I + J + T +R = 1. The detail of calculation is given in appendix

A.6


The domain of all the calculations is considered as [−2, 2]. Boundary and initial


∂S(−2, t)

∂x=

∂E(−2, t)

∂x=

∂I(−2, t)

∂x=

∂J(−2, t)

∂x=

∂T (−2, t)

∂x=

∂R(−2, t)

∂x= 0

(6.13)

114

Table 6.1: Biological definition of parameters


Λ Rate at which new recruits enter the population


(the complementary proportion are infective)

β Transmission coefficient



κ Rate of progression from exposed to the infectives




γ2 Recovery due to treatment

ζ Treatment rate

δ SARS-induced mortality rate

θ Effectiveness of drugs as a reduction factor in disease-induced death of

infectious individuals(0 ≤ θ ≤ 1)

115

Table 6.2: Parameters’ value for model

Parameter Value (per day) Source

Λ 0.00002 [175]

π 0.85 [175]

β 0.24 [175]

µ .000035 [95]

l 0.65 [175]

κ 0.195 [175]

q 0.1 [57]

α 0.238 [175]

γ1 0.046 [175]

γ2 0.05 [175]

ζ 0.2 [175]

δ 0.024 [175]

θ 0.25 [175]

116

∂S(2, t)

∂x=

∂E(2, t)

∂x=

∂I(2, t)

∂x=

∂J(2, t)

∂x=

∂T (2, t)

∂x=

∂R(2, t)

∂x= 0 (6.14)

(i)

S0 = 0.98Sech(5x− 1), − 2 ≤ x ≤ 2.

E0 = 0, − 2 ≤ x ≤ 2.

I0 = 0.02Sech(5x− 1), − 2 ≤ x ≤ 2.

J0 = 0, − 2 ≤ x ≤ 2.

T0 = 0, − 2 ≤ x ≤ 2.

R0 = 0, − 2 ≤ x ≤ 2.

(ii)

S0 = 0.97 exp(−5(x− 1)2), − 2 ≤ x ≤ 2.

E0 = 0, − 2 ≤ x ≤ 2.

I0 = 0.03 exp(−5(x+ 1)2), − 2 ≤ x ≤ 2.

J0 = 0, − 2 ≤ x ≤ 2.

T0 = 0, − 2 ≤ x ≤ 2.

R0 = 0, − 2 ≤ x ≤ 2.

(iii)

S0 ={

0.96Sech(15x), − 2 ≤ x ≤ 2,

E0 = 0, − 2 ≤ x ≤ 2.

I0 =

0, − 2 ≤ x < −0.6,

0.04, − 0.6 ≤ x ≤ 0.6,

0, 0.6 < x ≤ 2.

J0 = 0, − 2 ≤ x ≤ 2.

T0 = 0, − 2 ≤ x ≤ 2.

R0 = 0, − 2 ≤ x ≤ 2.

117

t = 0

t = 0(i)

-2 -1 0 1 2x

0.2

0.4

0.6

0.8

1.0S,I

t = 0

t = 0

(ii)

-2 -1 0 1 2x

0.2

0.4

0.6

0.8

1.0S,I

Figure 6.1: Initial conditions (i) and (ii).

t = 0(iii)

-2 -1 0 1 2x

0.2

0.4

0.6

0.8

1.0S,I

Figure 6.2: Initial condition (iii).

118

Figs. 6.1 and 6.2 show the initial “population distributions” for S and I. A larger

susceptible and a smaller infected proportion is concentrated towards the right

half of the main domain in initial condition (i). In initial condition (ii), I has

high concentration in the left half of the domain [−2, 2] and population S has

concentration on the right half of the domain [−2, 2]. In the initial condition (iii)

susceptible S exists in high concentration around the middle of domain [−2, 2]

with infected also around the middle but beyond the domain of S.


Operator splitting method has been used to solve the SEIJTR model. According

to this technique the system of equations is divided into non-linear reaction

equations and linear diffusion equations [244]. The non-linear reaction equations

to be used for the first half-time step are given as:

1

2

∂S

∂t= πΠ−ΠS − β(I + qE + lJ)S + δ(I + J + (1− θ1)T )S (6.15)

1

2

∂E

∂t= (1− π)Π + β(I + qE + lJ)S − (Π + κ)E + δ(I + J + (1− θ)T )E (6.16)

1

2

∂I

∂t= κE − (α + δ +Π)I + δ(I + J + (1− θ)T )I (6.17)

1

2

∂J

∂t= αI − (Π + γ1 + ζ)J + δ(I + J + (1− θ)T )J (6.18)

1

2

∂T

∂t= ζJ − (Π + γ2)T + δ(I + J + (θ − 1) + (1− θ)T )T (6.19)

1

2

∂R

∂t= γ1J + γ2T − ΠR + δ(I + J + (1− θ)T )R (6.20)



1

2

∂S

∂t= d1

∂2S

∂x2(6.21)

1

2

∂E

∂t= d2

∂2E

∂x2(6.22)

1

2

∂I

∂t= d3

∂2I

∂x2(6.23)

1

2

∂J

∂t= d4

∂2J

∂x2(6.24)

119

1

2

∂T

∂t= d5

∂2T

∂x2(6.25)

1

2

∂R

∂t= d6

∂2R

∂x2(6.26)

Applying the forward Euler scheme the non-linear equations transform to

Sj+ 1

2i = Sj

i +∆t(πΠ−ΠSji −β(Iji +qEj

i + lJ ji )S

ji +δ(Iji +J j

i +(1−θ)T ji )S

ji ) (6.27)

Ej+ 1

2i = Ej

i+∆t((1−π)Π+β(Iji+qEji+lJ j

i )Sji−(Π+κ)Ej

i+δ(Iji+J ji+(1−θ)T j

i )Eji )

(6.28)

Ij+ 1

2i = Iji +∆t(κEj

i − (α + δ +Π)Iji + δ(Iji + J ji + (1− θ)T j

i )Iji ) (6.29)

Jj+ 1

2i = J j

i +∆t(αIji − (Π+γ1+ ζ)J ji + δ(Iji +J j

i +(1− θ)T ji )J

ji ) (6.30)

Tj+ 1

2i = T j

i+∆t(ζJ ji−(Π+γ2)T

ji+δ(Iji+J j

i+(θ−1)+(1−θ)T ji )T

ji ) (6.31)

Rj+ 1

2i = Rj

i +∆t(γ1Jji +γ2T

ji −ΠRj

i +δ(Iji +J ji +(1−θ)T j

i )Rji ) (6.32)

where Sji , E

ji , I

ji , J

ji , T

ji and Rj

i are the approximated values of S, E, I, J , T and

R at position −2 + i∆x, for i = 0, 1, . . . and time j∆t, j = 0, 1, . . . and Sj+ 1

2i ,

Ej+ 1

2i , I

j+ 12

i , Jj+ 1

2i , T

j+ 12

i , and Rj+ 1

2i denote their values at the first half-time step.

Similarly, for the second half-time step, the linear equations transform as

Sj+1i = S

j+ 12

i + d1∆t

(∆x)2(S

j+ 12

i−1 − 2Sj+ 1

2i + S

j+ 12

i+1 ) (6.33)

Ej+1i = E

j+ 12

i + d2∆t

(∆x)2(E

j+ 12

i−1 − 2Ej+ 1

2i + E

j+ 12

i+1 ) (6.34)

Ij+1i = I

j+ 12

i + d3∆t

(∆x)2(I

j+ 12

i−1 − 2Ij+ 1

2i + I

j+ 12

i+1 ) (6.35)

J j+1i = J

j+ 12

i + d4∆t

(∆x)2(J

j+ 12

i−1 − 2Jj+ 1

2i + J

j+ 12

i+1 ) (6.36)

T j+1i = T

j+ 12

i + d5∆t

(∆x)2(T

j+ 12

i−1 − 2Tj+ 1

2i + T

j+ 12

i+1 ) (6.37)

Rj+1i = R

j+ 12

i + d6∆t

(∆x)2(R

j+ 12

i−1 − 2Rj+ 1

2i +R

j+ 12

i+1 ) (6.38)

The stability condition satisfied by the numerical method described above is

given as:dn∆t

(∆x)2≤ 0.5, n = 1, 2, 3, 4, 5, 6. (6.39)

In each case, ∆x = 0.1, d1 = 0.025, d2 = 0.01, d3 = 0.001, d4 = 0.0, d5 = 0.0,

d6 = 0.0 and ∆t = 0.03 are used.

120


6.4.1 Disease-free equilibrium (DFE)

The basic reproduction number RIT is considered to be the threshold parameter

for any DFE and is defined as “the number of secondary cases which one case

would produce in a completely susceptible population”. The probability of

infecting a susceptible individual during one contact, duration of the infectious

period, and the number of new susceptible individuals contacted per unit of time

are the main factors in calculation of the reproduction number. Therefore RIT

may vary remarkably for different infectious diseases and also for the same disease

in different populations. The variational matrix of the system of equations (6.1) -

(6.6) at the disease-free equilibrium P0 = (1, 0, 0, 0, 0, 0), giving:

V 0 =

−Π −qβ −β + δ −lβ + δ (1− θ)δ 0

0 (qβ − κ− Π) β lβ 0 0

0 κ −(α + δ +Π) 0 0 0

0 0 α −(γ1 + ζ +Π) 0 0

0 0 0 ζ −(γ2 − (θ − 1)δ +Π) 0

0 0 0 γ1 γ2 −Π

The variational matrix V 0 can be written as

V 0 =

A11 A12

A21 A22

where

A11 =

−Π −qβ

0 (qβ − κ− Π)

, A12 =

−β + δ −lβ + δ (1− θ)δ 0

β lβ 0 0

,

A21 =

0 κ

0 0

0 0

0 0

and

121

A22 =

−(α + δ +Π) 0 0 0

α −(γ1 + ζ +Π) 0 0

0 ζ −(γ2 − (θ − 1)δ +Π) 0

0 γ1 γ2 −Π

.

The stability of the point of equilibrium, P0(1, 0, 0, 0, 0, 0) depends on the

characteristic of the eigenvalues of the matrices A11 and A22. The eigenvalues of

matrix A22, −(α+ δ +Π), −(γ1 + ζ +Π), −(γ2 − δ(θ− 1) +Π), −Π are all clearly

negative. The eigenvalues of the matrix A11 are qβ − κ−Π, and −Π. The second

eigenvalue is clearly negative. Also shown in 3.4.2 RIT < 1 implies that

qβ − κ− Π < 0.

where RIT = β (q(α + δ + Π) (γ1 + ζ + Π) +κ (lα + γ1 + ζ + Π))(κ + Π) (α + δ + Π) (γ1 + ζ + Π)

This shows that P0 is stable for RIT < 1. In the same way we can illustrate the

stability of the endemic point for RIT > 1. The expression for RIT is derived in

the same way as given in appendix A.5.

6.4.2 Endemic equilibrium without diffusion

The variational matrix of the system of equations (6.1) - (6.6) at

P ∗(S∗, E∗, I∗, J∗, T ∗, R∗), is given by

V ∗ =

a11 a12 a13 a14 a15 a16

a21 a22 a23 a24 a25 a26

a31 a32 a33 a34 a35 a36

a41 a42 a43 a44 a45 a46

a51 a52 a53 a54 a55 a56

a61 a62 a63 a64 a65 a66

where

a11 = −β(I + qE + lJ) + δ(I + J + (1− θ)T )− Π, a12 = −qβS, a13 = −βS + δS,

a14 = −lβS + δS, a15 = δ(1− θ)S, a21 = β(I + qE + lJ),

a22 = qβS + δ(I + J + (θ − 1)T )− κ− Π,

a23 = βS + δE, a24 = lβS + δE, a25 = δ(1− θ)E, a32 = κ,

122

Table 6.3: Values of LHS of Routh-Hurwitz criteria of equilibrium without diffusion

Case Equil. Pt. C1 C2 C3 C4 C5 C6 Stability

1 P1 0.760 4.7× 10−13 0.167 0.011 0.2× 10−4 2.7× 10−8 Stable

2 P2 0.761 3.4× 10−13 0.127 0.012 2.4× 10−4 2.2× 10−8 Stable

3 P3 0.702 4.1× 10−13 0.009 0.009 1.4× 10−4 2.3× 10−8 Stable

4 P4 0.759 6.2× 10−13 0.126 0.011 1.7× 10−4 3.3× 10−8 Stable

5 P5 0.818 5.1× 10−13 0.156 0.014 2.6× 10−4 3.2× 10−8 Stable

a33 = −(Π + α + δ) + δI + δ(I + J + (1− θ)T ), a34 = δI, a35 = δ(1− θ)I,

a43 = α + δJ , a44 = −(Π + γ1 + ζ) + δJ + δ(I + J + (1− θ)T ), a45 = δ(θ − 1)J ,

a53 = δT , a54 = δT + ζ,

a55 = −(Π + γ2) + δ(1− θ)T + δ(I + J + (1− θ)T + (θ − 1)),

a63 = δR, a64 = γ1 + δR, a65 = γ2 + δ(1− θ)R, a66 = δ(I + J + (1− θ)T )− Π,

a16 = a26 = a31 = a36 = a41 = a42 = a46 = a51 = a52 = a56 = a61 = a62 = 0.

The characteristic equation for P ∗(S∗, E∗, I∗, J∗, T ∗, R∗) can be written as

λ6 + p1λ5 + p2λ

4 + p3λ3 + p4λ

2 + p5λ+ p6 = 0 (6.40)

where p1, p2, p3, p4, p5, p6 and the Routh-Hurwitz conditions are calculated on

the basis of [167]. This calculation is given in appendix A.6 whereas their

expressions are given as:

C1 : p1 > 0

C2 : p6 > 0

C3 :p1p2−p3

p1> 0

C4 :p1p2p3−p23−p21p4−p1p5

p1p2−p3> 0

C5 :=p23p4−p2p3p5+p25+p21(p

24−p2p6)+p1(p22p5−p2p3p4−2p4p5+p3p6)

p23+p21p4−p1(p2p3+p5)> 0

C6 :=

p23p4p5−p2p3p25+p35−p33p6+p31p26+p21(p

24p5−p3p4p6−2p2p5p6)+p1(p22p

25+p2p3(p3p6−p4p5)+p5(3p3p6−2p4p5))

p23p4−p2p3p5+p25+p21(p24−p2p6)+p1(p22p5−p2p3p4−2p4p5+p3p6)

>

0

Here, P1, P2, P3, P4, and P5 are the points of equilibrium given as:

P1 = (0.474068, 0.000068, 000050, 0.000049, 000143, 0.600255),

P2 = (0.586992, 0.000056, 0.000042, 0.000040, 0.000119, 0.471351),

123

P3 = (0.420033, 0.000073, 0.000071, 0.000051, 0.000151, 0.661921),

P4 = (0.388376, 0.000076, 0.000056, 0.000055, 0.000160, 0.698072),

P5 = (0.514339, 0.000063, 0.000038, 0.000046, 0.000136, 0.554296).

6.4.3 Endemic equilibrium with diffusion

To calculate the small perturbations S1(x, t), E1(x, t), I1(x, t), J1(x, t), T1(x, t)

and R1(x, t) the equations (6.1) - (6.6) are linearised about the point of

equilibrium P ∗(S∗, E∗, I∗, J∗, T ∗, R∗) as described in [35, 201], giving

∂S1

∂t= a11S1 + a12E1 + a13I1 + a14J1 + a15T1 + a16R1 + d1

∂2S1

∂x2(6.41)

∂E1

∂t= a21S1 + a22E1 + a23I1 + a24J1 + a25T1 + a26R1 + d2

∂2E1

∂x2(6.42)

∂I1∂t

= a31S1 + a32E1 + a33I1 + a34J1 + a35T1 + a36R1 + d3∂2I1∂x2

(6.43)

∂J1∂t

= a41S1 + a42E1 + a43I1 + a44J1 + a45T1 + a46R1 + d4∂2J1∂x2

(6.44)

∂T1

∂t= a51S1 + a52E1 + a53I1 + a54J1 + a55T1 + a56R1 + d5

∂2T1

∂x2(6.45)

∂R1

∂t= a61S1 + a62E1 + a63I1 + a64J1 + a65T1 + a66R1 + d6

∂2R1

∂x2(6.46)

where a11, a12, a13 ... etc are the elements of the variational matrix V ∗ calculated

using the method described in [198]. We assume the existence of a Fourier series

solution of equations (6.41) - (6.46), of form:

S1(x, t) =∑k


E1(x, t) =∑k


I1(x, t) =∑k


J1(x, t) =∑k


T1(x, t) =∑k

Tkeλt cos(kx) (6.51)

R1(x, t) =∑k


124

where k = nπ2, (n = 1, 2, 3, · · · · · · ) is the wave number for node n. Substituting

the values of S1, E1, I1, J1, T1 and R1 into the equations (6.41) -(6.46), the

equations are transformed into∑k

(a11 − d1k2 − λ)Sk +

∑k

a12Ek +∑k

a13Ik +∑k

a14Jk +∑k

a15Tk = 0 (6.53)

∑k

a21Sk +∑k

(a22 − d2k2 − λ)Ek +

∑k

a23Ik +∑k

a24Jk +∑k

a25Tk = 0 (6.54)

∑k

a32Ek+∑k

(a33−d3k2−λ)Ik+

∑k

a34Jk+∑k

a35Tk = 0 (6.55)

∑k

a43Ik+∑k

(a44−d4k2−λ)Jk+

∑k

a45Tk = 0 (6.56)

∑k

a53Ik+∑k

a54Jk+∑k

(a55−d5k2−λ)Tk = 0 (6.57)

∑k

a63Ik+∑k

a64Jk+∑k

a65Tk+∑k

(a66−d6k2−λ)Rk = 0 (6.58)

The Variational matrix V for the equations (6.53) - (6.58)

V =

a11 − d1k2 a12 a13 a14 a15 0

a21 a22 − d2k2 a23 a24 a25 0

0 a32 a33 − d3k2 a34 a35 0

0 0 a43 a44 − d4k2 a45 0

0 0 a53 a54 a55 − d5k2 0

0 0 a63 a64 a65 a66 − d6k2

The characteristic equation for the variational matrix V is given as

λ6 + q1λ5 + q2λ

4 + q3λ3 + q4λ+ q2 + q5λ+ q1 + q6 = 0 (6.59)

where q1, q2, q3, q4, q5 and q6 are calculated by the technique used in [198].

The Routh-Hurwitz Conditions are given as:

C1 : q1 > 0

C2 : q6 > 0

C3 :q1q2−q3

q1> 0

C4 :q1q2q3−q23−q21q4−q1q5

q1q2−q3> 0

C5 :=q23q4−q2q3q5+q25+q21(q

24−q2q6)+q1(q22q5−q2q3q4−2q4q5+q3q6)

q23+q21q4−q1(q2q3+q5)> 0

125

Table 6.4: Values of LHS of Routh-hurwitz criteria of equilibrium with diffusion

Case Equil. Pt. C1 C2 C3 C4 C5 C6 Stability

1 P1 0.849 4.5× 10−10 0.216 0.023 0.0011 2.2× 10−5 Stable

2 P2 0.850 5.1× 10−10 0.217 0.023 0.0012 2.5× 10−5 Stable

3 P3 0.790 3.3× 10−10 0.187 0.019 0.0008 1.6× 10−5 Stable

4 P4 0.849 4.2× 10−10 0.215 0.022 0.0010 1.9× 10−5 Stable

5 P5 0.908 5.9× 10−10 0.245 0.027 0.0013 2.8× 10−5 Stable

C6 :=

q23q4q5−q2q3q25+q35−q33q6+q31q26+q21(q

24q5−q3q4q6−2q2q5q6)+q1(q22q

25+q2q3(q3q6−q4q5)+q5(3q3q6−2q4q5))

q23q4−q2q3q5+q25+q21(q24−q2q6)+q1(q22q5−q2q3q4−2q4q5+q3q6)

> 0


The technique used in chapter 3 is used to calculate the the first excited mode of

the oscillation n. According to the description of mode of excitation for the curve

given in Fig. 6.3, n = 1 represents the first mode of excitation as being closest to

the β-axis.

f(β) =A+B + (CD + E)

F. (6.60)

where A = p23p4p5 − p2p3p25 + p35 − p33p6 + p31p

26, B = p21(p

24p5 − p3p4p6 − 2p2p5p6),

C = p1(p22p

25 + p2p3), D = −p4p5 + p3p6, E = p1p5(−2p4p5 + 3p3p6).

F = (p23p4 − p2p3p5 + p25 + p21(p24 − p2p6) + p1(−p2p3p4 + p22p5 − 2p4p5 + p3p6))

It is observed that the bifurcation value of transmission coefficient, β and rate of

progression from infective to diagnosed, α increases with diffusion as compared to

the system without diffusion. The corresponding bifurcation diagrams for β and

α are shown in appendix A.6.


Five cases with different values of β, the transmission coefficient and α, rate of

progression from infective to diagnosed, are summarised in Table 6.5. For all the

cases given in Table 6.5, numerical solutions are calculated for the SEIJTR

126

n=1

0.1 0.2 0.3 0.4 0.5 0.6Β

-0.00002

0.00002

0.00004

0.00006

fHΒL

n=2

0.2 0.4 0.6 0.8Β

-0.0001

0.0001

0.0002

0.0003

fHΒL

n=30.2 0.4 0.6 0.8Β

0.0002

0.0004

0.0006

0.0008

0.0010

fHΒL


Table 6.5: Bifurcation values of β and α.

Case Values Considered Bifurcation Values


β α β α β α

1 0.242 0.238 0.313 0.068 0.356 0.111

2 0.182 0.238 0.253 0.080 0.288 0.099

3 0.242 0.179 0.306 0.045 0.349 0.089

4 0.303 0.238 0.383 0.053 0.435 0.118

5 0.242 0.298 0.321 0.088 0.387 0.129

127

model both in the absence and presence of diffusion in the system.

6.5.1 Solutions of SEIJTR model in the absence of

diffusion (Case 1)

Fig. 6.4, shows the numerical solution for initial condition (i) in the absence of

diffusion. It can be observed that the susceptible population decreases abruptly in

the first five days of disease, with the concentration of the population fluctuating

near the edges of domain [−1, 1]. At t = 10, 15, 20 days, the number of susceptible

decays with the passage of time. The decrease in susceptible population continues

between days t = 5 and t = 20, but move slowly than in the first five days. In the

first five days of the onset of the disease, maximum population is exposed to

SARS within the domain [−.6, 1]. After t = 5 days, exposed population

proportion shows a rapid decrease till t = 10 days. With the passage of time, the

exposed population proportion keeps on decreasing slowly, with concentration

confined to the domain [−1, 1] at t = 20 days. In the first five days of onset of

SARS, most of the exposed get infected. There is a very large increase in the

proportion of infective in the first 5 days, concentrated in the domain [−.6, 1].

After t = 5 days, infective proportion decreases quickly till t = 15 days. From

t = 15 days to t = 20 days, the infected population proportion decreases slowly

but remains higher than the initial infected population proportion. Infectives, are

largely confined to the main concentration region [−1, 1] throughout the period of

prevailing disease. As soon as the infection spreads and some of the population

becomes infective, the proportion of diagnosed cases increases quite quickly from

t = 0 to t = 5 days. This trend continues but at a slower pace, for the next five

days within the domain of concentration [−1, 1]. The diagnosed population

proportion decreases quickly after t = 10 days of the disease, but there is still a

substantial proportion of diagnosed at t = 20. Once the infected are diagnosed,

they start entering the treatment compartment. A small increase in the

population being treated can be observed in the first five days of disease. From

t = 5 days to t = 10 days, there is a sharp increase in the proportion of

population in the treatment class. This proportion keeps on increasing but at a

128

low pace and reaches its maximum at t = 15 days. After that, the treated

population proportion decreases slowly with most of this population remaining

confined to the domain [−1, 1]. At t = 20 days, a large proportion of the

population is still in the treatment compartment. Recovery is slow in the first five

days of spread of SARS but increases rapidly in the next five days. A sharp

increase in the proportion of recovered individuals is noticeable between

t = 10, 15 and 20 days. The highest recovery is observed at t = 20 days.

Fig. 6.5, shows the output for initial condition (ii) in the absence of diffusion.

The susceptible population, which is mainly concentrated in the domain [0, 2],

shows sharp decline in the first five days of the spread of SARS. After five days,

concentration of susceptible is confined to the edges of the domain [0, 2]. A small

decrease is observed from t = 5 days onward (not visible in Fig. 6.5). The

exposed population grows in proportion remarkably fast in the first five days,

showing that the rate of infection is very high during this period. Almost half of

the susceptible population gets exposed to the disease in the first five days. Then

the proportion of the population that are exposed rapidly decreases in the next

five days. After t = 10 days, a slower decrease in exposed population proportion

occurs. Most of the population becomes infective in the first five days and the

domain of concentration of infective moves from [−2, 0] to [0, 2]. After t = 5 days,

the infective proportion starts decreasing slowly till t = 10 days in the domain

[0, 2]. A sharp decline in infective proportion is observed in the whole domain

[−2, 2] between t = 10 days and t = 15 days, which continues at slow pace

thereafter. The diagnosed proportion increases over the first five days. Between

t = 5 and t = 10 days, an increase in diagnosed population proportion is noticed

in the domain [0, 2]. In the same period, a much smaller proportion of diagnosed

is also observed in the domain [−2, 0]. A small proportion of the population is in

the treatment class after five days and this proportion increases quickly in the

next five days. The proportion of the population in treatment class reaches its

maximum in t = 15 days and after that it decreases. There is negligible recovery

in the first five days followed by a quicker recovery in the next five days in

domain [−1.5, 2], particularly the right half of the domain [−2, 2]. The recovery

129

t = 0

t = 5

t=10

t=10t=5

-2 -1 0 1 2x

0.2

0.4

0.6

0.8

1.0S

t = 5

t = 10

t = 20

t = 15

-2 -1 0 1 2x

0.1

0.2

0.3

0.4E

t = 0

t = 5

t = 10

t = 20

t = 15

t=0

-2 -1 0 1 2x

0.1

0.2

0.3

0.4I

t = 0

t = 5

t = 10

t = 20

t = 15

-2 -1 0 1 2x

0.1

0.2

0.3

0.4J

t = 0t = 5

t = 10

t = 20

t = 15

-2 -1 0 1 2x

0.1

0.2

0.3

0.4T

t = 0

t = 5

t = 10

t = 20

t = 15

-2 -1 0 1 2x

0.1

0.2

0.3

0.4

0.5R

Figure 6.4: Solutions for initial condition (i) without diffusion.

130

proportion is greatest at t = 20 days.

Fig. 6.6, shows the results with initial condition (iii) in the absence of diffusion.

In the first five days of the spread of SARS, the susceptible proportion reduces

quickly to very low concentration around the edges at x = −0.1 and x = 0.1. In

the first five days, most of the susceptible population gets exposed to the disease.

After that, there is a quick decrease in the exposed population proportion until

t = 10 days. This decreasing trend continues until t = 20 days in the domain

[−.1, .1]. The infected population proportion increases quickly at a slower pace in

the domain [−.1, .1] from its initial concentration in domain [−.5, .5] and reaches

its peak value at t = 5 days. After that, infection decreases until t = 20 days.

The diagnosed population proportion increases considerably in the first five days

and is concentrated in the domain [−.5, .5]. The maximum diagnosed proportion

occurs ten days after the onset of SARS. After t = 10 days, the diagnosed

proportion of the population decreases. In the beginning, diagnosed individuals

move slowly to the treatment class but a considerable increase in the proportion

of the treatment population is observed between t = 5 and t = 10 days. Recovery

is extremely slow in the first five days but during the next five days a substantial

increase in the recovered population is observed, which continues till t = 20 days.

6.5.2 Solutions of SEIJTR model in the presence of

diffusion (Case 1)

Fig. 6.7, shows the output for initial condition (i) in the presence of diffusion.

The impact of the diffusion on the solution is well observed, as the proportion of

the susceptible reduces rapidly in the first five days, with a negligible proportion

of susceptible at t = 5 days in the domain [−2,−.5] and [.5, 2]. Most of these

susceptible in domain [−1.5, 1.5] are exposed to SARS during the first five days.

A sharp decrease in the exposed proportion of population between t = 5 days and

t = 10 days is observed, with the spread of the exposed population in the domain

[−2, 2]. The infected proportion of the population is maximised within t = 5 days

of spread of SARS in the domain [−1.5, 1.5]. In the next five days, infection not

131

t = 0

t = 5t = 10

-2 -1 0 1 2x

0.2

0.4

0.6

0.8

1.0S

t = 5

t = 10

t = 20

t = 15

-2 -1 0 1 2x

0.1

0.2

0.3

0.4

0.5E

t = 0

t = 5

t = 10

t = 20

t = 15

t=15

-2 -1 0 1 2x

0.1

0.2

0.3

0.4I

t = 5

t = 10

t = 20

t = 15

t=5

t=10

-2 -1 0 1 2x

0.1

0.2

0.3

0.4J

t = 5

t = 10

t = 20

t = 15

t=10 t=15

-2 -1 0 1 2x

0.1

0.2

0.3

0.4T

t = 5

t = 10

t = 20

t = 15

t=15t=10

t=5

-2 -1 0 1 2x

0.1

0.2

0.3

0.4

0.5R

Figure 6.5: Solutions for initial condition (ii) without diffusion.

132

t = 0

t = 5

-2 -1 0 1 2x

0.2

0.4

0.6

0.8

1.0S

t = 5

t = 10

t = 20

t = 15

-2 -1 0 1 2x

0.1

0.2

0.3

0.4E

t = 0

t = 5

t = 10

t = 20

t = 15

-2 -1 0 1 2x

0.1

0.2

0.3

0.4I

t = 5

t = 10

t = 20

t = 15

-2 -1 0 1 2x

0.1

0.2

0.3

0.4J

t = 5

t = 10

t = 20

t = 15

-2 -1 0 1 2x

0.1

0.2

0.3

0.4T

t = 5

t = 10

t = 20

t = 15

-2 -1 0 1 2x

0.1

0.2

0.3

0.4R

Figure 6.6: Solutions for initial condition (iii) without diffusion.

133

only decreases but also spreads rapidly in the domain [−2, 2]. The diagnosed

population proportion peaks in the first ten days of the disease and is spread

across whole domain [−2, 2]. The treatment class attains its peak value fifteen

days after the spread of SARS. This is similar to the case without diffusion but

with, different peak values. Due to the presence of diffusion, recovery spreads in

the domain [−2, 2]. The recovered population proportion increases rapidly after

t = 5 days and spreads to the domain [−1.5, 1.5]. Maximum population is

recovered in t = 20 days.

Fig. 6.8, shows the solution for initial condition (ii) and with diffusion.

Susceptible move from the domain [0, 2] to the domain [−1.5, 0] at t = 5 days and

then to the domain [−2,−.5] at t = 10 days (not clearly visible in Fig. 6.8 ). The

exposed population spread in the domain [−.5, 2], attaining its maximum

proportion at t = 5 days and then declining sharply in the next five days,

spreading in the domain [−1, 2]. After that there is slow decrease during the next

ten days. Concentration of infected population remains mostly confined to the

domain [−.5, 2] with a peak value at t = 5 days. After that, the infection

proportion reduces gradually. At t = 20 days a very small proportion of the

population is infected. The diagnosed population proportion also spreads in the

domain [−.5, 2] with its maximum occurring at t = 10 days. A sharp increase in

the treated population is observed from t = 5 days to t = 10 days in the domain

[−2, 2], mostly in the domain [−.5, 2]. The treated proportion of the population

attains its peak value fifteen days after the spread of the disease and then reduces

slowly. After fifteen days recovery seems to propagate in the larger domain [−2, 2]

but mostly in the domain [−.5, 2]. As compared to without diffusion case for

recovered population, peak values are smaller but domain of concentration

spreads.

Fig. 6.9, shows the solution for initial condition (iii) with diffusion. There is a

quick decline in the susceptible population proportion during the initial five days

of the onset of disease. Maximum population exposure to SARS occurs in the

domain [−1, 1] in the first five days. Between t = 5 and t = 10 days, the peak

value of the exposed population proportion decreases while the exposed spread in

134

t = 0

t = 5t=5

-2 -1 0 1 2x

0.2

0.4

0.6

0.8

1.0

S

t = 5

t = 10

t = 15t = 20

-2 -1 0 1 2x

0.05

0.10

0.15

0.20

0.25E

t = 5

t = 10

t = 15t = 20

t=0

-2 -1 0 1 2x

0.05

0.10

0.15

0.20

0.25I

t = 5

t = 10

t = 20

t = 15

-2 -1 0 1 2x

0.05

0.10

0.15

0.20

0.25J

t = 5

t = 10

t = 20

t = 15

-2 -1 0 1 2x

0.05

0.10

0.15

0.20

0.25T

t = 5

t = 10

t = 15

t = 20

-2 -1 0 1 2x

0.05

0.10

0.15

0.20

0.25R

Figure 6.7: Solutions for initial condition (i) with diffusion.

135

t = 0

t = 5

-2 -1 0 1 2x

0.2

0.4

0.6

0.8

1.0S

t = 5

t = 10

t = 20 t = 15

-2 -1 0 1 2x

0.05

0.10

0.15

0.20

0.25

0.30E

t = 0

t = 5

t = 10

t = 20

t = 15

t=5

-2 -1 0 1 2x

0.05

0.10

0.15

0.20

0.25

0.30I

t = 5

t = 10

t = 20

t = 15

t=10

-2 -1 0 1 2x

0.05

0.10

0.15

0.20

0.25

0.30J

t = 5

t = 10

t = 20

t = 15

t=10 t=15t=5

-2 -1 0 1 2x

0.05

0.10

0.15

0.20

0.25

0.30T

t = 5

t = 10

t = 20

t = 15

t=20

t=15

t=10

-2 -1 0 1 2x

0.05

0.10

0.15

0.20

0.25

0.30R

Figure 6.8: Solutions for initial condition (ii) with diffusion.

136

Table 6.6: Peak values of susceptible(S), exposed(E), infective(I) and recovered(R)

(without diffusion)

Case t S(i) S(ii) S(iii) E(i) E(ii) E(iii) I(i) I(ii) I(iii) R(i) R(ii) R(iii)

1 00 0.980 0.970 0.960 0.000 0.000 0.000 0.020 0.030 0.040 0.000 0.000 0.000

05 0.034 0.039 0.009 0.406 0.455 0.393 0.324 0.313 0.322 0.024 0.014 0.026

10 0.009 0.007 0.004 0.133 0.149 0.128 0.199 0.213 0.194 0.145 0.120 0.150

15 0.004 0.003 0.002 0.041 0.047 0.040 0.082 0.090 0.079 0.305 0.275 0.309

20 0.002 0.002 0.001 0.013 0.014 0.012 0.029 0.032 0.028 0.437 0.408 0.441

2 00 0.980 0.970 0.960 0.000 0.000 0.000 0.020 0.030 0.040 0.000 0.000 0.000

05 0.053 0.070 0.009 0.423 0.479 0.409 0.324 0.309 0.322 0.021 0.012 0.024

10 0.012 0.014 0.007 0.138 0.157 0.133 0.204 0.219 0.199 0.139 0.113 0.145

15 0.005 0.006 0.002 0.043 0.049 0.042 0.085 0.094 0.082 0.299 0.267 0.305

20 0.003 0.002 0.001 0.013 0.015 0.013 0.029 0.034 0.029 0.433 0.403 0.437

3 00 0.980 0.970 0.960 0.000 0.000 0.000 0.020 0.030 0.040 0.000 0.000 0.000

05 0.034 0.039 0.009 0.407 0.373 0.394 0.456 0.352 0.372 0.019 0.011 0.022

10 0.009 0.007 0.004 0.134 0.271 0.129 0.150 0.284 0.266 0.124 0.102 0.129

15 0.004 0.003 0.004 0.042 0.131 0.041 0.048 0.141 0.127 0.276 0.247 0.281

20 0.002 0.002 0.001 0.013 0.053 0.013 0.015 0.058 0.052 0.411 0.382 0.415

4 00 0.980 0.970 0.960 0.000 0.000 0.000 0.020 0.030 0.040 0.000 0.000 0.000

05 0.008 0.037 0.008 0.217 0.439 0.383 0.191 0.315 0.322 0.025 0.016 0.028

10 0.007 0.006 0.002 0.129 0.144 0.125 0.195 0.208 0.191 0.149 0.125 0.154

15 0.003 0.002 0.001 0.040 0.045 0.039 0.079 0.087 0.077 0.308 0.279 0.313

20 0.002 0.001 0.001 0.012 0.014 0.012 0.028 0.031 0.027 0.439 0.412 0.443

5 00 0.980 0.970 0.960 0.000 0.000 0.000 0.020 0.030 0.040 0.000 0.000 0.000

05 0.035 0.039 0.009 0.406 0.456 0.393 0.284 0.279 0.281 0.028 0.016 0.031

10 0.009 0.007 0.004 0.132 0.148 0.127 0.152 0.147 0.165 0.161 0.134 0.166

15 0.004 0.003 0.002 0.041 0.046 0.039 0.056 0.054 0.061 0.323 0.293 0.328

20 0.002 0.002 0.001 0.012 0.014 0.012 0.018 0.017 0.020 0.452 0.424 0.456

the domain [−1.5, 1.5]. Infection peaks at t = 5 days and spreads in the domain

[−1.5, 1.5] at t = 10 days. Diffusion also causes diagnosed, treated and recovered

population proportions to spread in the domain [−1.5, 1.5] with peak values at

t = 10, 15, and 20 days respectively.

6.5.3 Other cases

The graphical output for Cases 2, 3, 4 and 5 is not shown because of its similarity

to Case 1. The numerical results of all cases are summarized in Tables 6.6 and

6.7. Here Sj, Ej, Ij and Rj for j = (i), (ii) and (iii) represent peak values of the

proportion of susceptible, exposed, infected and recovered population in the

domain [−2, 2] in the absence (Table 6.6) and presence (Table 6.7) of diffusion for

the initial population distributions (i), (ii) and (iii). The following description is

based on the results given in Tables 6.6 and 6.7.

Moving from Case 1 to Case 2, there is a decrease in the transmission coefficient

137

t = 0

t = 5

-2 -1 0 1 2x

0.2

0.4

0.6

0.8

1.0S

t = 5

t = 10t = 20

t = 15

-2 -1 0 1 2x

0.02

0.04

0.06

0.08

0.10E

t = 0

t = 5

t = 10

t = 20t = 15

-2 -1 0 1 2x

0.02

0.04

0.06

0.08

0.10I

t = 5t = 10

t = 20

t = 15

-2 -1 0 1 2x

0.02

0.04

0.06

0.08

0.10J

t = 5

t = 10

t = 15

t=20

-2 -1 0 1 2x

0.02

0.04

0.06

0.08

0.10T

t = 5

t = 10

t = 20

t=15

-2 -1 0 1 2x

0.02

0.04

0.06

0.08

0.10R

Figure 6.9: Solutions for initial condition (iii) with diffusion.

138

Table 6.7: Peak values of susceptible(S), exposed(E), infective(I) and recovered(R)

(with diffusion)

Case t S(i) S(ii) S(iii) E(i) E(ii) E(iii) I(i) I(ii) I(iii) R(i) R(ii) R(iii)

1 00 0.980 0.970 0.960 0.000 0.000 0.000 0.020 0.030 0.040 0.000 0.000 0.000

05 .00986 .01191 .00706 .22313 .29394 .06802 .18704 .20155 .05063 .01379 .00794 .00613

10 .00108 .00173 .00129 .05818 .07859 .01705 .09902 .12357 .02796 .08119 .07235 .02591

15 .00013 .00067 .00035 .01584 .02141 .00467 .03612 .04704 .01034 .16517 .16339 .04998

20 .00008 .00032 .00017 .00445 .00593 .00138 .01168 .01541 .00343 .23444 .24169 .07034

2 00 0.980 0.970 0.960 0.000 0.000 0.000 0.020 0.030 0.040 0.000 0.000 0.000

05 .01416 .01892 .01175 .23344 .31042 .07408 .18182 .19248 .04489 .01161 .00590 .00552

10 .00171 .00252 .00201 .06135 .08350 .01909 .10112 .12661 .02878 .07472 .06503 .02318

15 .00021 .00096 .00054 .01679 .02284 .00530 .03754 .04916 .01113 .15645 .15331 .04589

20 .00009 .00044 .00023 .00473 .00635 .00157 .01226 .01626 .00378 .22495 .23083 .06566

3 00 0.980 0.970 0.960 0.000 0.000 0.000 0.020 0.030 0.040 0.000 0.000 0.000

05 .00996 .01204 .00712 .22348 .29444 .06814 .21454 .22479 .05868 .01100 .00621 .00499

10 .00109 .00174 .00130 .05851 .07904 .01711 .13599 .16516 .03819 .06908 .06095 .02217

15 .00013 .00068 .00035 .01600 .02165 .00469 .05891 .07484 .01665 .14766 .14526 .04453

20 .00008 .00032 .00017 .00451 .00602 .00138 .02229 .02882 .00641 .21674 .22296 .06451

4 00 0.980 0.970 0.960 0.000 0.000 0.000 0.020 0.030 0.040 0.000 0.000 0.000

05 .00759 .00839 .00490 .21664 .28347 .06427 .19018 .20684 .05379 .01543 .00955 .00669

10 .00075 .00132 .00092 .05618 .07547 .01587 .09765 .12152 .02743 .08576 .07762 .02799

15 .00009 .00052 .00025 .01523 .02049 .00431 .03522 .04567 .00987 .17129 .17053 .05301

20 .00007 .00025 .00014 .00427 .00567 .00127 .01132 .01487 .00323 .24110 .24937 .07383

5 00 0.980 0.970 0.960 0.000 0.000 0.000 0.020 0.030 0.040 0.000 0.000 0.000

05 .00997 .01206 .00719 .22333 .29429 .06821 .16394 .18103 .043998 .01612 .00938 .00704

10 .00109 .00175 .00131 .05807 .07846 .01709 .07468 .09523 .02127 .09009 .08086 .02859

15 .00013 .00068 .00035 .01576 .02129 .00468 .02411 .03193 .00701 .17671 .17542 .05352

20 .00008 .00032 .00017 .00442 .00589 .00138 .00712 .00949 .00213 .24523 .25310 .07389

139

from β = 0.242 to β = 0.182, as shown in Table 6.5. As a result, the susceptible

population proportion show an increase in peak values for initial population

distributions (i)− (iii) with and without diffusion. There is quite a significant

increase in the first ten days of SARS as compared to Case 1. The proportions of

population exposed also show a significant increase from t = 10 to t = 15 days as

compared to Case 1. Infection grows in the last ten days of the disease without

diffusion but with diffusion infective population show an increase as compared to

Case 1 from t = 10 to t = 20 days. There is a moderate decrease in the peak

value of the recovered population proportion with all initial conditions.

In Case 3, the rate of progression from infective to diagnosed, α, is decreased

from α = 0.238 to α = 0.179 as compared to Case 1, while keeping values of β

same. There is no change in the susceptible population proportion in the absence

of diffusion but, with diffusion, a small increase appears at t = 5 and t = 10 days

of disease as compared to Case 1. Initial conditions (i) and (iii) show a small

increase in the proportion of exposed both with and without diffusion while

initial condition (ii) shows a decrease at t = 5 days and great increase in next

t = 10 days without diffusion and small increase with diffusion. In Case 3 infected

proportion values are higher than for Case 1 both with and without diffusion.

Peak values of the recovered population proportion are lower for Case 3 than

Case 1.

In Case 4, there is an increase in the transmission coefficient, β, from β = 0.242

to β = 0.303 as compared to Case 1. As a result, peak susceptible proportion

values in Case 4 decrease in comparison to Case 1 for initial population

distributions (i)− (iii), with and without diffusion. There is also a decrease in

exposed population proportion as a result of the reduction in the proportion of

susceptible under all initial conditions. In particular in the absence of diffusion it

occurs in the first five days of SARS, where population distribution with initial

condition (i) shows significant decrease in susceptible population as compared to

initial conditions (ii) and (iii) while in the presence of diffusion this decrease can

be observed on all days. A decrease in proportion of infected population has been

observed in the emergence of SARS as compared to Case 1, in the absence of

140

diffusion. This decrease is quite significant in the case of initial condition (i).

With diffusion in the system, the peak values of infected proportion are higher for

Case 4 than Case 1 under all initial conditions. Only very small increase in

recovered proportions from Case 1 to Case 4, has been observed under all three

conditions (i), (ii) and (iii), without diffusion because there is not much

movement in population due to absence of diffusion in the system. With diffusion

in the system, initial conditions (i) and (ii) reflect a significant increase in

recovery from Case 1 to Case 4, while initial condition (iii) shows a slight

increase. It is because of movement of population from higher to lower density

due to diffusion in the system. In comparison to Case 2, it has been observed that

there are lower proportions of susceptible and exposed in Case 4. Also in

comparison to Case 2, proportion of infective is slightly less with diffusion as

compared to without diffusion in the system during the first five days of disease.

As compared to Case 2, there is an increase in the recovered population both

with and without diffusion in the system.

In Case 5, as compared to Case 1 the rate of progression from infective to

diagnosed, α is increased from α = 0.238 to α = 0.298. In the absence of diffusion

this has not affected susceptible and exposed proportion values. With the

inclusion of diffusion in the system, a small increase is observed in the susceptible

and exposed population proportion in the early stage of disease. A significant

reduction in the proportion of infected individuals has been noticed both with

and without diffusion, showing that if infection is diagnosed earlier, the

population move to diagnosed compartment quickly for treatment. A large

increase in the recovered population proportion is observed both with and

without diffusion as compared to Case 1. In comparison to Case 3, a lower

proportion of exposed and infective are observed in Case 5. The proportion of the

population recovered, both with and without diffusion, has been observed to be

higher in Case 5 as compared to Case 3 .

141

Table 6.8: Basic reproduction number RIT

Case β α Value of RIT

1 0.242 0.238 1.6148

2 0.182 0.238 1.2212

3 0.242 0.179 1.8668

4 0.303 0.238 2.0354

5 0.242 0.298 1.4561

6.6 Discussion

The SEIJTR model for numerical study of the SARS epidemics is used with

diffusion and treatment included in the system to explore the effects of their

availability on the spread of disease. Three different initial conditions have been

used to examine the effects on transmission of the disease under different

population distributions. Operator splitting technique is used to calculate the

numerical solutions of the differential equations. The Routh-Hurwitz criterion is

used to check stability of points of equilibrium. The models under investigation

have two possible equilibria, namely the disease-free equilibrium and endemic

equilibria. Reproduction numbers RIT for the various cases considered (RIT > 1

) are given in Table 6.8. A study of bifurcation values of the transmission

coefficient, β and rate of progression from infective to diagnosed, α, as shown in

Table 6.5, indicates that the system remain stable with higher values of β and α

with diffusion in comparison to the system without diffusion.

It is observed from the values given in Table 6.8 that transmission coefficient, β

and rate of progression from infective to diagnosed, α, are quite sensitive

parameters having significant impact on the reproductive number RIT . If the

transmission coefficient, β is decreased as in Case 2, the reproduction number

decreases significantly even though it is still greater than 1. This causes slow

transmission of infection and thus higher peak values in the infected compartment

as shown in Tables 6.6 and 6.7 from t = 10 to t = 20 days. A decrease in infective

142

to diagnosed coefficient, α as in Case 3, causes a significant increase in the value

of the basic reproduction number RIT . Here a significant increase in the peak

value of the infected proportion can be seen at t= 20 days. An increased value of

transmission coefficient β, as in Case 4, gives the maximum value of the

reproduction number RIT . Here, transmission of the infection becomes fastest in

all initial conditions, as shown in the rate of decrease of values at various time

steps shown in Tables 6.6 and 6.7. There are also slightly lower peak values of

infected population proportion at t= 20 days, as compared to the original

situation depicted in Case 1. An increase in the rate of progression from infective

to diagnosed, α, as given in Case 5 causes a slight decrease in the value of the

basic reproduction number.

In initial condition (i), without diffusion, infectives are concentrated in the

domain [−1, 1]. With and without diffusion in the system, infection spreads

quickly in the first five days of onset of disease. With diffusion in the system,

infected population spreads to the edges of the domain [−2, 2] in the first ten

days, where initially there were no infectives. But a decrease in the peak values of

the infected proportion occurs, showing that diffusion causes a decrease in the

intensity of disease. Maximum population of diagnosed then enters treatment

class at the day fifteen of the disease. In initial condition (ii), susceptible and

infected proportions are in different domains initially. Here, infected not only

increase with the passage of time but also move from domain [−2, 0] to domain

[0, 2] without diffusion. With diffusion in the system, the infected spread to

almost in the whole domain [−2, 2]. Under initial condition (iii), there is a

significant shift as the infected population move to the domain [−0.1, 0.1]

significantly from the initial domain [−0.6, 0.6]. Diffusion again causes the

infection to spread, in the domain [−1.5, 1.5] but with reduced peak values.

The SEIJTR model considered here has been extended further in the next

chapter with the inclusion of quarantine of exposed population in the system of

equations. For that a model with the inclusion of another compartment named as

quarantined is considered. The new model is named as SEQIJTR model where

the suspected cases of SARS are kept under quarantined.

143

Chapter 7

Simulating the Effect of

Quarantine on Isolation

Treatment Model for SARS

Epidemic

7.1 Introduction

Mathematical models are the most efficient epidemiological tools for

understanding and determining the factors that cause and encourage epidemic.

These models allow us to estimate the effects of the measures and strategies

before their physical application, saving time and cost and helping to decide

which factors should be focused on more in order to control an epidemic. At the

time of emergence of a new infectious disease, when there is no effective

treatment and vaccine, isolation of known cases and quarantine of their contacts

are the two most effective strategies. The chances of spread of an infection

increase with the delay in diagnosis of infected individuals. In the history of

epidemiology, quarantine was first practised in 1377 for a plague epidemic, when

an official order of isolation for 30 days for ships and 40 days for land travelers

was passed by the Rector of the seaport of Ragusa [83]. According to Sattenspiel

[204] “In recent years the word quarantine has been used to refer to two related

144

but distinct activities, especially when applied to human activities and behaviors.

In most of the historical literature, quarantine usually refers to attempts to limit

flow of goods and people between different places. However, the same word has

also been used to refer to attempts to keep infectious individuals isolated from

everyone else. Both situations focus on limiting potential interactions between

infectious material or individuals and susceptible material or individuals, but the

intended targets of any rules or regulations are very different.” She also states

that “The term quarantine is derived from the Italian words quarantins and

quaranta giorni, which refer to the 40-days period during which ships, their

goods, crew, and passengers were isolated in the Port of Venice during the 14th

and 15th centuries [162]. The Italian authorities believed that an isolation period

of 40 days would be sufficient to dissipate the causes of infections.” The early

typical quarantine not only focused on sea travelers but also involved

arrangements to reduce population movement in and out of affected communities.

Over the last hundred years, quarantine has been used to control the transmission

of many infectious epidemics like cholera, leprosy, plague, yellow fever,

tuberculosis, diphtheria, smallpox, ebola, typhus, lassa fever, mumps and measles

[58]. During the outbreak of SARS epidemic in 2003, a new European institute

for research on controlling and developing quarantine and isolation measures for

SARS epidemic was designed [83]. Mathematical models including quarantine

effects answers numerous questions about using quarantine as a control strategy.

For example, does quarantine make any difference to the transmission of disease?

Is quarantine of the exposed population is more important than quarantine of the

infected population? Is there a difference in effectiveness between model with

quarantine and the model without quarantine and with treatment once the

infection is diagnosed? In deterministic compartmental models to study

quarantine effects, a new compartment usually referred to as Q is included in the

model. This compartment represents the isolated individuals that are removed

from the susceptible, exposed or infected (depending on model) compartment.

For some moderate infectious diseases, people who stay home from work or school

are considered to be in quarantine. But in the case of some deadly diseases,

145

quarantined individuals are those who are forced to live in isolation. Also, it is

assumed that there is no mixing of quarantined people with the rest of population

[112].

In 1995, Feng and Thieme [70] formulated a susceptible, infected, quarantined

and recovered (SIQR) model considering “quarantine-adjusted incidence”. They

concluded that addition of a quarantine compartment to the basic SIR

compartmental model can give rise to periodic solutions, explaining the oscillating

behavior of the disease considered in their studies. Five years later, Feng and

Thieme [71, 72] further developed a susceptible, exposed, infected, quarantined

and recovered (SEIQR) compartmental model with a general incidence term for

quarantine. They calculated the length of periods for quarantine in which

endemic equilibrium destablizes and can extinct and persist. A generalization of

the work of Feng and Thieme’s [70, 71, 72] was presented in the model of Hethcote

et al. [112]. They added quarantine to the basic SIR and SIS compartmental

models and compared the new SIQR and SIQS models on the basis of three

types of incidence. They investigated the equilibria, thresholds and stabilities of

the models. They found that only the cases with “quarantine adjusted incidence”

have unstable spiral endemic equilibria for few parameters and that periodic

solutions with “Hopf bifurcation surface” can arise in these cases.

At the time of the worldwide spread of severe acute respiratory syndrome

(SARS) in 2003 the major task for health authorities was to apply control

measures that could control the spread of the disease. Mass quarantine was

considered as the main control measure to reduce transmission of the infection.

The main concern in this respect was whether or not quarantine and isolation

were sufficient to decrease disease transmission? [50]. Many mathematical models

were formulated to answer this fundamental question afterwards. Fraser et al.

[78] investigated the time during infecting and for appearance of infection for an

individual, with the help of mathematical modelling. They also concluded that,

in the case of SARS simple public control measures are enough to control the

spread of disease. Many mathematical model studies have illustrated that

quarantined and isolation of a small number of individuals from the infected

146

population can lead to reduced transmission of SARS [184, 20, 189] but this is

economically expensive [216, 104, 96]. Day et al. [50] investigated useful

quarantine for SARS epidemic in all possible circumstances. They stated that “

Our results indicate that there are three main requirements for quarantine to

substantially reduce the number of infections that occur during a disease

outbreak. These are the following: 1) a large disease reproduction number in the

presence of isolation alone; 2) a large proportion of infections generated by an

individual that can be prevented through quarantine, q; and 3) a large probability

that an asymptomatic infected individual will get placed into quarantine before

he/she develops symptoms and is isolated, q.”

In this chapter a numerical study of the previously developed susceptible,

exposed, infected, diagnosed, treated and recovered (SEIJTR) model in Chapter

5 is extended to an SEQIJTR model including a quarantine compartment with a

view to study the effect of quarantine on disease transmission, for SARS

outbreak. Investigation is done with the inclusion of diffusion in the system. In

Sec. 2 the numerical scheme is derived. In Sec. 3 stability analyses for

disease-free and endemic equilibrium are performed. Bifurcation values, modes of

excitation and reproduction number in the absence and presence of diffusion is

also estimated. In Sec. 4 numerical solutions are calculated. The results are

summarised in Sec. 5.

7.2 The SEQIJTR epidemic model

7.2.1 Equations

This model of SARS consists of the following system of nonlinear partial

differential equations.

∂S

∂t= πΛ− β

(I + qE + pQ+ lJ)

NS − µS + d1

∂2S

∂x2(7.1)

∂E

∂t= (1− π)Λ + β

(I + qE + pQ+ lJ)

NS − (µ+ κ1 + κ2)E + d2

∂2E

∂x2(7.2)

∂Q

∂t= κ1E − (µ+ σ)Q+ d3

∂2Q

∂x2(7.3)

147

∂I

∂t= κ2E − (µ+ α + δ1)I + d4

∂2I

∂x2(7.4)

∂J

∂t= αI − (µ+ γ1 + δ2 + ζ)J + σQ+ d5

∂2J

∂x2(7.5)

∂T

∂t= ζJ − (µ+ γ2 + δ2(1− θ))T + d6

∂2T

∂x2(7.6)

∂R

∂t= γ1J + γ2T − µR + d7

∂2R

∂x2(7.7)

with initial conditions

S(0) = S0, E(0) = E0, Q(0) = Q0, I(0) = I0, J(0) = J0, T (0) = T0 and R(0) = R0

where S,E,Q, I, J, T and R represent susceptible, exposed, quarantined, infected,

diagnosed, treated and recovered classes, respectively, and N denotes the total

population, N = S + E +Q+ I + J + T +R. d1, d2, d3, d4, d5 ,d6 and d7 are the

diffusivity constants. Table 7.1 and Table 7.2, provide, respectively, the

description and values of the parameters. To scale the population size in each

compartment by the total population sizes by substituting s1 = S/N , e1 = E/N ,

q1 = Q/N , i1 = I/N , j1 = J/N , t1 = T/N , r1 = R/N Π = Λ/N giving the system

of equations (7.8) - (7.14). After simplification replacing s1 by S, e1 by E, q1 by

Q, i1 by I, j1 by J, t1 by T and r1 by R, the following dimensionless system of

equations is obtained:

∂S

∂t= −β(I+qE+pQ+ lJ)S−Π(S−π)+δ1IS+δ2(J+(1−θ)T )S+d1

∂2S

∂x2(7.8)

∂E

∂t= (1−π)Π+β(I+qE+pQ+lJ)S−(Π+κ1+κ2)E+δ1EI+δ2(J+(1−θ)T )E+d2

∂2E

∂x2

(7.9)∂Q

∂t= κ1E− (Π+ σ)Q+ δ1IQ+ δ2(J + (1− θ)T )Q+ d3

∂2J

∂x2(7.10)

∂I

∂t= κ2E − (Π + α + δ1)I + δ1I

2 + δ2(J + (1− θ)T )I + d4∂2I

∂x2(7.11)

∂J

∂t= σQ+αI−(Π+δ2+γ1+ζ)J+δ1IJ+δ2(J+(1−θ)T )J+d5

∂2J

∂x2(7.12)

∂T

∂t= ζJ−(Π+γ2+δ2)T+δ1IT +δ2(J+θ+(1−θ)T )T+d6

∂2T

∂x2(7.13)

∂R

∂t= γ1J + γ2T + δ1IR+ δ2(J + (1− θ)T )R−ΠR+ d7

∂2R

∂x2(7.14)

where S + E +Q+ I + J + T +R = 1

148

Table 7.1: Biological definition of parameters


Λ Rate at which new recruits enter the population


β Transmission coefficient




p Relative measure of reduced risk among quarantined individuals

κ1 Rate of progression from exposed to quarantined

κ2 Rate of progression from exposed to infective

σ Rate of progression from quarantine to diagnosed



γ2 Recovery due to treatment

ζ Treatment rate

δ1 SARS-induced mortality rate for infected

δ2 SARS-induced mortality rate for diagnosed and treated

θ Effectiveness of drugs as a reduction factor in disease-induced death of infected

149

Table 7.2: Parameters values for model

Parameter Value Source

Λ 0.00002 per day [175]

π 0.85000 [175]

β 0.24000 [175]

µ .000035 [114]

l 0.65000 [175]

q 0.10000 [57]

p 0.25000 [197]

κ1 0.10000 [95]

κ2 0.19500 [175]

α 0.23800 [175]

σ 0.15700 [57]

γ1 0.04600 [175]

γ2 0.05000 [175]

ζ 0.200000 [175]

δ1 0.04200 [150]

δ2 0.02400 [175]

θ 0.25000 [175]

150


The domain of all the calculations is taken as [−2, 2]. Boundary and initial


∂S(−2, t)

∂x=

∂E(−2, t)

∂x=

∂Q(−2, t)

∂x=

∂I(−2, t)

∂x=

∂J(−2, t)

∂x=

∂T (−2, t)

∂x=

∂R(−2, t)

∂x= 0

(7.15)∂S(2, t)

∂x=

∂E(2, t)

∂x=

∂Q(2, t)

∂x=

∂I(2, t)

∂x=

∂J(2, t)

∂x=

∂T (2, t)

∂x=

∂R(2, t)

∂x= 0

(7.16)

(i)

S0 = 0.98Sech(5x− 1), − 2 ≤ x ≤ 2.

E0 = 0, − 2 ≤ x ≤ 2.

Q0 = 0, − 2 ≤ x ≤ 2.

I0 = 0.02Sech(5x− 1), − 2 ≤ x ≤ 2.

J0 = 0, − 2 ≤ x ≤ 2.

T0 = 0, − 2 ≤ x ≤ 2.

R0 = 0, − 2 ≤ x ≤ 2.

(ii)

S0 = 0.96Sech(5x− 1), − 2 ≤ x ≤ 2.

E0 = 0, − 2 ≤ x ≤ 2.

Q0 = 0, − 2 ≤ x ≤ 2.

I0 =

0, − 2 ≤ x < −0.6,

0.04, − 0.6 ≤ x ≤ 0.6,

0, 0.6 < x ≤ 2.

J0 = 0, − 2 ≤ x ≤ 2.

T0 = 0, − 2 ≤ x ≤ 2.

R0 = 0, − 2 ≤ x ≤ 2.

The graphs of initial conditions are shown in Fig. 7.1

151

t = 0

(i)-2 -1 0 1 2

x

0.2

0.4

0.6

0.8

1.0S,I

t = 0

(ii)-2 -1 0 1 2

x

0.2

0.4

0.6

0.8

1.0S,I

Figure 7.1: Initial conditions (i) and (ii).

7.2.3 Numerical scheme

The SEQIJTR model is solved with operator splitting technique by dividing the

system into non-linear reaction equations and linear diffusion equations [244].

The non-linear reaction equations to be used for the first half-time are:

1

2

∂S

∂t= −β(I+qE+pQ+lJ)S−Π(S−π)+δ1IS+δ2(J+(1−θ)T )S (7.17)

1

2

∂E

∂t= (1−π)Π+β(I+qE+pQ+lJ)S−(Π+κ1+κ2)E+δ1EI+δ2(J+(1−θ)T )E

(7.18)1

2

∂Q

∂t= κ1E−(Π+σ)Q+δ1IQ+δ2(J+(1−θ)T )Q (7.19)

1

2

∂I

∂t= κ2E−(Π+α+δ1)I+δ1I

2+δ2(J+(1−θ)T )I (7.20)

1

2

∂J

∂t= σQ+αI−(Π+δ2+γ1+ζ)J+δ1IJ+δ2(J+(1−θ)T )J (7.21)

1

2

∂T

∂t= ζJ−(Π+γ2+δ2)T+δ1IT+δ2(J+θ+(1−θ)T )T (7.22)

1

2

∂R

∂t= γ1J+γ2T+δ1IR+δ2(J+(1−θ)T )R−ΠR (7.23)

The linear diffusion equations to be used for the second half-time step are:

1

2

∂S

∂t= d1

∂2S

∂x2(7.24)

1

2

∂E

∂t= d2

∂2E

∂x2(7.25)

1

2

∂Q

∂t= d3

∂2Q

∂x2(7.26)

1

2

∂I

∂t= d4

∂2I

∂x2(7.27)

152

1

2

∂J

∂t= d5

∂2J

∂x2(7.28)

1

2

∂T

∂t= d6

∂2T

∂x2(7.29)

1

2

∂R

∂t= d7

∂2R

∂x2(7.30)

Applying the forward Euler scheme, the non-linear equations transform to

Sj+ 1

2i = −β(Iji+qEj

i+pQji+lJ j

i )Sji−Π(Sj

i−π)+δ1IjiS

ji+δ2(J

ji+(1−θ)T j

i )Sji (7.31)

Ej+ 1

2i = (1−π)Π+β(Iji+qEj

i+pQji+lJ j

i )Sji−(Π+κ1+κ2)E

ji+δ1E

jiI

ji+δ2(J

ji+(1−θ)T )Ej

i

(7.32)

Qj+ 1

2i = κ1E

ji−(Π+σ)Qj

i+δ1IjiQ

ji+δ2(J

ji+(1−θ)T j

i )Qji (7.33)

Ij+ 1

2i = κ2E

ji − (Π+α+ δ1)I

ji + δ1I

2ji + δ2(J

ji +(1− θ)T j

i )Iji (7.34)

Jj+ 1

2i = σQj

i+αIji−(Π+δ2+γ1+ζ)J ji+δ1I

jiJ

ji+δ2(J

ji+(1−θ)T j

i )Jji (7.35)

Tj+ 1

2i = ζJ j

i − (Π+γ2+δ2)Tji +δ1I

jiT

ji +δ2(J

ji +θ+(1−θ)T j

i )Tji (7.36)

Rj+ 1

2i = γ1J

ji + γ2T

ji + δ1I

jiR

ji + δ2(J

ji + (1− θ)T j

i )Rji −ΠRj

i (7.37)

where Sji , E

ji , Q

ji , I

ji , J

ji , T

ji and Rj

i are the approximated values of S, E, Q, I, J ,

T and R at position −2+ iδx, for i = 0, 1, . . . and time jδt, j = 0, 1, . . . and Sj+ 1

2i ,

Ej+ 1

2i , Q

j+ 12

i , Ij+ 1

2i , J

j+ 12

i , Tj+ 1

2i and R

j+ 12

i denote their values at the first half-time

step. Similarly, for the second half-time step, the linear equations transform as

Sj+1i = S

j+ 12

i + d1δt

(δx)2(S

j+ 12

i−1 − 2Sj+ 1

2i + S

j+ 12

i+1 ) (7.38)

Ej+1i = E

j+ 12

i + d2δt

(δx)2(E

j+ 12

i−1 − 2Ej+ 1

2i + E

j+ 12

i+1 ) (7.39)

Qj+1i = Q

j+ 12

i + d3δt

(δx)2(Q

j+ 12

i−1 − 2Qj+ 1

2i +Q

j+ 12

i+1 ) (7.40)

Ij+1i = I

j+ 12

i + d4δt

(δx)2(I

j+ 12

i−1 − 2Ij+ 1

2i + I

j+ 12

i+1 ) (7.41)

J j+1i = J

j+ 12

i + d5δt

(δx)2(J

j+ 12

i−1 − 2Jj+ 1

2i + J

j+ 12

i+1 ) (7.42)

T j+1i = T

j+ 12

i + d6δt

(δx)2(T

j+ 12

i−1 − 2Tj+ 1

2i + T

j+ 12

i+1 ) (7.43)

Rj+1i = R

j+ 12

i + d7δt

(δx)2(R

j+ 12

i−1 − 2Rj+ 1

2i +R

j+ 12

i+1 ) (7.44)

153

The stability condition satisfied by the numerical method described above is

given as:dnδt

(δx)2≤ 0.5, n = 1, 2, 3, 4, 5, 6, 7. (7.45)

In each case, δx = 0.1, d1 = 0.025, d2 = 0.01, d3 = 0.0, d4 = 0.001, d5 = 0.0,

d6 = 0.0, d7 = 0.0 and δt = 0.03 are used.

7.3 Stability Analysis

7.3.1 Reproduction number without diffusion

For all the susceptible under consideration, RQIT represents the reproduction

number, i.e. the average number of secondary cases the infection is transmitted to

during a typical individuals infectious period in a situation where all persons are

susceptible. If RQIT < 1, the disease dies out and if RQIT > 1, then introduction

of one infected individual in susceptible, will spread the disease. The

reproduction number may vary remarkably for different infectious diseases and

also for the same disease in different populations. For a disease-free equilibrium

the reproduction number should always be less than one. The reproduction

number for the SEQIJTR model is calculated using the generation matrix

method [228]. The derivation is given in appendix A.7 whereas the expression for

of the reproduction number RQIT is given by:

RQIT = β(a+b+c+d)(κ1+κ2+Π)(α+δ1+Π)(γ1+δ2+ζ+Π)(σ+Π)

a = q(α + δ1 +Π)(γ1 + δ2 + ζ +Π)(σ +Π).

b = κ2(γ1 + δ2 + ζ +Π)(σ +Π).

c = pκ1(α + δ1 +Π)(γ1 + δ2 + ζ +Π).

d = l(κ1(δ1 +Π)σ + α(κ1σ + κ2(σ +Π))).

In the situation where there are no quarantine measures, but isolation and

treatment are still available i.e where p = 1, κ1 = 0 and σ = 0, the reproduction

number expression reduces to the following:

RIT = β(q(α+δ1+Π)(γ1+δ2+ζ+Π)+κ2(γ1+δ2+ζ+Π)+lακ2)(κ2+Π)(α+δ1+Π)(γ1+δ2+ζ+Π)

154

and in the absence of any treatment for disease, ζ = 0. Hence the expression for

the reproduction number is:

RI =β(q(α+δ1+Π)(γ1+δ2+Π)+κ2(γ1+δ2+Π)+lακ2)

(κ2+Π)(α+δ1+Π)(γ1+δ2+Π)

while in the absence of quarantine, treatment and isolation of infected individuals

after diagnosis i.e α = 0 and δ2 = 0, the RI changes to the basic reproduction

number R0, and is given as :

R0 =β(q(δ1+Π)+κ2)(κ2+Π)(δ1+Π)

The values of the quantities are given in Table 7.4

7.3.2 Disease-free equilibrium (DFE)

The variational matrix of the system of equations (7.8) - (7.14) at the disease-freeequilibrium P0 = (1, 0, 0, 0, 0, 0, 0), is:

V0

=

−Π −qβ −pβ (−β + δ1) (−lβ + δ2) (δ2(1 − θ)) 0

0 (qβ − κ1 − κ2 − Π) pβ β lβ 0 0

0 κ1 −(σ + Π) 0 0 0 0

0 κ2 0 −(α + δ1 + Π) 0 0 0

0 0 σ α −(γ1 + δ2 + ζ + Π) 0 0

0 0 0 0 ζ −(γ2 + δ2(θ − 1) + Π) 0

0 0 0 0 γ1 γ2 −Π

It is observed that the first eigenvalue −Π is negative and all entries below it arezero. This allows the elimination of the first row and column. The last eigenvalue−Π is also negative and all entries above it are zero, so the last row and columnmay be eliminated, giving:

V01 =

(qβ − κ1 − κ2 − Π) pβ β lβ 0

κ1 −(σ + Π) 0 0 0

κ2 0 −(α + δ1 + Π) 0 0

0 σ α −(γ1 + δ2 + ζ + Π) 0

0 0 0 ζ −(γ2 + δ2(θ − 1) + Π)

Now, the third eigenvalue of the variational matrix in the last column and last

row of the above matrix, −(γ2 + δ2(θ − 1) + Π) is negative and the entries above

are zero so, after eliminating the last row and column a fourth order variational

matrix is obtained:

V02 =

(qβ − κ1 − κ2 − Π) pβ β lβ

κ1 −(σ + Π) 0 0

κ2 0 −(α + δ1 + Π) 0

0 σ α −(γ1 + δ2 + ζ + Π)

The characteristic equation det(V 0 − λI) = 0 gives

λ4 + p1λ3 + p2λ

2 + p3λ1 + p4 = 0 (7.46)

155

The Routh-Hurwitz conditions used to check the stability here are (i)p1 > 0,

(ii)p4 > 0 and (iii)p2 − p3p1

> 0 and (iv)p1p2p3 − p23 − p21p4 > 0 where expressions

for p1, p2, p3 and p4 are given in appendix A.7.

(i)p1 = α− qβ + γ1 + δ1 + δ2 + ζ + κ1 + κ2 + 4Π + σ

(ii)p4 = (α + δ1 +Π)(γ1 + δ2 + ζ +Π)(κ1 + κ2 +Π)(Π + σ)− lβ(ακ2Π+ ακ1σ +

δ1κ1σ + ακ2σ + κ1Πσ)− qβ(α+ δ1 +Π)(γ1 + δ2 + ζ +Π)(σ +Π)− pβκ1(α+ δ1 +

Π)(γ1 + δ2 + ζ +Π)− β(κ2Π+ κ2σ)(γ1 + δ2 + ζ +Π)

(iii)p2 − p3p1

= qαβγ1 + qβγ1δ1+ qαβδ2+ qβδ1δ2 + qαβζ + qβδ1ζ + pαβκ1−αγ1κ1+

pβγ1κ1 + pβδ1κ1 − γ1δ1κ1 − αδ2κ1 + pβδ2κ1 − δ1δ2κ1 − αζκ1 + pβζκ1 − δ1ζκ1 +

lαβκ2−αγ1κ2+βγ1κ2− γ1δ1κ2− αδ2κ2+βδ2κ2− δ1δ2κ2−αζκ2+βζκ2− δ1ζκ2+

2qαβΠ−2αγ1Π+2qβγ1Π+2qβδ1Π−2γ1δ1Π−2αδ2Π+2qβδ2Π−2δ1δ2Π−2αζΠ+

2qβζΠ−2δ1ζΠ−2ακ1Π+2pβκ1Π−2γ1κ1Π−2δ1κ1Π−2δ2κ1Π−2ζκ1Π−2ακ2Π+

2βκ2Π− 2γ1κ2Π− 2δ1κ2Π− 2δ2κ2Π− 2ζκ2Π− 3αΠ2 + 3qβΠ2 − 3γ1Π2 − 3δ1Π

2 −

3δ2Π2 − 3ζΠ2 − 3κ1Π

2 − 3κ2Π2 − 4Π3 + qαβσ − αγ1σ + qβγ1σ + qβδ1σ − γ1δ1σ −

αδ2σ+qβδ2σ−δ1δ2σ−αζσ+qβζσ−δ1ζσ−ακ1σ+ lβκ1σ−γ1κ1σ−δ1κ1σ−δ2κ1σ−

ζκ1σ− ακ2σ+ βκ2σ− γ1κ2σ− δ1κ2σ− δ2κ2σ− ζκ2σ− 2αΠσ+ 2qβΠσ− 2γ1Πσ−

2δ1Πσ−2δ2Πσ−2ζΠσ−2κ1Πσ−2κ2Πσ−3Π2σ+(α− qβ+γ1+ δ1+ δ2+ ζ+κ1+

κ2+4Π+σ)(γ1δ1+δ1δ2+δ1ζ−pβκ1+γ1κ1+δ1κ1+δ2κ1+ζκ1−βκ2+γ1κ2+δ1κ2+

δ2κ2+ζκ2+3γ1Π+3δ1Π+3δ2Π+3ζΠ+3κ1Π+3κ2Π+6Π2+γ1σ+δ1σ+δ2σ+ζσ+

κ1σ+κ2σ+3Πσ−qβ(α+γ1+δ1+δ2+ζ+3Π+σ)+α(γ1+δ2+ζ+κ1+κ2+3Π+σ))

(iv)p1p2p3−p23−p21p4 = −(α− qβ+γ1+ δ1+ δ2+ ζ+ κ1+κ2+4Π+σ)2(αγ1κ1Π+

γ1δ1κ1Π+ αδ2κ1Π+ δ1δ2κ1 Π+ αζκ1Π+ δ1ζκ1Π− lαβκ2Π+ αγ1κ2Π− βγ1κ2Π+

γ1δ1κ2Π+αδ2κ2Π−βδ2κ2Π+δ1δ2κ2Π+αζκ2Π−βζκ2Π+δ1ζκ2Π+αγ1Π2+γ1δ1Π

2+

αδ2Π2 + δ1δ2Π

2 + αζΠ2 + δ1ζΠ2 + ακ1Π

2 + γ1κ1Π2 + δ1κ1Π

2 + δ2κ1Π2 + ζκ1Π

2 +

ακ2Π2−βκ2Π

2+ γ1κ2Π2+ δ1κ2Π

2+ δ2κ2Π2+ ζκ2Π

2+αΠ3+ γ1Π3+ δ1Π

3+ δ2Π3+

ζΠ3 + κ1Π3 + κ2Π

3 +Π4 − pβκ1(α+ δ1 +Π)(γ1 + δ2 + ζ +Π)− lαβκ1σ+αγ1κ1σ−

lβδ1κ1σ+γ1δ1κ1σ+αδ2κ1σ+δ1δ2κ1σ+αζκ1σ+δ1ζκ1σ−lαβκ2σ+αγ1κ2σ−βγ1κ2σ+

γ1 δ1κ2σ+αδ2κ2σ−βδ2κ2σ+δ1δ2κ2σ+αζκ2σ−βζκ2σ+δ1ζκ2σ+αγ1Πσ+γ1δ1Πσ+

αδ2Πσ+δ1δ2Πσ+αζΠσ+δ1ζΠσ+ακ1Πσ− lβκ1Πσ+γ1κ1Πσ+δ1κ1Πσ+δ2κ1Πσ+

ζκ1Πσ+ακ2Πσ−βκ2Πσ+γ1κ2Πσ+ δ1κ2Πσ+ δ2κ2Πσ+ ζκ2Πσ+αΠ2σ+γ1Π2σ+

δ1Π2σ+δ2Π

2σ+ζΠ2σ+κ1Π2σ+κ2Π

2σ+Π3σ−qβ(α+δ1+Π)(γ1+δ2+ζ+Π)(Π+

156

σ))+(α−qβ+γ1+δ1+δ2+ζ+κ1+κ2+4Π+σ)(γ1δ1+δ1δ2+δ1ζ−pβκ1+γ1κ1+δ1κ1+

δ2κ1+ζκ1−βκ2+γ1κ2+δ1κ2+δ2κ2+ζκ2+3γ1Π+3δ1Π+3δ2Π+3ζΠ+3κ1Π+3κ2Π+

6Π2+γ1σ+δ1σ+δ2σ+ζσ+κ1σ+κ2σ+3Πσ−qβ(α+γ1+δ1+δ2+ζ+3Π+σ)+α(γ1+

δ2+ζ+κ1+κ2+3Π+σ))(αγ1κ1+γ1δ1κ1+αδ2κ1+δ1δ2κ1+αζκ1+δ1ζκ1− lαβκ2+

αγ1κ2− βγ1κ2+γ1δ1κ2+αδ2κ2−βδ2κ2+ δ1δ2κ2+αζκ2−βζκ2+ δ1ζκ2+2αγ1Π+

2γ1δ1Π+2αδ2Π+2δ1δ2Π+2αζΠ+2δ1ζΠ+2ακ1Π+2γ1κ1Π+2δ1κ1Π+2δ2κ1Π+

2ζκ1Π+ 2ακ2Π− 2βκ2Π+ 2γ1κ2Π+ 2δ1κ2Π+ 2δ2κ2Π+ 2ζκ2Π+ 3αΠ2 + 3γ1Π2 +

3δ1Π2+3δ2Π

2+3ζΠ2+3κ1Π2+3κ2Π

2+4Π3−pβκ1(α+γ1+δ1+δ2+ζ+2Π)+αγ1σ+

γ1δ1σ+αδ2σ+δ1δ2σ+αζσ+δ1ζσ+ακ1σ− lβκ1σ+γ1κ1σ+δ1κ1σ+δ2κ1σ+ζκ1σ+

ακ2σ− βκ2σ+ γ1κ2σ+ δ1κ2σ+ δ2κ2σ+ ζκ2σ+2αΠσ+2γ1Πσ+2δ1Πσ+2δ2Πσ+

2ζΠσ+2κ1Πσ+2κ2Πσ+3Π2σ− qβ(δ1δ2+ δ1ζ +2δ1Π+2δ2Π+2ζΠ+3Π2+ δ1σ+

δ2σ+ζσ+2Πσ+γ1(δ1+2Π+σ)+α(γ1+δ2+ζ+2Π+σ)))−(αγ1κ1+γ1δ1κ1+αδ2 κ1+

δ1δ2κ1+αζκ1+ δ1ζκ1− lαβκ2+αγ1κ2−βγ1κ2+γ1δ1κ2+αδ2κ2−βδ2κ2+ δ1δ2κ2+

αζκ2−βζκ2+δ1ζκ2+2αγ1Π+2γ1δ1Π+2αδ2Π+2δ1δ2Π+2αζΠ+2δ1ζΠ+2ακ1Π+

2γ1κ1Π+2δ1κ1Π+2δ2κ1Π+2ζκ1Π+2ακ2Π−2βκ2Π+2γ1κ2Π+2δ1κ2Π+2δ2κ2Π+

2ζκ2Π+3αΠ2+3γ1Π2+3δ1Π

2+3δ2Π2+3ζΠ2+3κ1Π

2+3κ2Π2+4Π3− pβκ1(α+

γ1+ δ1+ δ2+ ζ+2Π)+αγ1σ+γ1δ1σ+αδ2σ+ δ1δ2σ+αζσ+ δ1ζσ+ακ1σ− lβκ1σ+

γ1κ1σ+δ1κ1σ+δ2κ1σ+ζκ1σ+ακ2σ−βκ2σ+γ1κ2σ+δ1κ2σ+δ2κ2σ+ζκ2σ+2αΠσ+

2γ1Πσ+2δ1Πσ+2δ2Πσ+2ζΠσ+2κ1Πσ+2κ2Πσ+3Π2σ− qβ(δ1δ2+ δ1ζ+2δ1Π+

2δ2Π+2ζΠ+3Π2+δ1σ+δ2σ+ζσ+2Πσ+γ1(δ1+2Π+σ)+α(γ1+δ2 +ζ+2Π+σ)))2

7.3.3 Endemic equilibrium without diffusion

The variational matrix of the system of equations (7.8)-(7.14) at

P ∗(S∗, E∗, Q∗, I∗, J∗, T ∗, R∗), is given by

V ∗ =

a11 a12 a13 a14 a15 a16 a17

a21 a22 a23 a24 a25 a26 a27

a31 a32 a33 a34 a35 a36 a37

a41 a42 a43 a44 a45 a46 a47

a51 a52 a53 a54 a55 a56 a57

a61 a62 a63 a64 a65 a66 a67

a71 a72 a73 a74 a75 a76 a77

157

where

a11 = (−I∗ − J∗l − E∗q − pQ∗)β + I∗δ1 + δ2(J∗ + T ∗(1− θ))− Π, a12 = −qS∗β,

a13 = −pS∗β, a14 = −S∗β + S∗δ1, a15 = −lS∗β + S∗δ2, a16 = S∗δ2(1− θ),

a21 = (I∗ + J∗l + E∗q + pQ∗)β,

a22 = qS∗β + I∗δ1 + δ2(J∗ + T ∗(1− θ))− κ1 − κ2 − Π,

a23 = pS∗β, a24 = S∗β + E∗δ1, a25 = lS∗β + E∗δ2, a26 = E∗δ2(1− θ),

a32 = κ1, a33 = I∗δ1 + δ2(J∗ + T ∗(1− θ))− Π− σ, a34 = δ1Q

∗,

a35 = δ2Q∗, a36 = δ2(1− θ)Q∗, a42 = κ2,

a44 = −α− δ1 + 2I∗δ1 + δ2(J∗ + T ∗(1− θ))− Π), a45 = I∗δ2, a46 = δ2(1− θ)I∗,

a53 = σ, a54 = α+ J∗δ1, a55 = −γ1 + I∗δ1 − δ2 + J∗δ2 − ζ + δ2(J∗ + T ∗(1− θ))−Π,

a56 = J∗δ2(1− θ), a64 = T ∗δ1, a65 = T ∗δ2 + ζ,

a66 = −γ2 + I∗δ1 − δ2 + T ∗δ2(1− θ) + δ2(J∗ + T ∗(1− θ) + θ)− Π,

a74 = R∗δ1, a75 = γ1 +R∗δ2, a76 = γ2 +R∗δ2(1− θ),

a77 = I∗δ1 + δ2(J∗ + T ∗(1− θ))− Π,

a17 = a27 = a31 = a37 = a41 = a43 = a47 = a51 = a52 = 0,

a57 = a61 = a62 = a63 = a67 = a71 = a72 = a73 = 0.

The characteristic equation for P ∗(S∗, E∗, Q∗, I∗, J∗, T ∗, R∗) can be written as

λ7 + p1λ6 + p2λ

5 + p3λ4 + p4λ

3 + p5λ2 + p6λ

1 + p7 = 0 (7.47)

where the expression for p1, p2, p3, p4, p5, p6 ,p7 and the Routh-Hurwitz conditions

are calculated using method described in [84]. The Routh-Hurwitz condition are:

C1 : p1 > 0,

C2 : p7 > 0,

C3 : p2 − p3p1

> 0,

C4 :p23+p21p4−p1(p2p3+p5)

p3−p1p2> 0,

C5 :p23p4+p25+p21(p

24−p2p6)−p3(p2p5+p7)+p1(p22p5−2p4p5+p3p6+p2(p7−p3p4))

p23+p21p4−p1(p2p3+p5)> 0,

C6 :A1+A2

(p23p4+p25+p21(p24−p2p6)−p3(p2p5+p7)+p1(p22p5−2p4p5+p3p6+p2(p7−p3p4+)))

> 0,

C7 :A1+A2+A3

B1+B2> 0.

where

A1 = p35−p33p6+p31p26−p3p5(p2p5+2p7)+p23(p4p5+p2p7)+p21(p

24p5−2p6(p2p5+p7),

A2 =

p4(p2p7−p3p6))+p1(3p3p5p6−2p4p25+p27+p22(p

25−p3p7)+p2(p

23p6−p3p4p5+p5p7)),

158

Table 7.3: Values for Routh-Hurwitz criteria of equilibrium

Diffusion C1 C2 C3 C4 C5 C6 C7 Stability

No 1.057 6.5× 10−14 0.340 0.051 3.1× 10−3 6.5× 10−5 4.9× 10−9 Stable

Yes 1.145 8.8× 10−11 0.405 0.070 6.2× 10−3 2.7× 10−4 5.3× 10−6 Stable

A3 = p35p6 − p33p26 + p31p

36 − p4p

25p7 + p2p5p

27 − p37 + p23(p4p5p6 − p24p7 + 2p2p6p7)−

p21(p34p7 − p24p5p6 + p26(2p2p5 + 3p7) + p4p6(p3p6 − 3p2p7)),

A4 = −p3(p22p

27 + p7(3p5p6 − 2p4p7) + p2p5(p5p6 − p4p7)) + p1(2p

24p5p7 + p32p

27 −

p4p6(2p25 + p3p7) + p22(p

25p6 − p4p5p7 − 2p3p6p7),

A5 = 3p6(p3p5p6 + p27) + p2(p23p

26 + p7(p5p6 − 3p4p7) + p3p4(p4p7 − p5p6)),

B1 = p35 − p33p6 + p31p26 − p3p5(p2p5 + 2p7) + p23(p4p5 + p2p7) + p21(p

24p5 − 2p6(p2p5 +

p7) + p4(p2p7 − p3p6)) ,

B2 = p1(3p3p5p6 − 2p4p25 + p27 + p22(p

25 − p3p7) + p2(p

23p6 − p3p4p5 + p5p7)).

The derivation of Routh-Hurwitz conditions is given in appendix A.7. The

numerical values for the Routh-Hurwitz conditions are given in Table 7.3 at the

endemic point of equilibrium

P1 = (0.55678, 3.86×10−5, 2.46×10−5, 2.69×10−5, 3.80×10−5, 1.12×10−4, 0.44298).

7.3.4 Endemic equilibrium with diffusion

To calculate the small perturbations S1(x, t), E1(x, t), Q1(x, t), I1(x, t), J1(x, t),

T1(x, t) and R1(x, t) the equations (7.8) - (7.14) are linearised about the point of

equilibrium P ∗(S∗, E∗, Q∗, I∗, J∗, T ∗, R∗) as described in [35, 201] giving

∂S1

∂t= a11S1 + a12E1 + a13Q1 + a14I1 + a15J1 + a16T1 + a17R1d1

∂2S1

∂x2(7.48)

∂E1

∂t= a21S1 + a22E1 + a23Q1 + a24I1 + a25J1 + a26T1 + a27R1 + d2

∂2E1

∂x2(7.49)

∂Q1


∂2Q1

∂x2(7.50)

∂I1∂t

= a41S1 + a42E1 + a43Q1 + a44I1 + a45J1 + a46T1 + a47R1 + d4∂2I1∂x2

(7.51)

∂J1∂t

= a51S1 + a52E1 + a53Q1 + a54I1 + a55J1 + a56T1 + a57R1 + d5∂2J1∂x2

(7.52)

159

∂T1


∂2T1

∂x2(7.53)

∂R1

∂t= a71S1 + a72E1 + a73Q1 + a74I1 + a75J1 + a76T1 + a77T1 + d7

∂2R1

∂x2(7.54)

where a11, a12, a13 ... are the elements of the variational matrix V ∗ as calculated in

the previous section 7.3.3. The existence of a Fourier series solution of the

following form for Eqs. (7.48) - (7.54) is assumed:

S1(x, t) =∑k


E1(x, t) =∑k


Q1(x, t) =∑k

Qkeλt cos(kx) (7.57)

I1(x, t) =∑k


J1(x, t) =∑k


T1(x, t) =∑k

Tkeλt cos(kx) (7.60)

R1(x, t) =∑k


where k = nπ2, (n = 1, 2, 3, · · · · · · ) is the wave number for node n. Substituting

the values of S1, E1, Q1 I1, J1, T1 and R1 into equations (7.48) -(7.54), the

equations are transformed into:∑k

(a11−d1k2−λ)Sk+

∑k

a12Ek+∑k

a13Qk+∑k

a14Ik+∑k

a15Jk+∑k

a16Tk = 0

(7.62)∑k

a21Sk+∑k

(a22−d2k2−λ)Ek+

∑k

a23Qk+∑k

a24Ik+∑k

a25Jk+∑k

a26Tk = 0

(7.63)∑k

a32Ek +∑k

(a33 − d3k2 − λ)Qk +

∑k

a34Ik +∑k

a35Jk∑k

a36Tk = 0 (7.64)

∑k

a42Ek +∑k

(a44 − d4k2 − λ)Ik +

∑k

a45Jk +∑k

a46Tk = 0 (7.65)

∑k

a53Qk +∑k

a54Ik +∑k

(a55 − d5k2 − λ)Jk +

∑k

a56Tk = 0 (7.66)

∑k

a64Ik +∑k

a65Jk +∑k

(a66 − d6k2 − λ)Tk = 0 (7.67)

160

∑k

a74Ik +∑k

a75Jk +∑k

a76Tk +∑k

(a77 − d7k2 − λ)Rk = 0 (7.68)

The Variational matrix Vd for the equations (7.62) - (7.68) is

Vd =

a11 − d1k2 a12 a13 a14 a15 a16 0

a21 a22 − d2k2 a23 a24 a25 a26 0

0 a32 a33 − d3k2 a34 a35 a36 0

0 a42 0 a44 − d4k2 a45 a46 0

0 0 a53 a54 a55 − d5k2 a56 0

0 0 0 a64 a65 a66 − d6k2 0

0 0 0 a74 a75 a76 a77 − d7k2

The characteristic equation for the variational matrix Vd is the same as given in

equation (7.47), where as p1, p2, p3..., p7 are calculated again, using the above

variational matrix Vd using method described in [84]. The numerical results for

the Routh-Hurwitz conditions for endemic equilibrium are given in Table 7.3. It

is observed that at the point of equilibrium, P1, Routh-Hurwitz conditions for

stability are satisfied in the presence of diffusion in the system.

7.3.5 Reproduction number with diffusion

Variational matrix method is used to calculate reproduction number with

diffusion RdQIT . The variational matrix with diffusion for P0 = (1, 0, 0, 0, 0, 0, 0) is

as follows:

V 0d =

[V 0d1

V 0d2

]

V 0d1

=

−(Π + d1k2) −qβ −pβ −(β − δ1)

0 qβ − κ1 − κ2 − Π− d2k2 pβ β

0 κ1 −(Π + σ)− d3k2 0

0 κ2 0 −(α + δ1 +Π)− d4k2

0 0 σ α

0 0 0 ζ

0 0 0 0

161

V 0d2

=

−(lβ − δ2) δ2(1− θ) 0

lβ 0 0

0 0 0

0 0 0

−(γ1 + δ2 + ζ +Π+ d5k2) 0 0

ζ −(γ2 + δ2(1− θ) + Π + d6k2) 0

γ1 γ2 −Π− d7k2

It is observed that the first eigenvalue −(d1k

2 +Π) of the variational matrix V 0d is

negative and all entries below it are zero. This allows us to eliminate the first row

and column. So the reduced matrix is:

V 0d =

[V 0d1

V 0d2

]

V 0d1

=

qβ − (κ1 + κ2 +Π+ d2k2) pβ β

κ1 −(Π + σ + d3k2) 0

κ2 0 −(α + δ1 +Π+ d4k2)

0 σ α

0 0 ζ

0 0 0

V 0d2

=

lβ 0 0

0 0 0

0 0 0

−(γ1 + δ2 + ζ +Π+ d5k2) 0 0

ζ −(γ2 + δ2(1− θ) + Π + d6k2) 0

γ1 γ2 −(Π + d7k2)

Now, the last eigenvalue −(Π + d7k

2) is again negative and all entries above it are

zero, so the last row and column can be eliminated. The reduced matrix is:

V0d =

(qβ − κ1 − κ2 − Π − d2k2) pβ β lβ 0

κ1 −(Π + σ + d3k2) 0 0 0

κ2 0 −(α + δ1 + Π + d4k2) 0 0

0 σ α −(γ1 + δ2 + ζ + Π + d5k2) 0

0 0 0 ζ −(γ2 + δ2(1 − θ) + Π + d6k2)

Again, the third eigenvalue of the variational matrix given in the last column

and last row −(d6k2 + γ2 + δ2(1− θ) + Π) is negative and the entries above it are

162

zero, so after eliminating the last row and column, a 4th order variational matrix

V 0d is obtained.

V0d =

qβ − κ1 − κ2 − Π − d2k

2 pβ β lβ

κ1 −(Π + σ) − d3k2 0 0

κ2 0 −(α + δ1 + Π) − d4k2 0

0 σ α −(γ1 + δ2 + ζ + Π) − d5k2

Characteristic equation det(V 0d − λI) = 0 is given as

λ4 + q1λ3 + q2λ

2 + q3λ1 + q4 = 0 (7.69)

Where q’s are given in appendix A.7. The Routh-Hurwitz condition p4 > 0 is

satisfied only for RdQIT < 1. So, the expression for the reproduction number with

diffusion in the system is given by:

RdQIT = β(A+B+C+D)

(d4k2+α+δ1+Π)(d5k2+γ1+δ2+ζ+Π)(d2k2+κ1+κ2+Π)(d3k2+Π+σ),

where A = l(d3k2ακ2 + κ1(d4k

2 + δ1 +Π)σ + α(κ1σ + κ2(Π + σ))),

B =

κ2(d3k2(d5k

2+γ1+δ2+ζ)+Π(1+d3k2+d5k

2+γ1+δ2+ζ+Π)+(d5k2+γ1+δ2+ζ)σ),

C = pκ1((d4k2 + α+ δ1 +Π)(d5k

2 + γ1 + δ2 + ζ +Π)),

D = q(((d5k2 + γ1 + δ2 + ζ +Π)(d3k

2(d4k2 + α + δ1 +Π) + (d4k

2 + α +Π)(Π +

σ) + δ1(Π + σ))) + δ1).

The numerical values of all reproduction numbers without diffusion are given in

Table 7.4. It is observed that, in the absence of any control measures and

treatment, the reproduction number of SARS, R0 is quite high. If isolation of

the infected population is introduced into the system, the reproduction number

RI halves. With the additional presence of any treatment of SARS, the

reproduction number RIT halves again. The value of the reproduction number

RQIT reduces further if quarantine measures are introduced.

In the presence of diffusion in the compartment, where the mixing of the

individuals is maximized, the reproduction number, RdQIT is 1.47506. This value

is higher than RQIT , the corresponding value without diffusion. The value of

RdQIT is smaller to that of RIT , where there is no quarantine. Hence, introducing

the diffusion at the beginning of the infection opposes the effects of quarantine

measures. Although it is assumed that there is no diffusion in the quarantine

163

compartment, the diffusion in the other compartments increases the reproduction

number and eventually reduces the effectiveness of the quarantine measures.

Table 7.4: Value of reproduction number (without diffusion)

Reproduction Number Value

R0 5.88256

RI 2.89739

RIT 1.4833

RQIT 1.30855


The technique from Chapter 3 is used to calculate the first excited mode of

oscillation n. According to the description of mode of excitation the curve.

f(β) =2.72611 ∗ 10−16 − 2.10836 ∗ 10−15β + 6.03344 ∗ 10−15β2 − 8.0138 ∗ 10−15β3 + 4.99653 ∗ 10−15β4 − 1.29674 ∗ 10−15β5 + 1.08455 ∗ 10−16β6

3.45506 ∗ 10−8 − 2.14726 ∗ 10−7β + 4.38499 ∗ 10−7β2 − 3.49595 ∗ 10−7β3 + 1.02242 ∗ 10−7β4 − 9.04955 ∗ 10−9β5

(7.70)

where n = 1 represents the first mode of excitation as being closest to the β-axis

as shown in Fig. 7.2. It is observed that bifurcation value of β, α, ζ and σ

increase in the presence of diffusion. So, the system remains stable for larger

values if diffusion included. On the other hand, bifurcation values of β is higher

than the bifurcation value of β for SEIJTR as given in Chapter 6 model but

lower in the presence of diffusion. The bifurcation values of α is lower in the

present SEQIJTR model than for SEIJTR (Chapter 6) both, with and without

diffusion. The corresponding bifurcation diagrams for β, α ,ζ and σ are shown in

appendix A.7.


Two initial population distributions are considered and solved numerically, with

and without diffusion. Graphs and descriptions of these solutions are given below:

164

n=1

0.2 0.4 0.6 0.8 1.0Β

-1.´ 10-8

-5.´ 10-9

5.´ 10-9

1.´ 10-8

1.5´ 10-8

fHΒL

n=2

0.2 0.4 0.6 0.8 1.0Β

-1.´10-8

-5.´10-9

5.´10-9

1.´10-8

1.5´10-8

f HΒL

n=3

0.2 0.4 0.6 0.8 1.0Β

-1.´ 10-8

-5.´ 10-9

5.´ 10-9

1.´ 10-8

1.5´ 10-8

fHΒL


Table 7.5: Bifurcation value of influential parameters

Parameters Value Considered Bifurcation Value


β 0.242 0.33224 0.362239

α 0.238 0.04879 0.07952

ζ 0.200 0.01619 0.05062

σ 0.157 0.01615 0.02636

165

7.4.1 Numerical solution without diffusion

Fig. 7.3, shows the output for initial condition (i) without diffusion for the

SEQIJTR system. At t = 0 a small proportion of infected are introduced to the

system. As soon as the disease spreads the proportion of susceptible reduces

dramatically, the peak proportion of susceptible dropping from 0.98 to 0.03488

within five days in the domain [−1, 1.5], where a large number of the population

get exposed to SARS. In the next fifteen days the peak proportions of susceptible

0.01102, 0.00459 and 0.00278 on days t = 10, t = 15 and t = 20 respectively. After

the first five days of disease the exposed population concentrated in the domain

[−0.6, 1.0] with peak value 0.25885. In the next five days a rapid decrease in the

exposed population is observed with peak proportion value 0.04860 in the domain

[−1, 1.5]. At t = 15 and t = 20 there are very small proportions of exposed left,

with peak values 0.00899 and 0.00167. Over time, from the exposed compartment

a proportion of the population that is not knowing to be infected yet is removed

and kept under observation in the quarantine compartment. Fig. 7.3, shows that

in the beginning of the disease, the proportion observed in the quarantine class is

maximized. At t = 5 days the peak proportion of the quarantined population in

the domain [−0.45, 0.85] is 0.17154. After five days the proportion of the

population quarantined decreases and the main concentration of the population

in the domain [−1.0, 1.5]. The peak values of proportion of population

quarantined at t = 10, t = 15 and t = 20 days are 0.11979, 0.05915 and 0.02601.

The proportion of the exposed population that becomes infected with SARS

moves to the infected compartment. The first five days of study of the SARS

disease are observed to be very important for exposed, quarantined and infected

compartments. In the first five days the proportion of infected increases rapidly

and attains its peak value, 0.25039 at t = 5 days. Thereafter, the proportion of

population infected decreases and the peak at t = 10 days is almost half of the

peak value in the first five days, being 0.11139 in the domain [−1, 1.5]. In the

next ten days, the proportion of infected keeps on decreasing with very low peaks

of 0.03350 and 0.00862 at t = 15 and t = 20 days, respectively. Once a

quarantined individual is diagnosed with SARS it is moved to the diagnosed

166

compartment as, for the simplification of the model, it is assumed that the

population in the quarantine compartment develop the disease at the end. The

individuals in the infected compartment diagnosed with SARS are isolated and

moved to the diagnosed compartment. The diagnosed population proportion

rapidly increase in the first five days of the disease and at t = 5 days, attains the

peak value 0.20568 across the domain [−0.4, 0.8]. This proportion keeps

increasing in the next five days and at t = 10 attains the peak value 0.23601 in

the domain [−1, 1.25]. After ten days the proportion diagnosed decreases rapidly,

with the peak values of the proportion having 0.13043 and 0.05602 at t = 15 and

t = 20 days, respectively. There is a only a small proportion of the population

admitted for treatment in the first five days of disease, so the peak value of the of

treated proportion of the population is 0.08539 at t = 5, with principal domain

[−0.4, 0.8]. Between t = 5 and t = 10 days, treated proportion increases further

and the peak proportion at t = 10 is 0.31772. This increase continues in the next

five days of study with a peak value of 0.41515 at t = 15. After that, the

proportion treated starts decreasig slowly, and at t = 20 days the peak of the

treated proportion is 0.38180. Negligible recovery is observed at t = 5 days across

the domain [−0.2, 0.6]. An increase in the recovered population is observed in the

next fifteen days with the recovered being concentrated in the domain [−0.8, 1.2].

The peak values of the recovered proportion of the population at t = 10, t = 15

and t = 20 days are 0.16649, 0.35276 and 0.52586.

Fig. 7.4, shows the output of initial condition (ii) in the absence of diffusion. The

peak values for the graphs in Fig. 7.4, are given in Table 7.6.

Table 7.6: Peak values for initial condition (ii) without diffusion

t S E Q I J T R

t = 00 0.96000 0.00000 0.00000 0.04000 0.00000 0.00000 0.00000

t = 05 0.00896 0.24929 0.16856 0.24887 0.21033 0.09204 0.03091

t = 10 0.00553 0.04681 0.11681 0.10886 0.23331 0.32254 0.17166

t = 15 0.00076 0.00867 0.05755 0.03257 0.12772 0.41538 0.35811

t = 20 0.00084 0.00161 0.02527 0.00835 0.05462 0.37974 0.53039

167

t = 0

t = 5t = 10

-2 -1 0 1 2x

0.2

0.4

0.6

0.8

1.0S

t = 5

t = 10

t = 15

-2 -1 0 1 2x

0.05

0.10

0.15

0.20

0.25

0.30E

t = 5

t = 10

t = 20

t = 15

-2 -1 0 1 2x

0.05

0.10

0.15

0.20Q

t = 0

t = 5

t = 10

t = 20

t = 15

-2 -1 0 1 2x

0.05

0.10

0.15

0.20

0.25

0.30I

t = 0

t = 5

t = 10

t = 20

t = 15

-2 -1 0 1 2x

0.05

0.10

0.15

0.20

0.25J

t = 0

t = 5

t = 10

t = 20

t = 15

-2 -1 0 1 2x

0.1

0.2

0.3

0.4

T

t = 0

t = 5

t = 10

t = 20

t = 15

-2 -1 0 1 2x

0.1

0.2

0.3

0.4

0.5

R

Figure 7.3: Solutions for initial condition (i) without diffusion.

168

t = 0

-2 -1 0 1 2x

0.2

0.4

0.6

0.8

1.0S

t = 5

t = 10

t = 15

-2 -1 0 1 2x

0.05

0.10

0.15

0.20

0.25

E

t = 5

t = 10

t = 20

t = 15

-2 -1 0 1 2x

0.05

0.10

0.15

0.20Q

t = 0

t = 5

t = 10

t = 20 t = 15

-2 -1 0 1 2x

0.05

0.10

0.15

0.20

0.25I

t = 0

t = 5

t = 10

t = 20

t = 15

-2 -1 0 1 2x

0.05

0.10

0.15

0.20

0.25J

t = 5

t = 10

t = 20

t=15

-2 -1 0 1 2x

0.1

0.2

0.3

0.4

T

t = 0

t = 5

t = 10

t = 20

t = 15

-2 -1 0 1 2x

0.1

0.2

0.3

0.4

0.5

R

Figure 7.4: Solutions for initial condition (ii) without diffusion.

169

7.4.2 Numerical solution with diffusion

Fig. 7.5, shows the output for initial condition (i), when the diffusion is

introduced in the SEQIJTR model. In the first five days of the spread of the

disease, susceptible move to the other compartment rapidly. At t = 5, t = 10,

t = 15 and t = 20 days the peak proportions in the susceptible compartment are

0.01069, 0.00154, 0.00034 and 0.00020, respectively. The exposed population

rapidly increases in the first five days across the domain [−1.4, 1.6] with peak

proportion value 0.14742 at t = 5 days. The next five days of the study show a

significant decrease in exposed with a peak proportion of 0.02186 across the

domain [−2, 2]. In the next ten days negligible proportions of exposed population

are observed, with peak values 0.00343 and 0.00586 at t = 15 and t = 20. The

proportion in the quarantined compartment rapidly increases in the first five

days. The peak value of the proportion is 0.10207 at t = 5 days, mainly

concentrated in the domain [−1.2, 1.5]. This is followed by rapid decrease in the

next fifteen days across the domain [−1.6, 2], with peak values 0.06492, 0.02984

and 0.01252 at t = 10, t = 15, and t = 20 days. The infected population

proportion quickly increases over the first five days, attaining peak value 0.14795

at t = 5, across the domain [−1.2, 1.6]. A major decrease is observed in the next

five days and at t = 10 the peak value of the infected proportion across the larger

domain [−1.6, 2] has dropped to 0.05708. At t = 15 and t = 20 lower peaks are

observed across the same domain i.e 0.01517 and 0.00354 respectively. The

proportion diagnosed has peak value 0.12121 at t = 5 days with principal domain

[−1, 1.2]. At t = 10 days the peak diagnosed proportion across the effective

domain [−1.5, 1.8] is 0.12882. That is the maximum diagnosed population

proportion of the study. After that, the values decrease rapidly and at t = 15 and

t = 20 the peak proportions of diagnosed are 0.06563 and 0.02655, respectively.

The treated proportions peak at t = 15 days, with these treated concentrated in

the domain [−1.6, 2]. The maximum of treated population proportions are

0.04955, 0.17665, 0.21799 and 0.19196 at t = 5, t = 10, t = 15 and t = 20 days,

respectively. Recovery increases significantly after t = 5 days and spreads across

the domain [−1.4, 1.6] within the next fifteen days. Maximum recovery is

170

observed at t = 20 days, where the peak value of the proportion is 0.268532.

Fig. 7.6, shows the output of the initial condition (ii) with diffusion. The peak

values for the graphs in Fig. 7.6, are given in Table 7.7.

Table 7.7: Peak values for initial condition (ii) without diffusion

t S E Q I J T R

t = 00 0.96000 0.00000 0.00000 0.04000 0.00000 0.00000 0.00000

t = 05 0.00789 0.05115 0.02311 0.04255 0.03207 0.01669 0.00618

t = 10 0.00169 0.00742 0.01677 0.01742 0.03512 0.04873 0.02686

t = 15 0.00064 0.00122 0.00777 0.00468 0.01794 0.05842 0.05133

t = 20 0.00039 0.00025 0.00325 0.00113 0.00722 0.05061 0.07173

7.5 Discussion

SEQIJTR model, with and without diffusion is defined and used to simulate the

transmission of SARS in 2003. A quarantined compartment is added to the

SEIJTR model given in Chapter 5 and 6, in order to construct the new model.

The numerical study of this model shows the effects of quarantine on disease

dynamics. At the time of the SARS epidemic, the two most influential

non-medical interventions were isolation of the infected and quarantine of the

exposed. So, it is important to study the effect of quarantine especially for

situations where medical interventions like treatment and vaccination are not

very effective. Two different initial conditions have been used to study

numerically their effects on transmission of the disease under different population

distributions. The operator splitting technique is used to calculate numerical

solutions of the differential equations. Expressions for the seventh-order

Routh-Hurwitz criteria are calculated and then used to check the stability of two

possible equilibria for the model, namely the disease-free and endemic equilibria.

The reproduction number RQIT is calculated with and without diffusion. For the

case without diffusion generation matrix method is used. While for with diffusion

case a new method involving the Routh-Hurwitz stability conditions and the

171

t = 0

-2 -1 0 1 2x

0.2

0.4

0.6

0.8

1.0S

t = 5

t = 10

t=15

-2 -1 0 1 2x

0.02

0.04

0.06

0.08

0.10

0.12

0.14

E

t = 5

t = 10

t = 20

t = 15

-2 -1 0 1 2x

0.02

0.04

0.06

0.08

0.10

0.12Q

t = 0

t = 5

t = 10

t = 20

t = 15

-2 -1 0 1 2x

0.05

0.10

0.15

I

t = 0

t = 5

t = 10

t = 15

t=20

-2 -1 0 1 2x

0.02

0.04

0.06

0.08

0.10

0.12

0.14

J

t = 0

t = 5

t = 10

t = 20

t = 15

-2 -1 0 1 2x

0.05

0.10

0.15

0.20

0.25T

t = 5

t = 10

t = 20

t = 15

-2 -1 0 1 2x

0.05

0.10

0.15

0.20

0.25

R

Figure 7.5: Solutions for initial condition (i) with diffusion.

172

t = 0

t = 5t=5

-2 -1 0 1 2x

0.2

0.4

0.6

0.8

1.0S

t = 5

t = 10

t =15

-2 -1 0 1 2x

0.01

0.02

0.03

0.04

0.05

0.06E

t = 5

t = 20

t = 15

t=10

-2 -1 0 1 2x

0.005

0.010

0.015

0.020

0.025Q

t = 0

t = 5

t = 10

t = 20

t = 15

-2 -1 0 1 2x

0.01

0.02

0.03

0.04

0.05I

t = 0

t = 5

t = 10

t = 15

t=20

-2 -1 0 1 2x

0.01

0.02

0.03

0.04J

t = 0

t = 5

t = 10

t = 30

-2 -1 0 1 2x

0.01

0.02

0.03

0.04

0.05

0.06T

t = 5

t = 10

t = 15

t=20

-2 -1 0 1 2x

0.02

0.04

0.06

0.08R

Figure 7.6: Solutions for initial condition (ii) with diffusion.

173

variational matrix is developed to calculate the expression for reproductive

number RdQIT . In disease-free equilibrium (DFE) the stability conditions are

satisfied for RQIT < 1. For endemic equilibrium RQIT and RdQIT are calculated.

The numerical value of the reproduction number without diffusion RQIT as given

in Table 7.4 is less than the numerical value of the reproduction number RdQIT

(with diffusion), showing the negative effect of diffusion on the system even

though it is assumed that there is no diffusion in the quarantined compartment.

The reproduction number without effective quarantine in the system is higher

than with quarantine measures as given in Table 7.4. This shows that the absence

of quarantine measures or the presence of diffusion both increase the infection, as

compared to the system with quarantine measures and no diffusion. The

bifurcation values of influential parameters are calculated, with and without

diffusion in the system, and are given in the Table 7.5. In the absence of

diffusion, the bifurcation value of β increases negligibly with the introduction of

quarantine compartment to the model, showing that the system SEQIJTR

system will be stable for larger transmission coefficient values than the SEIJTR

model studied earlier in Chapter 6. On the other hand, in the presence of

diffusion, SEIJTR system (Chapter 6) remains stable for higher values of β than

the SEQIJTR system. This shows the negative effect of diffusion on quarantine

measures. The bifurcation values of rate of progression from infected to

diagnosed, α, in the SEQIJTR model, both in the presence and absence of

diffusion, are lower as compared to the SEIJTR model.

The numerical solutions of the new SEQIJTR model are compared with those

for the previously-developed model SEIJTR (Chapter 6), in order to understand

the effects of quarantine on disease dynamics. In the absence of diffusion and

with initial condition (i), the proportion of population exposed is observed to be

significantly lower in the SEQIJTR model than in the model without a

quarantined class, SEIJTR. At t = 5 days, the peak value is nearly half of the

peak value under treatment model studied in Chapter 6 whereas, at t = 10 days,

the peak value of the proportion for SEQIJTR is even less than half of the

SEIJTR value. This trend continues in the following ten days of the disease

174

study. The present SEQIJTR model shows that a significant proportion of the

exposed population moves to quarantine compartment where infective but not

infectious population is isolated and quarantined. This immensely effect the

transmission of SARS by reducing the infected population by restricting the

entry to the infected class for those individuals who are not infectious yet.

Finally, it effects the recovery by increasing the recovery immensely. The

difference between the infected proportion of the population in the models grows

until the maximum difference occurs at t = 10 days and then, in the next ten

days, difference starts on decreasing, but the infected proportion in SEQIJTR

model is less than that of the SEIJTR model on all days under study. This

shows that using quarantine measures in the beginning of the disease affects the

prorogation of the disease. The proportion of the population under treatment is

higher than with the treatment model in Chapter 6. This difference keeps on

increasing with the passage of time until the maximum difference is observed at

t = 15 days. Recovery is higher in the present model than the model without

quarantine (Chapter 6) on all days under study, the largest difference in recovery

between the models occurring at day 20. Under initial condition (ii), very similar

results are observed whereas the difference in recovery under the two models is

even more pronounced, with SEQIJTR recovery even higher under condition (ii)

than under (i).

When diffusion is introduced to the system for initial conditions (i) and (ii), the

proportion of population exposed is lower than for the treatment isolation model

in Chapter 6. Although the proportion of exposed is lower in the model under

study, the difference between the proportions for the two models is less in the

presence of diffusion. It is very important to note that, although the diffusion in

the quarantined class is assumed to be zero, the diffusion in other compartments

affects the transmission of disease so much that it reduces the intensity of the

population in the quarantined compartment and spreads the population across a

larger domain than in the without diffusion case. The infected population

proportion is comparatively lower than that in the treatment model. As

compared to the without diffusion case for the same SEQIJTR model, the

175

intensity of infection is almost half. The differences are more significant in the

first ten days of the disease than the last ten days of disease under study. The

numerical results of the SEQIJTR model shown in Fig. 7.5, indicate that at

t = 5 the peak value of the diagnosed population is significantly higher than for

the SEIJTR model in Chapter 6, but at t = 10 the diagnosed population is

significantly lower than for the treatment model whereas at t = 15 and t = 20

there is negligible difference between diagnosed proportions. The proportion of

population recovered with diffusion, in both conditions, is greater under

SEQIJTR model than in the SEIJTR model (in Chapter 6). In the last ten

days the difference is higher than in the first ten days for condition (i) whereas

condition (ii) shows no difference in recovery in the first ten days and a negligible

difference in the last ten days for the two models in the presence of diffusion.

176

Chapter 8

Conclusions

The aim of this work is to study the transmission dynamics of the SARS disease.

First, a model consisting of susceptible, exposed, infected, diagnosed and

recovered (SEIJR) compartments is chosen. In Chapter 3, a SEIJR model is

considered with the inclusion of diffusion. Four different initial conditions are

taken for the population distribution. The equations governing the system are

solved numerically using the operator splitting method. The reproduction

number RI is calculated for the disease. It is shown that in the disease-free

equilibrium, the disease dies out for RI < 1, but prevails for endemic equilibrium,

where RI > 1, as shown in Table 3.11. Stability of solutions with and without

diffusion is established using the Routh-Hurwitz conditions. The parameters

transmission coefficient, β, recovery coefficient in infectious class, γ1, and recovery

coefficient in diagnosed, γ2, are varied to observe the effects on the spread of

disease. Effect on the spread of disease is examined for four cases involving

different values of transmission coefficient β, recovery rates γ1 and γ2. It is

observed that the system can be stable for large values of transmission coefficient

β and small values of recovery rates γ1 and γ2 in the presence of diffusion. Thus

the spread of disease will not increase in the presence of diffusion even with larger

value of transmission coefficient β and smaller value of recovery rates γ1 and γ2.

In order to examine the effects of diffusion further, cross-diffusion along with

self-diffusion are introduced to the SEIJR model in Chapter 4. Two different

177

initial conditions are taken for the population distribution. Stability of solutions

with and without cross-diffusion is also established here using the Routh-Hurwitz

conditions. Four different cases of cross-diffusion coefficients in the susceptible

and exposed compartments are chosen in order to see their effects on the spread

of disease. Bifurcation values of the transmission coefficient, β and recovery

coefficients γ1 and γ2 are obtained. It is observed that when both populations i.e

susceptible and exposed, cross diffuse, the system is destabilised for a smaller

value of β. It is also observed that with a positive value of cross-diffusion in the

susceptible and exposed compartments, the system stabilises for higher values of

the recovery coefficients γ1 and γ2.

In order to investigate the effect on the transmission dynamics of SARS with the

inclusion of a treatment compartment in the system a SEIJTR model has been

constructed in Chapter 5. First of all, parameters involved in the model are

estimated based on the best fit to the field data [121] published as daily reports

by the World Health Organization when the SARS epidemic outbreak occurred

in Hong Kong in 2003. A numerical method called the Dormand-Prince Pairs

method has been used as system solver for the non-linear differential model

SEIJTR, and the Levenberg-Marquardt technique has been used as the least

squares optimiser for determining the best fit to the field data. MATLAB has

been used for all the calculations. A large number of simulations and

demographic information based on the city of Hong Kong have been used to

estimate the parameters. Different graphical and numerical methods have been

used to verify the estimation. Autocorrelations of the residuals are within the

confidence interval, as shown in Fig. 5.4. The model considered here thus

captures the dynamical behavior of the SARS epidemic field data [121].

After estimating the required parameters in Chapter 5, a SEIJTR model is used

to obtain numerical solutions in Chapter 6, with the inclusion of diffusion and

treatment in the system. Three different initial population distributions have

been chosen to examine their effects on transmission of the disease. The models

178

under investigation have two possible equilibria, disease-free and endemic. The

reproduction numbers RIT for the various cases considered (RIT > 1 ) are given

in Table 6.8. A study of bifurcation values of the transmission coefficient, β and

rate of progression from infective to diagnosed, α, as shown in Table 6.5, indicates

that the system remains stable for higher values of β and α with diffusion than

without diffusion. It is observed from the values given in Table 6.8 that β and α,

have significant impact on the basic reproductive number RIT . If the value of β is

decreased, as in Case 2, the reproduction number RIT decreases significantly even

though it is still greater than 1. This causes slow transmission of infection and

thus higher peak values in the infected compartment, from t = 10 to t = 20 days

as shown in Tables 6.6 and 6.7. A decrease in the value of α, as in Case 3, causes

a significant increase in the value of the basic reproduction number RIT . Thus

there is an increase in transmission of the infection under all initial population

distributions considered, as shown in Tables 6.6 and 6.7. There are also slightly

lower peak values of infected population proportion at t= 20 days, as compared to

the original situation depicted in Case 1. An increase in the value of α, as given

in Case 5, causes a slight decrease in the value of the basic reproduction number.

Finally, quarantine of the exposed population is introduced in Chapter 7. A

“quarantined” compartment is added to SEIJTR model from Chapter 6, giving

a SEQIJTR model which is used to simulate transmission of the SARS

epidemic with and without diffusion in Hong Kong in 2003. The numerical study

of this model shows the effects of quarantine on the disease dynamics. At the

time of the SARS epidemic, the two most influential non-medical interventions

were isolation of the infected and quarantine of the exposed. Hence it is

important to study the effect of quarantine, especially when medical interventions

for disease like treatment and vaccination, are not very effective. Two different

initial population distributions have been used to study numerically the effects on

transmission of the disease. Expressions for the seventh order Routh-Hurwitz

criteria are calculated and then used to check the stability of two possible

equilibria for the model, namely disease-free and endemic. The reproduction

179

number RQIT is calculated with and without diffusion. For the case without

diffusion the generation matrix method is used. For the case with diffusion, the

Routh-Hurwitz stability conditions and the variational matrix are used to

calculate an expression for the reproductive number RdQIT . In the disease-free

equilibrium (DFE) the stability conditions are satisfied for RQIT < 1. For the

endemic equilibrium, RQIT and RdQIT are calculated. The numerical value of the

reproduction number without diffusion, RQIT is less than that of the

reproduction number RdQIT with diffusion, as given in Table 7.4. This shows the

negative effect of diffusion on the system, although it is assumed that there is no

diffusion in the quarantined compartment. The value of the reproduction number

RIT is similar to the value of RdQIT . This shows that the absence of quarantine

measures or presence of diffusion both increase infections as compared to the

system with quarantine measures and without diffusion. The bifurcation values of

influential parameters are calculated with and without diffusion in the system,

and are given in Table 7.5. In the absence of diffusion, the bifurcation value of β

increases negligibly with the introduction of a quarantine compartment to the

model, showing that system SEQIJTR is stable for large value of transmission

coefficient, β, than the SEIJTR model studied in Chapter 6. On the other hand,

in the presence of diffusion the SEIJTR system remains stable for larger values

of β than the SEQIJTR system, thus showing the negative effect of diffusion on

quarantine measures.

In summary it can be concluded that

• Initial population distribution plays a crucial role in the spread of disease.

• Diffusion is significant in reducing the intensity of disease.

• Increased recovery of the infectives through intervention is effective in

reducing the spread of disease during the initial days of the onset of disease.

• Increased recovery of the diagnosed is more effective in reducing spread of

disease during the last days of the disease.

180

• Introduction of positive and negative cross-diffusion to the system leads to,

respectively increase and decrease in the domains of the susceptible and

exposed proportions. This in turn affects the domains in other

compartments of the system. Thus, positive cross-diffusion in the system

expands and negative cross-diffusion restricts the spread of disease.

• Introduction of negative cross-diffusion to the system lowers the

reproduction number significantly. This indicates lower intensity of the

infection, thus making the transmission of disease slower.

• Implementation of quarantine measures, along with isolation and treatment,

affects disease dynamics by slowing disease transmission and thus reducing

the proportion infected.

• System without diffusion shows greater recovery with quarantine measures

than system with diffusion.

• Quarantine is most effective during the initial days of the onset of disease.

• Even though treatment and isolation are necessary for recovery, with the

addition of quarantine earlier recovery is possible in diseases like SARS

which occur for short periods.

Recommendations for possible future work related to the work done this thesis:

• The self-diffusion model can be further studied on the basis of age and

gender, as medical findings show SARS is sensitive to age and gender where

the diffusivity terms are functions of age and gender.

• Develop a model for SARS vaccination, so that the public health sector can

measure vaccines efficacy before the implementation of vaccines, should

they become available in the future.

• Study cross-diffusion among susceptible, exposed and infected in the model

to see the effects on transmission of disease. Also the effects of self and

cross-diffusion can be studied for a vaccination model.

181

• The models can be extended by introducing patches or edges in the spatial

domain, in order to study the effects on the spread of disease of restricted

movement of infected from one region to another within a boundary.

182

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Appendix

Appendix A.3

Bifurcation value and diagram for transmission

coefficient β for Case(1)

The Routh-Hurwitz criterion for stability gives:

p1 = 1.00317− 0.0352016β = f1(β),

p5 = 3.68074 ∗ 10−11 + 4.29617 ∗ 10−11β = f2(β),

p1p2 − p3 = 0.287979− 0.110303β + 0.00496585β2 = f3(β),

p1p2p3 + p1p5 − (p23 + p21p4) =

0.00917341− 0.0157348β + 0.00483904β2 − 0.00021073β3 = f4(β),

(p1p4 − p5)(p1p2p3 − p23 − p21p4) + p21p4p5 − (p5(p1p2 − p3)2 + p1p

25) =

1.99247 ∗ 10−8 − 3.65302 ∗ 10−8β +1.45989 ∗ 10−8β2 − 1.79598 ∗ 10−9β3 +8.44811 ∗

10−11β4 − 1.32948 ∗ 10−12β5 = f5(β).

f5(β) = 0, gives the bifurcation value of β for which the point of equilibrium

moves from stable to unstable equilibrium. For f5(β) = 0, β = 0.750435. Other

values of β either being large numbers or negative, are neglected. Thus for any

value greater than β = 0.750435 as shown in Fig. 1 for case 1, the point of

equilibrium will be unstable. Similarly the bifurcation value of β is calculated for

other cases with and without diffusion. Figs. 1 and 2, shows the bifurcation

diagrams for β with and without diffusion.

Bifurcation value and diagram for recovery rate

γ1 Case(1)

The Routh-Hurwitz criterion for stability gives: p1 = 0.851769 + γ1 = f1(γ1),

i

p5 = 4.10822 ∗ 10−11 + 2.23572 ∗ 10−10γ1 = f2(γ1),

p1p2 − p3 = 0.134729 + 0.522435γ1 + 0.512735γ21 = f3(γ1),

p1p2p3 + p1p5 − (p23 + p21p4) =

−0.00106088 + 0.00440029γ1 + 0.0289846γ21 + 0.0324131γ3

1 = f4(γ1),

(p1p4 − p5)(p1p2p3 − p23 − p21p4) + p21p4p5 − (p5(p1p2 − p3)2 + p1p

25) =

−8.50633 ∗ 10−10 − 5.36645 ∗ 10−9γ1 + 5.07547 ∗ 10−8γ21 + 3.06905 ∗ 10−7γ3

1 +

5.24123 ∗ 10−7γ41 + 2.82745 ∗ 10−7γ5

1 = f5(γ1).

f5(γ1) = 0, gives the bifurcation value of γ1 for which the point of equilibrium

moves from stable to unstable equilibrium. For f5(γ1) = 0, γ1 = 0.124708. Other

values of γ1 either being small numbers or negative, are neglected. Thus for any

value less than γ1 = 0.124708 as shown in Fig. 3, for case 1, the point of

equilibrium will be unstable. Similarly the bifurcation value of γ1 is calculated for

other cases with and without diffusion. Figs. 3 and 4, shows the bifurcation

diagrams for γ1 with and without diffusion.

Bifurcation value and diagram for recovery

coefficients γ2

The bifurcation values of γ2 are also calculated in same way for all cases as for γ1.

The bifurcation diagrams for γ2 with and without diffusion are given in Figs. 5

and 6 respectively.

The coefficient of characteristic equation 3.29

p1 = −a11 − a22 − a33 − a44 − a55;

p2 =

−a12a21+a11a22−a13a31−a23a32+a11a33+a22a33−a14a41−a24a42−a34a43+a11a44+

a22a44+a33a44−a15a51−a25a52−a35a53−a45a54+a11a55+a22a55+a33a55+a44a55;

p3 = a13a22a31 − a12a23a31 − a13a21a32 + a11a23a32 + a12a21a33 − a11a22a33 +

a14a22a41 − a12a24a41 + a14a33a41 − a13a34a41 − a14a21a42 + a11a24a42 + a24a33a42 −

a23a34a42 − a14a31a43 − a24a32a43 + a11a34a43 + a22a34a43 + a12a21a44 − a11a22a44 +

ii

a13a31a44 + a23a32a44 − a11a33a44 − a22a33a44 + a15a22a51 − a12a25a51 + a15a33a51 −

a13a35a51 + a15a44a51− a14a45a51 − a15a21a52 + a11a25a52 + a25a33a52 − a23a35a52 +

a25a44a52 − a24a45a52 − a15a31a53 − a25a32a53 + a11a35a53 + a22a35a53 + a35a44a53 −

a34a45a53 − a15a41a54 − a25a42a54 − a35a43a54 + a11a45a54 + a22a45a54 + a33a45a54 +

a12a21a55 − a11a22a55 + a13a31a55 + a23a32a55 − a11a33a55 − a22a33a55 + a14a41a55 +

a24a42a55 + a34a43a55 − a11a44a55 − a22a44a55 − a33a44a55;

p4 = a14a23a32a41 − a13a24a32a41 − a14a22a33a41 + a12a24a33a41 + a13a22a34a41 −

a12a23a34a41 − a14a23a31a42 + a13a24a31a42 + a14a21a33a42 − a11a24a33a42 −

a13a21a34a42+a11a23a34a42+a14a22a31a43−a12a24a31a43−a14a21a32a43+a11a24a32a43+

a12a21a34a43 − a11a22a34a43 − a13a22a31a44 + a12a23a31a44 + a13a21a32a44 −

a11a23a32a44 − a12a21a33a44 + a11a22a33a44 + a15a23a32a51 − a13a25a32a51 −

a15a22a33a51+a12a25a33a51+a13a22a35a51−a12a23a35a51+a15a24a42a51−a14a25a42a51+

a15a34a43a51−a14a35a43a51−a15a22a44a51+a12a25a44a51−a15a33a44a51+a13a35a44a51+

a14a22a45a51 − a12a24a45a51 + a14a33a45a51 − a13a34a45a51 − a15a23a31a52 +

a13a25a31a52+a15a21a33a52−a11a25a33a52−a13a21a35a52+a11a23a35a52−a15a24a41a52+

a14a25a41a52+a25a34a43a52−a24a35a43a52+a15a21a44a52−a11a25a44a52−a25a33a44a52+

a23a35a44a52−a14a21a45a52+a11a24a45a52+a24a33a45a52−a23a34a45a52+a15a22a31a53−

a12a25a31a53−a15a21a32a53+a11a25a32a53+a12a21a35a53−a11a22a35a53−a15a34a41a53+

a14a35a41a53−a25a34a42a53+a24a35a42a53+a15a31a44a53+a25a32a44a53−a11a35a44a53−

a22a35a44a53−a14a31a45a53−a24a32a45a53+a11a34a45a53+a22a34a45a53+a15a22a41a54−

a12a25a41a54+a15a33a41a54−a13a35a41a54−a15a21a42a54+a11a25a42a54+a25a33a42a54−

a23a35a42a54−a15a31a43a54−a25a32a43a54+a11a35a43a54+a22a35a43a54+a12a21a45a54−

a11a22a45a54 + a13a31a45a54 + a23a32a45a54 − a11a33a45a54 − a22a33a45a54 −

a13a22a31a55+a12a23a31a55+a13a21a32a55−a11a23a32a55−a12a21a33a55+a11a22a33a55−

a14a22a41a55+a12a24a41a55−a14a33a41a55+a13a34a41a55+a14a21a42a55−a11a24a42a55−

a24a33a42a55+a23a34a42a55+a14a31a43a55+a24a32a43a55−a11a34a43a55−a22a34a43a55−

a12a21a44a55+a11a22a44a55−a13a31a44a55−a23a32a44a55+a11a33a44a55+a22a33a44a55;

p5 = −a15a24a33a42a51 + a14a25a33a42a51 + a15a23a34a42a51 − a13a25a34a42a51 −

a14a23a35a42a51 + a13a24a35a42a51 + a15a24a32a43a51 − a14a25a32a43a51 −



iii


a14a22a33a45a51 + a12a24a33a45a51 + a13a22a34a45a51 − a12a23a34a45a51 +

a15a24a33a41a52 − a14a25a33a41a52 − a15a23a34a41a52 + a13a25a34a41a52 +





a14a21a33a45a52 − a11a24a33a45a52 − a13a21a34a45a52 + a11a23a34a45a52 −


















a13a21a32a44a55 + a11a23a32a44a55 + a12a21a33a44a55 − a11a22a33a44a55.

The coefficient of characteristic equation 3.45

q1 = −a11 − a22 − a33 − a44 − a55 + d1k2 + d2k

2 + d3k2 + d4k

2 + d5k2;

q2 = −a12a21 + a11a22 − a13a31 − a23a32 + a11a33 + a22a33 − a14a41 − a24a42 −

iv

a34a43 + a11a44 + a22a44 + a33a44 − a15a51 − a25a52 − a35a53 − a45a54 + a11a55 +

a22a55 + a33a55 + a44a55 − a22d1k2 − a33d1k

2 − a44d1k2 − a55d1k

2 − a11d2k2 −

a33d2k2 − a44d2k

2 − a55d2k2 − a11d3k

2 − a22d3k2 − a44d3k

2 − a55d3k2 − a11d4k

2 −

a22d4k2 − a33d4k

2 − a55d4k2 − a11d5k

2 − a22d5k2 − a33d5k

2 − a44d5k2 + d1d2k

4 +

d1d3k4 + d2d3k

4 + d1d4k4 + d2d4k

4 + d3d4k4 + d1d5k

4 + d2d5k4 + d3d5k

4 + d4d5k4;

q3 = a13a22a31−a12a23a31−a13a21a32+a11a23a32+a12a21a33−a11a22a33+a14a22a41−

a12a24a41 + a14a33a41 − a13a34a41 − a14a21a42 + a11a24a42 + a24a33a42 − a23a34a42 −

a14a31a43 − a24a32a43 + a11a34a43 + a22a34a43 + a12a21a44 − a11a22a44 + a13a31a44 +

a23a32a44 − a11a33a44 − a22a33a44 + a15a22a51 − a12a25a51 + a15a33a51 − a13a35a51 +

a15a44a51 − a14a45a51 − a15a21a52 + a11a25a52 + a25a33a52 − a23a35a52 + a25a44a52 −

a24a45a52 − a15a31a53 − a25a32a53 + a11a35a53 + a22a35a53 + a35a44a53 − a34a45a53 −

a15a41a54 − a25a42a54 − a35a43a54 + a11a45a54 + a22a45a54 + a33a45a54 + a12a21a55 −

a11a22a55 + a13a31a55 + a23a32a55 − a11a33a55 − a22a33a55 + a14a41a55 + a24a42a55 +

a34a43a55−a11a44a55−a22a44a55−a33a44a55−a23a32d1k2+a22a33d1k

2−a24a42d1k2−

a34a43d1k2 + a22a44d1k

2 + a33a44d1k2 − a25a52d1k

2 − a35a53d1k2 − a45a54d1k

2 +

a22a55d1k2 + a33a55d1k

2 + a44a55d1k2 − a13a31d2k

2 + a11a33d2k2 − a14a41d2k

2 −

a34a43d2k2 + a11a44d2k

2 + a33a44d2k2 − a15a51d2k

2 − a35a53d2k2 − a45a54d2k

2 +

a11a55d2k2 + a33a55d2k

2 + a44a55d2k2 − a12a21d3k

2 + a11a22d3k2 − a14a41d3k

2 −

a24a42d3k2 + a11a44d3k

2 + a22a44d3k2 − a15a51d3k

2 − a25a52d3k2 − a45a54d3k

2 +

a11a55d3k2 + a22a55d3k

2 + a44a55d3k2 − a12a21d4k

2 + a11a22d4k2 − a13a31d4k

2 −

a23a32d4k2 + a11a33d4k

2 + a22a33d4k2 − a15a51d4k

2 − a25a52d4k2 − a35a53d4k

2 +

a11a55d4k2 + a22a55d4k

2 + a33a55d4k2 − a12a21d5k

2 + a11a22d5k2 − a13a31d5k

2 −

a23a32d5k2 + a11a33d5k

2 + a22a33d5k2 − a14a41d5k

2 − a24a42d5k2 − a34a43d5k

2 +

a11a44d5k2+a22a44d5k

2+a33a44d5k2−a33d1d2k

4−a44d1d2k4−a55d1d2k

4−a22d1d3k4−

a44d1d3k4−a55d1d3k



4−a33d1d4k4−




4−a55d3d4k4−




4−a11d3d5k4−

a22d3d5k4 − a44d3d5k

4 − a11d4d5k4 − a22d4d5k

4 − a33d4d5k4 + d1d2d3k

6 + d1d2d4k6 +

d1d3d4k6+d2d3d4k

6+d1d2d5k6+d1d3d5k

6+d2d3d5k6+d1d4d5k

6+d2d4d5k6+d3d4d5k

6;

q4 = a14a23a32a41 − a13a24a32a41 − a14a22a33a41 + a12a24a33a41 + a13a22a34a41 −


v




a12a23a35a51+a15a24a42a51−a14a25a42a51+a15a34a43a51−a14a35a43a51−a15a22a44a51+




a11a25a44a52−a25a33a44a52+a23a35a44a52−a14a21a45a52+a11a24a45a52+a24a33a45a52−


a11a22a35a53−a15a34a41a53+a14a35a41a53−a25a34a42a53+a24a35a42a53+a15a31a44a53+

a25a32a44a53 − a11a35a44a53 − a22a35a44a53 − a14a31a45a53 − a24a32a45a53 +

a11a34a45a53+a22a34a45a53+a15a22a41a54−a12a25a41a54+a15a33a41a54−a13a35a41a54−

a15a21a42a54+a11a25a42a54+a25a33a42a54−a23a35a42a54−a15a31a43a54−a25a32a43a54+

a11a35a43a54+a22a35a43a54+a12a21a45a54−a11a22a45a54+a13a31a45a54+a23a32a45a54−

a11a33a45a54 − a22a33a45a54 − a13a22a31a55 + a12a23a31a55 + a13a21a32a55 −

a11a23a32a55−a12a21a33a55+a11a22a33a55−a14a22a41a55+a12a24a41a55−a14a33a41a55+

a13a34a41a55+a14a21a42a55−a11a24a42a55−a24a33a42a55+a23a34a42a55+a14a31a43a55+

a24a32a43a55 − a11a34a43a55 − a22a34a43a55 − a12a21a44a55 + a11a22a44a55 −

a13a31a44a55 − a23a32a44a55 + a11a33a44a55 + a22a33a44a55 + a24a33a42d1k2 −

a23a34a42d1k2 − a24a32a43d1k

2 + a22a34a43d1k2 + a23a32a44d1k

2 − a22a33a44d1k2 +

a25a33a52d1k2 − a23a35a52d1k

2 + a25a44a52d1k2 − a24a45a52d1k

2 − a25a32a53d1k2 +

a22a35a53d1k2 + a35a44a53d1k

2 − a34a45a53d1k2 − a25a42a54d1k

2 − a35a43a54d1k2 +

a22a45a54d1k2 + a33a45a54d1k

2 + a23a32a55d1k2 − a22a33a55d1k

2 + a24a42a55d1k2 +

a34a43a55d1k2 − a22a44a55d1k

2 − a33a44a55d1k2 + a14a33a41d2k

2 − a13a34a41d2k2 −

a14a31a43d2k2 + a11a34a43d2k

2 + a13a31a44d2k2 − a11a33a44d2k

2 + a15a33a51d2k2 −

a13a35a51d2k2 + a15a44a51d2k

2 − a14a45a51d2k2 − a15a31a53d2k

2 + a11a35a53d2k2 +

a35a44a53d2k2 − a34a45a53d2k

2 − a15a41a54d2k2 − a35a43a54d2k

2 + a11a45a54d2k2 +

a33a45a54d2k2 + a13a31a55d2k

2 − a11a33a55d2k2 + a14a41a55d2k

2 + a34a43a55d2k2 −

a11a44a55d2k2 − a33a44a55d2k

2 + a14a22a41d3k2 − a12a24a41d3k

2 − a14a21a42d3k2 +

a11a24a42d3k2 + a12a21a44d3k

2 − a11a22a44d3k2 + a15a22a51d3k

2 − a12a25a51d3k2 +

a15a44a51d3k2 − a14a45a51d3k

2 − a15a21a52d3k2 + a11a25a52d3k

2 + a25a44a52d3k2 −

vi

a24a45a52d3k2 − a15a41a54d3k

2 − a25a42a54d3k2 + a11a45a54d3k

2 + a22a45a54d3k2 +

a12a21a55d3k2 − a11a22a55d3k

2 + a14a41a55d3k2 + a24a42a55d3k

2 − a11a44a55d3k2 −

a22a44a55d3k2 + a13a22a31d4k

2 − a12a23a31d4k2 − a13a21a32d4k

2 + a11a23a32d4k2 +

a12a21a33d4k2 − a11a22a33d4k

2 + a15a22a51d4k2 − a12a25a51d4k

2 + a15a33a51d4k2 −

a13a35a51d4k2 − a15a21a52d4k

2 + a11a25a52d4k2 + a25a33a52d4k

2 − a23a35a52d4k2 −

a15a31a53d4k2 − a25a32a53d4k

2 + a11a35a53d4k2 + a22a35a53d4k

2 + a12a21a55d4k2 −

a11a22a55d4k2 + a13a31a55d4k

2 + a23a32a55d4k2 − a11a33a55d4k

2 − a22a33a55d4k2 +

a13a22a31d5k2 − a12a23a31d5k

2 − a13a21a32d5k2 + a11a23a32d5k

2 + a12a21a33d5k2 −

a11a22a33d5k2 + a14a22a41d5k

2 − a12a24a41d5k2 + a14a33a41d5k

2 − a13a34a41d5k2 −

a14a21a42d5k2 + a11a24a42d5k

2 + a24a33a42d5k2 − a23a34a42d5k

2 − a14a31a43d5k2 −

a24a32a43d5k2 + a11a34a43d5k

2 + a22a34a43d5k2 + a12a21a44d5k

2 − a11a22a44d5k2 +

a13a31a44d5k2 + a23a32a44d5k

2 − a11a33a44d5k2 − a22a33a44d5k

2 − a34a43d1d2k4 +

a33a44d1d2k4 − a35a53d1d2k

4 − a45a54d1d2k4 + a33a55d1d2k

4 + a44a55d1d2k4 −

a24a42d1d3k4 + a22a44d1d3k

4 − a25a52d1d3k4 − a45a54d1d3k

4 + a22a55d1d3k4 +

a44a55d1d3k4 − a14a41d2d3k

4 + a11a44d2d3k4 − a15a51d2d3k

4 − a45a54d2d3k4 +

a11a55d2d3k4 + a44a55d2d3k

4 − a23a32d1d4k4 + a22a33d1d4k

4 − a25a52d1d4k4 −

a35a53d1d4k4 + a22a55d1d4k

4 + a33a55d1d4k4 − a13a31d2d4k

4 + a11a33d2d4k4 −

a15a51d2d4k4 − a35a53d2d4k

4 + a11a55d2d4k4 + a33a55d2d4k

4 − a12a21d3d4k4 +

a11a22d3d4k4 − a15a51d3d4k

4 − a25a52d3d4k4 + a11a55d3d4k

4 + a22a55d3d4k4 −

a23a32d1d5k4 + a22a33d1d5k

4 − a24a42d1d5k4 − a34a43d1d5k

4 + a22a44d1d5k4 +

a33a44d1d5k4 − a13a31d2d5k

4 + a11a33d2d5k4 − a14a41d2d5k

4 − a34a43d2d5k4 +

a11a44d2d5k4 + a33a44d2d5k

4 − a12a21d3d5k4 + a11a22d3d5k

4 − a14a41d3d5k4 −

a24a42d3d5k4 + a11a44d3d5k

4 + a22a44d3d5k4 − a12a21d4d5k

4 + a11a22d4d5k4 −

a13a31d4d5k4 − a23a32d4d5k

4 + a11a33d4d5k4 + a22a33d4d5k

4 − a44d1d2d3k6 −

a55d1d2d3k6−a33d1d2d4k

6−a55d1d2d4k6−a22d1d3d4k


6−




6−




6−

a22d3d4d5k6 + d1d2d3d4k

8 + d1d2d3d5k8 + d1d2d4d5k

8 + d1d3d4d5k8 + d2d3d4d5k

8;

q5 = −a15a24a33a42a51 + a14a25a33a42a51 + a15a23a34a42a51 − a13a25a34a42a51 −



vii




























a25a34a43a52d1k2 − a24a35a43a52d1k

2 − a25a33a44a52d1k2 + a23a35a44a52d1k

2 +

a24a33a45a52d1k2 − a23a34a45a52d1k

2 − a25a34a42a53d1k2 + a24a35a42a53d1k

2 +

a25a32a44a53d1k2 − a22a35a44a53d1k

2 − a24a32a45a53d1k2 + a22a34a45a53d1k

2 +

a25a33a42a54d1k2 − a23a35a42a54d1k

2 − a25a32a43a54d1k2 + a22a35a43a54d1k

2 +

viii

a23a32a45a54d1k2 − a22a33a45a54d1k

2 − a24a33a42a55d1k2 + a23a34a42a55d1k

2 +

a24a32a43a55d1k2 − a22a34a43a55d1k

2 − a23a32a44a55d1k2 + a22a33a44a55d1k

2 +

a15a34a43a51d2k2 − a14a35a43a51d2k

2 − a15a33a44a51d2k2 + a13a35a44a51d2k

2 +

a14a33a45a51d2k2 − a13a34a45a51d2k

2 − a15a34a41a53d2k2 + a14a35a41a53d2k

2 +

a15a31a44a53d2k2 − a11a35a44a53d2k

2 − a14a31a45a53d2k2 + a11a34a45a53d2k

2 +

a15a33a41a54d2k2 − a13a35a41a54d2k

2 − a15a31a43a54d2k2 + a11a35a43a54d2k

2 +

a13a31a45a54d2k2 − a11a33a45a54d2k

2 − a14a33a41a55d2k2 + a13a34a41a55d2k

2 +

a14a31a43a55d2k2 − a11a34a43a55d2k

2 − a13a31a44a55d2k2 + a11a33a44a55d2k

2 +

a15a24a42a51d3k2 − a14a25a42a51d3k

2 − a15a22a44a51d3k2 + a12a25a44a51d3k

2 +

a14a22a45a51d3k2 − a12a24a45a51d3k

2 − a15a24a41a52d3k2 + a14a25a41a52d3k

2 +

a15a21a44a52d3k2 − a11a25a44a52d3k

2 − a14a21a45a52d3k2 + a11a24a45a52d3k

2 +

a15a22a41a54d3k2 − a12a25a41a54d3k

2 − a15a21a42a54d3k2 + a11a25a42a54d3k

2 +

a12a21a45a54d3k2 − a11a22a45a54d3k

2 − a14a22a41a55d3k2 + a12a24a41a55d3k

2 +

a14a21a42a55d3k2 − a11a24a42a55d3k

2 − a12a21a44a55d3k2 + a11a22a44a55d3k

2 +

a15a23a32a51d4k2 − a13a25a32a51d4k

2 − a15a22a33a51d4k2 + a12a25a33a51d4k

2 +

a13a22a35a51d4k2 − a12a23a35a51d4k

2 − a15a23a31a52d4k2 + a13a25a31a52d4k

2 +

a15a21a33a52d4k2 − a11a25a33a52d4k

2 − a13a21a35a52d4k2 + a11a23a35a52d4k

2 +

a15a22a31a53d4k2 − a12a25a31a53d4k

2 − a15a21a32a53d4k2 + a11a25a32a53d4k

2 +

a12a21a35a53d4k2 − a11a22a35a53d4k

2 − a13a22a31a55d4k2 + a12a23a31a55d4k

2 +

a13a21a32a55d4k2 − a11a23a32a55d4k

2 − a12a21a33a55d4k2 + a11a22a33a55d4k

2 +

a14a23a32a41d5k2 − a13a24a32a41d5k

2 − a14a22a33a41d5k2 + a12a24a33a41d5k

2 +

a13a22a34a41d5k2 − a12a23a34a41d5k

2 − a14a23a31a42d5k2 + a13a24a31a42d5k

2 +

a14a21a33a42d5k2 − a11a24a33a42d5k

2 − a13a21a34a42d5k2 + a11a23a34a42d5k

2 +

a14a22a31a43d5k2 − a12a24a31a43d5k

2 − a14a21a32a43d5k2 + a11a24a32a43d5k

2 +

a12a21a34a43d5k2 − a11a22a34a43d5k

2 − a13a22a31a44d5k2 + a12a23a31a44d5k

2 +

a13a21a32a44d5k2 − a11a23a32a44d5k

2 − a12a21a33a44d5k2 + a11a22a33a44d5k

2 +

a35a44a53d1d2k4 − a34a45a53d1d2k

4 − a35a43a54d1d2k4 + a33a45a54d1d2k

4 +

a34a43a55d1d2k4 − a33a44a55d1d2k

4 + a25a44a52d1d3k4 − a24a45a52d1d3k

4 −

a25a42a54d1d3k4 + a22a45a54d1d3k

4 + a24a42a55d1d3k4 − a22a44a55d1d3k

4 +

a15a44a51d2d3k4 − a14a45a51d2d3k

4 − a15a41a54d2d3k4 + a11a45a54d2d3k

4 +

a14a41a55d2d3k4 − a11a44a55d2d3k

4 + a25a33a52d1d4k4 − a23a35a52d1d4k

4 −

ix

0.2 0.4 0.6 0.8 1.0Β

5.´ 10-9

1.´ 10-8

1.5´ 10-8

2.´ 10-8

fHΒL

0.2 0.4 0.6 0.8 1.0Β

1.´ 10-8

2.´ 10-8

3.´ 10-8

fHΒL

0.2 0.4 0.6 0.8 1.0Β

-5.´ 10-9

5.´ 10-9

1.´ 10-8

1.5´ 10-8

2.´ 10-8

2.5´ 10-8

3.´ 10-8

fHΒL

0.2 0.4 0.6 0.8 1.0Β

5.´ 10-9

1.´ 10-8

1.5´ 10-8

2.´ 10-8

fHΒL

Figure 1: Bifurcation diagram for β without diffusion for Case(1)-(4)

a25a32a53d1d4k4 + a22a35a53d1d4k

4 + a23a32a55d1d4k4 − a22a33a55d1d4k

4 +

a15a33a51d2d4k4 − a13a35a51d2d4k

4 − a15a31a53d2d4k4 + a11a35a53d2d4k

4 +

a13a31a55d2d4k4 − a11a33a55d2d4k

4 + a15a22a51d3d4k4 − a12a25a51d3d4k

4 −

a15a21a52d3d4k4 + a11a25a52d3d4k

4 + a12a21a55d3d4k4 − a11a22a55d3d4k

4 +

a24a33a42d1d5k4 − a23a34a42d1d5k

4 − a24a32a43d1d5k4 + a22a34a43d1d5k

4 +

a23a32a44d1d5k4 − a22a33a44d1d5k

4 + a14a33a41d2d5k4 − a13a34a41d2d5k

4 −

a14a31a43d2d5k4 + a11a34a43d2d5k

4 + a13a31a44d2d5k4 − a11a33a44d2d5k

4 +

a14a22a41d3d5k4 − a12a24a41d3d5k

4 − a14a21a42d3d5k4 + a11a24a42d3d5k

4 +

a12a21a44d3d5k4 − a11a22a44d3d5k

4 + a13a22a31d4d5k4 − a12a23a31d4d5k

4 −

a13a21a32d4d5k4 + a11a23a32d4d5k

4 + a12a21a33d4d5k4 − a11a22a33d4d5k

4 −

a45a54d1d2d3k6 + a44a55d1d2d3k

6 − a35a53d1d2d4k6 + a33a55d1d2d4k

6 −


6 − a15a51d2d3d4k6 + a11a55d2d3d4k

6 −


6 − a24a42d1d3d5k6 + a22a44d1d3d5k

6 −


6 − a23a32d1d4d5k6 + a22a33d1d4d5k

6 −

a13a31d2d4d5k6+a11a33d2d4d5k

6−a12a21d3d4d5k6+a11a22d3d4d5k

6−a55d1d2d3d4k8−

a44d1d2d3d5k8 − a33d1d2d4d5k

8 − a22d1d3d4d5k8 − a11d2d3d4d5k

8 + d1d2d3d4d5k10.

x

0.2 0.4 0.6 0.8 1.0Β

0.00001

0.00002

0.00003

0.00004

fHΒL

0.2 0.4 0.6 0.8 1.0Β

0.00002

0.00004

0.00006

0.00008

fHΒL

0.2 0.4 0.6 0.8 1.0Β

0.00001

0.00002

0.00003

0.00004

0.00005

0.00006

fHΒL

0.2 0.4 0.6 0.8 1.0Β

0.00001

0.00002

0.00003

0.00004

fHΒL

Figure 2: Bifurcation diagram for β with diffusion Case(1)-(4)

0.05 0.10 0.15 0.20Γ1

-1.´ 10-9

1.´ 10-9

2.´ 10-9

3.´ 10-9

fHΓ1L

0.05 0.10 0.15 0.20Γ1

-1.´ 10-9

1.´ 10-9

2.´ 10-9

3.´ 10-9

4.´ 10-9

5.´ 10-9

6.´ 10-9

fHΓ1L

0.1 0.2 0.3 0.4Γ1

5.´ 10-9

1.´ 10-8

1.5´ 10-8

2.´ 10-8

2.5´ 10-8

fHΓ1L

0.1 0.2 0.3 0.4Γ1

1.´ 10-8

2.´ 10-8

3.´ 10-8

fHΓ1L

Figure 3: Bifurcation diagram for γ1 without diffusion Case(1)-(4)

xi

0.05 0.10 0.15 0.20Γ2

-2.´ 10-7

2.´ 10-7

4.´ 10-7

6.´ 10-7

8.´ 10-7

fHΓ2L

0.05 0.10 0.15 0.20Γ2

-4.´ 10-7

-3.´ 10-7

-2.´ 10-7

-1.´ 10-7

1.´ 10-7

fHΓ2L

0.05 0.10 0.15 0.20Γ2

-4.´ 10-7

-2.´ 10-7

2.´ 10-7

4.´ 10-7

6.´ 10-7

8.´ 10-7

1.´ 10-6

fHΓ2L

0.05 0.10 0.15 0.20Γ2

-2.´ 10-7

2.´ 10-7

4.´ 10-7

6.´ 10-7

8.´ 10-7

fHΓ2L

Figure 4: Bifurcation diagram for γ1 with diffusion Case(1)-(4)

0.05 0.10 0.15 0.20Γ2

-4.´ 10-10

-2.´ 10-10

2.´ 10-10

fHΓ2L

0.05 0.10 0.15 0.20Γ2

-8.´ 10-10

-6.´ 10-10

-4.´ 10-10

-2.´ 10-10

2.´ 10-10

fHΓ2L

0.05 0.10 0.15 0.20Γ2

-6.´ 10-10

-4.´ 10-10

-2.´ 10-10

2.´ 10-10

4.´ 10-10

fHΓ2L

0.05 0.10 0.15 0.20Γ2

-4.´ 10-10

-2.´ 10-10

2.´ 10-10

fHΓ2L

Figure 5: Bifurcation diagram for γ2 without diffusion Case(1)-(4)

xii

0.05 0.10 0.15 0.20Γ2

-2.´ 10-7

2.´ 10-7

4.´ 10-7

6.´ 10-7

8.´ 10-7

fHΓ2L

0.05 0.10 0.15 0.20Γ2

-4.´ 10-7

-3.´ 10-7

-2.´ 10-7

-1.´ 10-7

1.´ 10-7

fHΓ2L

0.05 0.10 0.15 0.20Γ2

-4.´ 10-7

-2.´ 10-7

2.´ 10-7

4.´ 10-7

6.´ 10-7

8.´ 10-7

1.´ 10-6

fHΓ2L

0.05 0.10 0.15 0.20Γ2

-2.´ 10-7

2.´ 10-7

4.´ 10-7

6.´ 10-7

8.´ 10-7

fHΓ2L

Figure 6: Bifurcation diagram for γ2 with diffusion Case(1)-(4)

Appendix A.4

The coefficient of characteristic equation (4.24)

q1 = −d1k2 − d2k

2 − d3k2 − d4k

2 − α + qβ − γ1 − γ2 − 2δ − κ− 4µ;

q2 = −d1d2k4−d1d3k

4−d2d3k4−d1d4k

4−d2d4k4−d3d4k

4+desdsek4−d1k

2α−d2k2α−

d4k2α+d1k

2qβ+d3k2qβ+d4k

2qβ+desk2qβ+qαβ−d1k

2γ1−d2k2γ1−d4k

2γ1+qβγ1−

d1k2γ2 − d2k

2γ2 − d3k2γ2 −αγ2 + qβγ2 − γ1γ2 − 2d1k

2δ− 2d2k2δ− d3k

2δ− d4k2δ−

αδ+2qβδ− γ1δ− γ2δ− δ2 − d1k2κ− d3k

2κ− d4k2κ−ακ+ βκ− γ1κ− γ2κ− 2δκ−

3d1k2µ−3d2k

2µ−3d3k2µ−3d4k

2µ−3αµ+3qβµ−3γ1µ−3γ2µ−6δµ−3κµ−6µ2;

q3 = −d1d2d3k6 − d1d2d4k

6 − d1d3d4k6 − d2d3d4k

6 + d3desdsek6 + d4desdsek

6 −

d1d2k4α− d1d4k

4α− d2d4k4α + desdsek

4α+ d1d3k4qβ + d1d4k

4qβ + d3d4k4qβ +

d3desk4qβ + d4desk

4qβ + d1k2qαβ + d4k

2qαβ + desk2qαβ − d1d2k

4γ1 − d1d4k4γ1 −

d2d4k4γ1 + desdsek

4γ1 + d1k2qβγ1 + d4k

2qβγ1 + desk2qβγ1 − d1d2k

4γ2 − d1d3k4γ2 −

d2d3k4γ2 + desdsek

4γ2 − d1k2αγ2 − d2k

2αγ2 + d1k2qβγ2 + d3k

2qβγ2 + desk2qβγ2 +

qαβγ2 − d1k2γ1γ2 − d2k

2γ1γ2 + qβγ1γ2 − 2d1d2k4δ− d1d3k

4δ− d2d3k4δ− d1d4k

4δ−

d2d4k4δ+2desdsek

4δ−d1k2αδ−d2k

2αδ+2d1k2qβδ+d3k

2qβδ+d4k2qβδ+2desk

2qβδ+

qαβδ− d1k2γ1δ− d2k

2γ1δ+ qβγ1δ− d1k2γ2δ− d2k

2γ2δ+ qβγ2δ− d1k2δ2− d2k

2δ2+

qβδ2−d1d3k4κ−d1d4k

4κ−d3d4k4κ−d1k

2ακ−d4k2ακ+d1k

2βκ+d4k2βκ+desk

2βκ+

xiii

lαβκ− d1k2γ1κ− d4k

2γ1κ− d1k2γ2κ− d3k

2γ2κ−αγ2κ+ βγ2κ− γ1γ2κ− 2d1k2δκ−

d3k2δκ−d4k

2δκ−αδκ+βδκ−γ1δκ−γ2δκ−δ2κ−2d1d2k4µ−2d1d3k

4µ−2d2d3k4µ−

2d1d4k4µ− 2d2d4k

4µ− 2d3d4k4µ+ 2desdsek

4µ− 2d1k2αµ− 2d2k

2αµ− 2d4k2αµ+

2d1k2qβµ+ 2d3k

2qβµ+ 2d4k2qβµ+ 2desk

2qβµ+ 2qαβµ− 2d1k2γ1µ− 2d2k

2γ1µ−

2d4k2γ1µ+2qβγ1µ−2d1k

2γ2µ−2d2k2γ2µ−2d3k

2γ2µ−2αγ2µ+2qβγ2µ−2γ1γ2µ−

4d1k2δµ− 4d2k

2δµ− 2d3k2δµ− 2d4k

2δµ− 2αδµ+4qβδµ− 2γ1δµ− 2γ2δµ− 2δ2µ−

2d1k2κµ− 2d3k

2κµ− 2d4k2κµ− 2ακµ+2βκµ− 2γ1κµ− 2γ2κµ− 4δκµ− 3d1k

2µ2−

3d2k2µ2 − 3d3k

2µ2 − 3d4k2µ2 − 3αµ2 +3qβµ2 − 3γ1µ

2 − 3γ2µ2 − 6δµ2 − 3κµ2 − 4µ3;

q4 = d4desdsek6α + d4desk

4qαβ + d4desdsek6γ1 + d4desk

4qβγ1 + desdsek4αγ2 +

desk2qαβγ2+ desdsek

4γ1γ2+ desk2qβγ1γ2+ d4desdsek

6δ+ desdsek4αδ+ d4desk

4qβδ+

desk2qαβδ + desdsek

4γ1δ + desk2qβγ1δ + desdsek

4γ2δ + desk2qβγ2δ + desdsek

4δ2 +

desk2qβδ2 + d4desk

4βκ+ desk2lαβκ+ desk

2βγ2κ+ desk2βδκ+ d4desdsek

6µ−

d2d4k4αµ+ desdsek

4αµ+ d4desk4qβµ+ d4k

2qαβµ+ desk2qαβµ− d2d4k

4γ1µ+

desdsek4γ1µ+ d4k

2qβγ1µ+ desk2qβγ1µ+ desdsek

4γ2µ− d2k2αγ2µ+ desk

2qβγ2µ+

qαβγ2µ− d2k2γ1γ2µ+ qβγ1γ2µ− d2d4k

4δµ+ 2desdsek4δµ− d2k

2αδµ+ d4k2qβδµ+

2desk2qβδµ+qαβδµ−d2k

2γ1δµ+qβγ1δµ−d2k2γ2δµ+qβγ2δµ−d2k

2δ2µ+qβδ2µ−

d4k2ακµ+ d4k

2βκµ+ desk2βκµ+ lαβκµ− d4k

2γ1κµ−αγ2κµ +βγ2κµ− γ1γ2κ µ−

d4k2δκµ−αδκ µ+βδκµ−γ1δκ µ−γ2δκµ−δ2κµ −d2d4k

4µ2+desdsek4µ2−d2k

2αµ2−

d4k2αµ2+d4k

2qβµ2+desk2qβµ2+qαβµ2−d2k

2γ1µ2−d4k

2γ1µ2+qβγ1µ

2−d2k2γ2µ

2−

αγ2µ2+ qβγ2µ

2− γ1γ2µ2− 2d2k

2δµ2− d4k2δµ2−αδµ2+2qβδµ2− γ1δµ

2− γ2δµ2−

δ2µ2−d4k2κµ2−ακµ2+β κµ2−γ1κµ

2−γ2κ µ2−2δκµ2−d2k2µ3−d4k

2µ3−αµ3+

qβµ3−γ1µ3−γ2µ

3−2δµ3−κµ3−µ4−d1k2(−d4k

2qαβ−d4k2qβγ1−qαβγ2−qβγ1γ2−

d4k2qβδ−qαβδ−qβγ1δ−qβγ2δ−qβδ2+d4k

2ακ−d4k2βκ−lαβκ+d4k

2γ1κ+αγ2κ−

β γ2κ+γ1γ2κ+d4k2δκ+αδκ−β δκ+γ1δκ+γ2δ κ+δ2κ+d4k

2αµ−d4k2qβµ−qαβµ+

d4k2γ1µ− qβγ1µ+αγ2µ− qβγ2µ+ γ1γ2µ+ d4k

2δµ+αδµ− 2qβδµ+ γ1δµ+ γ2 δµ+

δ2µ+ d4k2κµ+ακµ−β κµ+ γ1κµ+ γ2κµ +2δκµ+ d4k

2µ2+αµ2− qβµ2+ γ1µ2+

γ2µ2+2δµ2+κµ2+µ3−d3k

2(qβ−κ−µ)(d4k2+γ2+ δ+µ)+d2k

2(d3k2+α+γ1+δ+

µ)(d4k2 +γ2+δ+µ))+d3k

2(d4k2+γ2+δ+µ)(des(dsek

4+k2qβ)−µ(d2k2−qβ+κ+µ)).

Bifurcation diagrams for β, γ1 and γ2 with

cross-diffusion

xiv

(a)

0.5 1.0 1.5 2.0Β

0.00002

0.00004

0.00006

fHΒL

(b)

0.5 1.0 1.5 2.0Β

0.00002

0.00004

0.00006

fHΒL

(c)

0.5 1.0 1.5 2.0Β

5.´ 10-6

0.00001

0.000015

0.00002

fHΒL

(d)

0.5 1.0 1.5 2.0Β

5.´ 10-6

0.00001

0.000015

0.00002

fHΒL

Figure 7: Bifurcation diagram for β for Case(a)-(d)

The bifurcation diagrams for transmission rate β and the recovery rates γ1 and γ2

with cross-diffusion are given in Figs. 7, 8 and 9.

xv

(a)

0.02 0.04 0.06 0.08 0.10 0.12 0.14Γ1

5.´ 10-8

1.´ 10-7

1.5´ 10-7

fHΓ1L

(b)

0.02 0.04 0.06 0.08 0.10 0.12 0.14Γ1

5.´ 10-8

1.´ 10-7

1.5´ 10-7

fHΓ1L

(c)

0.05 0.10 0.15 0.20Γ1

-1.´ 10-7

-5.´ 10-8

5.´ 10-8

1.´ 10-7

fHΓ1L

(d)

0.05 0.10 0.15 0.20Γ1

-1.´ 10-7

-5.´ 10-8

5.´ 10-8

1.´ 10-7

1.5´ 10-7

fHΓ1L

Figure 8: Bifurcation diagram for γ1 for Case(a)-(d)

(a)

0.05 0.10 0.15 0.20Γ2

-2.´ 10-8

2.´ 10-8

4.´ 10-8

6.´ 10-8

8.´ 10-8

1.´ 10-7

fHΓ2L

(b)

0.05 0.10 0.15 0.20Γ2

-2.´ 10-8

2.´ 10-8

4.´ 10-8

6.´ 10-8

8.´ 10-8

1.´ 10-7

fHΓ2L

(c)

0.05 0.10 0.15 0.20 0.25 0.30Γ2

-1.´ 10-7

-5.´ 10-8

5.´ 10-8

1.´ 10-7

fHΓ2L

(d)

0.05 0.10 0.15 0.20 0.25 0.30Γ2

-1.´ 10-7

-5.´ 10-8

5.´ 10-8

1.´ 10-7

1.5´ 10-7

2.´ 10-7

fHΓ2L

Figure 9: Bifurcation diagram for γ2 for Case(a)-(d)

xvi

Appendix A.5

The expression of basic reproduction number RIT

for SEIJTR system

The derivation of the basic reproduction number is based on the linearization of

the ODE model about disease free equilibrium. Following Diekmann and

Heesterbeek [52] the reproduction number is calculated using the next generation

matrix of the system at the disease free equilibrium. The disease free equilibrium

for this model is (S = Λ/µ, E = 0, I = 0, J = 0, T = 0, R = 0). The infected

classes are E, J , I and T . Then the matrices F and V chosen for the system 5.1 -

5.6 are as follows:

F =

β (I+qE+lJ)

NS

0

0

0

and V =

−(1− π)Λ + (µ+ κ)E

−κE + (µ+ α + δ)I

−αI + (µ+ γ1 + δ + ζ)J

−ζJ + (µ+ γ2 + δ(1− θ))T

F =

qβ β lβ 0

0 0 0 0

0 0 0 0

0 0 0 0

and

V =

µ+ κ 0 0 0

−κ µ+ α + δ 0 0

0 −α µ+ γ1 + δ + ζ 0

0 0 −ζ µ+ γ2 + δ(1− θ)

Thus, the basic reproduction number is given by the dominant eigenvalue of

F.V −1

xvii

ρ(F.V −1) = RIT = Λβ(q(µ+δ+α)(µ+δ+γ1+ζ)+κ(µ+δ+γ1+ζ+lα))µ(µ+κ)(µ+δ+α)(µ+δ+γ1+ζ)

.

Expressions of Sensitivity indices of RIT based on

perturbation of fixed points estimations

As the reproduction number RIT is a function of ten parameters Λ, β, µ, l, κ, q,

α, γ1, ζ and δ, where

RIT = Λβ(q(µ+δ+α)(µ+δ+γ1+ζ)+κ(µ+δ+γ1+ζ+lα))µ(µ+κ)(µ+δ+α)(µ+δ+γ1+ζ)

.

So using Eq. (5.11)

ΥRITx = ∂RIT

∂x× x

RIT

ΥRITΛ = ∂RIT

∂Λ× Λ

RIT=

β(q(α+δ+µ)(γ1+δ+ζ+µ)+κ(lα+γ1+δ+ζ+µ))µ(α+δ+µ)(γ1+δ+ζ+µ)(κ+µ)

× µ(α+δ+µ)(γ1+δ+ζ+µ)(κ+µ)β(q(α+δ+µ)(γ1+δ+ζ+µ)+κ(lα+γ1+δ+ζ+µ))

;

ΥRITβ = ∂RIT

∂β× β

RIT=

Λq(α+δ+µ)(γ1+δ+ζ+µ)+κ(lα+γ1+δ+ζ+µ))µ(α+δ+µ)(γ1+δ+ζ+µ)(κ+µ)

× µ(α+δ+µ)(γ1+δ+ζ+µ)(κ+µ)Λ(q(α+δ+µ)(γ1+δ+ζ+µ)+κ(lα+γ1+δ+ζ+µ))

;

ΥRITµ = ∂RIT

∂µ× µ

RIT=

−1 + βΛ(κ+q(α+δ+µ)+q(γ1+δ+ζ+µ))µ(α+δ+µ)(γ1+δ+ζ+µ)(κ+µ)

− βΛ(q(α+δ+µ)(γ1+δ+ζ+µ)+κ(lα+γ1+δ+ζ+µ))µ(α+δ+µ)(γ1+δ+ζ+µ)(κ+µ)2

−βΛ(q(α+δ+µ)(γ1+δ+ζ+µ)+κ(lα+γ1+δ+ζ+µ))

µ(α+δ+µ)(γ1+δ+ζ+µ)2(κ+µ)− βΛ(q(α+δ+µ)(γ1+δ+ζ+µ)+κ(lα+γ1+δ+ζ+µ))

µ(α+δ+µ)2(γ1+δ+ζ+µ)(κ+µ);

ΥRITl = ∂RIT

∂l× l

RIT= lακ

q(α+δ+µ)(γ1+δ+ζ+ µ)+κ(lα+γ1+δ+ζ+µ);

ΥRITκ = ∂RIT

∂κ× κ

RIT= − κ(q(α+δ+µ)(γ1+δ+ζ+µ)−µ(lα+γ1+δ+ζ +µ]))

(κ+µ)(q(α+δ+µ)(γ1+ δ+ζ+µ)+κ(lα+γ1+δ+ζ+µ));

ΥRITq = ∂RIT

∂q× q

RIT= q(α+δ+µ)(γ1+δ+ζ+ µ)

q(α+δ+µ)(γ1+δ+ζ+ µ)+κ(lα+γ1+δ+ζ+ µ);

ΥRITα = ∂RIT

∂α× α

RIT= − ακ(γ1+δ−lδ+ζ+µ−lµ)

(α+δ+µ)(q(α+δ+µ)(γ1+δ+ζ+µ)+κ(lα+γ1+δ+ζ+µ));

ΥRITγ1

= ∂RIT

∂γ1× γ1

RIT= − lαγ1κ

(γ1+δ+ζ+µ)(q(α+δ+µ)(γ1+δ+ζ+µ)+κ(lα+γ1+δ+ζ+µ));

xviii

ΥRITζ = ∂RIT

∂ζ× ζ

RIT= − lαζκ

(γ1+δ+ζ+µ)(q(α+δ+µ)(γ1+δ+ζ+µ)+κ(lα+γ1+δ+ζ+µ));

ΥRITδ = ∂RIT

∂δ× δ

RIT= − δκ((γ1+δ+ζ+µ)2+lα(α+γ1+2δ+ζ+2µ))

(α+δ+µ)(γ1+δ+ζ+ µ)(q(α+δ+µ)(γ1+δ+ ζ+µ)+κ(lα+γ1+δ+ ζ+µ));

xix

Appendix A.6

Non-dimensionalization of SEIJTR system

(6.1)-(6.6)

In order to analyze in terms of proportions of susceptible, exposed, infected,

diagnosed, treated and recovered individuals it is assumed that

s =S

N(1)

e =E

N(2)

i =I

N(3)

j =J

N(4)

t1 =T

N(5)

r =R

N(6)

Differentiate Equations (1)-(6) with respect to time, t gives

Nds

dt=

dS

dt− s

dN

dt(7)

Nde

dt=

dE

dt− e

dN

dt(8)

Ndi

dt=

dI

dt− i

dN

dt(9)

Ndj

dt=

dJ

dt− j

dN

dt(10)

Ndt1dt

=dT

dt− t1

dN

dt(11)

Ndr

dt=

dR

dt− r

dN

dt(12)

where

N = S+E+I+J+T+R,dN

dt= Λ−µN−δ(I+J+(1−θ)T ) (13)

Now from Equation (7)

Nds

dt=

dS

dt− s

dN

dt

xx

= −(I + lJ + qE)Sβ

N+ πΛ− Sµ− s(Λ− µN − δ(I + J + (1− θ)T ))

= −(iN + ljN + qeN)sNβ

Nµ− s(Λ− µN − δ(iN + jN + (1− θ)t1N)) + πΛ

= N [−β(i+ lj + qe)s+ δ(i+ j + (1− θ)t1)s−πΛ

N− sΛ

N]

Therefore

ds

dt= −β(i+ lj + qe)s+ δ(i+ j + (1− θ)t1)s+

πΛ

N− sΛ

N(14)


Nde

dt=

dE

dt− e

dN

dt

= β(I + lJ + qE)

NS + (1− π)Λ− E(κ+ µ)− e(Λ− µN − δ(I + J + (1− θ)T )

= β(iN + ljN + qeN)

NsN+(1−π)Λ−eN(κ+µ)−e(Λ−µN−δ(iN+jN+(1−θ)t1N)

= N [β(i+ lj + eq)s+ δe(i+ j + (1− θ)t1)− eκ− eΛ

N+ (1− π)

Λ

N]

Therefore

de

dt= β(i+ lj + eq)s+ δe(i+ j + (1− θ)t1)− eκ− eΛ

N+ (1− π)

Λ

N(15)


Ndi

dt=

dI

dt− i

dN

dt

= κE − I(α + δ + µ)− i(Λ− µN − δ(I + J + (1− θ)T )

xxi

= κeN − iN(α + δ + µ)− i(Λ− µN − δ(iN + jN + (1− θ)t1N)

= N [−iα−iδ+i2δ+ijδ+it1δ−it1δθ+eκ−(iΛ

N−iµ+eiµ+i2µ+ijµ+irµ+isµ+it1µ]

Thereforedi

dt= κe− (α+ δ +

Λ

N)i+ δ(i+ j + (1− θ)t1)i (16)


Ndj

dt=

dJ

dt− j

dN

dt

= αI − (γ1 + δ + ζ + µ)J − j(Λ− µN − δ(I + J + (1− θ)T )

= αiN − (γ1 + δ + ζ + µ)jN − j(Λ− µN − δ(iN + jN + (1− θ)t1N)

= N [αi− (Λ

N+ γ1 + ζ)j + δ(i+ j + (1− θ)t1)j]

Thereforedj

dt= αi− (

Λ

N+ γ1 + ζ)j + δ(i+ j + (1− θ)t1)j (17)


Ndt1dt

=dT

dt− t1

dN

dt

= ζJ − (γ2 + δ(1− θ) + µ)T − t1(Λ− µN − δ(I + J + (1− θ)T )

= ζjN − (γ2 + δ(1− θ) + µ)t1N − t1(Λ− µN − δ(iN + jN + (1− θ)t1N)

= N [ζj − (Λ

N+ γ2)t1 + δ(i+ j + (θ − 1) + (1− θ)t1)t1]

xxii

Therefore

dt

dt= ζj − (

Λ

N+ γ2)t1 + δ(i+ j + (θ − 1) + (1− θ)t1)t1 (18)


Ndr

dt=

dR

dt− r

dN

dt

= γ1J + γ2T − µR− r(Λ− µN − δ(I + J + (1− θ)T )

= γ1jN + γ2t1N − µrN − r(Λ− µN − δ(iN + jN + (1− θ)tN)

= N [γ1j + γ2t1 −Λ

Nr + δ(i+ j + (1− θ)t1)r]

Thereforedr

dt= γ1j + γ2t1 −

Λ

Nr + δ(i+ j + (1− θ)t1)r (19)

After replacing s by S, e by E, i by I, j by J , t1 by T , r by R, and ΛN

by Π

Equations (14)-(19) can be written as

dS

dt= −β(I + qE + lJ)S + (π − S)Π + γ1IS + δ(I + J + (1− θ)T )S (20)

dE

dt= (1− π)Π + β(I + qE + lJ)S − (Π + κ)E + δ(I + J + (1− θ)T )E (21)

dI

dt= κE − (α + δ +Π)I + δ(I + J + (1− θ)T )I (22)

dJ

dt= αI − (Π + γ1 + ζ)J + δ(I + J + (1− θ)T )J (23)

dT

dt= ζJ − (Π + γ2)T + δ(I + J + (θ − 1) + (1− θ)T )T (24)

dR

dt= γ1J + γ2T − ΠR + δ(I + J + (1− θ)T )R (25)

and

S + E + I + J + T +R = 1 (26)

xxiii

The Routh-Hurwitz stability condition

Consider an nth order characteristic equation of the form:

λn + p1λn−1 + p2λ

n−2 + p3λn−3 + · · · · · ·+ pn = 0 (27)

Then the roots of the characteristic Equation (27) lie in the left half of the

complex plane iff the first column of the following table are nonzero and have the

same sign [167]. The Routh Table is:

Routh Table

r1,1 r1,2 r1,3 r1,4 r1,5 · · ·

r2,1 r2,2 r2,3 r2,4 r2,5 · · ·

r3,1 r3,2 r3,3 r3,4 · · ·

r4,1 r4,2 r4,3 · · ·...

......

rn+1,1

where

r1,1 = 1; r1,2 = p2; r1,3 = p4; · · · · · · · · ·

r2,1 = p1; r2,2 = p3; r2,3 = p5; · · · · · · · · ·

and other rows are

ri,1 = ri−2,2 − ri−2,1

ri−1,1ri−1,2; ri,2 = ri−2,3 − ri−2,1

ri−1,1ri−1,3; ri,3 = ri−2,4 − ri−2,1

ri−1,1ri−1,4;

· · · · · · · · · (i > 2)

Stability conditions for sixth order polynomial

equation

The sixth order characteristic equation is of the form:

λ6 + p1λ5 + p2λ

4 + p3λ3 + p4λ

2 + p5λ+ p6 = 0 (28)

Then the Routh Table is: where

xxiv

Routh Table

r1,1 r1,2 r1,3 r1,4 r1,5 r1,6

r2,1 r2,2 r2,3 r2,4 r2,5 r2,6

r3,1 r3,2 r3,3 r3,4 r3,5

r4,1 r4,2 r4,3 r4,4

r5,1 r5,2 r5,3

r6,1 r6,2

r7,1

r1,1 = 1; r1,2 = p2; r1,3 = p4; r1,4 = p6;r1,5 = 0; r1,6 = 0; r2,1 = p1; r2,2 = p3;

r2,3 = p5; r2,4 = 0; r2,5 = 0;r2,6 = 0;

r3,1 = r1,2 − r1,1r2,1

r2,2 =p1p2−p3

p1;

r3,2 = r1,3 − r1,1r2,1

r2,3 =p1p4−p5

p1;

r3,3 = r1,4 − r1,1r2,1

r2,4 = p6;

r3,4 = r1,5 − r1,1r2,1

r2,5 = 0;

r4,1 = r2,2 − r2,1r3,1

r3,2 =p23+p21p4−p1(p2p3+p5)

p3−p1p2;

r4,2 = r2,3 − r2,1r3,1

r3,3 =P1p2p5−p3p5−p21p6

p1p2−p3;

r4,3 = r2,4 − r2,1r3,1

r3,4 = 0;

r4,4 = r2,5 − r2,1r3,1

r3,5 = 0;

r5,1 = r3,2 −r3,1r4,1

r4,2

=p23p4 − p2p3p5 + p25 + p21(p

24 − p2p6) + p1(−p2p3p4 + p22p5 − 2p4p5 + p3p6)

p23 + p21p4 − p1(p2p3 + p5);

r5,2 = r3,3 −r3,1r4,1

r4,3 = p6;

r5,3 = r3,4 −r3,1r4,1

r4,4 = 0;

r6,1 = r4,2 −r4,1r5,1

r5,2 =

p23p4p5 − p2p3p25 + p35 − p33p6 + p31p

26 + p21(p

24p5 − p3p4p6 − 2p2p5p6) + p1(p

22p

25 + p2p3(−p4p5 + p3p6) + p5(−2p4p5 + 3p3p6))

(p23p4 − p2p3p5 + p25 + p21(p24 − p2p6) + p1(−p2p3p4 + p22p5 − 2p4p5 + p3p6))

;

xxv

0.2 0.4 0.6 0.8 1.0 1.2 1.4Β

-2.´ 10-8

2.´ 10-8

4.´ 10-8

fHΒL

0.1 0.2 0.3 0.4 0.5Α

1.´ 10-8

2.´ 10-8

3.´ 10-8

4.´ 10-8

5.´ 10-8

fHΑL

Figure 10: Bifurcation diagrams for without diffusion

r6,2 = r4,3 −r4,1r5,1

r5,3 = 0;

r7,1 = r5,2 −r5,1r6,1

r6,2 = p6;

Therefore the Routh-Hurwitz criterion [167] for stability gives

C1 : p1 > 0

C2 : p6 > 0

C3 :p1p2−p3

p1> 0

C4 :p1p2p3−p23−p21p4−p1p5

p1p2−p3> 0

C5 :=p23p4−p2p3p5+p25+p21(p

24−p2p6)+p1(p22p5−p2p3p4−2p4p5+p3p6)

p23+p21p4−p1(p2p3+p5)> 0

C6 :=

p23p4p5−p2p3p25+p35−p33p6+p31p26+p21(p

24p5−p3p4p6−2p2p5p6)+p1(p22p

25+p2p3(p3p6−p4p5)+p5(3p3p6−2p4p5))

p23p4−p2p3p5+p25+p21(p24−p2p6)+p1(p22p5−p2p3p4−2p4p5+p3p6)

>

0

Bifurcation diagrams for β and α with and

without diffusion

The bifurcation diagrams for transmission rate, β and rate of progression from

infective to diagnosed, α with and without diffusion are given in in Figs. (10)-

(11) for Case (1). The remaining cases shows similar results.

xxvi

0.2 0.4 0.6 0.8 1.0Β

-0.0001

-0.00005

0.00005

fHΒL

0.2 0.4 0.6 0.8 1.0Α

0.00005

0.0001

fHΑL

Figure 11: Bifurcation diagrams with diffusion

Appendix A.7

The expression of basic reproduction number

RQIT for SEIQJTR system

The disease free equilibrium for this model is (S = 1, E = 0, Q = 0 I = 0, J = 0,

T = 0, R = 0). The infected classes are E, Q J , I and T . Then the matrices F

and V chosen for the system 7.8 - 7.14 are as follows:

F =

β((I + qE + pQ+ lJ)S)

0

0

0

0

and V =

−(1− π)Π + (Π + κ1 + κ2)E − δ1EI − δ2(J + (1− θ)T )E

−κ1E + (Π + σ)Q− δ1IQ− δ2(J + (1− θ)T )Q

−κ2E + (Π + α + δ1)I − δ1I2 − δ2(J + (1− θ)T )I

−σQ− αI + (δ2 +Π+ γ1 + ζ)J − δ1IJ − δ2(J + (1− θ)T )J

−ζJ + (Π + γ2 + δ2)T − δ1IT − δ2(J + (1− θ)T + θ)T

xxvii

F =

qβ pβ β lβ 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

and

V =

κ1 + κ2 +Π 0 0 0 0

−κ1 Π+ σ 0 0 0

−κ2 0 α + δ1 +Π 0 0

0 −σ −α γ1 + δ2 + ζ +Π 0

0 0 0 −ζ γ2 + δ2 − δ2θ +Π

Thus, the basic reproduction number is given by the dominant eigenvalue of

F.V −1

ρ(F.V −1) = RQIT = β(a+b+c+d)(κ1+κ2+Π)(α+δ1+Π)(γ1+δ2+ζ+Π)(σ+Π)

a = q(α + δ1 +Π)(γ1 + δ2 + ζ +Π)(σ +Π).

b = κ2(γ1 + δ2 + ζ +Π)(σ +Π).

c = pκ1(α + δ1 +Π)(γ1 + δ2 + ζ +Π).

d = l(κ1(δ1 +Π)σ + α(κ1σ + κ2(σ +Π))).

The coefficient of characteristic equation (7.46)

p1 = α− qβ + γ1 + δ1 + δ2 + ζ + κ1 + κ2 + 4Π + σ;

p2 = γ1δ1+δ1δ2+δ1ζ−pβκ1+γ1κ1+δ1κ1+ δ2κ1+ζκ1−βκ2+ γ1κ2+δ1κ2+δ2κ2+

ζκ2+3γ1Π+3δ1Π+3δ2Π+3ζΠ+3κ1Π+3κ2Π+6Π2+γ1σ+δ1σ+ δ2σ+ζσ+κ1σ+

κ2σ+3Πσ− qβ(α+ γ1 + δ1 + δ2+ ζ +3Π+σ)+α(γ1 + δ2+ ζ +κ1 +κ2 +3Π+σ);

p3 = αγ1κ1 + γ1δ1κ1 + αδ2κ1 + δ1δ2κ1 + αζκ1 + δ1ζκ1 − lαβκ2 +αγ1κ2 − βγ1κ2 +

γ1δ1κ2 + αδ2κ2 − βδ2κ2 + δ1δ2κ2 + αζκ2 − β ζκ2 + δ1ζκ2 + 2αγ1Π+ 2γ1δ1Π+

2αδ2Π+2δ1δ2Π+2αζΠ+2δ1ζΠ+2ακ1Π+2γ1κ1Π+2δ1κ1Π+2δ2κ1Π+2ζκ1Π+

xxviii

2ακ2Π− 2βκ2Π+ 2γ1κ2Π+ 2δ1κ2Π+ 2δ2κ2Π+ 2ζκ2Π+ 3αΠ2 + 3γ1Π2 + 3δ1Π

2 +

3δ2Π2 + 3ζΠ2 + 3κ1Π

2 + 3κ2Π2 + 4Π3 − pβκ1(α+ γ1 + δ1 + δ2 + ζ + 2Π)+ αγ1σ+

γ1 δ1σ + αδ2σ + δ1 δ2σ + αζσ + δ1ζ σ + ακ1σ − lβκ1σ + γ1κ1σ + δ1κ1σ + δ2κ1σ +

ζ κ1σ + ακ2σ − βκ2 σ + γ1κ2σ + δ1κ2 σ + δ2κ2σ + ζκ2σ + 2αΠσ + 2γ1Πσ +

2δ1Πσ+2δ2Πσ+2ζΠσ+2κ1Πσ+2κ2Πσ+3Π2σ− qβ(δ1δ2 + δ1ζ +2δ1Π+ 2δ2Π+

2ζΠ+ 3Π2 + δ1σ + δ2σ + ζσ + 2Πσ + γ1(δ1 + 2Π+ σ) + α(γ1 + δ2 + ζ + 2Π+ σ))

p4 = αγ1κ1Π+γ1δ1κ1Π+αδ2κ1Π+δ1δ2κ1Π+αζκ1Π+δ1ζκ1Π−lαβκ2Π+αγ1κ2Π−

βγ1κ2Π+γ1δ1κ2Π+αδ2κ2Π−βδ2κ2Π+δ1δ2κ2Π+αζκ2Π−βζκ2Π+δ1ζκ2Π+αγ1Π2+

γ1δ1Π2 + αδ2Π

2 + δ1δ2Π2 + αζΠ2 + δ1ζΠ

2 + ακ1Π2 + γ1κ1Π

2 + δ1κ1Π2 + δ2κ1Π

2 +

ζκ1Π2+ακ2Π

2−βκ2Π2+γ1κ2Π

2+δ1κ2Π2+δ2κ2Π

2+ζ κ2Π2+αΠ3+γ1Π

3+δ1Π3+

δ2Π3 +ζΠ3+κ1Π

3+κ2Π3+Π4−pβκ1(α+δ1+Π)(γ1+δ2+ζ+Π)−lαβκ1σ+αγ1κ1σ−

lβδ1κ1σ+γ1δ1κ1σ+αδ2κ1σ+ δ1δ2κ1σ+αζκ1σ+δ1ζκ1σ−lαβκ2σ+αγ1κ2σ−βγ1κ2σ+

γ1δ1κ2σ+αδ2κ2σ−β δ2κ2σ+δ1δ2κ2σ+αζκ2σ−βζκ2σ+δ1ζκ2σ+αγ1Πσ+γ1δ1Πσ+

αδ2Πσ+δ1δ2Πσ+αζΠσ+δ1ζΠσ+ακ1Πσ− lβκ1Πσ+γ1κ1Πσ+δ1κ1Πσ+δ2κ1Πσ+

ζκ1Πσ+ακ2Πσ−βκ2Πσ+γ1κ2Πσ+ δ1κ2Πσ+ δ2κ2Πσ+ ζκ2Πσ+αΠ2σ+γ1Π2σ+

δ1Π2σ+δ2Π

2σ+ζΠ2σ+κ1Π2σ+κ2Π

2σ+Π3σ−qβ(α+δ1+Π)(γ1+δ2+ζ+Π)(Π+σ)

Stability conditions for seventh order polynomial

equation

The sixth order characteristic equation is of the form:

λ7 + p1λ6 + p2λ

5 + p3λ4 + p4λ

3 + p5λ2 + p6λ+ p7 = 0 (29)

Then the Routh Table is: where

r1,1 = 1; r1,2 = p2; r1,3 = p4; r1,4 = p6;r1,5 = 0; r1,6 = 0;r1,7 = 0;r2,1 = p1; r2,2 = p3;

r2,3 = p5; r2,4 = p7; r2,5 = 0;r2,6 = 0;r2,7 = 0;

r3,1 = r1,2 − r1,1r2,1

r2,2 =p1p2−p3

p1;

r3,2 = r1,3 − r1,1r2,1

r2,3 =p1p4−p5

p1;

r3,3 = r1,4 − r1,1r2,1

r2,4 =p1p6−p7

p1;

r3,4 = r1,5 − r1,1r2,1

r2,5 = 0;

r3,5 = r1,6 − r1,1r2,1

r2,6 = 0;

r3,6 = r1,7 − r1,1r2,1

r2,7 = 0;

xxix

Routh Table

r1,1 r1,2 r1,3 r1,4 r1,5 r1,6 r1,7

r2,1 r2,2 r2,3 r2,4 r2,5 r2,6 r2,7

r3,1 r3,2 r3,3 r3,4 r3,5 r3,6

r4,1 r4,2 r4,3 r4,4 r4,5

r5,1 r5,2 r5,3 r5,4

r6,1 r6,2 r6,3

r7,1 r7,2

r8,1

r4,1 = r2,2 − r2,1r3,1

r3,2 =p23+p21p4−p1(p2p3+p5)

p3−p1p2;

r4,2 = r2,3 − r2,1r3,1

r3,3 =(p1p2−p3)p5+p1(p7−p1p6)

p1p2−p3;

r4,3 = r2,4 − r2,1r3,1

r3,4 = p7;

r4,4 = r2,5 − r2,1r3,1

r3,5 = 0;

r4,5 = r2,6 − r2,1r3,1

r3,6 = 0;

r5,1 = r3,2 −r3,1r4,1

r4,2

=p23p4 + p25 + p21(p

24 − p2p6)− p3(p2p5 + p7) + p1(p

22p5 − 2p4p5 + p3p6 + p2(−p3p4 + p7))

p23 + p21p4 − p1(p2p3 + p5);

r5,2 = r3,3 −r3,1r4,1

r4,3 = p6 −p7p1

+(−p1p2 + p3)

2p7p1(p23 + p21p4 − p1(p2p3 + p5))

;

r5,3 = r3,4 −r3,1r4,1

r4,4 = 0;

r5,4 = r3,5 −r3,1r4,1

r4,5 = 0;

r6,1 = r4,2 −r4,1r5,1

r5,2

=A1 + A2

(p23p4 + p25 + p21(p24 − p2p6)− p3(p2p5 + p7) + p1(p22p5 − 2p4p5 + p3p6 + p2(p7 − p3p4+)))

;

xxx

r6,2 = r4,3 −r4,1r5,1

r5,3 = p7;

r6,3 = r4,4 −r4,1r5,1

r5,4 = 0;

r7,1 = r5,2 −r5,1r6,1

r6,2

=A1 + A2 + A3

B1 +B2

;

r7,2 = r5,3 −r5,1r6,1

r6,3 = 0;

r8,1 = r6,2 −r6,1r7,1

r7,2 = p7.

Therefore the Routh-Hurwitz criterion [167] for stability gives

C1 : p1 > 0,

C2 : p7 > 0,

C3 : p2 − p3p1

> 0,

C4 :p23+p21p4−p1(p2p3+p5)

p3−p1p2> 0,

C5 :p23p4+p25+p21(p

24−p2p6)−p3(p2p5+p7)+p1(p22p5−2p4p5+p3p6+p2(p7−p3p4))

p23+p21p4−p1(p2p3+p5)> 0,

C6 :A1+A2

(p23p4+p25+p21(p24−p2p6)−p3(p2p5+p7)+p1(p22p5−2p4p5+p3p6+p2(p7−p3p4+)))

> 0,

C7 :A1+A2+A3

B1+B2> 0.

where

A1 = p35−p33p6+p31p26−p3p5(p2p5+2p7)+p23(p4p5+p2p7)+p21(p

24p5−2p6(p2p5+p7),

A2 =

p4(p2p7−p3p6))+p1(3p3p5p6−2p4p25+p27+p22(p

25−p3p7)+p2(p

23p6−p3p4p5+p5p7)),

A3 = p35p6 − p33p26 + p31p

36 − p4p

25p7 + p2p5p

27 − p37 + p23(p4p5p6 − p24p7 + 2p2p6p7)−

p21(p34p7 − p24p5p6 + p26(2p2p5 + 3p7) + p4p6(p3p6 − 3p2p7)),

A4 = −p3(p22p

27 + p7(3p5p6 − 2p4p7) + p2p5(p5p6 − p4p7)) + p1(2p

24p5p7 + p32p

27 −

p4p6(2p25 + p3p7) + p22(p

25p6 − p4p5p7 − 2p3p6p7),

A5 = 3p6(p3p5p6 + p27) + p2(p23p

26 + p7(p5p6 − 3p4p7) + p3p4(p4p7 − p5p6)),

B1 = p35 − p33p6 + p31p26 − p3p5(p2p5 + 2p7) + p23(p4p5 + p2p7) + p21(p

24p5 − 2p6(p2p5 +

p7) + p4(p2p7 − p3p6)) ,

xxxi

0.2 0.4 0.6 0.8 1.0Β

-1.´ 10-8

-5.´ 10-9

5.´ 10-9

1.´ 10-8

1.5´ 10-8

fHΒL

0.02 0.04 0.06 0.08 0.10Α

-1.´ 10-9

-5.´ 10-10

5.´ 10-10

1.´ 10-9

fHΑL

0.01 0.02 0.03 0.04 0.05Ζ

-4.´ 10-10

-2.´ 10-10

2.´ 10-10

4.´ 10-10

6.´ 10-10

8.´ 10-10

fHΖL

0.01 0.02 0.03 0.04 0.05Σ

-5.´ 10-10

5.´ 10-10

1.´ 10-9

fHΣL

Figure 12: Bifurcation diagrams without diffusion

B2 = p1(3p3p5p6 − 2p4p25 + p27 + p22(p

25 − p3p7) + p2(p

23p6 − p3p4p5 + p5p7)).

Bifurcation diagrams for β, α ζ and σ with and

without diffusion for SEQIJTR system

The bifurcation diagrams for transmission rate, β, rate of progression from

infective to diagnosed, α, treatment rate, ζ and rate of progression from

quarantine to diagnosed, σ with and without diffusion are given in in Figs. 12

and 13.

xxxii

0.2 0.4 0.6 0.8 1.0Β

-0.00002

-0.00001

0.00001

fHΒL

0.02 0.04 0.06 0.08 0.10 0.12 0.14Α

-2.´ 10-6

-1.´ 10-6

1.´ 10-6

2.´ 10-6

fHΑL

0.02 0.04 0.06 0.08 0.10Ζ

-1.5´ 10-6

-1.´ 10-6

-5.´ 10-7

5.´ 10-7

1.´ 10-6

1.5´ 10-6

fHΖL

0.01 0.02 0.03 0.04 0.05Σ

-1.´ 10-6

-5.´ 10-7

5.´ 10-7

1.´ 10-6fHΣL

Figure 13: Bifurcation diagrams with diffusion

xxxiii

List of Publication and Conference Presentations

⋆ Naheed A., Singh M., Lucy D., Numerical study of SARS epidemic model

with the inclusion of diffusion in the system, Applied Mathematics and

Computation 229 (2014), 480-498.

⋆ Naheed A., Singh M. and Lucy D., Parameter estimation with uncertainty

and sensitivity analysis for the SARS outbreak in Hong Kong, 4th IMA

Conference on numerical linear algebra and optimization, (2014),

Birmingham University, United Kingdom.

⋆ Naheed A., Singh M., Richards D. and Lucy D., Numerical study of SARS

model with treatment (SEIJTR) and diffusion in the System, ANZIAM

Conference, (2014), The Millennium Hotel, Rotorua, New Zealand.

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