Dynamical Models of Dynamical Models of Epidemics: from Black Death Epidemics: from Black Death to SARSto SARS

D. GurarieD. Gurarie

CWRUCWRU

Epidemics in History – Plague in 14th Century Europe killed 25 million – Aztecs lost half of 3.5 million to smallpox – 20 million people in influenza epidemic of 1919

Diseases at Present – 1 million deaths per year due to malaria – 1 million deaths per year due to measles – 2 million deaths per year due to tuberculosis– 3 million deaths per year due to HIV – Billions infected with these diseases

History of Epidemiology. Hippocrates's On the Epidemics (circa 400 BC). John Graunt's Natural and Political Observations made upon the Bills of Mortality (1662). Louis Pasteur and Robert Koch (middle 1800's)

History of Mathematical Epidemiology. Daniel Bernoulli studied the effect of vaccination with cow pox on life expectancy (1760). Ross's Simple Epidemic Model (1911). Kermack and McKendrick's General Epidemic Model (1927)

HistoryHistory

SchistosomiasisSchistosomiasis Chronic parasitic trematode infectionChronic parasitic trematode infection 200-300 million people worldwide200-300 million people worldwide Significant morbidity (esp. anemia) Significant morbidity (esp. anemia) Premature mortalityPremature mortality Life-cycle is complex, requiring species-specific Life-cycle is complex, requiring species-specific

intermediate snail host intermediate snail host Optimal control strategies have not been Optimal control strategies have not been

established.established.

Geographic Distribution -1990

Smallpox: XVIII centurySmallpox: XVIII century

Known facts:Known facts:– Short duration (10 days), high Short duration (10 days), high

mortality (75%)mortality (75%)– Life-long immunity for survivorsLife-long immunity for survivors– Prevention: immunity by inoculation Prevention: immunity by inoculation

(??)(??) Problem: could public health (life Problem: could public health (life

expectancy) be improved by expectancy) be improved by inoculation?inoculation? Daniel Bernoulli

1700-1782“I simply wish that, in a matter which so closely concerns the well-being of mankind, no decision shall be made without all the knowledge which a little analysis and calculation can provide.”

Daniel Bernoulli, on smallpox inoculation, 1766

Bernoulli smallpox model Bernoulli smallpox model (1766)(1766)

1) Population cohort of age a, n(a), mortality (a)

2 4 6 8 10a

0.2

0.4

0.6

0.8

1

adnda

n; n0 n0 na n0a=(a) natural mortality

(a) exp-0a a survival function

L0 0 a life expectacy

Special cases :

1. = 0; 0 a L0; a L0

= 1; 0 a L00; a L0

2. Const = e a; L0 1

X - susceptible pop. (infected pop. Z n , rapidly die

or recover with immunity)

- death incidence (1 - survival rate)

- force of infection/capitadnda

n X ; n0 n0

dXda

X ; X0 n0

Solution: na n0a1 ea 1a

Life spans:L0 0 a 1

;

L1 0 1 a 1

;

Bernoulli estimates: 1/8year; =1/8; L1 26.5 years

Life gained: L0 L1 2.54years

2) Small pox effect

Caveat: if inoculation mortality is includedone would need <.5% for success!

Modeling issues and Modeling issues and strategiesstrategies State variables for host/parasiteState variables for host/parasite

– ““mean” or “distributed” (deterministic/stochastic)mean” or “distributed” (deterministic/stochastic)– Prevalence or level/intensityPrevalence or level/intensity– Disease stages (latent,…)Disease stages (latent,…)– Susceptibility and infectiousnessSusceptibility and infectiousness

TransmissionTransmission– Homogeneous (uniformly mixed populations): “mass action”Homogeneous (uniformly mixed populations): “mass action”– Heterogeneous: age/gender/ behavioral strata, spatially Heterogeneous: age/gender/ behavioral strata, spatially

structured contactsstructured contacts– Environmental factorsEnvironmental factors

Multi-host systems, parasites with complex life cycles, …. Multi-host systems, parasites with complex life cycles, …. Goals of epidemic modelingGoals of epidemic modeling

– PredictionPrediction– Risk assessmentRisk assessment– Control (intervention, prevention)Control (intervention, prevention)

Box (compartment) Box (compartment) diagramsdiagrams

S – Susceptible E – Exposed

I – Infectious R - Removed

V – Vaccinated …

S ISI

S E I R

V

SEIR

S I RSIR S E I R

V

SEIR

Birth Death

recruitment

Total population: N = S+I+…

SIR-type SIR-type modelsmodelsRoss, Kermak-Ross, Kermak-McKendrickMcKendrick

•Population size is large and constant•No birth, death, immigration or emigration •No recovery •No latency•Homogeneous mixing

SI

1 2 3 4 5 6S

0.5

1

1.5

2

2.5

3

3.5

4I SI phaseplot

5 10 15 20t

0.2

0.4

0.6

0.8

1SIR

N

RtItSt

Residual S(∞)>0

S I

2 4 6 8t

0.2

0.4

0.6

0.8

1

S,I

S

S I

I

S I Logistic

I

N IIIt N I0N I0I0e N t

Transmission : S rate of new infection

per I

SIR with immunity

S I R

S

S I

I

S I I

R

I

Reductions :

d Id S

1 S

d Sd R

S

S transmission; recovery rate

Basic Reproductionnumber: R0=N/R– endemicR0<1 - eradication

S R

S

S I R

I

S I I

R

I R

ReductionS

S I N S II

S I :

S transmission; recovery rate; loss of immunity

Equilibria Jacobian

I , N

A B N B

0

II N, 0 B0 N

Saddle node bifurcation in N , or R0 N

:

R0 1 stable endemic equilibriumIR0 1 eradicationstable IIendemic epidemic

100 200 300 400

0.2

0.4

0.6

0.8

1

RIS

0.4 0.6 0.8 10

0.025

0.05

0.075

0.1

0.125

0.15

S

I

Control(i) R0=“transmissiom”x”pop. density”/”recovery”<1. Hence critical density N>/b

to sustain endemic level(ii) Vaccination removes a fraction of N from transmission cycle: so eradication

(equilibrium I<0) requires (1-1/R0) fraction of N vaccinated

SIR with loss of immunity

““Smallpox cohort” SIRSmallpox cohort” SIR

X Y

X

X

XZ

X ZY

1 Z Y

n n ZtotalHigh recovery rateshort illness duration: ,

implies near equilibrated Z Z

X,

Hence Bernoulli equation :X

Xn n X

Growth models: variable population Growth models: variable population N(t)N(t)

Const recruitment

.01

100 200 300 400

10

20

30

40 NIS

0 5 10 15 20 25 300

5

10

15

20

25

30

S

I

Linear growth due to S(Voltera-Lotka)

Linear growth rate due to S,I

S

S I a

I

S I I N

a I;SI

a ;SIREquilibrium :

,

a. Jacobian : a

a

0

sinkspiral sink.Nt, Itstabilize at N, IendemicS

S I a S

I

S I I N

a S I;SI

a S ;SIRa growth rate of S

Equilibrium :,

a. Jacobian :0

a 0 centercyclesSt, Itdo in cycles.

S

S I a S b I

I

S I I N

a S b I;SI

a S ;SIRa, b growth rates of S, contributed by S, Ib Equilibrium :

,

a b. Jacobian : ab

b b

ab

0

sinkspiral sinkSt, Itstabilize at endemic levels, Nt

HIV/AIDS and HIV/AIDS and STD STD

• Variable population N=S+I• Natural growth a for S• Mortality =10/year for I• Transmission: S I/(S+I)

= mean number of partners/per IS/(S+I) probability of infecting S (S-fraction of N)

10 20 30 40tyears0.2

0.4

0.6

0.8

1

s,i,n

isnm

Typical collapse

Conclusion: a Transm. treatment

Treatment w/o prevention of spread can only increase (collapse!)

SI wo recoveryAIDS: S

a S S ISI

I

S ISI I

Basicparameters: a or R0

a

transmission

birth recovery

Analytic solution based on function : mt IS m0e

t

Case : 0;mtm0

uncontroled epidemics: SS0

eat; II0

e tislow natural growth : a aiiincrease IN fraction S decays faster than IiiiCollapse of S, I, N

Case : 0;mtm0

, butSS0

ea't;II0

ea' t;igrow at rate rate : a' a m01 m0

aiimaintain constant IN fraction endemicCase : 0;

mtm0

0 :SS0

ea t;II0

eat,irestore natural growth rate aiidecrease IN fraction eradication

20 40 60 80 100 120 140

0.25

0.5

0.75

1

1.25

1.5

1.75

IfemIhetIhomSfemShetShom

bhH bhT btF bfT f H.1 .01 .4 .6 5 .01

Parameters: Initial stateShom Shet Sfem Ihom Ihet Ifem.1 1 1 .01 0 0

AIDS for behavioral groups: 6D model

Data (trends) of several African countries

Heterogeneous transmission Heterogeneous transmission for distributed populationsfor distributed populations

• SIR type are only conceptual models• Idealize transmissions and individual characteristics (susceptibilities)• Real epidemics requires heterogeneous models:

• age structure • spatial/behavioral heterogeneity, etc.

Age structured models Age structured models (smallpox)(smallpox)

t n a n n a 0, t 0;BCna0 bana, t a;ICnt0 n0a;

ba birth rate

Equilibriumsurvival f n: na n0exp0a a,provided :

0

baexp0

a

a a 1

Continuous population strata n(a,t), age “a”, time “t”

Discrete population bins: n=(na)

ODS t n An with Leslie matrix

A

b1 1 1b2 ... bn1 1 20 0

0 1 ... 0

1 1 n

Equilibrium: nk jk11 j, provided: k1

n bkj1k1 j 1

t n a n n X

t X a X X a 0, t 0

n Xa0 ba'na', t a'n Xt0 n0a all newborn susceptible

Force of infection : =a, a'na' Xa' a't n

A n .n X

Xt X

A X

.n X

X

11 12 ...21 22 ...

Structured transmission

"susceptibility" "contact rate"

Example :

" contact matrixij" "suscept. vector 1 1 ... P"

Normal growth Infection

Example: 15-bin system with linear Example: 15-bin system with linear growth and structured transmissiongrowth and structured transmission

2 4 6 8 10 12 14

0.045

0.05

0.055

0.06

0.065

Natural mortality

2 4 6 8 10 12 14

0.2

0.4

0.6

0.8

1Survival function : 1.13

Fatality rate 10%

10 20 30 40

10

20

30

40

Total

10 20 30 40

0.2

0.4

0.6

0.8

1

1.2

1.4Susceptible

Fatality rate 75%

10 20 30 40

1

2

3

4

Total

10 20 30 40

0.5

1

1.5

2

2.5Susceptible

Age bins: red (young) to blue (old)

High survival

Low survival

Fisher’s Equation (1937)

Infection: S(x,t), I(x,t) – (distributed) susceptibles and infectives

• Population density is constant N • No birth or death • No recovery or latent period • Only local infection • Infection rate is proportional to the number of infectives• Individuals disperse diffusively with constant D

1t S S I Dx2 S

tI S I Dx2 I t I IN I Dx

2 I

2t S IS Dx2 S

tI S I Dx2 I t I IN

I Dx

2 I ; N NxEquilibria solutions of BVP :

uxx fu 0; 0 x Lu0 uL 0

Elliptic function u0xor 0Linearized stability for S L operator : M = x2 f'u0

Original motivation: spread of a genetype in a population)

-4-2

02

4 0

5

10

15

00.250.5

0.751

-4-2

02

4-4 -2 2 4

0.2

0.4

0.6

0.8

1

-2

0

20

20

40

60

80

100

0

0.05

0.1

-2

0

2

-3 -2 -1 1 2 3

-0.1

-0.05

0.05

0.1

0.15

Spreading wave in uniformmedium with const pop. density

Spreading wave with variable pop. density (red)

Solutions: propagating density Solutions: propagating density waveswaves

Problems:•Equilibrium, Basic Reproduction Number?•Speed of propagation (traveling waves)?•Parameters for control, prevention?

Some current modeling Some current modeling issues and approachesissues and approaches

Spatial/temporal patterns of Spatial/temporal patterns of

outbreaks and spreadoutbreaks and spread

Stochastic modelingStochastic modeling

Cellular Automata and Agent-Cellular Automata and Agent-

Based ModelsBased Models

Network Models (STD)Network Models (STD)