Stability analysis of a delayed sir epidemic model with ... · ORIGINAL PAPER Stability analysis of...

ORIGINAL PAPER

Stability analysis of a delayed sir epidemic modelwith diffusion and saturated incidence rate

Abdelhadi Abta1 • Salahaddine Boutayeb1 • Hassan Laarabi2 • Mostafa Rachik2 •

Hamad Talibi Alaoui3

Received: 13 January 2020 / Accepted: 17 May 2020 / Published online: 16 June 2020� Springer Nature Switzerland AG 2020

AbstractIn this paper, we investigate the effect of spatial diffusion and delay on the dynamical

behavior of the SIR epidemic model. The introduction of the delay in this model makes it

more realistic and modelizes the latency period. In addition, the consideration of an SIR

model with diffusion aims to better understand the impact of the spatial heterogeneity of

the environment and the movement of individuals on the persistence and extinction of

disease. First, we determined a threshold value R0 of the delayed SIR model with diffusion.

Next, By using the theory of partial functional differential equations, we have shown that if

R0\1, the unique disease-free equilibrium is asymptotically stable and there is no endemic

equilibrium. Moreover, if R0 [ 1, the disease-free equilibrium is unstable and there is a

unique, asymptotically stable endemic equilibrium. Next, by constructing an appropriate

Lyapunov function and using upper–lower solution method, we determine the threshold

parameters which ensure the the global asymptotic stability of equilibria. Finally, we

presented some numerical simulations to illustrate the theoretical results.

Keywords SIR epidemic model � SEIR epidemic model � Incidence rate � Ordinary

differential equations � Delayed differential equations � Partial differential equations �Lyapunov function � Global stability

Mathematics Subject Classification 34K20 � 34K25 � 34K05 � 35B09 � 35B40 � 35B35

This article is part of the section ‘‘Theory of PDEs’’ edited by Eduardo Teixeira.

& Abdelhadi [email protected]

1 Department of Mathematics and Computer Science, Poly-disciplinary Faculty, Cadi AyyadUniversity, P.O. Box 4162, Safi, Morocco

2 Department of Mathematics and Computer Science, Faculty of Sciences Ben M’Sik, Hassan IIUniversity, P.O. Box 7955, Sidi Othmane, Casablanca, Morocco

3 Department of Mathematics, Faculty of Sciences El Jadida, Chouaib Doukkali University,P.O. Box 20, El Jadida, Morocco

SN Partial Differential Equations and Applications

SN Partial Differ. Equ. Appl. (2020) 1:13https://doi.org/10.1007/s42985-020-00015-1(0123456789().,-volV)(0123456789().,-volV)

http://crossmark.crossref.org/dialog/?doi=10.1007/s42985-020-00015-1&domain=pdf

https://doi.org/10.1007/s42985-020-00015-1

1 Introduction

The Kermack–McKendrick model is the first one to provide a mathematical description of

the kinetic transmission of an epidemic in an unstructured population [9]. In this model the

total population is assumed to be constant and divided into three classes: susceptible,

infected (and infective), and removed (recovered with permanent immunity) and assuming

that the transfers between these classes are instantaneous. The spread of an infection

governed by this simple model that integrates neither diseases that have a latency period

nor the influence of space on the dynamics of this model, has allowed many scientists to

participate in the improvement of this model and to present more realistic models to

describe the evolution of various types of epidemics.

Recently, several extensions of the Kermack–McKendrick model have been proposed

and analyzed, trying to take into consideration diseases that have a latency period. In

reality, the transfers between the different classes (susceptible, infected and removed) are

not instantaneous, because many diseases such as influenza and tuberculosis have an

incubation period, that is to say the time elapsing between the moment when a susceptible

individual is infected and the moment when he becomes infectious and can transmit this

disease. Motivated by these reasons that characterize most diseases, Cooke [3] proposed a

mathematical model formulated by delay differential equations (DDEs) to describe the

spread of communicable diseases. This delayed model is an extension of [9] that incor-

porates a bilinear incidence function. The bilinear incidence is based on the law of mass

action, which is more appropriate for communicable diseases, such as influenza, but not

suitable for sexually transmitted diseases. This prompted researchers to improve the

incidence function by considering a more general function. Several authors have con-

tributed to this improvement by proposing a delayed SIR model with a more general

incidence function (see, e.g., [2, 22] and references cited therein).

The models mentioned above have concentrated only on the temporal dimension with

out diffusion. As we know, in many case the spatial variation of population plays an

important role in the disease spreading model and the time variation governs the dynamical

behavior of the disease spreading, see [12]. Just as pointed in [12], an infectious case is first

found at one location and then the disease spreads to other areas. However, due to the large

mobility of people within a country or even worldwide, spatially uniform models are not

sufficient to give a realistic picture of disease diffusion. For this reason, the spatial effects

cannot be neglected in studying the spread of epidemics. Focusing on the influence of

space on the qualitative behavior of the SIR epidemic model, several improvements are

made (see,e.g., [19, 20] and references cited therein).

In this paper, we generalize all the DDE and DDE models PDE presented in [1, 24] by

proposing the following delayed SIR epidemic model with spatial diffusion and saturated

incidence function:

oSðx; tÞot

¼ dDSðx; tÞ þ A� lSðx; tÞ � bSðx; tÞIðx; tÞ1 þ a1Sðx; tÞ þ a2Iðx; tÞ

;

oIðx; tÞot

¼ dDIðx; tÞ þ be�lsSðx; t � sÞIðx; t � sÞ1 þ a1Sðx; t � sÞ þ a2Iðx; t � sÞ � ðlþ aþ cÞIðx; tÞ;

oRðx; tÞot

¼ dDRðx; tÞ þ cIðx; tÞ � lRðx; tÞ;

8>>>>>>><

>>>>>>>:

ð1:1Þ

where D denotes the Laplacian operator, S(x, t) , I(x, t) , R(x, t) are the numbers of

susceptible, infectious and recovered individuals at location x and time t, respectively. A is


13 of 25 SN Partial Differ. Equ. Appl. (2020) 1:13

the recruitment rate of new individuals into the susceptible class. l and a are positive

constants representing the natural mortality rate of the population and the death rate due to

disease, respectively. The positive constant d indicates the diffusion rate, b is the trans-

mission rate, a1 and a2 are the parameters that measure the inhibitory effect, c is the

recovery rate of the infective individuals. Further, we assume that when a susceptible

individual contacts an infectious individual at time t , the susceptible individual becomes

infectious at time t þ s, where s is the incubation period. The term e�ls denotes the

mortality rate during the incubation period. In addition, in order to improve the disease

transmission process, the following saturated incidence functionbSðx;tÞIðx;tÞ

1þa1Sðx;tÞþa2Iðx;tÞ has been

proposed. This is important because the number of effective contacts between susceptible

and infectious individuals can saturate at high transmutation levels due to crowding or to

appropriate preventive measures taken by the susceptible and infectious individuals to limit

the spread of disease.

Throughout this paper, we consider system (1.1) with initial conditions

Sðx; tÞ ¼ w1ðx; tÞ� 0; Iðx; tÞ ¼ w2ðx; tÞ� 0; Rðx; tÞ ¼ w3ðx; tÞ� 0; ðx; tÞ 2 X� ½�s; 0�;ð1:2Þ

and zero-flux boundary conditions

oS

om¼ oI

om¼ oR

om¼ 0; t� 0; x 2 oX; ð1:3Þ

where X is a bounded domain in Rn with a smooth boundary oX and oom represents the

outside normal derivative on oX. The boundary condition in (1.3) implies that susceptible,

infectious and recovered individuals do not across the boundary oX.

The paper is organized as follows. In next section, we study the well-posedness for

model (1.1). Section 3 is devoted to investigate to the local stability of the disease-free

equilibrium and the endemic through the study of associated characteristic equations.

equilibrium. In Sect. 4, we prove the global asymptotical stability of the endemic equi-

librium. In Sect. 5, to support our theoretical predictions, some numerical simulations are

given. Finally, a brief discussion is given to conclude this work.

2 The well-posedness

In this section, we focus on the well-posedness of solutions for (1.1) by establishing the

global existence, uniqueness, nonnegativity and boundedness of solutions. In the follow-

ing, we need some notations. Let X ¼ CðX;R3Þ be the Banach space of continuous

functions from X into R3, and CX ¼ Cð½�s; 0�;XÞ denotes the Banach space of continuous

X-valued functions on ½�s; 0� equipped with the supremum norm. For any real numbers

a� b; t 2 ½a; b� and any continuous function u : ½a� s; b� ! X, ut is the element of CXgiven by utðhÞ ¼ uðt þ hÞ for h 2 ½�s; 0�. Moreover, we identify any element w 2 CX as a

function from X� ½�s; 0� in R3 defined by wðx; tÞ ¼ wðtÞðxÞ.The next theorem gives us the existence and uniqueness of the global positive solution.

Theorem 2.1 For any given initial condition w 2 CX satisfying (1.2), the system (1.1)–

(1.3) admits a unique nonnegative solution. Moreover, this solution is global and remainsnon-negative.


SN Partial Differ. Equ. Appl. (2020) 1:13 of 25 13

Proof Let w ¼ ðw1;w2;w3Þ 2 CX and x 2 X. We define f ¼ ðf1; f2; f3Þ : CX ! X by

f1ðwÞðxÞ ¼ A� lw1ðx; 0Þ �bw1ðx; 0Þw2ðx; 0Þ

1 þ a1w1ðx; 0Þ þ a2w2ðx; 0Þ;

f2ðwÞðxÞ ¼be�lsw1ðx;�sÞw2ðx;�sÞ

1 þ a1w1ðx;�sÞ þ a2w2ðx;�sÞ � ðlþ aþ cÞw2ðx; 0Þ;

f3ðwÞðxÞ ¼ cw2ðx; 0Þ � lw3ðx; 0Þ:

Then system (1.1)–(1.3) can be rewritten as an abstract differential equation in the phase

space CX in the form

u: ¼ Buþ f ðutÞ; t� 0;

uð0Þ ¼ w 2 CX;

�

ð2:1Þ

where uðtÞ ¼ ðSð:; tÞ; Ið:; tÞ;Rð:; tÞÞT, w ¼ ðw1;w2;w3Þ and Bu ¼ ðdDS; dDI; dDRÞ: We can

easily show that f is locally Lipschitz in CX. According to [5, 10, 11, 18, 21], we deduce

that system (2.1) admits a unique local solution on its maximal interval of existence

½0; tmaxÞ.Since 0 ¼ ð0; 0; 0Þ is a lower-solution of the problem (1.1)–(1.3), we have Sðx; tÞ� 0,

Iðx; tÞ� 0, and Rðx; tÞ� 0.

In the following, our goal is to show that the maximum solution of problem (1.1)–(1.3)

is global. Let’s first consider the first equation of the system (1.1), then we have

oSðx; tÞot

� dDSðx; tÞ�A� lSðx; tÞ;oS

om¼ 0;

Sðx; 0Þ ¼ w1ðx; 0Þ� 0:

8>>>><

>>>>:

ð2:2Þ

By the comparison principle [17], we have Sðx; tÞ� ~SðtÞ. ~SðtÞ ¼ ~Sð0Þe�lt þ Al ð1 � e�ltÞ is

the solution of the following ordinary equation:

d ~S

dt¼ A� l ~S;

~Sð0Þ ¼ maxx2X

w1ðx; 0Þ:

8><

>:ð2:3Þ

Hence,

Sðx; tÞ� maxA

l;maxx2X

w1ðx; 0Þ� �

; 8ðx; tÞ 2 X� ½0:tmaxÞ:

This implies that S is bounded. Let

Tðx; tÞ ¼ e�lsSðx; t � sÞ þ Iðx; tÞ þ Rðx; tÞ:

Thus,

oTðx; tÞot

¼ e�lsdDSðx; t � sÞ þ dDIðx; tÞ þ dDRðx; tÞ þ e�lsA� lTðx; tÞ:

Then, we have



oTðx; tÞot

� dDTðx; tÞ� e�lsA� lTðx; tÞ;oT

om¼ 0;

Tðx; 0Þ ¼ e�lsw1ðx;�sÞ þ w2ðx; 0Þ þ w3ðx; 0Þ:

8>>>><

>>>>:

ð2:4Þ

Applying the comparison principle to system (2.4), we obtain

Tðx; tÞ� maxe�lsA

l;maxx2X

Tðx; 0Þ� �

; 8ðx; tÞ 2 X� ½0:tmaxÞ:

Therefore, I and R are bounded. So, we proved that S; I; R are bounded on X� ½0; tmaxÞ:By the standard theory for semilinear parabolic systems [7], we deduce that tmax ¼ þ1.

This completes the proof. h

3 Local stability of the equilibria

In the rest of this work, to simplify the notation we set l1 :¼ lþ a. Thereafter, in this

section, we will discuss the local stability of system (1.1)–(1.3) by analyzing the corre-

sponding characteristic equations.

System (1.1)–(1.3) always has a disease-free equilibrium P ¼ Al ; 0; 0� �

. Further, if

R0 :¼ Abe�ls

ða1Aþ lÞðl1 þ cÞ [ 1;

system (1.1)–(1.3) admits in addition another endemic equilibrium point P� ¼ ðS�p; I�p ;R�pÞ,

where

S�P ¼ A½ðl1 þ cÞ þ a2Ae�ls�

ðl1 þ cÞ½a1AðR0 � 1Þ þ lR0� þ a2Ae�ls;

I�P ¼ AðR0 � 1Þe�lsða1Aþ lÞðl1 þ cÞ½a1AðR0 � 1Þ þ lR0� þ a2Ae�ls

;

R�P ¼ AðR0 � 1Þe�lsða1Aþ lÞc

lðl1 þ cÞ½a1AðR0 � 1Þ þ lR0� þ la2Ae�ls:

Let ~S ¼ S� S�; ~I ¼ I � I�; ~R ¼ R� R�, where ðS�; I�;R�ÞT is an arbitrary equilibrium

point, and drop bars for simplicity. Then system (1.1) can be transformed into the fol-

lowing form

oSðx; tÞot

¼ dDSðx; tÞ þ A� lðSðx; tÞ þ S�Þ � bðSðx; tÞ þ S�ÞðIðx; tÞ þ I�Þ1 þ a1ðSðx; tÞ þ S�Þ þ a2ðIðx; tÞ þ I�Þ ;

oIðx; tÞot

¼ dDIðx; tÞ þ be�lsðSðx; t � sÞ þ S�ÞðIðx; t � sÞ þ I�Þ1 þ a1ðSðx; t � sÞ þ S�Þ þ a2ðIðx; t � sÞ þ I�Þ � ðl1 þ cÞðIðx; tÞ þ I�Þ;

oRðx; tÞot

¼ dDRðx; tÞ þ cðIðx; tÞ þ I�Þ � lðRðx; tÞ þ R�Þ:

8>>>>>>><

>>>>>>>:

ð3:1Þ



Thus, the arbitrary equilibrium point E� ¼ ðS�; I�;R�ÞT of system (1.1) is transformed into

the zero equilibrium point ð0; 0; 0ÞT of system (3.1).

In the following, we will analyze stability of the zero equilibrium point of system (3.1).

Denote uðtÞ ¼ ðSð:; tÞ; Ið:; tÞ;Rð:; tÞÞT and w ¼ ðw1;w2;w3Þ 2 CX; then system (3.1) can be

rewritten as an abstract differential equation in the phase space CX of the form

u� ðtÞ ¼ DDuðtÞ þ LðutÞ þ gðutÞ; ð3:2Þ

where D ¼ diagfd; d; dg, L : CX ! X and g : CX ! X are given, respectively, by

LðwÞðxÞ

¼

�ðlþ bI�ð1 þ a2I�Þ

ð1 þ a1S� þ a2I�Þ2Þw1ðx;0Þ �

bS�ð1 þ a1S�Þ

ð1 þ a1S� þ a2I�Þ2w2ðx;0Þ

bI�ð1 þ a2I�Þe�ls

ð1 þ a1S� þ a2I�Þ2w1ðx;�sÞ þ bS�ð1 þ a1S

�Þe�ls

ð1 þ a1S� þ a2I�Þ2w2ðx;�sÞ � ðl1 þ cÞw2ðx;0Þ

cw2ðx;0Þ � lw3ðx;0Þ

0

BBBBB@

1

CCCCCA

ð3:3Þ

and

gðwÞðxÞ ¼g1ðwÞðxÞg2ðwÞðxÞg3ðwÞðxÞ

0

B@

1

CA;

g1ðwÞðxÞ ¼bI�ð1 þ a2I

�Þð1 þ a1S� þ a2I�Þ2

w1ðx; 0Þ þbS�ð1 þ a1S


w2ðx; 0Þ

þ A� bðw1ðx; 0Þ þ S�Þðw2ðx; 0Þ þ I�Þ1 þ a1ðw1ðx; 0Þ þ S�Þ þ a2ðw2ðx; 0Þ þ I�Þ � lS�;

g2ðwÞðxÞ ¼ � bI�ð1 þ a2I�Þe�ls

ð1 þ a1S� þ a2I�Þ2w1ðx;�sÞ � bS�ð1 þ a1S

�Þe�ls

ð1 þ a1S� þ a2I�Þ2w2ðx;�sÞ

þ be�lsðw1ðx;�sÞ þ S�Þðw2ðx;�sÞ þ I�Þ1 þ a1ðw1ðx;�sÞ þ S�Þ þ a2ðw2ðx;�sÞ þ I�Þ � ðl1 þ cÞI�;

g3ðwÞðxÞ ¼ cI� � lR�:

ð3:4Þ

For w ¼ ut , w ¼ ðw1;w2;w3ÞT 2 CX, the linearized system of (3.2) at the zero equilibrium

point is

u: ¼ DDuðtÞ þ LðUtÞ ð3:5Þ

and its characteristic equation is

kx� DDx� Lðek�xÞ ¼ 0; ð3:6Þ

where x 2 domðDÞ, and x 6¼ 0; domðDÞ X.

Let 0 ¼ g0\g1\ � � � be the sequence of eigenvalues for the elliptic operator �D subject

to the Neumann boundary condition on X, and EðgiÞ be the eigenspace corresponding to giin L2ðXÞ. Let f/ij; j ¼ 1; . . .; dimEðgiÞg be an orthonormal basis of EðgiÞ, and

Yij ¼ fa/ij; a 2 Rg. Then



L2ðXÞ ¼ aþ1

i¼0

Yi and Yi ¼ adimEðgiÞ

j¼1

Yij:

Moreover, we put

b1ij ¼

/ij

0

0

0

B@

1

CA; b2

ij ¼0

/ij

0

0

B@

1

CA and b3

ij ¼0

0

/ij

0

B@

1

CA; i ¼ 0; 1; 2; . . .; j ¼ 1; 2; . . .; dimEðgiÞ:

ð3:7Þ

Clearly, the family b1ij; b

2ij; b

3ij

� �

ijis a basis of ðL2ðXÞÞ3

. Therefore, any element x of X can

be written in the in the following form

x ¼ ðx1;x2;x3Þ ¼Xþ1

i¼0

XdimEðgiÞ

j¼1

hx1;/ijib1ij þ hx2;/ijib2

ij þ hx3;/ijib3ij: ð3:8Þ

Next, from a straightforward analysis and using (3.7) and (3.8), we show that (3.6) is

equivalent to

kI3 þ giD�Mð Þhx1;/ijihx2;/ijihx3;/iji

0

B@

1

CA ¼

0

0

0

0

B@

1

CA; i ¼ 0; 1; 2; . . .; j ¼ 1; 2; . . .; dimEðgiÞ;

ð3:9Þ

where M is given by

M ¼

�l� bI�ð1 þ a2I�Þ

ð1 þ a1S� þ a2I�Þ2� bS�ð1 þ a1S


0

bI�ð1 þ a2I�Þe�ls

ð1 þ a1S� þ a2I�Þ2e�ks �ðl1 þ cÞ þ bS�ð1 þ a1S

�Þe�ls

ð1 þ a1S� þ a2I�Þ2e�ks 0

0 c �l

0

BBBBB@

1

CCCCCA

:

Thus the characteristic equation is

ðkþ dgi þ lÞðk2 þ pkþ r þ ðskþ qÞe�ksÞ ¼ 0; i ¼ 0; 1; . . .; ð3:10Þ

where

p ¼ 2gid þ lþ ðl1 þ cÞ þ bI�ð1 þ a2I�Þ

ð1 þ a1S� þ a2I�Þ2;

s ¼ � bS�ð1 þ a1S�Þe�ls

ð1 þ a1S� þ a2I�Þ2;

r ¼ dgi þ lþ bI�ð1 þ a2I�Þ

ð1 þ a1S� þ a2I�Þ2

" #

dgi þ l1 þ cð Þ;

q ¼ �ðdgi þ lÞbS�ð1 þ a1S�Þe�ls

ð1 þ a1S� þ a2I�Þ2:



3.1 Stability of disease-free equilibrium P

Using the above analysis, in this part, we take ðS�; I�;R�Þ ¼ P ¼ ðAl; 0; 0Þ: Thus, the

characteristic equation (3.10) becomes

ðkþ dgi þ lÞ2 kþ dgi �bAe�ls

lþ a1Aexpð�ksÞ þ l1 þ c

� �

¼ 0; i ¼ 0; 1; . . .: ð3:11Þ

Theorem 3.1 If R0\1, then the disease-free equilibrium P is locally asymptoticallystable for all s� 0.

Proof For s ¼ 0, the Eq. (3.11) is equivalent to the following cubic equation

ðkþ dgi þ lÞ2 kþ dgi � ðl1 þ cÞðR0 � 1Þ½ � ¼ 0; i ¼ 0; 1; . . .: ð3:12Þ

Clearly, (3.12) has two roots k1 ¼ �dgi � l\0, and k2 ¼ �dgi þ ðl1 þ cÞðR0 � 1Þ.Therefore, if R0\1, then the disease-free equilibrium P is locally asymptotically

stable when s ¼ 0.

Next we discuss the effect of the delay s on the stability of disease-free equilibrium P.

Assume that (3.11) has a purely imaginary root ix, with x[ 0. Then x should satisfy the

following equation for gi.

l1 þ cþ dgi ¼bAe�ls

lþ a1AcosðxsÞ;

x ¼ � bAe�ls

lþ a1AsinðxsÞ:

8>><

>>:

ð3:13Þ

Taking square on both sides of the equations of (3.13) and summing them up, we obtain

x2 ¼ ðl1 þ cþ dgiÞ þbAe�ls

lþ a1A

� �

ðl1 þ cÞðR0 � 1Þ � dgi½ �: ð3:14Þ

For R0\1, Eq. (3.14) has no positive roots. Thus, Eq. (3.11) has no purely imaginary

roots. Moreover, since the disease-free equilibrium P is locally asymptotically stable for

s ¼ 0, then P remains locally asymptotically stable for all s� 0:

3.2 Stability of endemic equilibrium P*

In this part, we will discuss the local stability of the endemic equilibrium P�. First, we take

ðS�; I�;R�Þ ¼ P�. Thus, the characteristic equation (3.10) becomes

ðkþ dgi þ lÞðk2 þ pkþ r þ ðskþ qÞe�ksÞ ¼ 0; i ¼ 0; 1; . . .; ð3:15Þ

where



p ¼ 2dgi þ lþ ðl1 þ cÞ þ bI�Pð1 þ a2I�PÞ

ð1 þ a1S�P þ a2I�PÞ2;

s ¼ � bS�Pð1 þ a1S�PÞe�ls

ð1 þ a1S�P þ a2I

�PÞ

2;

r ¼ dgi þ lþ bI�Pð1 þ a2I�PÞ


�PÞ

2

" #

dgi þ l1 þ cð Þ;

q ¼ �ðdgi þ lÞbS�Pð1 þ a1S�PÞe�ls


�PÞ

2:

Theorem 3.2 If R0 [ 1, then the endemic equilibrium P� is locally asymptoticallystable for all s� 0.

Proof For s ¼ 0, the characteristic equation (3.15) is transformed into the following form

ðkþ dgi þ lÞðk2 þ ðpþ sÞkþ r þ qÞ ¼ 0; i ¼ 0; 1; . . .; ð3:16Þ

where

pþ s ¼ 2dgi þ lþ bI�Pð1 þ a2I�PÞ


�PÞ

2þ ba2S

�PI

�Pe

�ls


�PÞ

2[ 0;

r þ q ¼ ðdgi þ lÞ dgi þba2S

�PI

�Pe

�ls

ð1 þ a1S�P þ a2I�PÞ2

" #

þ ðdgi þ l1 þ cÞ bI�Pð1 þ a2I�PÞ

ð1 þ a1S�P þ a2I�PÞ2

!

[ 0:

According to the Routh–Hurwitz criteria, all the roots of equation (3.16) have negative real

parts. Therefore, when s ¼ 0, the endemic equilibrium point P� is locally asymptotically

stable.

Next, Since all the roots of equation (3.16) have negative real parts for s ¼ 0. it follows

that if instability occurs for a particular value of the delay s, a characteristic root of (3.15)

must intersect the imaginary axis. If (3.15) has a purely imaginary root ix, with x[ 0,

then, by separating real and imaginary parts in (3.15), we have

�sx sinðxsÞ þ q cosðxsÞ ¼ x2 � r;

sx cosðxsÞ � q sinðxsÞ ¼ �px:

�

ð3:17Þ

Taking square on both sides of the equations of (3.17) and summing them up, we obtain

x4 þ ðp2 � s2 � 2rÞx2 þ r2 � q2 ¼ 0: ð3:18Þ

It is easy to see that r � q[ 0 and as R0 [ 1 we deduce that r2 � q2 [ 0.

Moreover, we have

p2 � s2 þ 2r ¼ ðdgiÞ2 þ 2dgiðl1 þ cÞ þ ba2S�PI

�Pe

�ls


�PÞ

2� l1 þ cþ bS�Pð1 þ a1S

�PÞe�ls


�PÞ

2

" #

þ dgi þ lþ bI�Pð1 þ a2I�PÞ


�PÞ

2

" #2

[ 0:

Therefore, Eq. (3.18) has no positive roots and characteristic equation (3.15) does not



admit any purely imaginary root for all gi. Since P� is asymptotically stable for s ¼ 0, it

remains asymptotically stable for all s� 0: h

4 Global stability

In this section, by constructing an appropriate Lyapunov function, we prove that when

R0\1, the disease-free equilibrium P is globally asymptotically stable. On the other hand,

we utilize the upper–lower solution method in [13, 14] to prove the global asymptotic

stability of the endemic equilibrium P�.

Theorem 4.1 If R0 � 1, then the disease-free equilibrium P of system (1.1)–(1.3) isglobally asymptotically stable for all s� 0.

Proof We consider the following Lyapunov functional

L1 ¼Z

Xe�ls

Z Sðx;t�sÞ

Al

1 � Að1 þ a1uÞðlþ a1AÞu

�

duþ Iðx; tÞ þ ðl1 þ cÞZ s

0

Iðx; t � uÞdu" #

dx:

ð4:1Þ

Calculating the time derivative of L1 along solution of system (1.1)–(1.3), we obtain

dL1ðtÞdt

¼Z

Xe�ls 1 � Að1 þ a1Sðx; t � sÞ

ðlþ a1AÞSðx; t � sÞ

� �dDSðx; t � sÞ þ A� lSðx; t � sÞ

�

� bSðx; t � sÞIðx; t � sÞ1 þ a1Sðx; t � sÞ þ a2Iðx; t � sÞ

þ dDIðx; tÞ

þ e�lsbSðx; t � sÞIðt � sÞ1 þ a1Sðx; t � sÞ þ a2Iðx; t � sÞ � ðl1 þ cÞIðx; tÞ

þ ðl1 þ cÞ½Iðx; tÞ � Iðx; t � sÞ��

dx

¼Z

X

�

e�ls 1 � Að1 þ a1Sðx; t � sÞÞðlþ a1AÞSðx; t � sÞ

�

dDSðx; t � sÞ þ A� lSðx; t � sÞð Þ

þ dDIðx; tÞ þ ðl1 þ cÞ Abe�lsð1 þ a1Sðx; t � sÞÞðl1 þ cÞðlþ a1AÞð1 þ a1Sðx; t � sÞ þ a2Iðx; t � sÞÞ � 1

�

Iðx; t � sÞ�

dx

¼Z

X

�

e�ls 1 � Að1 þ a1Sðx; t � sÞÞðlþ a1AÞSðx; t � sÞ

� �

dDSðx; t � sÞ

� e�lsðA� lSðx; t � sÞÞ2


þ dDIðx; tÞ þ ðl1 þ cÞ R0ð1 þ a1Sðx; t � sÞÞð1 þ a1Sðx; t � sÞ þ a2Iðx; t � sÞÞ � 1

�

Iðx; t � sÞ�

dx:



Recall thatR

X DIðx; tÞdx ¼ 0 and using Green’s formula, we have

dL1ðtÞdt

¼Z

X�e�ls dA

ðlþ a1AÞkrSðx; t � sÞk2

S2ðx; t � sÞ � e�lsðA� lSðx; t � sÞÞ2


(

þðl1 þ cÞ R0ð1 þ a1Sðx; t � sÞÞð1 þ a1Sðx; t � sÞ þ a2Iðx; t � sÞÞ � 1

�

Iðx; t � sÞ�

dx

�Z

X�e�ls dA

ðlþ a1AÞkrSðx; t � sÞk2

S2ðx; t � sÞ � e�lsðA� lSðx; t � sÞÞ2


(

þðl1 þ cÞ R0 � 1ð ÞIðx; t � sÞgdx:

Therefore, R0 � 1 ensuresdL1

dt� 0 for all t� 0. In addition, it can be shown that the largest

compact invariant set in ðS; I;RÞdL1

dt¼ 0

� �

is the singleton fPg. Therefore, it follows

from LaSalle’s invariant principle [6] that P is globally asymptotically stable when R0 � 1.

This completes the proof. h

Next, we show that the endemic equilibrium P� is globally asymptotically stable when

R1 :¼ R0 � ba2ða1AþlÞ [ 1. We first give the following lemma.

Lemma 4.2 Suppose that u(x, t) satisfies the following system

ouðx; tÞot

¼ dDuðx; tÞ þ A� luðx; tÞ;ou

om¼ 0; t[ 0; x 2 X;

uðx; 0Þ ¼ u0ðxÞ; x 2 X:

8>>>><

>>>>:

ð4:2Þ

Then limt!þ1

uðx; tÞ ¼ Al for any x 2 X.

Proof It is obvious that system (4.2) always has a constant solution Al. Let u be the positive

solution of (4.2), and define the Lyapunov function

V ¼Z

Xuðx; tÞ � A

l� A

lln

uðx; tÞAl

" # !

dx;

then

dVðtÞdt

¼Z

X1 � A

luðx; tÞ

�

_uðx; tÞdx

¼Z

X1 � A

luðx; tÞ

�

dDuðx; tÞ þ A� luðx; tÞð Þdx

¼ �Z

X

ðA� luðx; tÞ2

luðx; tÞ

!

dx� dA

l

Z

X

kruðx; tÞk2

u2ðx; tÞ dx:



Thus, it follows thatdVðtÞdt � 0 with equality only in A

l. Furthermore, the largest compact

invariant set in u

dVðtÞdt ¼ 0

� �

is the singleton Al. Hence, it follows from LaSalle’s invariant

principle that uðx; tÞ ¼ Al is globally asymptotically stable. h

Theorem 4.3 If that R1 [ max 1;2a1A

a1Aþ l

� �

, then the endemic equilibrium P� ¼

ðS�P; I�P;R�PÞ of system (1.1)–(1.3) is globally asymptotically stable for all s� 0.

Proof From the first equation of system (1.1), we have

oSðx; tÞot


� dDSðx; tÞ þ A� lSðx; tÞ;ð4:3Þ

then from the comparison principle [17] and Lemma 4.2, for an arbitrary e[ 0, there exists

t1 [ 0 such that for any t[ t1,

Sðx; tÞ�C1; ð4:4Þ

where C1 ¼ A

lþ e. This implies that lim sup

t!þ1maxx2X

Sðx; tÞ� A

l:

Moreover, from the second equation of system (1.1) and (4.4), we obtain

oIðx; tÞot

¼ dDIðx; tÞ þ be�lsSðx; t � sÞIðx; t � sÞ1 þ a1Sðx; t � sÞ þ a2Iðx; t � sÞ � ðl1 þ cÞIðx; tÞ

� dDIðx; tÞ þbe�ls A

l þ e� �

Iðx; t � sÞ

1 þ a1Al þ e� �

þ a2Iðx; t � sÞ� ðl1 þ cÞIðx; tÞ;

for t� t1 þ s. Therefore, there exists t2 [ t1 such that for any t[ t2,

Iðx; tÞ�C2; ð4:5Þ

where C2 ¼ ðlþ a1AÞla2

ðR0 � 1Þ þ be�ls � ðl1 þ cÞa2ðl1 þ cÞ þ 1

�

e: This leads to

lim supt!þ1

maxx2X

Iðx; tÞ� ðlþ a1AÞla2

ðR0 � 1Þ:

Next, from the third equation of system (1.1), we have

oRðx; tÞot

¼ dDRðx; tÞ þ cIðx; tÞ � lRðx; tÞ

�DRðx; tÞ þ cC2 � lRðx; tÞ;

for t[ t2. From where, there exists t3 [ t2 such that, for any t[ t3,

Rðx; tÞ�C3; ð4:6Þ

where C3 ¼ cC2

lþ e: This implies lim sup

t!þ1maxx2X

Rðx; tÞ� cðlþ a1AÞl2a2

ðR0 � 1Þ:



Thereafter, we determine a triplet ðC1;C2;C3Þ of lower solutions for system (1.1). From

the first equation of system (1.1) and (4.5), we have

oSðx; tÞot


� dDSðx; tÞ þ A� lSðx; tÞ � bSðx; tÞC2

1 þ a2C2

� dDSðx; tÞ þ A� Sðx; tÞ lþ bC2

1 þ a2C2

�

;

ð4:7Þ

for t[ t2. Therefore, there exists t4 [ t2 such that for any t[ t4,

C1 � Sðx; tÞ; ð4:8Þ

where C1 ¼ Að1 þ a2C2Þlð1 þ a2C2Þ þ bC2

� e[ 0, for e[ 0 small enough. Hence

lim inft!þ1

maxx2X

Sðx; tÞ� Að1 þ a2C2Þlð1 þ a2C2Þ þ bC2

:

Furthermore, From the second equation of system (1.1) and (4.8), we get

oIðx; tÞot

¼ dDIðx; tÞ þ be�lsSðx; t � sÞIðx; t � sÞ1 þ a1Sðx; t � sÞ þ a2Iðx; t � sÞ � ðl1 þ cÞIðx; tÞ

� dDIðx; tÞ þ be�lsC1Iðx; t � sÞ1 þ a1C1 þ a2Iðx; t � sÞ � ðl1 þ cÞIðx; tÞ;

for t þ s[ t4: Then there exists t5 [ t4 such that, for any t[ t5,

C2 � Iðx; tÞ; ð4:9Þ

where

C2 ¼ða1Aþ lÞ ðR0 � 1Þ þ a2C2ðR1 � 1Þ

� �

a2 lð1 þ a2C2Þ þ bC2

� � � be�ls � a1ðl1 þ cÞa2ðl1 þ cÞ þ 1

�

e[ 0;

for e[ 0 small enough. This implies

lim inft!þ1

maxx2X

Iðx; tÞ�ða1Aþ lÞ ðR0 � 1Þ þ a2C2ðR1 � 1Þ

� �

a2 lð1 þ a2C2Þ þ bC2

� � :

Finally, From the third equation of system (1.1) and (4.9), we have

oRðx; tÞot

¼ dDRðx; tÞ þ cIðx; tÞ � lRðx; tÞ

�DRðx; tÞ þ cC2 � lRðx; tÞ;

for t[ t5. Then there exists t6 [ t5 such that for any t[ t6,

C3 �Rðx; tÞ; ð4:10Þ

where C3 ¼ clC2 � e[ 0, for e[ 0 small enough. Therefore



lim inft!þ1

maxx2X

Rðx; tÞ�cða1Aþ lÞ ðR0 � 1Þ þ a2C2ðR1 � 1Þ

� �

la2 lð1 þ a2C2Þ þ bC2

� � :

In conclusion, we have

C1 � Sðx; tÞ�C1; C2 � Iðx; tÞ�C2; C3 �Rðx; tÞ�C3:

On the other hand, it is easy to show that C1; C1; C2 C2;C3; C3 satisfy the following

inequalities

A� lC1 �bC1C2

1 þ a1C1 þ a2C2

� 0�A� lC1 �bC1C2

1 þ a1C1 þ a2C2

e�lsbC1C2

1 þ a1C1 þ a2C2

� ðl1 þ cÞC2 � 0� e�lsbC1C2

1 þ a1C1 þ a2C2

� ðl1 þ cÞC2

cC2 � lC3 � 0� cC2 � lC3

ð4:11Þ

Thus, according to the definition given in [15, 16], ðC1;C2;C3Þ and ðC1;C2;C3Þ form a

pair of coupled upper and lower solutions of system (1.1). It is easy to show that there are

positive constants K1; K2 and K3 such that the following Lipschitz condition is satisfied:

A� lS1 �bS1I1

1 þ a1S1 þ a2I1

�

� A� lS2 �bS2I2

1 þ a1S2 þ a2I2

�

�K1 jS1 � S2j þ jI1 � I2jð Þ;

e�lsbS1I11 þ a1S1 þ a2I1

� ðl1 þ cÞI1�

� e�lsbS2I21 þ a1S2 þ a2I2

� ðl1 þ cÞI2�

�K2 jS1 � S2j þ jI1 � I2jð Þ;

cI1 � lR1ð Þ � cI2 � lR2ð Þj j �K3 jI1 � I2j þ jR1 � R2jð Þ:ð4:12Þ

We now define two sequences ðCðnÞ1 ;C

ðnÞ2 ;C

ðnÞ3 Þ and ðCðnÞ

1 ;CðnÞ2 ;C

ðnÞ3 Þ from the following

recursion relation:

CðnÞ1 ¼ C

ðn�1Þ1 þ 1

K1

A� lCðn�1Þ1 � bC

ðn�1Þ1 C

ðn�1Þ2

1 þ a1Cðn�1Þ1 þ a2C

ðn�1Þ2

!

;

CðnÞ1 ¼ C

ðn�1Þ1 þ 1

K1

A� lCðn�1Þ1 � bCðn�1Þ

1 Cðn�1Þ2


ðn�1Þ2

!

;

CðnÞ2 ¼ C

ðn�1Þ2 þ 1

K2

e�lsbCðn�1Þ1 C

ðn�1Þ2


ðn�1Þ2

� ðl1 þ cÞCðn�1Þ2

!

;

CðnÞ2 ¼ C

ðn�1Þ2 þ 1

K2

e�lsbCðn�1Þ1 C

ðn�1Þ2


ðn�1Þ2


!

;

CðnÞ3 ¼ C

ðn�1Þ3 þ 1

K3

cCðn�1Þ2 � lC

ðn�1Þ3

� �;

CðnÞ3 ¼ C

ðn�1Þ3 þ 1

K3

cCðn�1Þ2 � lCðn�1Þ

3

� �;

ð4:13Þ



where ðCð0Þ1 ;C

ð0Þ2 ;C

ð0Þ3 Þ ¼ ðC1;C2;C3Þ and ðCð0Þ

1 ;Cð0Þ2 ;C

ð0Þ3 Þ ¼ ðC1;C2;C3Þ. It is clear that

these sequences are well defined. Moreover, these sequences process the following

monotone property

ðC1;C2;C3Þ� ðCðnÞ1 ;C

ðnÞ2 ;C

ðnÞ3 Þ� ðCðnþ1Þ

1 ;Cðnþ1Þ2 ;C

ðnþ1Þ3 Þ� ðCðnþ1Þ

1 ;Cðnþ1Þ2 ;C

ðnþ1Þ3 Þ

� ðCðnÞ1 ;C

ðnÞ2 ;C

ðnÞ3 Þ� ðC1;C2;C3Þ; n ¼ 1; 2; . . .

ð4:14Þ

Since ðCð0Þ1 ;C

ð0Þ2 ;C

ð0Þ3 Þ ¼ ðC1;C2;C3Þ, ðCð0Þ

1 ;Cð0Þ2 ;C

ð0Þ3 Þ ¼ ðC1;C2;C3Þ and by (4.11),

(4.13) we have

Cð0Þ1 � C

ð1Þ1 ¼ � 1

K1

A� lCð0Þ1 � bC

ð0Þ1 C

ð0Þ2

1 þ a1Cð0Þ1 þ a2C

ð0Þ2

!

� 0;

Cð1Þ1 � C

ð0Þ1 ¼ 1

K1

A� lCð0Þ1 � bCð0Þ

1 Cð0Þ2


ð0Þ2

!

� 0;

Cð0Þ2 � C

ð1Þ2 ¼ � 1

K2

e�lsbCð0Þ1 C

ð0Þ2


ð0Þ2

� ðl1 þ cÞCð0Þ2

!

� 0;

Cð1Þ2 � C

ð0Þ2 ¼ 1

K2

e�lsbCð0Þ1 C

ð0Þ2


ð0Þ2


!

� 0;

Cð0Þ3 � C

ð1Þ3 ¼ � 1

K3

cCð0Þ2 � lC

ð0Þ3

� �� 0;

Cð0Þ3 � C

ð1Þ3 ¼ 1

K3

cCð0Þ2 � lCð0Þ

3

� �� 0:

This yields ðCð0Þ1 ;C

ð0Þ2 ;C

ð0Þ3 Þ� ðCð1Þ

1 ;Cð1Þ2 ;C

ð1Þ3 Þ and ðCð1Þ

1 ;Cð1Þ2 ;C


1 ;Cð0Þ2 ;C

ð0Þ3 Þ.

Similarly by (4.12) and (4.13) we have

K1ðCð1Þ1 � C

ð1Þ1 Þ ¼ K1ðC

ð0Þ1 � C

ð0Þ1 Þ þ A� lC

ð0Þ1 � bC

ð0Þ1 C

ð0Þ2


ð0Þ2

!

� A� lCð0Þ1 � bCð0Þ

1 Cð0Þ2


ð0Þ2

!

�K1ðCð0Þ1 � C

ð0Þ1 Þ þ A� lC

ð0Þ1 � bC

ð0Þ1 C

ð0Þ2


ð0Þ2

!

� A� lCð0Þ1 � bCð0Þ

1 Cð0Þ2


ð0Þ2

!

� 0;

K2ðCð1Þ2 � C

ð1Þ2 Þ ¼ K2ðC

ð0Þ2 � C

ð0Þ2 Þ þ e�lsbC

ð0Þ1 C

ð0Þ2


ð0Þ2


!



� e�lsbCð0Þ1 C

ð0Þ2


ð0Þ2


!


ð0Þ2 Þ þ e�lsbCð0Þ

1 Cð0Þ2


ð0Þ2


!

� e�lsbCð0Þ1 C

ð0Þ2


ð0Þ2


!

� 0;

K3ðCð1Þ3 � C

ð1Þ3 Þ ¼ K3ðC

ð0Þ3 � C

ð0Þ3 Þ þ cC

ð0Þ2 � lC

ð0Þ3

� �� cCð0Þ

2 � lCð0Þ3

� �


ð0Þ3 Þ þ cCð0Þ

2 � lCð0Þ3

� �� cCð0Þ

2 � lCð0Þ3

� �� 0:

This gives ðCð1Þ1 ;C

ð1Þ2 ;C


1 ;Cð1Þ2 ;C

ð1Þ3 Þ. From the above conclusions we can con-

clude that ðCð0Þ1 ;C

ð0Þ2 ;C


1 ;Cð1Þ2 ;C


1 ;Cð1Þ2 ;C


1 ;Cð0Þ2 ;C

ð0Þ3 Þ:

Thereafter, we assume by induction that

ðCðn�1Þ1 ;C

ðn�1Þ2 ;C

ðn�1Þ3 Þ� ðCðnÞ

1 ;CðnÞ2 ;C

ðnÞ3 Þ� ðCðnÞ

1 ;CðnÞ2 ;C

ðnÞ3 Þ� ðCðn�1Þ

1 ;Cðn�1Þ2 ;C

ðn�1Þ3 Þ for

some n[ 1. Then by (4.12) and (4.13), we have

K1ðCðnÞ1 � C

ðnþ1Þ1 Þ ¼ K1ðC

ðn�1Þ1 � C

ðnÞ1 Þ þ A� lC

ðn�1Þ1 � bC

ðn�1Þ1 C

ðn�1Þ2


ðn�1Þ2

!

� A� lCðnÞ1 � bC

ðnÞ1 C

ðnÞ2

1 þ a1CðnÞ1 þ a2C

ðnÞ2

!

�K1ðCðn�1Þ1 � C

ðnÞ1 Þ þ A� lC

ðn�1Þ1 � bC

ðn�1Þ1 C

ðn�1Þ2


ðn�1Þ2

!

� A� lCðnÞ1 � bC

ðnÞ1 C

ðn�1Þ2


ðn�1Þ2

!

� 0;

K2ðCðnÞ2 � C


ðn�1Þ2 � C

ðnÞ2 Þ þ e�lsbC

ðn�1Þ1 C

ðn�1Þ2


ðn�1Þ2


!

� e�lsbCðnÞ1 C

ðnÞ2


ðnÞ2

� ðl1 þ cÞCðnÞ2

!


ðnÞ2 Þ þ e�lsbC

ðnÞ1 C

ðn�1Þ2


ðn�1Þ2


!

� e�lsbCðnÞ1 C

ðnÞ2


ðnÞ2


!

� 0;

K3ðCðnÞ3 � C


ðn�1Þ3 � C

ðnÞ3 Þ þ cC

ðn�1Þ2 � lC

ðn�1Þ3

� �� cC

ðnÞ2 � lC

ðnÞ3

� �


ðnÞ3 Þ þ cC

ðnÞ2 � lC

ðn�1Þ3

� �� cC

ðnÞ2 � lC

ðnÞ3

� �� 0;



K1ðCðnþ1Þ1 � C

ðnÞ1 Þ ¼ K1ðCðnÞ

1 � Cðn�1Þ1 Þ þ A� lCðnÞ

1 � bCðnÞ1 C

ðnÞ2


ðnÞ2

!

� A� lCðn�1Þ1 � bCðn�1Þ

1 Cðn�1Þ2


ðn�1Þ2

!

�K1ðCðnÞ1 � C

ðn�1Þ1 Þ þ A� lCðnÞ

1 � bCðnÞ1 C

ðnÞ2


ðnÞ2

!

� A� lCðn�1Þ1 � bCðn�1Þ

1 CðnÞ2


ðnÞ2

!

� 0;

K2ðCðnþ1Þ2 � C


2 � Cðn�1Þ2 Þ þ e�lsbCðnÞ

1 CðnÞ2


ðnÞ2


!

� e�lsbCðn�1Þ1 C

ðn�1Þ2


ðn�1Þ2


!

� K2ðCðnÞ2 � C

ðn�1Þ2 Þ þ e�lsbCðn�1Þ

1 CðnÞ2


ðnÞ2


!

� e�lsbCðn�1Þ1 C

ðn�1Þ2


ðn�1Þ2


!

�0;

K3ðCðnþ1Þ3 �C


3 �Cðn�1Þ3 Þ þ cCðnÞ

2 � lCðnÞ3

� �� cCðn�1Þ

2 � lCðn�1Þ3

� �

� K3ðCðnÞ3 � C

ðn�1Þ3 Þ þ cCðn�1Þ

2 � lCðnÞ3

� �� cCðn�1Þ

2 � lCðn�1Þ3

� ��0:

This leads to ðCðnÞ1 ;C

ðnÞ2 ;C

ðnÞ3 Þ� ðCðnþ1Þ

1 ;Cðnþ1Þ2 ;C

ðnþ1Þ3 Þ and ðCðnþ1Þ

1 ;Cðnþ1Þ2 ;

Cðnþ1Þ3 Þ� ðCðnÞ

1 ;CðnÞ2 ;C

ðnÞ3 Þ: Similarly, we can show that ðCðnþ1Þ

1 ;Cðnþ1Þ2 ;C

ðnþ1Þ3 Þ

� ðCðnþ1Þ1 ;C

ðnþ1Þ2 ;C

ðnþ1Þ3 Þ. An application of the principle of induction gives the monotone

property (4.14). In view of the monotone property (4.14) the constant limits

limn!þ1

CðnÞ1 ¼ eC1; lim

n!þ1CðnÞ2 ¼ eC2;

limn!þ1

CðnÞ1 ¼ bC1; lim

n!þ1CðnÞ2 ¼ bC2;

ð4:15Þ

exist and satisfy the relation

ðC1;C2;C3Þ� ðCðnÞ1 ;C

ðnÞ2 ;C

ðnÞ3 Þ� ð bC1; bC2; bC3Þ� ð eC ; eC2; eC3Þ

� ðCðnÞ1 ;C

ðnÞ2 ;C

ðnÞ3 Þ� ðC1;C2;C3Þ:

ð4:16Þ

Letting n tend to infinity in (4.13) shows that the limits satisfy the following equations



A� l eC1 �b eC1

bC2

1 þ a1eC1 þ a2

bC2

¼ 0;

A� l bC1 �b bC1

eC2

1 þ a1bC1 þ a2

eC2

¼ 0;

e�lsb eC1eC2

1 þ a1eC1 þ a2

eC2

� ðl1 þ cÞ eC2 ¼ 0;

e�lsb bC1bC2

1 þ a1bC1 þ a2

bC2

� ðl1 þ cÞ bC2 ¼ 0;

c eC2 � l eC3 ¼ 0;

c bC2 � l bC3 ¼ 0:

ð4:17Þ

On the other hand, it is clear that the limits ð bC1; bC2; bC3Þ and ð eC1; eC2; eC3Þ of the iterative

sequences ðCðnÞ1 ;C

ðnÞ2 ;C

ðnÞ3 Þ and ðCðnÞ

1 ;CðnÞ2 ;C

ðnÞ3 Þ being defined by (4.15) satisfy

ð bC1; bC2; bC3Þ[ 0 and ð eC1; eC2; eC3Þ[ 0 . Thus, Eq. (4.17) are reduced to

A� l eC1

� �1 þ a1

eC1 þ a2bC2

� �¼ b eC1

bC2; A� l bC1

� �1 þ a1

bC1 þ a2eC2

� �¼ b bC1

eC2;

ðl1 þ cÞ 1 þ a1eC1 þ a2

eC2

� �¼ e�lsb eC1; ðl1 þ cÞ 1 þ a1

bC1 þ a2bC2

� �¼ e�lsb bC1;

c eC2 � l eC3 ¼ 0; c bC2 � l bC3 ¼ 0:

ð4:18Þ

Subtraction of the corresponding equations (4.18) gives

a1A� l� la1ð bC1 þ eC1Þ� �

ð eC1 � bC1Þ þ a2Að bC2 � eC2Þ þ ðla2 þ bÞ bC1eC2 � eC1

bC2

� �¼ 0;

a1ðl1 þ cÞ � e�lsbð Þð eC1 � bC1Þ þ a2ðl1 þ cÞð eC2 � bC2Þ ¼ 0;

cð eC2 � bC2Þ þ lð bC3 � eC3Þ ¼ 0:

ð4:19Þ

In addition, from the third and fourth equations of (4.17), we find

ð eC1 � bC1Þ þ a2ð eC1bC2 � bC1

eC2Þ ¼ 0: ð4:20Þ

Furthermore, from (4.19) and (4.20), we have

2a1A� ða1Aþ lÞR1 � la1ð bC1 þ eC1Þ� �

ð eC1 � bC1Þ ¼ 0: ð4:21Þ

Therefore, if R1 [ max 1;2a1A

a1Aþ l

� �

, then eC1 ¼ bC1. Moreover, from (4.19) we can easily

get eC2 ¼ bC2 and eC3 ¼ bC3. This shows that ð eC1; eC2; eC3Þ (or ð bC1; bC2; bC3Þ) is a positive

steady-state solution of (1.1)–(1.3). The uniqueness of the positive solution ðS�P; I�P;R�PÞ

ensures that eC3 ¼ bC3 ¼ ð eC1; eC2; eC3Þ ¼ ðS�P; I�P;R�PÞ. Then from the results in [15, 16], the

solution (S(x, t), I(x, t), R(x, t)) of system (1.1)–(1.3) satisfies

limt!þ1

Sðx; tÞ ¼ S�P; limt!þ1

Iðx; tÞ ¼ I�P; limt!þ1

Rðx; tÞ ¼ R�P; ð4:22Þ



uniformly for x 2 X. So the endemic equilibrium P� ¼ ðS�P; I�P;R�PÞ of system (1.1)–(1.3) is

globally asymptotically stable for all s� 0. h

5 Numerical simulations

In this section, we perform some numerical simulations to illustrate the theoretical results.

For the sake of simplicity, we consider a one-dimensional bounded spatial domain

X ¼ ½0; 1�. Thus, we propose system (1.1) with Neumann boundary conditions

oS

om¼ oI

om¼ oR

om¼ 0; t� 0; x ¼ 0; 1; ð5:1Þ

and initial conditions

Sðx; tÞ ¼ j cosð3pxÞj � 0; Iðx; tÞ ¼ j sinð2pxÞj � 0;

Rðx; tÞ ¼ j sinð2pxÞj � 0; ðx; tÞ 2 ½0; 1� � ½�s; 0�:ð5:2Þ

Moreover, to solve system (1.1) using a numerical algorithm, we must discretize each

equation of system (1.1) as a finite difference equation. The Crank–Nicolson method [4] is

a finite difference method used for numerically solving a partial differential equation. It is a

second-order method in time and space, and is numerically stable. Thereafter, a brief

description of the Crank–Nicolson method applied to our problem will be provided below.

We first start by partitioning the spatial interval [0, 1] and temporal interval ½0; tf � into

respective finite grids as follows.

xi ¼ ði� 1ÞDx; i ¼ 1; 2; . . .;Nx þ 1 where Dx :¼ 1

Nx:

tj ¼ ðj� 1ÞDt; j ¼ 1; 2; . . .;Nt þ 1 where Dt :¼ tfNt

:

Therefore, using discretization, we can describe S(x, t) as

Si;jði ¼ 1; . . .;Nx þ 1; j ¼ 1; . . .;Nt þ 1Þ, I(x, t) as Ii;jði ¼ 1; . . .;Nx þ 1; j ¼ 1; . . .;Nt þ1Þ and R(x, t) as Ri;jði ¼ 1; . . .;Nx þ 1; j ¼ 1; . . .;Nt þ 1Þ, respectively. In addition, we

can discretize the system (1.1) as follows:

Si;jþ1 � Si;jDt

¼ d

2

Siþ1;jþ1 � 2Si;jþ1 þ Si�1;jþ1

Dx2þ Siþ1;j � 2Si;j þ Si�1;j

Dx2

� �

þ A� lSi;j �bSi;jIi;j

1 þ a1Si;j þ a2Ii;j;

Ii;jþ1 � Ii;jDt

¼ d

2

Iiþ1;jþ1 � 2Ii;jþ1 þ Ii�1;jþ1

Dx2þ Iiþ1;j � 2Ii;j þ Ii�1;j

Dx2

� �

þe�lsbSi;j�s=DtIi;j�s=Dt

1 þ a1Si;j�s=Dt þ a2Ii;j�s=Dt� ðl1 þ cÞIi;j;

Ri;jþ1 � Ri;j

Dt¼ d

2

Riþ1;jþ1 � 2Ri;jþ1 þ Ri�1;jþ1

Dx2þ Riþ1;j � 2Ri;j þ Ri�1;j

Dx2

� �

þ cIi;j � lRi;j:

ð5:3Þ

Applying the central difference formula to approximate the Neumann boundary condition

(1.3), we see that (5.3) yields the following system:



MSjþ1 ¼ NSj þ Uj;

MIjþ1 ¼ NIj þ Vj;

MRjþ1 ¼ NRj þWj;

ð5:4Þ

where

Sj ¼

S1;j

S2;j

..

.

SNx ;j

SNxþ1;j

2

66666664

3

77777775

; Ij ¼

I1;j

I2;j

..

.

INx;j

INxþ1;j

2

66666664

3

77777775

; Rj ¼

R1;j

R2;j

..

.

RNx;j

RNxþ1;j

2

66666664

3

77777775

; Uj ¼ 2Dt:

A� lS1;j �bS1;jI1;j

1 þ a1S1;j þ a2I1;j

A� lS2;j �bS2;jI2;j

1 þ a1S2;j þ a2I2;j

..

.

A� lSi;j �bSNx;jINx;j

1 þ a1SNx ;j þ a2INx;j

A� lSNxþ1;j �bSNxþ1;jINxþ1;j

1 þ a1SNxþ1;j þ a2INxþ1;j

2

6666666666666664

3

7777777777777775

;

Vj ¼ 2Dt:

e�lsbS1;j�s=DtI1;j�s=Dt

1 þ a1S1;j�s=Dt þ a2I1;j�s=Dt� ðl1 þ cÞI1;j

e�lsbS2;j�s=DtI2;j�s=Dt

1 þ a1S2;j�s=Dt þ a2I2;j�s=Dt� ðl1 þ cÞI2;j

..

.

e�lsbSNx ;j�s=DtINx ;j�s=Dt

1 þ a1SNx;j�s=Dt þ a2INx;j�s=Dt� ðl1 þ cÞINx;j

e�lsbSNxþ1;j�s=DtINxþ1;j�s=Dt

1 þ a1SNxþ1;j�s=Dt þ a2INxþ1;j�s=Dt� ðl1 þ cÞINxþ1;j

2

66666666666666664

3

77777777777777775

; Wj ¼ 2Dt:

cI1;j � lR1;j

cI2;j � lR2;j

..

.

cINx;j � lRNx;j

cINxþ1;j � lRNxþ1;j

2

66666664

3

77777775

;

and we take r :¼ dDtDx2

, then the tridiagonal matrices M and N are given by:

M ¼

2 þ 2r � 2r 0 0 � � � 0

�r 2 þ 2r � r 0 . .. ..

.

0 � r . .. . .

. . ..

0

0 . .. . .

. . ..

� r 0

..

. . ..

0 � r 2 þ 2r � r

0 � � � 0 0 � 2r 2 þ 2r

2

666666666664

3

777777777775

;

N ¼

2 � 2r 2r 0 0 � � � 0

r 2 � 2r r 0 . .. ..

.

0 r . .. . .

. . ..

0

0 . .. . .

. . ..

r 0

..

. . ..

0 r 2 � 2r r

0 � � � 0 0 2r 2 � 2r

2

666666666664

3

777777777775

:

Consequently, it follows from (5.4) that



Sjþ1 ¼ M�1 NSj þ Uj

�;

Ijþ1 ¼ M�1 NIj þ Vj

�;

Rjþ1 ¼ M�1 NRj þWj

�:

ð5:5Þ

Therefore, we get a recursive schema, with is numerically stable. The parameters

employed in the numerical simulations are summarized in Table 1.

Now, if we choose A ¼ 0:04, a2 ¼ 0:04 and b ¼ 0:04, then we have R0 ¼ 0:2661. By

Theorem 3.1, the disease-free equilibrium P(1, 0, 0) is locally asymptotically stable. This

means that the disease dies out (see Fig. 1).

Next, if we choose A ¼ 0:2, a2 ¼ 0:8 and b ¼ 0:07, then we get R0 ¼ 2:0177. It follows

from Theorem 3.2 that the endemic equilibrium P�ð3:1734; 0:5212; 0:6010Þ is locally

asymptotically stable (see Fig. 2).

Finally, to illustrate numerically the global stability of equilibrium points, we must

disturb the initial conditions of the system (1.1). To do this, we give the evolution of the

solutions of system (1.1) for four different initial conditions. We start with the free-disease

equilibrium. Here, if we choose the same values of A, a and b used in the first simulation,

we have R0 ¼ 0:2661� 1. Then by Theorem 4.1, the disease-free equilibrium P(1, 0, 0) is

globally asymptotically stable. This means that regardless of the initial densities of sus-

ceptible, infectious and recovered individuals, there are no infectious and recovered

individuals except susceptible individuals in the end. That is to say, in this situation disease

cannot continue to spread (see Fig. 3).

Next, for the endemic equilibrium P� if we choose A ¼ 0:2, a2 ¼ 0:8 and b ¼ 0:4, we

obtain R1 ¼ 2:3882 and2a1A

a1Aþ l¼ 0:01. From Theorem 4.3, we know that

P�ð0:6813; 1:2870; 1:5833Þ is globally asymptotically stable. In this situation, we will

conclude that whatever the initial densities of susceptible, infectious and recovered indi-

viduals, the disease will settle in the population (see Fig. 4).

6 Discussion

It is well known that the spatial component has been identified as an important factor in

understanding the spread of infectious diseases. Recently, many studies show that the

spatial epidemic model is an appropriate tool for investigating the fundamental mechanism

of complex spatiotemporal epidemic dynamics [23]. In this article, we investigate a

delayed reaction–diffusion epidemic model with saturated incidence rate. Our model is

Table 1 List of parameters andtheir values used in numericalsimulations

Parameter Description Value

A Recruitment rate of the population Varied

l Natural death of the population 0.04

a Death rate due to disease 0.04

a1 Parameter that measure the inhibitory effect 0.001

a2 Parameter that measure the inhibitory effect Varied

b Transmission rate Varied

c Recovery rate 0.05

d Rate of diffusion 0.005

s Time incubation 0.8



based on incorporating the population diffusion into the SIR epidemic model and the

assumption that the diffusion rates for the susceptible, infectious and recovered individuals

are equal to d. Thanks to this new assumption, we can show that the solutions of system

(1.1)–(1.3) are global. In addition, we introduce the delay in this model in order to

modulate the latency period. Moreover, we assume that the transmission function between

susceptible and infectious individuals is saturated. This hypothesis widens the spectrum of

diseases studied, seen that the bilinear incidence rate is not adequate for sexually trans-

mitted diseases for example, because the contact between the susceptible and the infectious

individuals does not happen by chance.

Fig. 1 Spatiotemporal solution found by numerical integration of system (1.1) under conditions (5.1) and(5.2) when R0 ¼ 0:2661� 1

Fig. 2 Spatiotemporal solution found by numerical integration of system (1.1) under conditions (5.1) and(5.2) when R0 ¼ 2:0177[ 1



By comparing the results in Theorems , and 4.1 with the propositions 1, 2 of [8] and the

proposition 2 of [1], we affirm that we have obtained the same results, but for a more

general class of population models. In reality, we have extended these results to contain our

model of reaction–diffusion epidemic. Firstly, by analyzing the corresponding character-

istic equations, we discussed the local stability of the disease-free equilibrium P and the

endemic equilibrium P� of system (1.1) under homogeneous Neumann boundary

Fig. 4 The endemic equilibrium P�ð0:6813; 1:2870; 1:5833Þ is globally asymptotically stable

Fig. 3 The free-disease equilibrium point P(1, 0, 0) is globally asymptotically stable



conditions. Since R0 has no relation to the diffusion coefficient d, we have shown in

Theorem 3.1 and Theorem 3.2 that spatial diffusion has no effect on the local stability of

the steady states of our SIR model. Which indicates that, whatever the choice of the

diffusion coefficient d, the stability of the equilibrium points remains invariant when the

system passes from the dynamics governed by the ordinary differential equations ODE [1]

to that governed by the partial differential equations PDE. Furthermore, by constructing an

appropriate Lyapunov function, we have shown in Theorem 4.3 the global stability of

disease-free equilibrium P when R0 � 1. Secondly, using the upper–lower solution method

developed in [13, 15], we have proven the global asymptotic stability of the endemic

equilibrium P� when R0 [ 1 and R1 � max 1;a1A

a1Aþ l

� �

. Since R0 and R1 do not depend

on the diffusion coefficient d, then the spatial diffusion coefficient has no influence on the

study of stability of the equilibrium points. Here, we have extended the result obtained in

[1, Proposition 2.2] to our diffusive SIR epidemic model (1.1)–(1.3), by adding an addi-

tional condition on R1 and using a different technique. Finally, we have given the

numerical simulations to illustrate the theoretical analysis.

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