stability analysis of a delayed sir epidemic model with ... · original paper stability analysis of...
TRANSCRIPT
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)
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
SN Partial Differential Equations and Applications
13 Page 2 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 Differential Equations and Applications
SN Partial Differ. Equ. Appl. (2020) 1:13 Page 3 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
SN Partial Differential Equations and Applications
13 Page 4 of 25 SN Partial Differ. Equ. Appl. (2020) 1:13
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
SN Partial Differential Equations and Applications
SN Partial Differ. Equ. Appl. (2020) 1:13 Page 5 of 25 13
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
�Þð1 þ a1S� þ a2I�Þ2
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
SN Partial Differential Equations and Applications
13 Page 6 of 25 SN Partial Differ. Equ. Appl. (2020) 1:13
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
�Þð1 þ a1S� þ a2I�Þ2
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:
SN Partial Differential Equations and Applications
SN Partial Differ. Equ. Appl. (2020) 1:13 Page 7 of 25 13
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
SN Partial Differential Equations and Applications
13 Page 8 of 25 SN Partial Differ. Equ. Appl. (2020) 1:13
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
ð1 þ a1S�P þ a2I
�PÞ
2
" #
dgi þ l1 þ cð Þ;
q ¼ �ðdgi þ lÞbS�Pð1 þ a1S�PÞe�ls
ð1 þ a1S�P þ a2I
�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
ð1 þ a1S�P þ a2I
�PÞ
2þ ba2S
�PI
�Pe
�ls
ð1 þ a1S�P þ a2I
�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
ð1 þ a1S�P þ a2I
�PÞ
2� l1 þ cþ bS�Pð1 þ a1S
�PÞe�ls
ð1 þ a1S�P þ a2I
�PÞ
2
" #
þ dgi þ lþ bI�Pð1 þ a2I�PÞ
ð1 þ a1S�P þ a2I
�PÞ
2
" #2
[ 0:
Therefore, Eq. (3.18) has no positive roots and characteristic equation (3.15) does not
SN Partial Differential Equations and Applications
SN Partial Differ. Equ. Appl. (2020) 1:13 Page 9 of 25 13
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
ðlþ a1AÞSðx; t � sÞ
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:
SN Partial Differential Equations and Applications
13 Page 10 of 25 SN Partial Differ. Equ. Appl. (2020) 1:13
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
ðlþ a1AÞSðx; t � sÞ
(
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
ðlþ a1AÞSðx; t � sÞ
(
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:
SN Partial Differential Equations and Applications
SN Partial Differ. Equ. Appl. (2020) 1:13 Page 11 of 25 13
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
¼ dDSðx; tÞ þ A� lSðx; tÞ � bSðx; tÞIðx; tÞ1 þ a1Sðx; tÞ þ a2Iðx; tÞ
� 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:
SN Partial Differential Equations and Applications
13 Page 12 of 25 SN Partial Differ. Equ. Appl. (2020) 1:13
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
¼ dDSðx; tÞ þ A� lSðx; tÞ � bSðx; tÞIðx; tÞ1 þ a1Sðx; tÞ þ a2Iðx; tÞ
� 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
SN Partial Differential Equations and Applications
SN Partial Differ. Equ. Appl. (2020) 1:13 Page 13 of 25 13
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
1 þ a1Cðn�1Þ1 þ a2C
ðn�1Þ2
!
;
CðnÞ2 ¼ C
ðn�1Þ2 þ 1
K2
e�lsbCðn�1Þ1 C
ðn�1Þ2
1 þ a1Cðn�1Þ1 þ a2C
ðn�1Þ2
� ðl1 þ cÞCðn�1Þ2
!
;
CðnÞ2 ¼ C
ðn�1Þ2 þ 1
K2
e�lsbCðn�1Þ1 C
ðn�1Þ2
1 þ a1Cðn�1Þ1 þ a2C
ðn�1Þ2
� ðl1 þ cÞCðn�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
SN Partial Differential Equations and Applications
13 Page 14 of 25 SN Partial Differ. Equ. Appl. (2020) 1:13
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
1 þ a1Cð0Þ1 þ a2C
ð0Þ2
!
� 0;
Cð0Þ2 � C
ð1Þ2 ¼ � 1
K2
e�lsbCð0Þ1 C
ð0Þ2
1 þ a1Cð0Þ1 þ a2C
ð0Þ2
� ðl1 þ cÞCð0Þ2
!
� 0;
Cð1Þ2 � C
ð0Þ2 ¼ 1
K2
e�lsbCð0Þ1 C
ð0Þ2
1 þ a1Cð0Þ1 þ a2C
ð0Þ2
� ðl1 þ cÞCð0Þ2
!
� 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
ð1Þ3 Þ� ðCð0Þ
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
1 þ a1Cð0Þ1 þ a2C
ð0Þ2
!
� A� lCð0Þ1 � bCð0Þ
1 Cð0Þ2
1 þ a1Cð0Þ1 þ a2C
ð0Þ2
!
�K1ðCð0Þ1 � C
ð0Þ1 Þ þ A� lC
ð0Þ1 � bC
ð0Þ1 C
ð0Þ2
1 þ a1Cð0Þ1 þ a2C
ð0Þ2
!
� A� lCð0Þ1 � bCð0Þ
1 Cð0Þ2
1 þ a1Cð0Þ1 þ a2C
ð0Þ2
!
� 0;
K2ðCð1Þ2 � C
ð1Þ2 Þ ¼ K2ðC
ð0Þ2 � C
ð0Þ2 Þ þ e�lsbC
ð0Þ1 C
ð0Þ2
1 þ a1Cð0Þ1 þ a2C
ð0Þ2
� ðl1 þ cÞCð0Þ2
!
SN Partial Differential Equations and Applications
SN Partial Differ. Equ. Appl. (2020) 1:13 Page 15 of 25 13
� e�lsbCð0Þ1 C
ð0Þ2
1 þ a1Cð0Þ1 þ a2C
ð0Þ2
� ðl1 þ cÞCð0Þ2
!
�K2ðCð0Þ2 � C
ð0Þ2 Þ þ e�lsbCð0Þ
1 Cð0Þ2
1 þ a1Cð0Þ1 þ a2C
ð0Þ2
� ðl1 þ cÞCð0Þ2
!
� e�lsbCð0Þ1 C
ð0Þ2
1 þ a1Cð0Þ1 þ a2C
ð0Þ2
� ðl1 þ cÞCð0Þ2
!
� 0;
K3ðCð1Þ3 � C
ð1Þ3 Þ ¼ K3ðC
ð0Þ3 � C
ð0Þ3 Þ þ cC
ð0Þ2 � lC
ð0Þ3
� �� cCð0Þ
2 � lCð0Þ3
� �
�K3ðCð0Þ3 � C
ð0Þ3 Þ þ cCð0Þ
2 � lCð0Þ3
� �� cCð0Þ
2 � lCð0Þ3
� �� 0:
This gives ðCð1Þ1 ;C
ð1Þ2 ;C
ð1Þ3 Þ� ðCð1Þ
1 ;Cð1Þ2 ;C
ð1Þ3 Þ. From the above conclusions we can con-
clude that ðCð0Þ1 ;C
ð0Þ2 ;C
ð0Þ3 Þ� ðCð1Þ
1 ;Cð1Þ2 ;C
ð1Þ3 Þ� ðCð1Þ
1 ;Cð1Þ2 ;C
ð1Þ3 Þ� ðCð0Þ
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
1 þ a1Cðn�1Þ1 þ a2C
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
1 þ a1Cðn�1Þ1 þ a2C
ðn�1Þ2
!
� A� lCðnÞ1 � bC
ðnÞ1 C
ðn�1Þ2
1 þ a1CðnÞ1 þ a2C
ðn�1Þ2
!
� 0;
K2ðCðnÞ2 � C
ðnþ1Þ2 Þ ¼ K2ðC
ðn�1Þ2 � C
ðnÞ2 Þ þ e�lsbC
ðn�1Þ1 C
ðn�1Þ2
1 þ a1Cðn�1Þ1 þ a2C
ðn�1Þ2
� ðl1 þ cÞCðn�1Þ2
!
� e�lsbCðnÞ1 C
ðnÞ2
1 þ a1CðnÞ1 þ a2C
ðnÞ2
� ðl1 þ cÞCðnÞ2
!
�K2ðCðn�1Þ2 � C
ðnÞ2 Þ þ e�lsbC
ðnÞ1 C
ðn�1Þ2
1 þ a1CðnÞ1 þ a2C
ðn�1Þ2
� ðl1 þ cÞCðn�1Þ2
!
� e�lsbCðnÞ1 C
ðnÞ2
1 þ a1CðnÞ1 þ a2C
ðnÞ2
� ðl1 þ cÞCðnÞ2
!
� 0;
K3ðCðnÞ3 � C
ðnþ1Þ3 Þ ¼ K3ðC
ðn�1Þ3 � C
ðnÞ3 Þ þ cC
ðn�1Þ2 � lC
ðn�1Þ3
� �� cC
ðnÞ2 � lC
ðnÞ3
� �
�K3ðCðn�1Þ3 � C
ðnÞ3 Þ þ cC
ðnÞ2 � lC
ðn�1Þ3
� �� cC
ðnÞ2 � lC
ðnÞ3
� �� 0;
SN Partial Differential Equations and Applications
13 Page 16 of 25 SN Partial Differ. Equ. Appl. (2020) 1:13
K1ðCðnþ1Þ1 � C
ðnÞ1 Þ ¼ K1ðCðnÞ
1 � Cðn�1Þ1 Þ þ A� lCðnÞ
1 � bCðnÞ1 C
ðnÞ2
1 þ a1CðnÞ1 þ a2C
ðnÞ2
!
� A� lCðn�1Þ1 � bCðn�1Þ
1 Cðn�1Þ2
1 þ a1Cðn�1Þ1 þ a2C
ðn�1Þ2
!
�K1ðCðnÞ1 � C
ðn�1Þ1 Þ þ A� lCðnÞ
1 � bCðnÞ1 C
ðnÞ2
1 þ a1CðnÞ1 þ a2C
ðnÞ2
!
� A� lCðn�1Þ1 � bCðn�1Þ
1 CðnÞ2
1 þ a1Cðn�1Þ1 þ a2C
ðnÞ2
!
� 0;
K2ðCðnþ1Þ2 � C
ðnÞ2 Þ ¼ K2ðCðnÞ
2 � Cðn�1Þ2 Þ þ e�lsbCðnÞ
1 CðnÞ2
1 þ a1CðnÞ1 þ a2C
ðnÞ2
� ðl1 þ cÞCðnÞ2
!
� e�lsbCðn�1Þ1 C
ðn�1Þ2
1 þ a1Cðn�1Þ1 þ a2C
ðn�1Þ2
� ðl1 þ cÞCðn�1Þ2
!
� K2ðCðnÞ2 � C
ðn�1Þ2 Þ þ e�lsbCðn�1Þ
1 CðnÞ2
1 þ a1Cðn�1Þ1 þ a2C
ðnÞ2
� ðl1 þ cÞCðnÞ2
!
� e�lsbCðn�1Þ1 C
ðn�1Þ2
1 þ a1Cðn�1Þ1 þ a2C
ðn�1Þ2
� ðl1 þ cÞCðn�1Þ2
!
�0;
K3ðCðnþ1Þ3 �C
ðnÞ3 Þ ¼ K3ðCðnÞ
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
SN Partial Differential Equations and Applications
SN Partial Differ. Equ. Appl. (2020) 1:13 Page 17 of 25 13
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
SN Partial Differential Equations and Applications
13 Page 18 of 25 SN Partial Differ. Equ. Appl. (2020) 1:13
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:
SN Partial Differential Equations and Applications
SN Partial Differ. Equ. Appl. (2020) 1:13 Page 19 of 25 13
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
SN Partial Differential Equations and Applications
13 Page 20 of 25 SN Partial Differ. Equ. Appl. (2020) 1:13
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
SN Partial Differential Equations and Applications
SN Partial Differ. Equ. Appl. (2020) 1:13 Page 21 of 25 13
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
SN Partial Differential Equations and Applications
13 Page 22 of 25 SN Partial Differ. Equ. Appl. (2020) 1:13
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
SN Partial Differential Equations and Applications
SN Partial Differ. Equ. Appl. (2020) 1:13 Page 23 of 25 13
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.
References
1. Abta, A., Kaddar, A., Talibi Alaoui, H.: Global stability for delay SIR epidemic models with saturatedincidence rates. Electron. J. Differ. Equ. 2012(23), 1–13 (2012)
2. Capasso, V., Serio, G.: A generalization of the Kermack–Mckendrick deterministic epidemic model.Math. Biosci. 42, 41–61 (1978)
3. Cooke, K.: Stability analysis for a vector disease model. Rocky Mt. J. Math. 9, 31–42 (1979)4. Crank, J., Nicolson, P.: A practical method for numerical evaluation of solutions of partial differential
equations of the heat-conduction type. Adv. Comput. Math. 6, 207–226 (1996)5. Fitzgibbon, W.E.: Semilinear functional differential equations in Banach space. J. Differ. Equ. 29, 1–14
(1978)6. Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer-Verlag,
New York, NY (1993)7. Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol.
840. Springer-Verlag, Berlin (1993)8. Kaddar, A., Abta, A., Talibi Alaoui, H.: A comparison of delayed SIR and SEIR epidemic models.
Nonlinear Anal.: Model. Control 16(2), 181–190 (2011)9. Kermack, W.O., McKendrick, A.G.: A contribution to the mathematical theory of epidemics. Proc.
R. Soc. Lond. A 115, 700–721 (1927)10. Martin, R.H., Smith, H.L.: Abstract functional differential equations and reaction–diffusion systems.
Trans. Am. Math. Soc. 321, 1–44 (1990)11. Martin, R.H., Smith, H.L.: Reaction–diffusion systems with time delays: monotonicity, invariance,
comparison and convergence. J. Reine Angew. Math. 413, 1–35 (1991)12. Murray, J.D.: Mathematical Biology, I and II, 3rd edn. Springer, New York (2002)13. Pao, C.V.: Dynamics of nonlinear parabolic systems with time delays. J. Math. Anal. Appl. 198(3),
751–779 (1996)14. Pao, C.V.: Convergence of solutions of reaction–diffusion systems with time delays. Nonlinear Anal.
Theory Methods Appl. 48(3), 349–362 (2002)15. Pao, C.V.: On nonlinear reaction–diffusion systems. J. Math. Anal. Appl. 87(1), 165–198 (1982)16. Pao, C.V.: Nonlinear Parabolic and Elliptic Equations. Springer, New York (1992)17. Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. Englewood Cliffs,
Prentice Hall, NJ (1967)18. Travis, C.C., Webb, G.F.: Existence and stability for partial functional differential equations. Trans.
Am. Math. Soc. 200, 395–418 (1974)19. Wang, W., Cai, Y., Wu, M., Wang, K., Li, Z.: Complex dynamics of a reaction–diffusion epidemic
model. Nonlinear Anal. Real World Appl. 22(5), 2240–2258 (2012)
SN Partial Differential Equations and Applications
13 Page 24 of 25 SN Partial Differ. Equ. Appl. (2020) 1:13
20. Wang, X.S., Wang, H., Wu, J.: Travelling waves of diffusive predator–prey systems: disease outbreakpropagation. Discrete Contin. Dyn. Syst. 32, 3303–3324 (2012)
21. Wu, J.: Theory and Applications of Partial Functional Differential Equations. Springer, New York, NY(1996)
22. Xu, R., Ma, Z.: Global stability of a SIR epidemic model with nonlinear incidence rate and time delay.Nonlinear Anal. RWA 10, 3175–3189 (2009)
23. Xu, Z., Chen, D.: An SIS epidemic model with diffusion. Appl. Math. J. Chin. Univ. 32, 127–146(2017)
24. Yang, J., Siyang Liang, S., Zhang, Y.: Travelling waves of a delayed SIR epidemic model withnonlinear incidence rate and spatial diffusion. PLoS ONE 10, 1371 (2011)
SN Partial Differential Equations and Applications
SN Partial Differ. Equ. Appl. (2020) 1:13 Page 25 of 25 13