Correction
30 Aug 2013: Yang J, Liang S, Zhang Y (2013) Correction: Travelling Waves of a Delayed SIR Epidemic Model with Nonlinear Incidence Rate and Spatial Diffusion. PLOS ONE 8(8): 10.1371/annotation/308a02cb-c4a8-46db-a785-b218f197bba3. https://doi.org/10.1371/annotation/308a02cb-c4a8-46db-a785-b218f197bba3 View correction
Figures
Abstract
This paper is concerned with the existence of travlelling waves to a SIR epidemic model with nonlinear incidence rate, spatial diffusion and time delay. By analyzing the corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state to this system under homogeneous Neumann boundary conditions is discussed. By using the cross iteration method and the Schauder's fixed point theorem, we reduce the existence of travelling waves to the existence of a pair of upper-lower solutions. By constructing a pair of upper-lower solutions, we derive the existence of a travelling wave connecting the disease-free steady state and the endemic steady state. Numerical simulations are carried out to illustrate the main results.
Citation: Yang J, Liang S, Zhang Y (2011) Travelling Waves of a Delayed SIR Epidemic Model with Nonlinear Incidence Rate and Spatial Diffusion. PLoS ONE 6(6): e21128. https://doi.org/10.1371/journal.pone.0021128
Editor: Mike B. Gravenor, University of Swansea, United Kingdom
Received: January 10, 2011; Accepted: May 20, 2011; Published: June 24, 2011
Copyright: © 2011 Yang et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This work is supported by the National Natural Science Foundation of China (No. 31071002). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Introduction
Let represent the number of individuals who are susceptible to the disease, that is, who are not yet infected at time
;
represent the number of infected individuals who are infectious and are able to spread the disease by contact with susceptible individuals, and
represent the number of individuals who have been infected and then removed from the possibility of being infected again or of spreading at time
. In [1], Cooke formulated a SIR model with time delay effect by assuming that the force of infection at time t is given by
, where
is the average number of contacts per infective per day and
is a fixed time during which the infectious agents develop in the vector, and it is only after that time that the infected vector can infect a susceptible human. Cooke considered the following model
(0.1)where parameters
,
,
are positive constants representing the death rates of susceptibles, infectives, and recovered, respectively. It is natural biologically to assume that
. The parameters
and
are positive constants representing the birth rate of the population and the recovery rate of infectives, respectively. Much attention has been paid to the analysis of the stability of the disease free equilibrium and the endemic equilibrium of system (1.1) (see, for example, [2], [3], [4], [5]).
Incidence rate plays an important role in the modelling of epidemic dynamics. It has been suggested by several authors that the disease transmission process may have a nonlinear incidence rate. In [6], Xu and Ma considered the following SIR epidemic model with time delay and nonlinear incidence rate(0.2)By analyzing the corresponding characteristic equations, they studied the local stability of an endemic equilibrium and a disease free equilibrium. It was proved that if the basic reproductive number
, the system was permanent. By comparison arguments, it was shown that if
, the disease free equilibrium was globally asymptotically stable. If
, by means of an iteration technique and Lyapunov functional technique, respectively, sufficient conditions were obtained for the global asymptotic stability of the endemic equilibrium.
We note that the spatial content of the environment has been ignored in the models aforementioned. The models have been traditionally formulated in relation to the time evolution of uniform population distributions in the habitat and are as such governed by ordinary differential equations. 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 can not be neglected in studying the spread of epidemics. Noble [7] applied reaction-diffusion theory to describe the spread of plague through Europe in the mid-14th century. By using the linear theory of semigroups, Saccomandi [8] investigated the existence and uniqueness of the solution for an SIR model with spatial inhomogeneity, nonlocal interactions, and an open population. In recent times, many investigators have introduced population movements into related equations for epidemiological modeling and simulations in efforts to understand the most basic features of spatially distributed interactions (see, for example, [9], [10], [11], [12], [13]).
Motivated by the work of Xu and Ma [6] and Noble [7], in the present paper, we are concerned with the following delayed SIR epidemic model with nonlinear incidence rate and spatial diffusion(0.3)with initial conditions
(0.4)In problem (3)-(4), the positive constants
,
and
denote the corresponding diffusion rates for the susceptible, infected and removed populations, respectively;
is a bounded domain in
with smooth boundary
. The functions
are nonnegative and Hölder continuous and satisfy
in
. In this paper, we assume that
.
In the biological context, it is important to analyze the epidemic waves which is described by traveling wave solutions propagating with a certain speed. In this paper, we are interested in the existence of travelling wave solutions to SIRS epidemic model (0.3). The main tool to study the existence of travelling wave solutions for the reaction-diffusion equations with delays is the sub- and supersolution technique due to Atkinson and Reuter [14]. Wu and Zou [15], [16] studied the existence of travelling wavefronts for delayed reaction-diffusion systems with reaction terms satisfying the so-called quasi-monotonicity or exponential quasi-monotonicity conditions, where the well-known monotone iteration techniques for elliptic systems with advanced arguments [17], [18] were used. Ge and He [19] and Ge et al. [20] used the iteration technique developed by Wu and Zou [15] to investigate the existence of travelling wave solutions for two-species predator-prey system with diffusion terms and stage structure, respectively. However, we note that the nonlinear reaction terms of system (0.3) do not satisfy either the quasi-monotonicity or the exponential quasi-monotonicity conditions. Therefore, the method of upper-lower solutions and its associated monotone iteration scheme developed by Wu and Zou [15], [16] can not be used to study the existence of travelling wave solutions to system (0.3). Recently, by constructing a pair of desirable upper-lower solutions, Huang and Zou [21] got a subset, and employed the Schauder's fixed point theorem in this subset to investigate the existence of travelling wave solutions of a class of delayed reaction diffusion systems with two equations in which the nonlinear reaction terms satisfy the partial quasi-monotonicity and partial exponential quasi-monotonicity, respectively. Li et al. [22] investigated the existence of travelling wave solutions of a class of delayed reaction diffusion systems with two equations in which the reaction terms satisfy weak quasi-monotonicity and weak exponential quasi-monotonicity conditions, respectively. Sazonov et al. [23], [24] studied the problems of travelling waves in the SIR model. Clearly, all the results above can not directly be applied to a system with more than three equations. Therefore, it remains important and challenging to study the existence of travelling wave solutions for delayed reaction diffusion systems with more than three equations in which the nonlinear reaction terms do not satisfy either the quasi-monotonicity or the exponential quasi-monotonicity conditions.
Methods
Preliminaries
Throughout this paper, we adopt the usual notations for the standard ordering in . Thus, for
and
, we denote
if
,
;
if
but
; and
if
but
,
. If
, we also denote
,
, and
. We use
to denote the Euclidean norm in
and
to denote the supremum norm in
.
Before proving the existence of travelling wave solutions to system (0.3), we first investigate the following general delayed reaction-diffusion system:(0.5)
On substituting ,
,
and denote the travelling wave coordinate
still by
, we derive from (0.5) that
(0.6)satisfying the following partial quasi-monotonicity conditions
:
There exist three positive constants
such that
(0.7)where
, with
,
,
,
,
are positive constants,
,
,
, and the functions
are defined by
If, for some
, system (5) has a solution defined on
satisfying
(0.8)where
and
are steady states of (0.5). Then
,
,
is called a travelling wave solution of system (0.5) with speed
. Without loss of generality, we can assume
= (0,0,0) and
, and seek for travelling wave solution connecting these two steady states.
Corresponding to (0.5), we make the following hypotheses:
(A1) ,
.
(A2) There exist three positive constants
such that
for
,
with
,
,
,
.
In the next section, we will apply the Schauder's fixed point theorem, which requires the continuity of the operator under consideration. For this purpose, we need to introduce a topology in . Let
and equipped
with the exponential decay norm defined by
Define
Then it is easy to check that
is a Banach space.
We look for travelling wave solutions to system (0.5) in the following profile set:Obviously,
is non-empty, convex, closed and bounded.
We also need the following definition of upper and lower solutions to system (0.5).
Definition 0.1.
A pair of continuous functions and
are called a pair of upper-lower solutions of system (0.5) if
and
are twice differentiable almost everywhere in
and they are essentially bounded on
, and there hold
(0.9)
(0.10)
(0.11)and
(0.12)
(0.13)
(0.14)
Unlike the standard upper and lower solutions defined in23 [15], is evaluated in a cross iteration scheme given in (0.10) and (0.13).
Local stability
In this section, by analyzing the corresponding characteristic equations, we discuss the local stability of an endemic equilibrium and a disease-free equilibrium of system (0.3) with the initial conditions (0.4) and the homogeneous Neumann boundary conditions(0.15)respectively, where
denotes the outward normal derivative on
, the homogeneous Neumann boundary conditions imply that the populations do not move across the boundary
.
System (0.3) always has a disease-free steady state . Further, if
, then system (0.3) has a unique endemic steady state
, where
Let is called the basic reproductive number (sometimes called basic reproductive rate or basic reproductive ratio) of system (0.3), which describes the average number of newly infected cells generated from one infected cell at the beginning of the infectious process. It is easy to show that if
, the endemic steady state
exists; if
,
is not feasible.
Let be the eigenvalues of the operator
on
with the homogeneous Neumann boundary conditions, and
be the eigenspace corresponding to
in
. Let
,
be an orthonormal basis of
, and
. Then
Let
,
,
, where
and
represents any feasible uniform steady state of system (0.3). The linearization of system (0.3) at
is of the form
. For each
,
is invariant under the operator
, and
is an eigenvalue of
if and only if it is an eigenvalue of the matrix
for some
, in which case, there is an eigenvector in
.
The characteristic equation of takes the form
(0.16)where
Clearly, for any
, Eq. (0.16) always has one negative real root
. Its other roots are determined by the following equation
(0.17)When
, Eq. (0.17) becomes
(0.18)It is readily seen that if
, then
Hence, if
, the endemic steady state
of system (0.3) is locally stable when
.
If is a solution of (0.16), separating real and imaginary parts, we derive that
(0.19)
Squaring and adding the two equations of (0.19), it follows that(0.20)Let
, then Eq. (0.20) becomes
(0.21)By calculation it follows that for all
Hence, we can know that Eq. (0.21) has no positive roots. Therefore, if
,
is locally asymptotically stable for all
.
The characteristic equation of is of the form
(0.22)Clearly, for any
, Eq. (0.22) always has a negative real root
. All other roots are given by the roots of equation
(0.23)Let
If
, it is readily seen that for
real,
Hence, when
, (23) has a positive real root. Therefore, there is a characteristic root
with positive real part in the spectrum of
. Accordingly, if
,
is unstable.
We therefore obtain the following results.
Results
Existence of travelling waves for system (5)
In this section, we study the existence of travelling wave solutions to system (0.5) with nonlinear reaction terms satisfying .
We assume that a pair of upper-lower solutions and
are given such that
(P1) ,
(P2) .
For the constants in
, define
by
(0.24)
(0.25)
(0.26)
The operators ,
and
admit the following properties:
Lemma 0.1.
Assume that and
hold, then
for
with
,
,
.
Proof. By , direct calculation shows that
This completes the proof.
Lemma 0.2.
([15]) Assume that and
hold. Then for any
, we have
(i) ,
.
(ii) and
for
with
,
,
.
In terms of ,
and
, system (0.34) can be rewritten as
(0.27)Define
Let
and define
by
for
. It is easy to see that
,
and
satisfy
(0.28)
Corresponding to Lemmas and
, we have the following result for
.
Lemma 0.3.
Assume that and
hold. For any
, we have
,
,
,
,
for
with
,
,
.
We next verify the continuity of .
Lemma 0.4.
Assume holds, then
is continuous with respective to the norm
in
.
Proof. For any fixed , let
, then for
with
direct calculation shows that
For
, we see that
Similarly, for
, we have
which implies that
is continuous with respect to the norm
in
.
By using a similar argument as above, it can be shown that are continuous. Thus, we obtain that
is continuous with respect to the norm
in
. This completes the proof.
Lemma 0.5.
Assume that and
hold, then
Proof. For any
with
, it follows from Lemma
that
(0.29)
By the definition of upper-lower solutions, we have(0.30)Choosing
in the first equation of (0.28), and denoting
, we get
(0.31)Setting
and combining (0.30) and (0.31), we have
(0.32)Repeating the proof of Lemma
in Wu and Zou [15] shows that
, which implies that
.
By a similar argument, we know that ,
,
,
,
, then
. This completes the proof.
Lemma 0.6.
Assume holds, then
is compact.
Proof. Noting thatit follows from in Lemma
that
. By
in Lemma
and the fact that
, we have
Hence,
implies that there exists a constant
such that
.
For any ,
Thus, we have
If
, we get
If
, we obtain
Noting that , it follows from Lemma
that
is bounded by a positive number. Therefore, there exists a constant
such that
.
Similar to the proof of , we have that there exists a constant
such that
.
The above estimate for shows that
is equicontinuous. It follows from the proof of Lemma
that
is uniformly bounded.
Next, we defineThen, for each
,
is also equicontinuous and uniformly bounded. Now, in the interval
, it follows from Ascoli-Arzela Theorem that
is compact. On the other hand,
in
as
, since
By Proposition
in [25], we have that
is compact. This completes the proof.
Theorem 0.2.
Assume that ,
and
hold. Suppose there is a pair of upper-lower solutions
and
for (0.5) satisfying
and
. Then, system (0.5) has a travelling wave solution.
Proof. Combining Lemmas -
with the Schauder's fixed point theorem, we know that there exists a fixed point
of
in
, which gives a solution to (0.5).
In order to prove that the solution is a travelling wave solution, we need to verify the asymptotic boundary conditions (0.8).
By and the fact that
we get that
and
Therefore, the fixed point
satisfies the asymptotic boundary conditions (0.8). This completes the proof.
Existence of travelling waves for system (0.3)
In this section, we use the results developed in Section 4 to study the existence of travelling wave solutions to system (0.3).
Denoting , then system (0.3) is equivalent to the following system
(0.33)
By making change of variables ,
,
and dropping the tildes, system (0.33) becomes
(0.34)
It is easy to show that if , system (0.34) has two steady states
and
, where
On substituting ,
,
and denote the travelling wave coordinate
still by
, we derive from (34) that
(0.35)satisfying the following asymptotic boundary conditions
Lemma 0.7.
The nonlinear reaction terms of system (0.34) satisfy .
Proof. For any , with
,
,
,
, we have
Let
. We derive that
.
For , it follows that
Let
. We know that
.
For
, we have
Let
. We obtain
. This completes the proof.
LetThere exist
such that
We can find that there exist
satisfying
(0.36)
For the above constants, suitable constants
and
satisfying
,
and
, we define the continuous functions
and
as follows
where
is a constant to be chosen later. It is easy to know that
,
,
,
and
satisfy (36) and the following conditions:
(P1) ,
(P2) .
Lemma 0.8.
is an upper solution of system (0.35).
Proof. If ,
,
and
. It follows that
If
,
. We have
where
It follows from (0.36) that
and there exists
such that
for all
.
If ,
. It follows that
If
,
. We get
where
It follows from (0.36) that
and there exists
such that
for all
.
If ,
and
. It follows that
If
,
. We get
where
It follows from (0.36) that
and there exists
such that
for all
.
Taking , we see that the conclusion is true. This completes the proof.
Lemma 0.9.
is a lower solution of system (0.35).
Proof. If ,
. It follows that
If
,
and
. We have
where
It follows from (0.36) that
and there exists
such that
for all
.
If ,
. It follows that
If
,
. We get
If
,
. We know
where
It follows from (0.36) that
and there exists
such that
for all
.
If ,
. It follows that
If
,
. We get
where
It follows from (0.36) that
and there exists
such that
for all
.
Letting , we see that our conclusion is true. This completes the proof.
Applying Lemmas -
, we know that if
, system (0.34) has a travelling wave solution with speed
connecting the steady states
and
. Accordingly, we have the following conclusion.
Numerical simulations
In this section, by using the classical implicit format solving the partial differential equations and the method of steps for differential difference equations, we give the numerical simulations to illustrate the theoretical results above.
Example 1.
In system (0.3), let ,
,
,
,
,
,
,
and
. System (0.3) with above coefficients has a unique disease-free steady state
. It is easy to show that the basic reproduction number of system (0.3) is
. By Theorem
we see that
is locally stable. Numerical simulation illustrates our result (see Fig. 1).
Example 2.
In system (0.3), we set ,
,
,
,
,
,
,
and
. System (0.3) with above coefficients has a disease-free steady state
and an endemic steady state
. It is easy to show that the basic reproduction number of system (0.3) is
. Theorem
shows that
is unstable and
is locally stable. It follows from Theorem
that system (0.3) always has a travelling wave solution with speed
connecting
and
. The fact is illustrated by the numerical simulation in Fig. 2. Clearly, Fig. 2 shows that the travelling wave solution does not possess the monotonicity, this seems to be due to the feature of the prey-predator interaction.
Discussion
In this paper, we have dealt with the existence of travelling wave solutions for an SIR epidemic model with nonlinear incidence rate, spatial diffusion and time delay. At first, by analyzing the corresponding characteristic equations, we discussed the local stability of a disease-free steady state and an endemic steady state to system (0.3) under homogeneous Neumann boundary conditions. We have shown in Theorem 0.1 that time delay and spatial diffusion are negligible for the local stability of the steady states to system (0.3). By using the cross iteration method and the Schauder's fixed point theorem, we reduced the existence of travelling wave solutions to the existence of a pair of upper-lower solutions. By constructing a pair of upper-lower solutions, we derived the existence of a travelling wave solution connecting the uninfected steady state and the endemic steady state
.
In fact, we also find that time delay can influence the monotone of the travelling wave solution connecting the disease-free steady state
and the endemic steady state
. The effect of time delay on the travelling wave solution is illustrated by comparing the numerical simulations in Figs. 2 and 3.
Author Contributions
Conceived and designed the experiments: JY. Performed the experiments: SL. Analyzed the data: YZ. Contributed reagents/materials/analysis tools: YZ. Wrote the paper: JY.
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