The geographic transmission of infectious diseases is an important issue in mathematical epidemiology and various epidemic models have been proposed and analyzed by many researchers   . For instance, Capasso and Paveri-Fontana  formulated and studied a non-spatial model to investigate the cholera epidemic which spread in the European Mediterranean regions. To discover the impact of spatial spread of a class of bacterial and viral diseases, Capasso and Maddalena  proposed the following reaction-diffusion system
where and represent the spatial density of an infectious agent and an infected human population at point x in the habitat and time ,
respectively. Here, , , , and are positive constants, means the lifetime of the agent in the environment, denotes the mean infectious period of the human infections, is the multiplicative factor of the mean
infectious agent to the human population, and is the force of infection on the human population due to a concentration u of the infectious agent.
The dynamics of the spatially dependent model (1) and its corresponding Cauchy problem have been considered by many scholars. For example, the traveling waves’ solutions were studied in . Wang et al.  studied entire solutions in a time-delayed and diffusive epidemic model. In addition, Ahn et al.  discussed the corresponding free boundary problem. Recently, the corresponding strongly coupled elliptic system has been proposed in  to investigate the existence and non-existence of coexistence states.
In recent years, a great deal of mathematical models has been developed to study the impact of seasonal periodicity and environmental heterogeneity on the dynamics of infectious diseases   . On the other hand, the periodicity of environmental factors is realistic and highly important for the dynamics of infectious diseases.
Taking this fact into account, we extend model (1) to the following reaction-diffusion system
with the periodic condition
where is a bounded domain in with smooth boundary , denotes the outward unit vector on . Here, we assumed that the boundary is of a class for some ; and , , , and are all sufficiently smooth and strictly positive functions defined in and are T-periodic.
Moreover, in order to study the attractivity of T-periodic solutions of problem (2), (3), we consider model (2) under the initial condition
and the initial functions are nonnegative.
Furthermore, we make the following hypothesis on the function g
(H2) is decreasing and
Interest in problems (2), (3) and (2), (4) is motivated by valuable results about the periodic solution of a weakly-parabolic systems    , all those researchers have used the upper and lower solutions method developed by Pao , which is powerful and effective to derive the periodic solutions. For instance, various computation algorithms for numerical solutions of the periodic boundary problem were studied in . In  , the stability and attractivity analysis, which are for quasimonotone nondecreasing and mixed quasimonotone reaction functions by the monotone iteration scheme, were given. It is important to mentioned that there are other standard approaches to derive the periodic solution of a weakly-parabolic systems, Hopf bifurcation theorem and Lyapunov functional  , numerical methods  , Poincaré index , Schauder fixed point theorem  , etc.
The remainder of this paper is organized as follows. In the next section, we deal with the basic reproduction number of problem (2), (3) and its properties. The existence and non-existence of T-periodic solutions of problem (2), (3) and the long time behavior of problem (2), (4) are given in Section 3. Section 4 is devoted to numerical simulations and a brief discussion. Finally, we give some conclusions and future considerations in Section 5.
2. Basic Reproduction Numbers
The focus of this section is to present the basic reproduction number and its properties for the corresponding system in . It is worth emphasis that the basic reproduction number is defined as the expected number of secondary cases produced, in a completely susceptible population, by a typical infected individual during its entire interval of infectiousness . For spatially-independent epidemic models, which are described by ordinary differential systems, the numbers are usually calculated by the next generation matrix method , while for the models constructed by reaction-diffusion systems, the numbers are formulated as the spectral radius of next infection operator induced by a new infection rate matrix and an evolution operator of an infective distribution , and the numbers could be expressed in the term of the principal eigenvalues of relevant eigenvalue problems  .
Considering the linearized problem of (2), (3), we have
then we have
Consider the following problem
and let be the evolution operator of (7), then according to the standard semigroup theory there exist positive constants such that
for any .
Suppose that is the density distribution of u at the spatial location and time s. We use the same idea in   to introduce the linear operator
where . According to the definition, we can easily know that L is continuous, strong positive and compact on . We now define the basic reproduction number of system (5) by the spectral radius of L as follows
To ensure the existence of the basic reproduction numbers, we consider now the following linear periodic-parabolic eigenvalue problem
where . Letting
then the corresponding abstract eigenvalue problem of (8) becomes
in the space
and the domain of the operator is denoted by
Since the system (9) is strongly cooperative, it follows from    that for any , there are a unique value , and called the principal eigenvalue, such that problem (8) admits a unique solution pair (subject to constant multiples) with and in . That is, the solution pair is called the principal eigenfunction. Besides, is algebraically simple and dominant, and the following result hold.
Lemma 2.1 is continuous and strictly increasing with respect to R.
In light of the above discussion, we can further have the following relation between the two eigenvalues.
Theorem 2.1 , and is the principal eigenvalue of the eigenvalue problem
and is the unique principal eigenvalue of problem
Proof: According to (8) and (10), it is obvious that . Meanwhile, it follows from the monotonicity of coefficients in (8), one can easily deduce that and , therefore is the unique positive root of the equation . One finally gets the desired result by the monotonicity of with respect to R. □
Remark 2.1 Recalling the monotone non-increasing of with respect to in the sense that , if and in , one can deduce from Theorem 2.1 that is monotonically nondecreasing with respect to and if is big enough.
In what follows, we provide an explicit formula for if all coefficients are constants.
Theorem 2.2 Suppose that , and . Then the principal eigenvalue for (11), or a threshold parameter for model (2), (4), is expressed by
Proof: Let in and
Then we can check that is a positive solution of problem (11) with , and directly lead to (12) from the uniqueness of the principal eigenvalue of problem (11).
Define and for any given continuous T-periodic function f, we state the following estimate after combining Theorem 2.2 and the monotonicity of with respect to all coefficients in (11).
Corollary 2.1 The principal eigenvalue for (11) satisfies
3. T-Periodic Solution
In this section, we discuss the existence and non-existence of a positive T-periodic solution of problem (2), (3) and the attractivity of the initial-boundary value problem (2), (4) in relation to the maximal and minimal T-periodic solutions of problem (2), (4). To begin with, the following theorem gives the non-existence of a T-periodic solution for problem (2), (3).
Theorem 3.1 If , problem (2), (3) has no positive T-periodic solution.
Proof: If the assertion is not hold, then one can suppose that is a positive T-periodic solution of problem (2), (3), that is, in and satisfy
From (14), we can obtain that
On the other hand, the principal eigenvalue in problem (10) meets
It is follows from Lemma 2.1 that is monotone non-increasing with respect to , then by comparing (15) and (16) we can deduce that . Consequently, by Theorem 2.1. This leads to a contradiction. □
Before approaching the existence result of a positive T-periodic solution to problem (2), (3), we need some preliminaries. Set , , , and
where and are given in the following definition. Then we have an equivalent form of problem (2), (3) as
With regard to problem (17) we state the following definition about the upper and lower solutions.
Definition 3.1 , are ordered upper and lower solutions of problem (17), if and
It is worth emphasizing that the upper and lower solutions in the above definition are not required to be T-periodic with respect to t.
Now to verify the existence of a positive T-periodic solution to problem (2), (3), it suffices to find a pair of ordered upper and lower solutions to problem (2), (3). One can seek such as in the form , where and are positive constant with sufficiently small, is (normalized) positive eigenfunction corresponding to , and is the principal eigenvalue of periodic-parabolic eigenvalue problem (10). Then it is easy to verify that and satisfy (18) if
hold, where .
Recalling that , there exits constant such that , for . This implies that the first two inequalities of (19) will hold, if we take , where , .
Since , the principal eigenvalue of problem (10) is , so one can select small enough such that the last two inequalities of (19) hold.
Therefore, the function pair , are ordered upper and lower solutions of problem (17), respectively.
To summarize the above conclusions, we have the following result.
Theorem 3.2 If , then problem (2), (3) admits at least one positive T-periodic solution .
Remark 3.1 Suppose that all coefficients of (2) are constants. The corresponding basic reproduction ratio is represented by (12). If is big enough, then problem (2), (3) admits at least one positive T-periodic solution. On the other hand, if small enough, then problem (2), (3) has no positive T-periodic solution.
We can now construct the true solutions of problem (17) by applying the monotone iterative scheme associated with the method of upper and lower solutions. Due to
then problem (17) is equivalent to
With respect to (20), and are quasimonotone nondecreasing with respect to u and v, respectively. Choose and as an initial iteration, we can construct a sequence from the iteration process
Then we can easily see that the sequences and governed by (21) are well-defined. Consequently, to show the monotone property of these sequences we have the following result.
Lemma 3.1. The maximal and minimal sequences and are well-defined and possess the monotone property
for every .
According to above lemma, the pointwise limits
exist and satisfy the relation
for every .
Therefore, the maximal and minimal property of and is in the sense that is any other T-periodic solution of (17) in , then for .
Next, to shows that and are the maximal and minimal solutions of (2), (3), respectively, we have the following theorem.
Theorem 3.3 Let and be a pair of ordered upper and lower solutions of (2), (3), respectively, then the sequences and provided from (21) converge monotonically from above to a maximal T-periodic solution and from below to a minimal T-periodic solution in S, respectively, and satisfy relation
Moreover, if or then and is the unique T-periodic solution of (2), (3) in S.
In what follows, we will study the attractivity of problem (2), (4) in relation to the maximal and minimal T-periodic solutions of problem (2), (3). The following lemma plays an important role in the establishment of attractivity result and its proof is similar to that in , so we omit it here.
Lemma 3.2 let be the solution of (2), (4). Then
on for every .
With the help of above lemma and Theorem 3.1 of , the solution of the problem (2) under the initial condition (4) possesses the following convergence
Consequently, the function pair and are positive T-periodic solutions of problem (2), (3) and satisfy the following relation on .
On one hand, for any nonnegative and nontrivial the solution with is positive in for any . So for any , we can find , such that
The above derivations lead to the following theorem.
Theorem 3.4 Let be the solution of (2), (4) for with , i = 1, 2 and let condition be satisfied. Then we have
1) problem (2), (3) has maximal and minimal positive T-periodic solutions and such that on ;
2) the solution of (2), (4) possesses the convergence properties (23) and (24);
3) if , then
4. Numerical Simulation and Discussion
In this section, we provide some simulations for problem (2), (4) to illustrate our analytical results. We first fix the following diffusion coefficients and parameters:
Therefore, now we will change the value of parameter and then observe the long time behavior of problem (2), (4).
Example 1: Chose big so that
and from Figure 1 it is easy to see that the solutions and of (2), (4) tends to a positive T-periodic solution.
Example 2: Take small , then
hence the solutions and of (2), (4) decays to zero and no positive periodic solution exists to problem (2), (3) (see Figure 2).
To understand the impact of seasonal periodicity and environmental heterogeneity on the fecally-orally epidemic model, we consider a T-periodic
Figure 1. The solution of (2), (4) with big presents a state of periodic oscillation.
Figure 2. The solution of (2), (4) with small decays quickly to zero.
solution of problem (2), (3) and the attractivity of problem (2), (4) in relation to the maximal and minimal T-periodic solutions of problem (2), (3). Firstly, we introduce the basic reproduction number by using the next generation operator and associated eigenvalue problems, which is known as a threshold parameter of problem (2), (3) (Theorem 2.1). In the case that all coefficients are constants, we provide an explicit formula for (Theorem 2.2). Secondly, the existence of a T-periodic solution of problem (2), (3) is investigated by combining the method of upper and lower solutions, the eigenvalue problems and associated monotone iterative schemes. Our results indicate that if and big enough problem (2), (3) admits at least one positive T-periodic solution (Theorem 3.4, Theorem 3.5 and Figure 1), while if and sufficiently small, problem (2), (3) has no positive T-periodic solution (Theorem 3.3 and Figure 2).
Moreover, we discuss the attractivity of problem (2), (4) in relation to the maximal and minimal T-periodic solutions of problem (2), (3) by applying the monotone convergence property (Lemma 3.3 and Theorem 3.6). As result, it shown that the long time state of the solution to problem (2), (4) is between the maximal and minimal T-periodic solutions of problem (2), (3).
To better understand the impact of environmental heterogeneity and seasonal periodicity on the spatial spread of a class of bacterial and viral diseases, the fecally-orally epidemic model in heterogeneous environment has been considered. By means of next generation infection operator and associated eigenvalue problems, the spatial-temporal basic reproduction number is defined. We use the number to study the existence and non-existence of T-periodic solutions. The attractivity of our problem in relation to the maximal and minimal T-periodic solutions is discussed. We conclude that if the environment is T-periodic, then the solution of any initial-boundary system will present T-periodic phenomena gradually. Moreover, we believe that the investigation of the transmission dynamics of infectious diseases in heterogeneous environment is more close to reality than in homogeneous environment. However, here we considered the periodic solutions on specified period T. In the future, we will consider the time-periodic solutions for various possible periods. Moreover, the uniqueness of T-periodic solutions is still unclear.
We thank the Editor and the referee for their comments. The work is supported by People Republic of China grant National Natural Science Foundation (11872189, 11472116).
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