Theoretical Analysis on a Diffusive SIR Epidemic Model with Nonlinear Incidence in a Heterogeneous Environment

In this paper, we deal with a diffusive SIR epidemic model with nonlinear incidence of the form \begin{document}$I^pS^q$\end{document} for \begin{document}$0 in a heterogeneous environment. We establish the boundedness and uniform persistence of solutions to the system, and the global stability of the constant endemic equilibrium in the case of homogeneous environment. When the spatial environment is heterogeneous, we determine the asymptotic profile of endemic equilibrium if the diffusion rate of the susceptible or infected population is small. Our theoretical analysis shows that, different from the studies of [ 1 , 28 , 38 , 44 ] for the SIS models, restricting the movement of the susceptible or infected population can not lead to the extinction of infectious disease for such an SIR system.


1.
Introduction. In order to model disease dynamics, in the pioneering work of Kermack and McKendrick [24], according to the principle of mass action, the bilinear incidence βIS was used to describe the spread of an infection between susceptible and infected individuals.
Nevertheless, as explained in several research works (see, for example, [8,9,17,18,19,20,30,31]), such a standard bilinear incidence rate carries some shortcomings and may require modification. As such, in certain situations a nonlinear incidence rate is used to govern the spread of infectious disease. The most commonly used nonlinear incidence takes the form βI p S q , where S and I are the number of susceptible and infective individuals in the population, respectively, and β, 0 < p ≤ 1 and q are positive constants. In recent years, epidemic models with this incidence rate have received extensive investigation; see [8,9,16,17,18,19,20,30,31,32,33] among many others.
In particular, Korobeinikov and Maini [25] considered the following SIR ODE model Ṡ (t) = b − βI p S q − µS, t > 0, In (1), the population is divided into two classes: susceptible and infective, denoted by S and I, respectively. Here, b, µ, β, δ, 0 < p ≤ 1 and q are positive constants: b is the birth rate, β is the transmission rate, µ is the death rate of the susceptible, and δ is the removal rate (including the mortality rate) of the infective. Korobeinikov and Maini established the global stability of the unique endemic equilibrium (if it exists) by constructing suitable Lyapunov functions.
To take into account the inhomogeneous distribution of the population in different spatial locations within a fixed bounded domain Ω ⊂ R N (N ≥ 1) at any given time, and the natural tendency of each class of population to diffuse to areas of smaller population concentration, we are led to the following PDE system of reaction-diffusion type: x ∈ Ω, t > 0, x ∈ Ω, t > 0, ∂ ν S = ∂ ν I = 0, x ∈ ∂Ω, t > 0, S(x, 0) = S 0 (x) ≥ 0, I(x, 0) = I 0 (x) ≥, ≡ 0. (2) In (2), S and I are the density of susceptible and infected individuals at location x and time t, respectively, and the positive constants d S and d I are the respective movement (or diffusive) rates. The habitat Ω is a bounded domain with smooth boundary ∂Ω, and the homogeneous Neumann boundary conditions biologically mean that there is no population flux across the boundary ∂Ω. The functions b, β, µ and δ with the same interpretations as in (1) are positive and Hölder continuous on Ω.
The steady state problem corresponding to (2) is governed by the following elliptic system: We call a solution (S, I) ∈ C 2 (Ω) × C 2 (Ω) to (3) as an endemic equilibrium (EE, for short) if (S, I) solves (3) pointwisely and S ≥ 0 and I ≥, ≡ 0 on Ω. By the well-known strong maximum principle and Hopf lemma for elliptic equations (refer to [15]), one can easily show that any EE (S, I) satisfies S(x) > 0 and I(x) > 0 for all x ∈ Ω.
As in [1,5,6,7,14,28,38,44], in the present work we are interested in the qualitative properties of solutions to (2) as well as its EE.
Firstly, we derive the boundedness of solutions and then the uniform persistence property in the two cases: (i) 0 < p < 1 and (ii) p = 1 and the disease-free equilibrium (S, 0) is unstable, whereS will be given below; see our Proposition 1 and Theorem 2.1. When p = 1, the disease-free equilibrium (S, 0) is stable, it is shown by Proposition 2 that (S, 0) is the global attractor. Furthermore, when the spatial environment becomes homogeneous (that is, b, β, µ and δ are positive constants), Theorem 2.2 below asserts that the unique constant EE is globally attractive in cases (i) and (ii).
Then, the rest of our paper is devoted to the study of the asymptotic profile of the EE (when exists) with respect to the small movement (migration) rate of the susceptible or infected population. In Theorems 4.1 and 4.2, we will show that the densities of the susceptible and infected populations converge to the positive functions on the whole habitat Ω as d S → 0 for any fixed 0 < p ≤ 1 or as d I → 0 for any fixed 0 < p < 1. We conjecture that, as long as the EE exists, this is also true when d I → 0 for p = 1. These limiting behaviors are very different from those observed in the SIS models. In the SIS models studied by [1,28,38,44], it has been proved that the infected population can become extinct in the whole or part of the habitat provided that the small movement rate of the susceptible or infected population is controlled to be small. On the other hand, we want to emphasize that the qualitative behaviors of the solution to (1) and its EE do not depend on the value of q.
The paper is organized as follows. In section 2, we obtain the boundedness and uniform persistence of solutions to (2) and the global stability of the unique constant EE when the parameters b, β, µ, δ are constants. In section 3, we establish the positive upper and lower estimates for any EE. Such a priori estimates are crucial in the analysis of the asymptotic profile of EE for the small susceptible or infected diffusion rate which is addressed in section 4.

2.
The uniform persistence and the global stability of EE. In this section, we are concerned with the uniform persistence property of solutions to (2) and the global attractivity of the constant EE in the case that all the parameters in (2) are assumed to be positive constants.
By the standard theory for parabolic equations, given continuous and nonnegative initial data (S 0 , I 0 ), (2) admits a unique classical solution (S(x, t), I(x, t)) which exists for all positive time, and S(x, t) > 0 and I(x, t) ≥ 0 for all x ∈ Ω and t > 0. Moreover, if I 0 ≥, ≡ 0, then I(x, t) > 0 for all x ∈ Ω and t > 0.
For later purpose, let us denote byS the unique positive solution of Let λ 1 be the principal eigenvalue of the eigenvalue problem: By [40, Lemma 3.2], we know that uniformly on Ω, as d S → 0.
Thus, as d S → 0, where λ * 1 is the principal eigenvalue of the eigenvalue problem On the other hand, by a folklore fact (see, for instance, [1, Lemma 2.2]), we also know that 2.1. The uniform persistence. We start with the boundedness of solutions to (2). Indeed, we can state the following result. Proposition 1. There exists a positive constant C 1 depending on initial data such that the solution (S, I) of (2) satisfies Furthermore, there exists a positive constant C 2 independent of initial data such that for some large time T > 0.
Proof. By the first equation in (2), we see that Consider the following problem Denote by w the unique solution to (8). The standard comparison principle for parabolic equations gives Moreover, it is also known that , uniformly for all x ∈ Ω.
Thus, there is a large time t 0 > 0 such that We next set m = min{min and define From (2), it immediately follows that

A DIFFUSIVE SIR EPIDEMIC MODEL IN A HETEROGENEOUS ENVIRONMENT 4503
Thus, and so That is, Together with (9), we then apply [12, Lemma 2.1] (due to [34] with σ = p 0 = 1) to the second equation in (2) to conclude that (6) holds. In addition, by (12) one has Resorting to [12, Lemma 2.1] to the second equation in (2) again, we are able to assert that there exists a positive constant C 2 independent of initial data such that for some large time T 0 > 0. This and (11) imply (7). Hence, (2) has no EE in this case.
Proof. In light of (10), for any small > 0, there exists a large time t * > 0, such that 1 q for all x ∈ Ω, we can take > 0 to be smaller so that Thus, a simple comparison argument gives I(x, t) → 0 uniformly on Ω, as t → ∞.
Using this fact, it is easily seen from the first equation in (2) that lim inf In particular, this implies that (2) admits no EE.

Remark 1. In fact, it is easily seen thatS
µ(x) on Ω (by Lemma 3.1(i) below). Thus, the assertion of Proposition 2 holds provided that The main aim of this subsection is to establish the uniform persistence of solutions to (2). Specifically, we can state that Theorem 2.1. Assume that 0 < p < 1 or p = 1 and λ 1 < 0. Then there exists a real number η > 0 independent of the initial data, such that any solution (S, I) of uniformly for x ∈ Ω, and hence, the disease persists uniformly. Furthermore, (2) admits at least one EE.
Proof. We shall use the abstract dynamical systems theory to obtain the desired First of all, if ϕ ∈ ∂X 0 , the uniqueness of solutions, we notice that I(t, x, ϕ) ≡ 0 for all t ≥ 0, and so S(t, x, ϕ) solves (8). Hence, the analysis of Proposition 2 implies that S(t, x, ϕ) →S(x) uniformly on Ω, as t → ∞. Let M 0 = (S, 0), we next conclude that there is a positive constant η 0 such that To the end, we have to treat two cases: 0 < p < 1; p = 1 and λ 1 < 0. We first consider the case of 0 < p < 1. Clearly, in such a situation, one can find a sufficiently small η 0 > 0 such that Then Without loss of generality, we may assume that d(φ(t)ϕ, M 0 ) < η 0 , ∀t ≥ 0; otherwise, we can use φ(t 0 )ϕ as a new initial data for a large t 0 > 0. Then it is easily seen that

This in turn gives
Since m > 0, a simple comparison argument deduces that I(t, x) → ∞ uniformly on Ω as t → 0, leading to a contradiction with Proposition 1. Therefore, the claim (13) holds.
When p = 1 and λ 1 < 0, the claim (13) remains valid by using the similar analysis to that of [ This implies that for all ϕ ∈ X 0 , there exists a large T 1 = T 1 (ϕ) > 0 such that Making use of this fact and a comparison argument, one can easily get from the first equation of (2) that for some positive constant η 2 > 0 and for some T 2 ≥ T 1 . As a consequence, the uniform persistence holds by taking η = min{η 1 /2, η 2 /2}. Furthermore, appealing to the theory developed by Magal and Zhao (see [37,Theorem 4.5] or [45]), we also see that the system (2) admits at least one EE under our assumption. The proof is complete.

2.2.
Global stability of the EE. This subsection is devoted to the study of the global stability of the EE of (2) when b, β, µ, and δ are positive constants.
We first solve the following equations to find the constant EE of (2): If 0 < p < 1 and q > 0, according to the second equation of (14), we have Inserting (15) into (14) results in the following equality Next we define a function: If p = 1, q > 0 and b > µ δ β 1 q , it is also easily seen that (2) has a unique positive solution Our global stability result of the constant EE of (2) reads as follows.
Proof. We shall construct suitable Lyapunov functionals to verify the global attractivity of (S * , I * ). The Lyapunov functionals to be employed below are inspired by [25,26].
We first consider the case of 0 < p < 1, q = 1. Define This function L(S, I) satisfies We observe that (S * , I * ) is the only extremum and the global minimum of the function in the positive of octant R 2 + . Hence, the function (16) is indeed a Lyapunov function. By (14), we have b = δI * + µS * , β(I * ) p (S * ) q = δI * .

A DIFFUSIVE SIR EPIDEMIC MODEL IN A HETEROGENEOUS ENVIRONMENT 4509
Here u is given as before.
When p = 1 and q = 1, we then take and when 0 < p < 1 and q = 1 we take In these cases, the same computation as above shows that dL(t) dt < 0, ∀t > 0 along all trajectories except at (S * , I * ) where dL(t) dt = 0, ∀t > 0. Thus, we can assert that (S * , I * ) is globally attractive.

3.
A priori estimates for EE of (2). In this section, we establish a priori upper and lower bounds for solutions to (3). As it will be seen in the forthcoming section 4, such results play a fundamental role in obtaining the asymptotic behavior of EE.
We begin by recalling a known fact, which concerns a basic convexity property of a C 2 function on Ω at its local extrema; see, for instance, [23].

CHENGXIA LEI, FUJUN LI AND JIANG LIU
Clearly, one can find a positive constant C 3 , independent of d S , d I , such that Finally, by taking I(y 2 ) = min x∈Ω I(x) for some y 2 ∈ Ω, it follows from Lemma 3.1 and (21) that which in turn yields Next we will derive the upper and lower bounds for any EE of (3) if p = 1 and d S is small. Such kind of results will be used in Section 4 in the analysis of the asymptotic behavior of EE as d S goes to zero. In view of (4), Proposition 2 and Theorem 2.1, we have to assume that λ * 1 < 0 so that EE of (3) exists when p = 1 and d S is small.
Proof. From the proof of Theorem 3.2, we see that Our next aim is to establish the desired upper bound for the component I. To the end, let W = d S S + d I I. According to (3), we have Take W (x 0 ) = max x∈Ω W (x) for some x 0 ∈ Ω. Then, an application of Lemma 3.1 implies that ∆W (x 0 ) ≤ 0 and so b(x 0 ) ≥ µ(x 0 )S(x 0 ) + δ(x 0 )I(x 0 ), which gives As a result, we obtain This is what we wanted. Next, by letting S(y 0 ) = min x∈Ω S(x) for some y 0 ∈ Ω, we have ∆S(y 0 ) ≥ 0 due to Lemma 3.1. Thus,

This clearly indicates that
S(x) ≥ S(y 0 ) ≥C 2 , ∀x ∈ Ω for some positive constantC 2 , which is independent of all 0 < d S ≤ 1.
In what follows, we are going to derive the positive lower bound for the component I. First of all, from (22) and the equation of I: we can use the Harnack inequality (Lemma 3.3) to obtain where the positive constant C > 0 does not depend on d S with 0 < d S ≤ 1. We proceed by contradiction and suppose that there is a sequence {d Si } satisfying d Si → 0 as i → ∞ and the corresponding positive solution sequence {(S i , I i )} of (3) with d S = d Si , such that min x∈Ω I i → 0 as i → ∞. By (24), there holds I i → 0 uniformly on Ω, as i → ∞.
Arguing as in the proof of [11,Lemma 2.4] (or refer to [35], [40,Lemma 3.2]), one can easily obtain from the first equation of (3) that On the other hand, for any i ≥ 1, if denoting λ 1,i by the principal eigenvalue of the eigenvalue problem: x ∈ ∂Ω, then (23) tells us that λ 1,i = 0 for all i ≥ 1. However, in view of the continuous dependence of the principal eigenvalue of the weight function, (25), combined with our assumption asserts that λ 1,i → λ * 1 < 0 as i → ∞, leading to a contradiction. This shows the existence of positive lower bound for I, which is independent of d S with 0 < d S ≤ 1.
4. Asymptotic profiles of the EE. In this section, we are going to determine the asymptotic behavior of the EE of (2). We will treat two cases: d S → 0 and d I → 0.
Proof. In view of Theorem 3.2 and Theorem 3.4, under our assumption, one can find two constants C 1 , C 2 > 0, independent of all small d S > 0, such that any EE (S d S , I d S ) of (3) satisfies By the well-known L p -theory for elliptic equations, we see that where C is allowed to vary from place to place but does not depend on any small d S > 0. Choosing r to be sufficiently large, it follows from the embedding theorem that I d S C 1+α (Ω) ≤ C for some 0 < α < 1.
where I ∈ C 1 (Ω) and I > 0 on Ω. Then for any given small > 0, it holds 0 < I − ≤ I i ≤ I + on Ω, for all large i.
Clearly for any fixed large i, S i is a subsolution of and a supersolution of It is easy to see that any large positive constant is a supersolution to (27), and any small positive constant is a subsolution to (28). Therefore, both (27) and (28) admit at least one positive solution. We next show the uniqueness of positive solution to both (27) and (28). We only consider (27) since (28) can be treated in the same way. Suppose that (27) have two positive solutions u * and u * * . Denote Ω + = {x ∈ Ω | u * (x) > u * * (x)}. We will show that Ω + = ∅. Otherwise, Ω + is a nonempty open set. To produce a contradiction, we define U = u * − u * * and choose a component of Ω + , say Ω 1 + . Then, for any fixed large i, elementary calculation gives that The maximum principle for elliptic equations ensures that U must attain its maximum at some point x + ∈ ∂Ω 1 + . Thus, x + ∈ ∂Ω ∩ ∂Ω 1 + . But the Hopf boundary lemma implies ∂U ∂ν (x + ) > 0, a contradiction. Hence, it is necessary that u * ≤ u * * on Ω. Similarly, one can conclude that u * ≥ u * * on Ω. Thus, u * ≡ u * * . This verifies the uniqueness.
Denote by u i and v i , respectively, by the unique positive solution to (27) and (28). Since S i is a subsolution and a large constant M with M > max x∈Ω S i is a supersolution of (27), together with the uniqueness, we know S i ≤ u i on Ω, for all large i.

Similar analysis shows that
By the similar argument to that [11,Lemma 2.4], one has Thus, due to the arbitrariness of , it is clear that By virtue of (26) and (29), it is not hard to see that I is a positive solution to the following elliptic problem Thus, we complete the proof.  where and S is the unique positive solution to the following elliptic equation Proof. By Theorem 3.2, there exist two positive constants C 1 , C 2 , independent of d I > 0, such that any EE (S d I , I d I ) of (3) satisfies So the standard L p -theroy for elliptic equations guarantees that S W 2,r (Ω) ≤ C for any given r ≥ 1, where C does not depend on d I > 0. Taking r to be properly large, we see from the embedding theorem that (Ω) ≤ C for some 0 < α < 1.
Hence, one can find a subsequence of d I → 0, say d i := d Ii , satisfying d i → 0 as i → ∞, and a corresponding positive solution sequence (S i , where S ∈ C 1 (Ω) and S > 0 on Ω. Then, given any small > 0, for all large i, we have 0 < S − ≤ S i ≤ S + , on Ω.
It is easily observed that, for fixed large i, I i is a subsolution of and is a supersolution of −d i ∆w = β(x)w p (S − ) q − δ(x)w, x ∈ Ω, ∂ ν w = 0, x ∈ ∂Ω.
As 0 < p < 1, it is further seen that any large positive constant is a supersolution to (32), and any small positive constant is a subsolution to (33), and hence (32) and (33) admit at least one positive solution. Next we will show the uniqueness of positive solution of (32). Suppose that there are two positive solutions u and u. We may choose a constant M 0 > 1 such that M −1 0 u < u, u < M 0 u on Ω. Thanks to 0 < p < 1, clearly M 0 u is a supersolution of (32) and M −1 0 u is a subsolution. Hence, the iteration argument of sub-supersolutions implies that there exist a minimal and a maximal solution in the order interval [M −1 0 u, M 0 u], which we denote by u min and u max , respectively. Thus u min ≤ u, u ≤ u max on Ω.
Thus it suffices to show that u min ≡ u max .
Notice that −d i ∆u min = β(x)u p min (S + ) q − δ(x)u min −d i ∆u max = β(x)u p max (S + ) q − δ(x)u max . We multiply the two equations by u max and u min , respectively, and then integrate by parts to obtain Ω β(x)(S + ) q [u p min u max − u min u p max ] = 0.

A DIFFUSIVE SIR EPIDEMIC MODEL IN A HETEROGENEOUS ENVIRONMENT 4515
That is, Ω β(x)(S + ) q u min u max [u p−1 min − u p−1 max ] = 0.
As u min ≤ u max on Ω, u p−1 min ≥ u p−1 max on Ω. The equality (34) implies that u min ≡ u max , and in turn u ≡ u on Ω. Denote by u i the unique positive solution of (32).
By a similar argument, we see that (33) has a unique positive solution, denoted by v i . A simple sub-supersolution analysis, together with the uniqueness, guarantees that v i ≤ I i ≤ u i on Ω, for all large i.
The proof of [11,Lemma 2.4] with some obvious modifications can be used to conclude that Thus, letting i → ∞ in (35) gives Combined with this and (31), one can easily see from the equation satisfied by S i that S solves (30). The same analysis as in the proof of uniqueness of positive solutions to (27) shows that (30) has a unique positive solution. The proof is thus complete.