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Concentration phenomenon of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic model with spontaneous infection

In this paper, we investigate the effect of spontaneous infection and advection for a susceptible-infected-susceptible epidemic reaction-diffusion-advection model in a heterogeneous environment. The existence of the endemic equilibrium is proved, and the asymptotic behaviors of the endemic equilibrium in three cases (large advection; small diffusion of the susceptible population; small diffusion of the infected population) are established. Our results suggest that the advection can cause the concentration of the susceptible and infected populations at the downstream, and the spontaneous infection can enhance the persistence of infectious disease in the entire habitat.


1.
Introduction. Classical epidemic models assume that the entire population lives in one area and is well mixed. This assumption is not necessarily satisfied; for instance, the population may be living on isolated islands or in different countries while traveling from one location to another. This spatial heterogeneity affects the transmission of the disease. Spatially explicit models are also more effective in evaluating control strategies, particularly those applied to movement of individuals; one may refer to [32] for related discussion. To understand the impact of spatial heterogeneity on the dynamics of a disease, an increasing number of reaction-diffusion models have been proposed and studied in recent years. In particular, the SIS epidemic reaction-diffusion models have been developed to investigate the effects of the spatial heterogeneity and the movement of populations on the persistence or extinction of diseases, see, for instance, [1,4,10,13,15,18,22,24,25,26,35,36,37,38,39,41,43].
In the SIS epidemic models as mentioned above, infection is considered to be only transmitted by the contact between the infected and the susceptible populations. However, according to [16,17], the social infection can also arise because of spontaneous factors other than transmission. For such a reason, the term whereby uninfected individuals turn into infected at the rate can be added to extend the primary SIS models, so that the effect of spontaneous infection can be incorporated. On the other hand, in some heterogeneous environments, the movement of populations may be subjected to some passive directions definitely as in [5,33], which can be considered as external natural forces like the wind as [9] or streamflow as [21,27,28,29,30], etc. Therefore, it is important to explore how the advection and spontaneous infection affect the persistence and extinction of the infectious diseases modeled by an SIS system under consideration.
In this paper, we are concerned with the following reaction-diffusion-advection system with spontaneous infection over the interval habitat [0, L] in a spatially heterogeneous environment: (S(x, 0), I(x, 0)) = (S 0 (x), I 0 (x)) ≥ (0, 0), I 0 (x) ≡ 0, 0 < x < L. (1) Here, the unknown functions S(x, t) and I(x, t) represent, respectively, the population densities at position x ∈ [0, L] and time t > 0 of the susceptible individuals and infected individuals, and the positive constants d S and d I are their respective diffusion coefficients; the functions β(x) and γ(x) are Hölder continuous positive functions in [0, L] and stand for the rates of epidemic transmission and recovery at position x, respectively; the function η(x) is also positive and Hölder continuous while it stands for the spread of the disease according to the imported cases of the infectious disease. It is noteworthy that this kind of import can be considered as brief contacts with individuals who are not included in the population in [17,19,34]; meanwhile, one can refer to [2,16,40,44] regarding some other different explanations for η. The positive constant q represents the advection speed of a stream or wind, which carries the susceptible and infected populations from the upstream x = L to the downstream x = 0. This boundary value condition assumes the no-flux situation at the downstream and the upstream. Given initial data S 0 , I 0 , thanks to the boundary value conditions, one can integrate the sum of the equations of (1) to yield the conservation law: We would like to mention that the non-advection version (q = 0) of (1) was proposed and studied by Tong and Lei [42] and the non-spontaneous-infection version (η = 0) was investigated in [6,20]. More precisely, the systems in [42] and [6,20] read, respectively, as the following and The SIS reaction-diffusion-advection model with mass action infection mechanism was considered in [7]. In the present paper, our main goal is to understand the combined influence of the advection and spontaneous infection on the persistence, extinction, and spatial distribution of the infectious diseases modeled by system (1). To study the spatial distribution of the infectious diseases, we are then led to analyze the steady-state (i.e., stationary) solution problem corresponding to (1). The steady-state problem of (1) consists of a pair of nonlinear ordinary differential equations: constrained with the boundary value conditions and the conservation law associated with (2) L 0 (S(x) + I(x)) dx = N.
From now on, we set N to be a given positive constant. From the ecological point of view, we are only interested in nonnegative steadystate solutions. With regard to (5) and (6), a solution (S, I) with I = 0 is called a disease-free equilibrium (DFE), whereas there also is an endemic equilibria (EE), which is the componentwise positive solution. However, it is easy to see that no DFE can exist in (5) and (6) due to the presence of spontaneous infection term in (1).
First of all, we will prove the existence of EE via the abstract theory dynamical system combined with a priori estimate of solutions to (1), which indeed shows the uniform persistence of system (1); see Theorem 2.1. Then, we study the spatial profile of EE in the following three cases: Our first main result in this regard concerns the asymptotic behavior of EE as q → ∞, and our conclusion reads as follows.
Theorem 1.1. Fix d S , d I and denote by (S(x, q), I(x, q)) any EE of (5)- (6). Then, as q → ∞, we have • (S(x, q), I(x, q)) → (0, 0) locally uniformly in (0, L]; • at the downstream, Our second main result on the asymptotic behavior of EE as d S → 0 can be stated as follow. Theorem 1.2. Fix q, d I and denote by (S(x, d S ), I(x, d S )) any EE of (5)- (6). Then, as d S → 0, we have Dirac measure centered at 0. Here (Ŝ,Î) is a positive solution of the following system: with L 0 [Ŝ(x) +Î(x)] dx +N = N and 0 <N < N . Our last main result focuses on the asymptotic behavior of EE as d I → 0. Indeed, we can state the following. Theorem 1.3. Fix q, d S and denote by (S(x, d I ), I(x, d I )) any EE of (5)- (6). Then, as d I → 0, we have • d I I(d I y, d I ) → C I e −qy (0 < C I < N q) uniformly for y in any compact subset of [0, ∞); • (S(·, d I ), I(·, d I )) → ( S(·), I(·) + N δ 0 ) weakly in L 1 ((0, L)), where δ 0 is the Dirac measure centered at 0. And ( S, I) is a positive solution of the following system: The implication of Theorems 1.1-1.3 in the control strategy of infectious disease will be discussed in the conclusion section.
Indeed, compared to [6,20], the presence of the spontaneous infection causes several mathematical difficulties in the establishment of fine lower and upper positive bounds of the two components S and I with respect to large advection or small diffusion rates, as well as in the analysis of the asymptotic behaviors of S and I. We have to introduce some new techniques to overcome those difficulties. More interestingly, the incorporation of the spontaneous infection can produce some very different spatial distribution behaviors of the disease, as discussed in the final section.
The paper is organized as follows. In section 2, we will prove the existence of EE. Section 3 is devoted to the analysis of the asymptotic profiles of the EE, and Theorems 1.1-1.3 are proved there. Section 4 ends the paper with a discussion.
2. The existence of EE. In this section, we establish the uniform persistence of solution to system (1) for any given parameters d S , d I , β, γ, η and q, which in turn yields the existence of EE of (5)-(6).
Theorem 2.1. There exists a positive constant 0 independent of the initial data uniformly for x ∈ [0, L], which means that the disease persists uniformly. Furthermore, (5)-(6) admits at least one EE.
Proof. For any positive solution (S, I) of (1), we make the transformations u(x, t) = e q d S x S(x, t), v(x, t) = e q d I x I(x, t). Then system (1) becomes equivalent to the following one: and (u, v) satisfies By (10), for any solution (u(x, t), v(x, t)), it is clear that u(·, t) L 1 ((0,L)) + v(·, t) L 1 ((0,L)) is bounded, independent of t ∈ [0, ∞). Now, in view of [11, Lemma 2.1] with p 0 = 1, q = 2 and = 0, we can deduce the bound of u(·, t) L ∞ ((0,L)) and v(·, t) L ∞ ((0,L)) , which is independent of t ∈ [0, ∞). We want to point out that [11,Lemma 2.1] holds in the presence of the advection terms; this is a straightforward consequence of [12, Theorem 1 and Corollary 1]. That is, there is a positive constant C which is independent of the initial data (u(x, 0), u(x, 0)) with Based on such a priori estimates, one can easily follow the abstract theory developed in [31,45] to assert the uniform persistence property and the existence of EE for system (9); see, for example, [23, Theorem 3.1, Theorem 5.1]. The details are omitted here. Equivalently, we have obtained the desired conclusion of the theorem due to the transformation u( 3. Asymptotic profiles of EE: proof of Theorems 1.1-1.3. In this section, we first present a priori estimates for any EE of (5)-(6) and then prove the three main results: Theorems 1.1-1.3.

3.1.
A priori estimates. This subsection is devoted to a priori estimates for any EE of (5)- (6). We start with the estimates of the lower and upper bounds of the integral of the S-component and I-component for any EE (S, I).
Lemma 3.1. Let (S, I) be any EE of (5)-(6). Then we have and 0 < N min Proof. According to the boundary condition (5c), we integrate (5a) or (5b) over (0, L) and find that Similarly, we get Using (13), we can apply a similar analysis as above to assert that The proof is complete. Now we will establish the pointwise estimate for the S-component of any EE (S, I) of (5)-(6). Lemma 3.2. Let (S, I) be any EE of (5)- (6). Then the following assertions hold.
(i) For any q, d S , d I > 0, it holds that Proof. We first verify (i). We apply a similar technique introduced by Cui and Lou [8,Lemma 2.5]. Let where A is a positive constant which will be determined later. Hence, Thus (5a) reduces to the following equation for x ∈ (0, L). By (14) and (15), the boundary condition (5c) becomes In (14), we choose where C is a positive constant to be determined below. Then by (17), we see that w(x). We prove x * = 0 when C is suitably chosen. Clearly, the boundary condition (17) implies x * = L. To produce a contradiction, we suppose that 0 < x * < L. Since w xx (x * ) ≥ 0 and w x (x * ) = 0, one deduces from (16) that This infers that Therefore, a contradiction occurs if we take C = max η(x). This contradiction enables us to conclude that x * = 0, that is, As a result, by (14), we derive the desired estimate: We now verify (ii). Integrating (5a) over (0, x), we obtain By (6), we should also notice that That is, we have Then the function Integrating the above inequality over (0, x), we obtain which means, In light of the expression of M , the desired estimate is proved.
Using a similar analysis to that of Lemma 3.2, we can establish the pointwise estimate for the I-component of any EE (S, I) of (5)- (6). More precisely, we have Lemma 3.3. Let (S, I) be any EE of (5)- (6). Then the following assertions hold.
(i) For any q, d S , d I > 0, it holds With the help of Lemmas 3.2 and 3.3, we can obtain the upper estimates for any EE (S, I) of (5)- (6). Precisely, we can conclude that Lemma 3.4. Let (S, I) be any EE of (5)- (6). Then the following assertions hold.
(i) For any q, d S , d I > 0, we have (ii) For any q, d S , d I > 0, we have Proof. According to (i) of Lemma 3.2, we obtain Integrating this inequality over (0, L), we then get Applying the same procedure to the inequality (i) of Lemma 3.3 leads to Due to the conservation law (6), we can easily find Hence, for any d I , d S , q > 0, we obtain and Now, substituting (18) and (19) into the inequalities in Lemma 3.2(ii) and Lemma 3.3(ii) respectively, we deduce the desired results.

3.2.
Profile of EE as q → ∞: proof of Theorem 1.1. In order to explore the profile of any EE (S(x, q), I(x, q)) of (5)-(6) as q → ∞, we make use of the following transformations: u(y, q) := 1 q S y q , q , v(y, q) := 1 q I y q , q , 0 ≤ y ≤ qL.
Now we present the proof of Theorem 1.1.
Proof of Theorem 1.1. First of all, by (i) and (ii) of Lemma 3.4, it is easily seen that S(x, q) and I(x, q) converge to zero locally uniformly for x ∈ (0, L] as q → ∞, respectively. It remains to explore the limiting behaviors of S(x, q) and I(x, q) at the downstream x = 0. Our proof consists of four steps as follows.
Step 1: Convergence of {(u(y , q), v(y , q))}. It follows from (i), (ii) of Lemma 3.4 that for any fixed K > 0, when q satisfies q > K/L, Therefore, for any fixed K > 0, both {u(y , q)} and {v(y , q)} are uniformly bounded in [0, K] with respect to q satisfying q > K/L. Thus, for a given constant K > 0, the standard L p -estimates (up to the boundary y = 0) for elliptic equations (see [14]) ensure that for some positive constant M = M (K, p) and any p > 1. Moreover, the constant M does not depend on all q > K/L and is allowed to vary when necessary. Due to the embedding theorem (by choosing p to be properly large), we have for some α ∈ (0, 1). Because of the above estimates, we can apply the Ascoli-Arzelà theorem, combined with a diagonal argument, to find a positive sequence {q n } with lim n→∞ q n = ∞ and a pair of nonnegative functions (u * , v * ) such that u n (y) := u(y, q n ) and v n (y) := v(y, q n ) satisfy As (u n , v n ) solves (20)-(21), we let n → ∞ to find that (u * , v * ) satisfies the limiting system (in the weak sense and then in the classical sense) with the boundary conditions By solving the whole limiting system (23), one has for some constants C S , C I ≥ 0. That is, it follows from (22) that Step 2: C S , C I > 0. Employing the change of variable x = y q n in the second inequality of (11), we obtain This shows Similarly, we can prove C I > 0 using the equations satisfied by v n .
Letting n → ∞ in the above equality allows us to know that In what follows, we will present the proof of Theorem 1.2.
Proof of Theorem 1.2. For any EE (S(x, d S ), I(x, d S )) of (5)-(6), we make the transformations: Then, for y ∈ (0, d −1 S L), (5)-(6) will become the following with the boundary conditions and the conservation law For our purpose, we also make the following transformations: Thus the original system (5) is reduced to the equivalent one: The remaining argument consists of five steps.
Step 1: Estimates of u and v for small d S . Firstly, from (i) of Lemma 3.4, we can immediately notice that d S S(·, d S ) is uniformly bounded on [0, L] with respect to d S for 0 < d S ≤ 1. Thus, according to the transformations (26), for any fixed constant K > 0, one can find a positive constant C such that Hereafter, the positive constant C may be different from line to line but is independent of all 0 < d S < L/K. Fix K > 0, we need to estimate v(· , d S ) C([0,K]) . Using system (29), we know that Appealing to [14,Theorem 8.18], we can see that and so there holds Hence, in view of (30) and (32), one can apply the standard regularity of elliptic equations to (27a), (27b) and (27c) to conclude that, for any fixed constant K > 0, for some constant α ∈ (0, 1).
Step 3: lim Since we now have the positive lower bound δ of S(·, d S ), we will use the similar technique as in the proof of Lemmas 3.2 and 3.3 to establish the improved upper bound of S(·, d S ) when d S > 0 is small. By taking where the positive constant C to be chosen later, we notice that w satisfies for x ∈ (0, L) with the boundary conditions w(x, d S ). It is easily seen that x * = L. Suppose 0 < x * < L. Because of w xx (x * , d S ) ≤ 0 and w x (x * , d S ) = 0, the equation (39) gives Remember that max for all small d S > 0. Hence, we have By choosing and requiring it can lead to a contradiction against (40). This implies that x * = 0 for any d S > 0 satisfying (42) with C being defined in (41). In conclusion, for all small d S , we have w(x, d S ). This and (38) lead to Finally, making use of (18), we derive for all small d S and x ∈ [0, L]. Therefore, (43) implies that S(x, d S ) converges to zero locally uniformly in (0, L] as d S → 0, which contradicts our previous hypothesis that S(x, d S ) ≥ δ > 0, ∀x ∈ [0, L], for all small d S > 0. So it is necessary that min We next provex = L. Since in Step 2 we have already proved that it is easy to seex > 0. Supposex ∈ (0, L). Thenx d S ∈ (0, L) for all small d S > 0. Due to the definition ofx d S , S xx (x d S , d S ) ≥ 0 and S x (x d S , d S ) = 0, we are able to deduce from (5a) that as d S → 0. This is impossible because of (31). Thus,x = L. In fact,x d S ∈ (0, L) for all small d S > 0. We apply the inequality (44) to derive a contradiction. Hence, it must hold thatx d S = L. This implies thatŜ(L) = 0.
Step 4: (Ŝ,Î) is a classical solution of (8). To find the equations satisfied by (Ŝ,Î), we fix any small ε > 0. Take any ϕ ∈ C 2 ([ε, L]) such that ϕ = 0 near ε and L. Multiplying the S-component equation by ϕ and integrating by parts, we can deduce Sending d S → 0 and according to the arbitrariness of ε, we find thatŜ is a nonnegative nontrivial solution (in the weak sense and then in the classical sense) of Here, we used the fact thatÎ ∈ C([0, L]),Î > 0 on [0, L], and the claim proved in Step 3. Similarly, one can show thatÎ is the classical solution of Next, we will determine the boundary condition ofÎ at 0. To this end, we denotê Here we used (37). For any fixed small ε ∈ (0, L), we integrate the equation of I-component over [0, ε] to deduce It follows from the mean value theorem that for some x ε ∈ (0, ε). Here, we see from (48) and (49) that by first sending d S → 0 and then ε → 0. Now, given ϕ ∈ C([0, L]), for any small ε > 0, it is clear that Then, letting d S → 0 first and then ε → 0, and using (47), we find That is, S(·, d S ) →Ŝ(·) +N δ 0 weakly in L 1 ((0, L)), where δ 0 is the Dirac measure centered at 0. Furthermore, we use the fact thatÎ is positive on [0, L] and (45) to conclude that S > 0 on [0, L).
Step 5:Ĉ S < N q andN > 0. By virtue of (28) and (34), one can observe that Hence,Ĉ S < N q. Moreover, adding up the first two equations in (45) and (46) and integrating over [x, L], together with the conditions ofŜ,Î at x = L, we have By (50) and (51), it holds that which yields that One can see from this equality thatN > 0 due toŜ(0),Î(0) > 0. The proof of Theorem 1.2 is now complete.
For y ∈ (0, d −1 I L), (5)-(6) will become the following with the boundary conditions and the conservation law Henceforth, the positive constant C may be different from line to line but is independent of all 0 < d I < L/K.
Fix K > 0. From system (29) it follows that [14,Theorem 8.18], it holds that S(x, d I ) ≤ C, and so max Hence, by (55) and (57), one can apply the standard regularity of elliptic equations to (53a),(53b) and (53c) to deduce that, for any fixed constant K > 0, for some constant α ∈ (0, 1). In view of (58), there exists a sequence of d I , denoted by itself, and the corresponding positive solution (u d I (y), v d I (y)) := (u(y, d I ), v(y, d I )) such that Since (u d I , v d I ) solves (53)-(54), we can easily find that ( u, v) satisfies with the boundary conditions Solving (60a), (60b) and (60c), we deduce u(y) = C S , v(y) = C I e −qy , y ≥ 0 for some nonnegative constants C S , C I . Arguing as in Step 2 in the proof of Theorem 1.1, one can see that C S > 0, C I > 0. Especially, S(0, d I ) → C S > 0 as d I → 0 because of (52) and (59). Using the equation of S * and the L 1 -estimate theory for elliptic equations (see, for instance, [3]), we can obtain that, for any given p ∈ [1, ∞), S * (·, d I ) W 1,p ((0,L)) ≤ C, and then S(·, d I ) W 1,p ((0,L)) ≤ C. Thus, passing to a subsequence of d I , denoted still by itself for convenience, we have As I is locally uniformly bounded in (0, L] due to Lemma 3.4, for any fixed small ε > 0, after passing to a subsequence of d I → 0, one may assume that I(·, d I ) → I(·) weakly in L 2 ((ε, L)), where I ∈ L 2 ((ε, L)) and I ≥ 0 a.e. in (ε, L). Since and S is the classical solution of For any fixed small ε ∈ (0, L), we integrate the equation of S-component over [0, ε] to deduce It follows from the mean value theorem that for some x ε ∈ (0, ε). Owing to the uniform convergence above, (65) and (66) imply that , for any small ε > 0, it is clear that Letting d I → 0. Then ε → 0, and using (64), we derive This means that I(·, d I ) → I(·) + N δ 0 weakly in L 1 ((0, L)), where δ 0 is the Dirac measure centered at 0. Moreover, by the fact that S is positive on [0, L], one sees that I > 0 in [0, L) using (62).
In light of(54) and (59), one can further obtain that This yields C I < N q. Finally, adding up the first two equations in (62)

Conclusion.
In this paper, we study an SIS epidemic reaction-diffusion model (1) with spontaneous infection and advection in one-dimensional space. We first prove the uniform persistence property of the solution using the abstract theory of dynamical systems, which then implies the existence of endemic equilibrium (EE). We focus on the asymptotic behavior of EE in three cases: large advection; small diffusion of the susceptible population; small diffusion of the infected population. Our results suggest that the asymptotic profiles of the susceptible and infected populations behave differently from that of the corresponding system without advection and that of the system without spontaneous infection. In what follows, we would like to compare the results of those models and reveal the effect of advection and spontaneous infection of either the susceptible or infected population on the spatial distribution of disease.
(i) Case of large advection. When the advection coefficient q becomes sufficiently large, Theorem 1.1 shows that the densities of both the susceptible and infected populations only concentrate at the downstream x = 0 behaving like delta functions with some masses at the downstream. These masses are determined by the conservation law and the integral equality in (7). Similar results hold for (4); see [20,Theorem 3.1] and [6,Theorem 1.2]. This shows that increasing the advection rate can not help to extinguish the infectious disease.
(ii) Case of small diffusion of the susceptible. When the mobility of the susceptible population is controlled to be small, we know from Theorem 1.2 that for model (1), the density of the susceptible population is positive on the domain except at the upstream, where it can reach its minimum value zero, and also concentrates at the downstream. But the density of the infected population tends to be positive on the entire habitat. Comparing to [42,Theorem 4.1] for (3), we see that the existence of advection will trigger the concentration phenomenon of the susceptible at the downstream while leading to its extinction at the upstream. In addition, the infected population for models (1) and (3) persist in the entire habitat.
On the other hand, when comparing to [20,Theorem 3.2] for model (4), one will notice that if the spontaneous infection is incorporated, the infected population persists in the entire habitat if the mobility of the susceptible population is controlled to be small. Thus, the presence of the spontaneous infection enhances the persistence of the disease.
(iii) Case of small diffusion of the infected. When the diffusion rate of the infected is restricted to be small, the density of the infected population modeled by system (1) is positive everywhere except at the upstream and concentrate at the downstream while the density of the susceptible population will stay positive on the whole domain. As far as system (3) is concerned, [42,Theorem 4.2] shows that the densities of both susceptible and infected populations remain positive everywhere. Hence, whether there is advection or not, the disease can not be eradicated if the diffusion of the infected is controlled to be small for both models (1) and (3). However, for (4), the disease will become extinct once we restrict the mobility of the infected, as shown by [20,Theorem 3.3]. In other words, the appearance of spontaneous infection can enhance the persistence of infectious disease.
In view of the above discussion, we may conclude that both spontaneous infection and advection play important roles in the spatial distribution of both the susceptible and infected populations: the advection can cause the concentration of the susceptible and infected populations at the downstream, and the spontaneous infection can enhance the persistence of infectious disease.