DYNAMICS OF A NONLOCAL SIS EPIDEMIC MODEL WITH FREE BOUNDARY

This paper is concerned with the spreading or vanishing of a epidemic disease which is characterized by a diffusion SIS model with nonlocal incidence rate and double free boundaries. We get the full information about the sufficient conditions that ensure the disease spreading or vanishing, which exhibits a detailed description of the communicable mechanism of the disease. Our results imply that the nonlocal interaction may enhance the spread of the disease.


1.
Introduction. As seen in its long history, epidemic disease has caused orders of magnitude more deaths through out the world than wars or famines. In order to get a better understanding of the transmission mechanism, many researchers devoted to the mathematical modelling and dynamical behaviors of epidemic diseases [21]. We find that aforementioned models are strongly based on the homogeneity of the space. Concerning the fact that many physical aspects of the environment such as climate, chemical composition or physical structure can vary from place to place, as early as in 1957, Kendall [23] proposed the spatially dependent integro-differential SIR epidemic system to describe the spread of an epidemic disease in a one-dimensional habitat, where S, I and R obey the susceptible-infective-removed (SIR) scheme just as that showed in the Kermack-McKendrick epidemic equations; the constant β represents the contact rate and γ is the recovery rate. In addition, K(x − y) ≥ 0 denotes the probability density that weights the contributions of infectious at location y to the infection of susceptible individuals at location x. Moreover, Kendall [24] established that if βS0 γ > 1 and c ≥ c * , system (1) admits a traveling wavefront solution of the form 248 JIA-FENG CAO, WAN-TONG LI AND FEI-YING YANG (S(x + ct), I(x + ct), R(x + ct)) and has no such solution for c < c * , for which c denotes the wave speed, S 0 is the population density at the beginning of the epidemic with everyone susceptible and c * > 0 is called the minimal wave speed. We refer readers to Aronson [3] and Mollison [32] for some relevant progress on epidemic waves of (1). In particular, we refer to Ai et al. [1] about the existence and uniqueness (for sufficient large wave speed) of traveling wavefront solutions for spatial SIRS epidemic models by assuming that the kernel function K(·) is nonnegative, symmetric and satisfies +∞ −∞ K(x)dx = 1 and K(x) = 0 for |x| ≥ , where > 0 is small.
Concerning the fact that population individuals are distributed in space randomly, and typically interact with the physical environment and other organisms in their spatial neighborhood randomly, too, Mottoni et al. [33] added diffusion term into the Kendall model as K(x, y)I(y, t)dy, I t = d∆I + S Ω K(x, y)I(y, t)dy − γI (2) with Neumann boundary condition, where Ω ⊂ R N (N ≥ 1) is an open bounded domain with piecewise smooth boundary, K(·, ·) :Ω ×Ω → [0, ∞) is smooth, nonnegative (typically, with compact support), symmetric (i.e. K(x, y) = K(y, x) for any x, y ∈Ω), not identically equals to zero, and satisfies Ω K(x, y)dx ≤ 1 for any y ∈Ω. For (2), Mottoni et al. established the stability and attractivity of the stationary solutions, especially in the case that the initial data are spatially inhomogeneous.
It is worth pointing out that excepting the nonlocal interaction as in (2), Wang et al. [42] incorporated the time delay as a latent period into the spread of the diffusive Kermack-McKendrick epidemic disease, that is, the incidence rate is characterized by βS(x, t) t −∞ +∞ −∞ K(x−y, t−s)I(y, t)dyds, and obtained the threshold dynamics for the spread of the disease (see [42] for the detailed description about the kernel function K). For more relevant work on the existence of traveling waves of reactiondiffusion equations with nonlocal interaction and time delay, we refer readers to Ducrot et al. [15], Faria et al. [16] and references cited therein. We also refer to Lou [31] for some challenging mathematical problems in evolution of dispersal and population dynamics.
Recently, free boundary problems have been studied intensively in many fields. In particular, the well-known Stefan condition has been used to describe the spreading front in many applied problems. For example, it was used to describe the melting of ice in contact with water [35], the wound healing [6], the tumor growth [7] and so on. In order to get a more precise prediction of the location of the spreading front of an invading species, Du et al. [10] firstly studied the spreading-vanishing dichotomy of some invasion species which is described by a diffusive logistic model in the homogenous environment of one dimensional space. Since then, more results for more general free boundary problems have been obtained, for example, see [8,9,14,12,13,34,41] for single species model and [29,37,11,18,19,36,39] for Lotka-Vottera systems. In particular, Kim et al. [26] investigated the following free boundary diffusion SIR epidemic model where r = |x| and x ∈ R N , h(t) is the moving boundary to be determined, τ 1 , τ 2 , τ 3 are positive constants that denote the death rates of each class respectively, and µ 1 has the same meaning as that in (4). They proved some sufficient conditions that ensure the disease vanishing or spreading. For other related results, one refer to [22,30,17] for other epidemic models with free boundary and [28,38,46,43,44] for more relevant theoretical advances. It is noted that for the nonlocal diffusive SI epidemic model (2), if we add the recovery class term, that is, making (2) becomes following with µ > 0 denotes the death rate and γ > 0 represents the recovery rate, then we arrive at a nonlocal diffusive SIS epidemic model. Moreover, assume that N = S +I, and then adding the two equations in (4) reduces to x ∈ Ω, t > 0, Motivated by above, we will discuss the dynamics of the following nonlocal epidemic model with free boundary as x ∈ R, t > 0, where x = g(t) and x = h(t) are moving boundaries that represent the spreading frontiers of the disease, and will be determined together with (N, I); the parameter µ 1 > 0 can be understood as the diffusivity of the disease, i.e., the larger µ 1 is, then the easier that the disease can transmit to a new area; the initial functions N 0 and We see that problem (6)- (7) indicates that the whole class exhibit themselves in the whole area R, while the individuals which are infected occupy initial region [−h 0 , h 0 ] at the beginning of the stage and spread further into the environment from two ends of the initial region. The spreading frontiers expand at a speed that is proportional to the infectious gradient at the front, which gives rise to the Stefan conditions g (t) = −µ 1 I x (g(t), t) and h (t) = −µ 1 I x (h(t), t). Furthermore, S(x, t) R K(x, y)I(y, t)dy implies that the infectious at location y can contact with the susceptible individuals at location x and then make them become infectious, which gives rise to the nonlocal effect. This work can be regarded as a continuation of our previous paper [5] in which we investigated a diffusion SIRS model with double free boundaries and bilinear incidence, and got the sufficient conditions that ensure disease spreading or vanishing and a estimate for the spreading speed if the disease spreading happens.
The organization of this paper is as follows. In Section 2, we prove the general existence and uniqueness result, which implies in particular that problem (6)-(7) has a unique positive solution defined for all t > 0, the method is inspired by [10,8,12,26]. In Section 3, we firstly analyze an eigenvalue problem and discuss the property of its principal eigenvalue λ 1 . Then we define R F 0 (t) as a critical function and run our discussions by comparing R F 0 (0) with 1. Meanwhile, we propose a comparison principle for our free boundary problem which will be frequently used in this paper. Section 4 is concerned with some sufficient conditions that ensure the disease vanishing and Section 5 exhibits some conditions that make the disease spread. Finally, we give a brief discussion.
2. Solutions of (6) and (7). It is quite well understood now that the global existence and uniqueness of solutions of (6) are deduced from the local existence, uniqueness and a priori estimates, see [10,29,37,38]; and the local existence and uniqueness are obtained by contraction mapping theorem. Firstly, we introduce a standard hypotheses on kernel K(·, ·) as follows: x, y ∈ R with x = y), and satisfies R K(x, y)dx = 1 for any y ∈ R. We prove the following local existence and uniqueness result by the contraction mapping theorem and then use a priori estimates to show that the solution is defined for all t > 0. The proof can be done by modifying the arguments of [10] (see also [37,38]).
Noticing that for h 1 , h 2 ∈ H T and g 1 , (9) Next, we prove the existence and uniqueness result by using the contraction mapping theorem. First, for any given (u, v; g, h) ∈ D, we have which lead to 7 8 h 0 ≤ h(t), −g(t) ≤ 9 8 h 0 . Therefore, the transformation (r, t) → (x, t) as well as A, B and C are well defined. Applying standard L p theory and then the Sobolev embedding theorem [27], we can find that for any (u, v; g, h) ∈ D, the following initial boundary value problem Then we havẽ In what follows, we define a map F : It is clear that (u, v; g, h) ∈ D is a fixed point of F if and only if it solves (8). By (11) and (12), we see that Now, we are in the position to prove that F is a contraction mapping on D for T > 0 small. Just as in the proof of Theorem 2.1 in Du et al. [10], the left part relies on the L p estimates for parabolic equations and Sobolev embedding theorem, we will not repeat them again. Therefore, we obtain a unique fixed point (u, v; g, h) of operator F in D. Moreover, by the Schauder's estimate, we get additional regularity for (u, v; g, h) as a solution of (8). Namely, In other words, (u, v; g, h) is the unique classical solution of problem (8). This completes the proof.
To show that the local solution obtained in Theorem 2.1 can be extended to all t > 0, we need the following estimate.
Theorem 2.2. Assume that (K) holds and let (N, I; g, h) be the solution of problem (6) for t ∈ (0, T 0 ) with some T 0 ∈ (0, +∞]. Then, we have Therefore, we obtain that Note that the bound of N (x, t) immediately deduces that I(x, t) satisfies

JIA-FENG CAO, WAN-TONG LI AND FEI-YING YANG
whereĪ(t) is the solution of It is clear thatĪ(t) is continuous on t ≥ 0. In addition, we see thatĪ(t) is nondecreasing in t if I 0 L ∞ ≤ M 1 and decreasing if I 0 L ∞ > M 1 . Therefore, for any x ∈ (g(t), h(t)) and t ∈ (0, T 0 ), there is Applying the strong maximum principle and the Hopf lemma to the equations of N and I resulting that , h(t)) and t ∈ (0, T0).
It remains to show that h (t), −g (t) ≤ M 3 for all t ∈ (0, T 0 ) with some M 3 independent of T 0 . We just give the proof of h(t), the proof for g(t) is similar. As in Du et al. [10], define in which M > 0 is a constant that will be chosen later. For any (x, t) ∈ Ω M , we haveĪ On the other hand, for x ∈ [h 0 − M −1 , h 0 ], there arē then the maximum principle deduces thatĪ(x, t) ≥ I(x, t) for (x, t) ∈ Ω M . Therefore, we acquire that I x (h(t), t) ≥Ī x (h(t), t) = −2M M 2 and then This completes the proof. Proof. It follows from the uniqueness of the solutions of (6) that there is some T max > 0 such that [0, T max ) is the maximal time interval that the solution exists. It remains to show T max = ∞. We will get our conclusion by deriving a contradiction. Suppose that T max < ∞, then there exist positive constants M 1 , M 2 and M 3 independent of T max such that with the help of Theorem 2.2. According to the standard L p estimates, the Sobolev embedding theorem and the Hölder estimates for parabolic equations, for some fixed γ * ∈ (0, T max ), we can find a positive constant M 4 depending on γ * , M 1 , M 2 and for all t ∈ [γ * , T max ). Then, according to the proof of Theorem 2.1, there is a τ > 0 depending on M i (i = 1, 2, 3, 4) such that the solution of problem (6) with initial time T max − τ 2 can be extended uniquely to T max − τ 2 + τ , which contradicts to the definition of T max . Thus, our results follow.
We note that Theorem 2.2 implies that free boundaries h(t) and −g(t) are increasing in time and bounded, then 3. Critical function and comparison principle. In this section, we mainly analyze an eigenvalue problem and discuss the property of its principal eigenvalue, and then propose a function R F 0 (t) as a critical term to determine the spreading or vanishing of the disease. These results will play an important role in discussing the main results and have their own interests.
Firstly, let us consider the following eigenvalue problem where Ω ⊂ R is a bounded domain with ∂Ω of class C 2+α . It is well-known that (13) admits a unique principal eigenvalue (Krein-Rutman theorem), denoted by λ 1 (Ω), and there is a positive eigenfunction φ 1 with φ 1 L ∞ = 1 corresponding to λ 1 (Ω). By the variational method, λ 1 (Ω) can be characterized by Thus, the following property holds.
It follows from Theorems 2.2 and 3.1 that R F 0 (t) enjoys the following property.
Remark 2. Note that in [25,28], they could give the exact limit of R F 0 (t) which is defined in as t → +∞ and h ∞ = ∞. For (19), we have Hence, we obtain that R F 0 (∞) < βσ µ(µ+γ) due to Remark 1. In other words, we have R F 0 (t) < βσ µ(µ+γ) for all t > 0, which is different from the one that the infective class is just in contact with the adjacent individuals, see for example [25].
In what follows, we will regard R F 0 (t) as a critical term to run our discussions. Many conclusions below mainly depend on the constructions of some suitable upper and lower solution and then the comparison principle for free boundary problem is essential here. The following comparison principle is a similar result just as in [10]. 4. Conditions for vanishing. In this section, we give some sufficient conditions that ensure the disease vanishing. If h ∞ , −g ∞ < ∞ and lim t→+∞ I(·, t) C([g(t),h(t)]) = 0, we say that the disease vanishing happens; while if h ∞ , −g ∞ = ∞, then the domain (g(t), h(t)) becomes the whole region R, in this case, we say that the disease spreading happens.
Proof. First of all, we show that for any integer n ≥ 0 and 0 < α < 1, there exists a constantC =C(µ 1 , g ∞ , h ∞ , M 3 ) with M 3 is defined in Theorem 2.2 such that Our proof here is motivated by Theorem 2.1 in Wang [41] (see also Theorem 4.1 in [40]). We straighten the free boundaries by the following transformations. That is, let and K(x, y) = K( (h(t)−g(t)) 2 s + (h(t)+g(t)) 2 , y) =K(s, y) withK satisfying the condition (K). Thus, we arrive at and Then direct calculations show that v(s, t) satisfies where E = E(g(t), h(t), s) and F = F (g(t), h(t)). It is clear that (22) is an initialboundary value problem with fixed boundary. For any integer n ≥ 0, define u n (s, t) = u(s, t + n) and v n (s, t) = v(s, t + n).

Then problem (22) becomes following
where E n = E(t + n) and F n = F (t + n). With the help of Theorem 2.2, we note that u n , v n , E n and F n are bounded uniformly on n, and max 0≤t1,t2≤3, |t1−t2|≤τ with h n (t) = h(t + n) and g n (t) = g(t + n). In addition, we have F n ≥ 4 (h∞−g∞) 2 for all n ≥ 0 and 0 ≤ t ≤ 3 as if h(t) ≤ h ∞ < ∞ and g(t) ≥ g ∞ > −∞.
Choosing p 1, we can use the interior L p estimate to derive that there exists a positive constant C * independent of n such that v n and 0 < h (t), −g (t) ≤ M 3 allows us to derive (21). Since these rectangles I n overlap and constant C * is independent of n, it follows that Moreover, we arrive at g (t) → 0, h (t) → 0 as t → ∞ since g(t) and h(t) are bounded. Now we return to the proof of lim t→+∞ I(·, t) C([g(t),h(t)]) = 0. Suppose to the contrary that lim sup t→+∞ I(·, t) C([g(t),h(t)]) = > 0. Then there exists a sequence (x k , t k ) in (g(t), h(t)) × (0, +∞) such that I(x k , t k ) ≥ 2 for all k ∈ N and t k → ∞ as k → ∞. Since −∞ < g ∞ < g(t) < x k < h(t) < h ∞ < ∞, we then have that a subsequence of {x k } converges to x 0 ∈ (g ∞ , h ∞ ). Without loss of generality, we assume x k → x 0 as k → ∞. Define N k (x, t) = N (x, t k + t) and I k (x, t) = I(x, t k + t)

DYNAMICS OF A NONLOCAL SIS EPIDEMIC MODEL 259
for (x, t) ∈ (g(t k + t), h(t k + t)) × (−t k , ∞). It follows from the parabolic regularity theory that {(N k , I k )} has a subsequence {(N ki , I ki )} such that (N ki ,
, where M 1 is the super bounds of N (x, t) obtained in Theorem 2.2 and M * = 4 3 I 0 L ∞ . Proof. We are going to construct a suitable upper solution to (6) and then apply Lemma 3.3 to prove this theorem. As in [10,26], we define where η and M * are positive constants which will be chosen later. We notice that . On the other hand, we have Then Thus, following from Lemma 3.3, we have g(t) ≥ ϑ(t) and h(t) ≤ δ(t) for t > 0. An immediate result is h ∞ , −g ∞ < lim t→∞ δ(t) = 4h 0 < ∞. This completes the proof.
Remark 3. Compared Theorem 4.2 with Lemma 4.1 in [26], due to the effect of the nonlocal term, we find that our result shows that the disease will die out eventually with a smaller h 0 but with no restriction on the relation between βσ µ(µ+γ) and 1.
By the monotonicity and continuity of R F 0 (t), we find that if R F 0 (0) < 1, there exists some t 0 > 0 such that R F 0 (t 0 ) = 1. Thus, for any fixed t * ∈ (0, t 0 ), we have R F 0 (t * ) < 1 and λ 1 (t * ) > 0. By repeating the proof of Theorem 4.3 in which h 0 and λ 1 (0) are replaced by h(t * ) and λ 1 (t * ) respectively, we can obtain that h ∞ , −g ∞ < ∞ if R F 0 (t * ) < 1 as well as I 0 L ∞ and h 0 are sufficiently small. In addition, theorems 4.2 and 4.3 show that the disease vanishes in the long run with an exponential decay as long as the initial values and initial region are sufficiently small.

5.
Conditions for spreading. This section is devoted to the sufficient conditions that make the disease spread. For the case of R F 0 (0) ≥ 1, we have the following spreading result.
6. Discussion. As far as we concerned that the threshold for many epidemiology models is the so-called basic reproduction number R 0 , which is defined as the average number of secondary infections produced when one infected individual is introduced into a host population where everyone is susceptible. It is shown in [2] (the basic reproduction number is defined for the involved SIS PDE model there) that regardless of the initial values and initial habit region, if R 0 < 1, the unique disease-free equilibrium is globally asymptotic stable and if R 0 > 1, the disease-free equilibrium is unstable and there is a unique endemic equilibrium, see also [20] and the references therein.
Our results here indicate that the spreading of the disease is not only determined by the reproduction number, but also the initial size [−h 0 , h 0 ] as well as the initial functions S 0 (resp. N 0 ) and I 0 , which seems more reasonable and acceptable. Meanwhile, we note that R F 0 (t) defined as a critical function and R F 0 (0) defined as a threshold are not bigger than βσ µ(µ+γ) , which gives rise to a phenomenon that the disease spreads more easily due to the free boundary in the senses that: if R F 0 (0) ≥ 1, then lim t→+∞ I(·, t) C([g(t),h(t)]) > 0 and h ∞ , −g ∞ = ∞; if R F 0 (0) < 1, then the disease will spread as time elapses if N 0 L ∞ and I 0 L ∞ are sufficiently large. Precisely, due to the appearance of the nonlocal effect, we obtain that the disease becomes more easier to spread, see Remark 1. That is, the nonlocal interaction may enhance the spread of the disease.