TRAVELING WAVES FOR A NONLOCAL DISPERSAL SIR MODEL WITH GENERAL NONLINEAR INCIDENCE RATE AND SPATIO-TEMPORAL DELAY

. In this paper, we study a nonlocal dispersal SIR model with general nonlinear incidence rate and spatio-temporal delay. By Schauder’s ﬁxed point theorem and Laplace transform, we show that the existence and nonexistence of traveling wave solutions are determined by the basic reproduction number and the minimal wave speed. Some examples are listed to illustrate the theoretical results. Our results generalize some known results.


1.
Introduction. As the pioneering work, Schaaf [30] studied the existence of traveling wave solutions in two scalar reaction-diffusion equations with delay. Then, the traveling wave solutions of reaction-diffusion equations with delay have attracted significant attention, see [5,18,24,28,32,44].
Since population takes time to move in space and usually is not at the same position in space at previous time, Britton [6,7] proposed nonlocal delay or spatiotemporal delay. Later, some researchers paid attention to the existence of traveling wave solutions for reaction-diffusion equations with spatio-temporal delays, see [2,16,26,36]. For example, Gourley and Ruan [17] investigated the existence of traveling front solutions for a two-species competition model with nonlocal delays by employing linear chain techniques and geometric singular perturbation theory. Wang et al. [33] established the existence of traveling wave fronts in reactiondiffusion systems with spatio-temporal delays by using monotone iterations. Yu and Yuan [42] considered more general reaction-diffusion systems with distributed and spatio-temporal delays by a new monotone iterations. Recently, Wang et al. [35] incorporated spatio-temporal delay into the classical Kermack-McKendrick model and considered the following nonlocal model ∂I(x, t) ∂t = d2 I(x, t) + βS(x, t)K * I(x, t) − γI(x, t), J(x − y)(u(y, t) − u(x, t))dy, which gives the probability that a particle or agent at location y will jump to location x. For more results on traveling wave solutions of nonlocal dispersal problems, see [40,25,29,38,39]. To our knowledge, there are few works on nonlocal dispersal SIR model with spatio-temporal delay. More rencently, Cheng and Yuan [10] investigated the existence and nonexistence of traveling wave solutions for a nonlocal dispersal SIR model with bilinear incidence rate and spatio-temporal delay. Wang et al. [34] considered the existence and nonexistence of traveling wave solutions of a nonlocal dispersal SIR model with standard incidence rate and spatio-temporal delay.
The organization of this paper is as follows. In Section 2, we establish the existence of traveling wave solutions as R 0 > 1 and c > c * . In Section 3, we prove the nonexistence of traveling wave solutions. In Section 4, some examples are provided to illustrate the main results. The paper ends with a brief conclusion in Section 5.
2. Existence of traveling waves. In this section, we discuss the existence of traveling wave solutions of (2).
Since R does not appear in the first two equations of (2), we let ξ = x + ct and consider the reduced system where Also, we need the following asymptotic boundary conditions where S(±∞) = lim ξ→±∞ S(ξ) and I(±∞) = lim ξ→±∞ I(ξ).
Suppose that the kernel functions J(x) and K(x, t) satisfy the following conditions.
(ii) For any c > c * , we have In the following, we assume that R 0 > 1 and c > c * . Define some continuous functions as follows: where σ, α, η, M are positive constants to be determined in the following lemmas. By the conditions (H1) and (H2), we conclude that there exists constant K such that f (S 0 )g(K) = γK.
Proof. For any given (φ(·), ϕ(·)) ∈ Γ X , we first show It is easy to check that S 0 is the upper solution of (9). Thus, for ξ = ξ 3 . Using the Sturm Comparison Theorem, we conclude that Lemma 2.6. The operator F : Γ X → Γ X is completely continuous.
Theorem 2.7. For the operator F : Γ X → Γ X , there exists a fixed point in Γ X .
Proof. It is obvious that Γ X is closed, bounded and convex. By Schauder's fixed point theorem, Lemma 2.5 and 2.6, there exists (S X (·), I X (·)) ∈ Γ X such that Define where C is a constant. Then we have the following result.
From (14) and (15), we know that there exists a constant C 1 such that for any ξ, η ∈ (−X, X).
By (16), we have Using the assumption (A1), we obtain where L J is the Lipschitz constant of kernel J and J L ∞ (R) = sup x∈R |J(x)|.
From (16), we get By calculation, we have where L K is the Lipschitz constant of kernel K(y, t) about y. Since Similarly, there exists a constant C i such that Therefore, there exists a constant C > 0 such that for any X > max{ξ 1 , ξ 2 , ξ 3 }.
Also, (33) can be re-written as where 0 < λ < λ c . By the definition of ∆(λ, c) > 0 and conditions (A1) and (A2), we obtain ∆(λ, c) → +∞ as λ → λ − c . This yields that the integral function is positive for sufficiently large λ. Hence, it is impossible that the integral is zero. 4. Examples. In this section, we present some examples to illustrate the theoretical results.

5.
Conclusion. In this paper, we have investigated the existence and nonexistence of traveling wave solutions for a nonlocal dispersal SIR model with general nonlinear incidence rate and spatio-temporal delay. From Theorems 2.9, 3.1 and 3.2, we conclude that whether the disease can spread or not depends on R 0 and c * .
Here the minimal wave speed c * is determined by the following equations ∆(λ, c) = d 2 ( For the case f (S) = βS and g(I) = I, system (1) is similar to that considered by Cheng and Yuan [10]. The model could also be improved by further generalizing the incidence rate, for example, according to Korobeinikov [23], Huang and Takeuchi [21]. Furthermore, we can study the asymptotic speed of propagation, the uniqueness and stability of traveling wave solutions. We leave this for future work.