Existence of traveling wave solutions for a delayed nonlocal dispersal SIR epidemic model with the critical wave speed

Abstract: This paper is about the existence of traveling wave solutions for a delayed nonlocal dispersal SIR epidemic model with the critical wave speed. Because of the introduction of nonlocal dispersal and the generality of incidence function, it is difficult to investigate the existence of critical traveling waves. To this end, we construct an auxiliary system and show the existence of traveling waves for the auxiliary system. Employing the results for the auxiliary system, we obtain the existence of traveling waves for the delayed nonlocal dispersal SIR epidemic model with the critical wave speed under mild conditions.


Introduction
In the biological context, to better understand the spatial spread of infectious diseases, epidemic waves in all kinds of epidemic models are attracting more and more attention, for instance, in Wu et al. [1], Wang et al. [2] and Zhang et al. [3][4][5]. Biologically speaking, the existence of an epidemic wave suggests that the disease can spread in the population. The traveling wave describes the epidemic wave moving out from an initial disease-free equilibrium to the endemic equilibrium with a constant speed. Various theoretical results, numerical algorithms and applications have been studied extensively for traveling waves about epidemic models in the literature; for instance, we refer the reader to [6][7][8][9]. More precisely, Hosono and Ilyas [10] studied the existence of traveling wave solutions for a reactiondiffusion model. In view of the fact that individuals can move freely and randomly and can be exposed to the infection from contact with infected individuals in different spatial location, Wang and Wu [11] investigated the existence and nonexistence of non-trivial traveling wave solutions of a general class of diffusive Kermack-Mckendrick SIR models with nonlocal and delayed transmission, see also [12]. Incorporating random diffusion into epidemic model, then the dynamics of disease transmission between species in a heterogeneous habitat can be described by a variety of reaction-diffusion models (see, for example, [13][14][15] and the references therein). Random diffusion is essentially a local behavior, which depicts the individuals at the location x can only be influenced by the individuals in the neighborhood of the location x. In real life, individuals can move freely. One way to solve such problems is to introduce nonlocal dispersal, which is the standard convolution with space variable. Recently, Yang et al. [16] studied a nonlocal dispersal Kermack-McKendrick epidemic model. Cheng and Yuan [17] investigated the traveling waves of a nonlocal dispersal Kermack-McKendrick epidemic model with delayed transmission, Zhang et al. [18] discussed the traveling waves for a delayed SIR model with nonlocal dispersal and nonlinear incidence, and Zhou et al. [19] proved the existence and non-existence of traveling wave solutions for a nonlocal dispersal SIR epidemic model with nonlinear incidence rate. As we know, there are many existence of traveling wave solutions for reaction-diffusion models when the wave speed is greater than the minimum wave speed (see, e.g. [20][21][22]). However, there are few discussions on the existence of traveling wave solutions when the wave speed is equal to the minimum wave speed (the critical wave speed), see [23][24][25][26].
In this paper, we focus on the delayed SIR model with the nonlocal dispersal and nonlinear incidence which proposed by Zhang et al. [18] as follows: ∂I(x, t) ∂t = d 2 (J * I(x, t) − I(x, t)) + f (S (x, t))g(I(x, t − τ)) − γI(x, t), ∂R(x, t) ∂t = d 3 (J * R(x, t) − R(x, t)) + γI(x, t), where S (x, t), I(x, t) and R(x, t) denote the densities of susceptible, infective and removal individuals at time t and location x, respectively. The parameters d i > 0(i = 1, 2, 3) are diffusion rates for susceptible, infected and removal individuals, respectively. The removal rate γ is positive number and τ > 0 is a given constant. Moreover, J * S (x, t), J * I(x, t) and J * R(x, t) represent the standard convolution with space variable x, namely, where u can be either S , I or R. Throughout this paper, assume that the nonlinear functions f and g, and the dispersal kernel J satisfy the following assumptions: (A1) f (S ) is positive and continuous for all S > 0 with f (0) = 0 and f (S ) is positive and bounded for all S ≥ 0 with L := max S ∈[0,∞) f (S ) ; (A2) g(I) is positive and continuous for all I > 0 with g(0) = 0, g (I) > 0 and g (I) ≤ 0 for all I ≥ 0; (A3) J ∈ C 1 (R), J(y) = J(−y) ≥ 0, R J(y)dy = 1 and J is compactly supported. Since the third equation in (1.1) is decoupled with the first two equations, it is enough to consider the following subsystem of (1.1): (1.2) We recall that, a traveling wave solution of system (1.2) is a solution of form (S (ξ), I(ξ)) for system (1.2), where ξ = x + ct. Substituting (S (ξ), I(ξ)) with ξ = x + ct into system (1.2) yields the following system: Clearly, if τ = 0, then system (1.2) becomes which was considered by Zhou et al. [19]. Combining the method of auxiliary system, Schauder's fixed point theorem and three limiting arguments, they proved the following result.
γ is the reproduction number of (1.4), then system (1.4) admits a nontrivial and nonnegative traveling wave solution (S (x+ct), I(x+ct)) satisfying the following asymptotic boundary conditions: where S 0 > 0 is a constant representing the size of the susceptible individuals before being infected.
We note that the assumption (H) plays a key role in the proof of Theorem 1.2 ( [18, Theorem 2.7]). However, we should pointed out here that (H) cannot be applied for some incidence, such as bilinear incidence, see [27]. Therefore, one natural question is: can we obtain the existence of traveling wave solutions for system (1.2) without assumption (H)? This constitutes our first motivation of the present paper. In addition, as was pointed out in [28] that, epidemic waves with the minimal/critical speed play a significant role in the study of epidemic spread. However, it is very challenging to investigate traveling waves with the critical wave speed. Herein, we should point out that Zhang et al. [29] defined a minimal wave speed c * := inf λ>0 d 2 R J(y)e −λy dy−d 2 + f (S 0 )g (0)e −λcτ −γ λ and then studied the existence of critical traveling waves for system (1.1). They took a bit lengthy analysis to derive the boundedness of the density of infective individual I. Unlike [29], we will apply the auxiliary system to obtain the existence of critical traveling waves, since the method is first applied in nonlocal dispersal epidemic model in 2018, see [19] for more details. Our second motivation is to make an attempt in this direction. The rest of this paper is organized as follows. In Section 2, we propose an auxiliary system and establish the existence of traveling wave solutions for the auxiliary system. In Section 3, we prove the existence of traveling waves under the critical wave speed. The paper ends with an application for our general results and a brief conclusion in Section 4.

Existence of traveling wave solutions for an auxiliary system
In this section, we will derive the existence of traveling wave solutions for the following auxiliary system on R: where ε > 0 is a constant. Clearly, (A1) and (A2) imply that f (0) = g(0) = 0. Thus, linearizing the second equation in (2.1) at the initial disease free point (S 0 , 0) yields Obviously, ∆(λ, c) = 0 also has the following properties: (i) If c > c * , then ∆(λ, c) = 0 has two different positive roots λ 1 := λ 1 (c) < λ 2 := λ 2 (c) with We now present some lemmas for our main results. Throughout this section, we always assume that R 0 > 1 and c > c * .
Next, we wish to obtain the existence of traveling wave solutions of (2.1) on R. Before doing this, we need to give some estimates for S X (ξ) and I X (ξ) in the following space: Proposition 2.4. Let (S X (ξ), I X (ξ)) be the fixed point of the operator F which is guaranteed by Proposition 2.3. Then there exists a positive constant C 1 independent of X such that Proof. First, we know that (S X (ξ), I X (ξ)) satisfies and cI X (ξ) = d 2 R J(y) I X (ξ − y)dy + f (S X (ξ))g(I X (ξ − cτ)) − (d 2 + γ)I X (ξ) − εI 2 X (ξ), ξ ∈ [−X, X], where By the facts that S X (ξ) ≤ S 0 , 0 ≤ S X (ξ) ≤ S 0 , 0 ≤ I X (ξ) ≤ K ε and I X (ξ − cτ) ≤ K ε for ξ ∈ [−X, X], it follows from (A1), (A3), (2.5) and (2.29) that Thus, there exists a positive constant C 2 independent of X such that Similar arguments apply to the case I X (ξ), we have Next, we intend to show that S X (ξ), I X (ξ), S X (ξ) and I X (ξ) are Lipschitz continuous. For any ξ, η ∈ [−X, X], it follows from (2.31) and (2.32) that and so S X (ξ) and I X (ξ) are Lipschitz continuous. In view of (2.29), we have From (A3), we know that the kernel function J is Lipschitz continuous and compactly supported. Let L J be the Lipschitz constant of J and R be the radius of supp J. Then, (2.35) in which we have used the mean-value theorem, the assumptions (A1), (A2) and inequality (2.5). Combining (2.33), (2.34) and (2.35), there exists some positive constant L 1 independent of X such that and so S X is Lipschitz continuous. It follows from (2.30) that Analogously, we have |I X (ξ) − I X (η)| ≤ L 1 |ξ − η| and so I X is Lipschitz continuous. Thus, there is a constant C 1 independent of X such that This ends the proof. Now, we are in a position to derive the existence of solutions for (2.1) on R by a limiting argument. (2.36) Proof. Choose a sequence {X n } ∞ n=1 satisfying and lim n→+∞ X n = +∞. Then, for each n ∈ N, the solution (S X n (ξ), I X n (ξ)) ∈ Γ X n ,τ satisfies Propositions 2.3 and 2.4, Eqs.(2.29) and (2.30) in ξ ∈ [−X n − cτ, X n ] for every c > c * . According to the estimates in Proposition 2.4, for the sequence {(S X n (ξ), I X n (ξ))}, we can extract a subsequence by a standard diagonal argument, denoted by {(S X n k (ξ), I X n k (ξ))} k∈N , such that S X n k (ξ) → S (ξ), I X n k (ξ) → I(ξ) in C 1 loc (R) as k → ∞ (2.37) and        cS X n k (ξ) = d 1 R J(y) S X n k (ξ − y)dy − d 1 S X n k (ξ) − f (S X n k (ξ))g(I X n k (ξ − cτ)), ξ ∈ [−X n k , X n k ], S X n k (ξ) = S (−X n k ), ξ ∈ [−X n k − cτ, −X n k ] (2.39) and S (ξ) ≤ S X n k (ξ) ≤ S 0 , I(ξ) ≤ I X n k (ξ) ≤ I(ξ), ξ ∈ (−X n k , X n k ), (2.40) where S X n k (ξ) and I X n k (ξ) are defined analogously as the φ(ξ) and ϕ(ξ) in (2.15) and (2.16), respectively. Sine J is compactly supported (see(A3)), by the Lebesgue dominated convergence theorem, one has Furthermore, in light of the continuity of f and g, we obtain lim k→+∞ f (S X n k (ξ))g(I X n k (ξ − cτ)) = f (S (ξ))g(I(ξ − cτ)), ∀ξ ∈ R.
where we have used (3.8) and (3.9). Consequently, it follows from (3.10) that Upon combining with the fact that I (ξ) is bounded on R (see (3.4)), we have This completes the proof.
That is, the longer the delay τ, the slower the spreading speed. It is known that the existence and non-existence of the traveling wave solution to nonlinear partial equations have been investigated extensively since they can predict whether or not the disease spread in the individuals and how fast a disease invades geographically. In the present paper, we have studied the traveling wave solutions for a delayed nonlocal dispersal SIR epidemic model with the critical wave speed. It has been found that the existence of traveling wave solutions are totally determined by the basic reproduction number and the minimal wave speed c * . More precisely, if R 0 > 1 and c ≥ c * , then system (1.2) admits a nontrivial and nonnegative traveling wave solution (S (x + ct), I(x + ct)) satisfying (1.5). Results on this topic may help one predict how fast a disease invades geographically, and accordingly, take measures in advance to prevent the disease, or at least decrease possible negative consequences. The approaches applied in this paper have prospects for the study of the existence and non-existence of traveling wave solutions for nonlocal dispersal epidemic models with more general nonlinear incidences. Finally, we remark that there are quite a few spaces to deserve further investigations. For example, we can study the asymptotic speed of propagation, the uniqueness and stability of traveling wave solutions. Moreover, the exact boundary behavior of susceptible S (ξ) at +∞ is not obtained although the existence of S (+∞) is established. We leave these problems for future work.