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

1.   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 reaction-diffusion 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:

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 $\gamma$ is positive number and $\tau > 0$ 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 $S\geq0$ with $L: = \max_{S\in [0, \infty)} f'(S)$;

(A2) $g(I)$ is positive and continuous for all $I > 0$ with $g(0) = 0, g'(I) > 0$ and $g''(I)\leq 0$ for all $I\geq 0$;

(A3) $J\in C^1(\mathbb{R}), J(y) = J(-y)\geq 0, \int_{\mathbb{R}}J(y)dy = 1$ 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):

We recall that, a traveling wave solution of system (1.2) is a solution of form $(S(\xi), I(\xi))$ for system (1.2), where $\xi = x+ct$. Substituting $(S(\xi), I(\xi))$ with $\xi = x+ct$ into system (1.2) yields the following system:

Clearly, if $\tau = 0$, 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.

Theorem 1.1. ([19,Theorem 2.3]) Assume that (A1)-(A3) hold. If $\mathcal{R}_0 > 1$ and $c\geq c^*$, where $c^* > 0$ is the minimal wave speed and $\mathcal{R}_0 = \frac{f(S_0)g'(0)}{\gamma}$ 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.

For (1.3) satisfying (1.5), Zhang et al.^[18] obtained the following result.

Theorem 1.2. ([18,Theorem 2.7]) Assume that (A1)-(A3) hold. In addition, suppose that

$(H)$ there exists $I_0 > 0$ such that $f(S_0)g(I_0)-\gamma I_0\leq 0$.

If $\mathcal{R}_0 > 1$ and $c > c^*$, where $c^* > 0$ is the minimal wave speed and $\mathcal{R}_0 = \frac{f(S_0)g'(0)}{\gamma}$ is the reproduction number of $(1.3)$, then system $(1.3)$ admits a traveling wave solution $(S(\xi), I(\xi))$ satisfying $(1.5)$.

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 $c^*: = \inf_{\lambda > 0}{\frac{d_2\int_{\mathbb{R}}J(y)e^{-\lambda y}dy-d_2+f(S_0)g'(0)e^{-\lambda c\tau}-\gamma}{\lambda}}$ 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.

2.   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 $\mathbb{R}$:

where $\varepsilon > 0$ 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

Substituting $I(\xi) = e^{\lambda\xi}$ into (2.2) leads to the corresponding characteristic equation:

Lemma 2.1. (^[18]) Suppose that $\mathcal{R}_0: = \frac{f(S_0)g'(0)}{\gamma} > 1$. Then there exist $c^* > 0$ and $\lambda^* > 0$ such that

Obviously, $\Delta(\lambda, c) = 0$ also has the following properties:

(i) If $c > c^*$, then $\Delta(\lambda, c) = 0$ has two different positive roots $\lambda_1: = \lambda_1(c) < \lambda_2: = \lambda_2(c)$ with

(ii) If $0 < c < c^*,$ then $\Delta(\lambda, c) > 0$ for all $\lambda\geq0$.

We now present some lemmas for our main results. Throughout this section, we always assume that $\mathcal{R}_0 > 1$ and $c > c^*$.

For our purpose, we define the following functions on $\mathbb{R}$:

where $K_\varepsilon = \frac{f(S_0)g'(0)-\gamma}{\varepsilon}$, $\lambda_1$ is the smallest positive real root of Eq.(2.3), and $\alpha, \rho, \eta$, $M$ are four positive constants to be determined in the following lemmas.

Lemma 2.2. The function $\overline{I}(\xi)$ satisfies the following inequality:

Proof. The concavity of $g(I)$ with respect to $I$ implies that $g(I)\leq g'(0)I$ and so

When $\overline{I}(\xi) = e^{\lambda_1\xi}$, it follows from (2.5) that

When $\overline{I}(\xi) = K_\varepsilon = \frac{f(S_0)g'(0)-\gamma}{\varepsilon}$, we derive from (A3) and (2.5) that

and the lemma follows.

Lemma 2.3. Suppose that $\alpha\in(0, \lambda_1)$ is sufficiently small. Then the function $\underline{S}(\xi)$ satisfies

for any $\xi\neq\xi_1: = \frac{1}{\alpha}\ln(\frac{S_0}{\rho})$ and $\rho > S_0$ large enough.

Proof. If $\xi > \xi_1$, then $\underline{S}(\xi) = 0$ and (2.6) holds. If $\xi < \xi_1$, then $\underline{S}(\xi) = S_0-\rho e^{\alpha\xi}$ and $\overline{I}(\xi) = e^{\lambda_1\xi}$. According to the assumptions (A1), (A2) and (2.5), one has

Since $0 < \alpha < \lambda_1$ and $e^{(\lambda_1-\alpha)\xi} < (\frac{S_0}{\rho})^{\frac{\lambda_1-\alpha}{\alpha}}$ for $\xi < \xi_1$, it follows from (2.7) that

Taking $\rho = \frac{1}{\alpha}$ in (2.8) and noting that

for sufficiently small $\alpha > 0$, we have

Owing to (2.8) and (2.9), we find

This completes the proof.

Lemma 2.4. Assume that $0 < \eta < \min\{\lambda_2-\lambda_1, \lambda_1\}$. Then for sufficiently large $M > 1$, the function $\underline{I}(\xi)$ satisfies

for any $\xi\neq\xi_2: = \frac{1}{\eta}\ln\frac{1}{M}$.

Proof. If $\xi > \xi_2$, then $\underline{I}(\xi) = 0, J*\underline{I}(\xi)\geq0$ and $g(\underline{I}(\xi-c\tau))\geq0$ (by (A2)), Thus (2.10) holds. If $\xi < \xi_2$, then we can take $M_1 > 1$ such that $\frac{1}{\eta}\ln\frac{1}{M_1}+1 = \xi_1$ and choose large enough $M\geq M_1$ with $\underline{I}(\xi) = e^{\lambda_1\xi}(1-Me^{\eta\xi})$ and $\underline{S}(\xi) = S_0-\rho e^{\alpha\xi}$. Thus, (2.10) is equivalent to

and so

For any $\widehat{\varepsilon}\in(0, g'(0))$, noting that $\lim\limits_{I\rightarrow 0^+}{\frac{g(I)}{I}} = g'(0)$, there exists a small positive number $\delta_0$ such that

Choose $M$ large enough such that $0 < \underline{I}(\xi) < \delta_0$. Then, it follows from (2.12) that

Since (2.13) holds for arbitrary sufficiently small $\widehat{\varepsilon}\in(0, g'(0))$ and $\underline{S}(\xi)\rightarrow S_0$ for sufficiently large $M$, one can conclude from (2.13) that

Then, to prove (2.11), we only need to show that

Noting $\underline{I}^2(\xi-c\tau)\leq e^{2\lambda_1\xi}$ and $\underline{I}^2(\xi)\leq e^{2\lambda_1\xi}$, it suffices to show that

Due to $0 < \eta < \lambda_2-\lambda_1$, we have $\Delta(\lambda_1+\eta, c) < 0$ (by Lemma 2.1). Then (2.14) leads to

The facts that $\xi < \xi_2 < 0$ and $0 < \eta < \lambda_1$ imply that $e^{(\lambda_1-\eta)\xi} < 1$. To end the proof, we only need to take

This completes the proof.

Next we define a bounded set as follows:

where

For any $(\phi(\cdot), \varphi(\cdot))\in C([-X-c\tau, X], \mathbb{R}^2)$, we define

and

We consider the following initial value problems:

and

with

By the existence theorem of ordinary differential equations, problems (2.17)-(2.19) admit a unique solution $(S_X(\cdot), I_X(\cdot))$ satisfying $S_X(\cdot)\in C^1([-X, X])$ and $I_X(\cdot)\in C^1([-X, X])$. Thus, we can define an operator $\mathcal{F} = (\mathcal{F}_1, \mathcal{F}_2):\Gamma_{X, \tau}\rightarrow C([-X-c\tau, X])$ by

and

Proposition 2.1. The operator $\mathcal{F} = (\mathcal{F}_1, \mathcal{F}_2)$ maps $\Gamma_{X, \tau}$ into $\Gamma_{X, \tau}$.

Proof. For any $(\phi(\cdot), \varphi(\cdot))\in\Gamma_{X, \tau}$, we should show that

and

By the definition of the operator $\mathcal{F}$, it is easy to see that the last two equalities hold.

For $\xi\in[-X, X]$, we first consider $\mathcal{F}_1[\phi, \varphi](\xi)$. By the definition of the operator $\mathcal{F}$, it is sufficient to prove $\underline{S}(\xi)\leq S_X(\xi)\leq S_0$. Note that $f(0) = 0$ (see (A1)). Then it is obvious that $0$ is a sub-solution of (2.17). It follows from the maximum principle that $S_X(\xi)\geq0$ for $\xi\in[-X, X]$. From the definition of $\widehat{\phi}(\xi)$, (A1) and (A2), we obtain

which implies that $\overline{S}(\xi) = S_0$ is a super-solution of (2.17). Thus, we have $S_X(\xi)\leq S_0$ for $\xi\in[-X, X]$. Clearly, $\underline{S}(\xi) = S_0-\rho e^{\alpha\xi}$ for $\xi\in[-X, \xi_1)$. Thus, utilizing Lemma 2.3 and (A2),

for any $\xi\in(-X, \xi_1)$. Since $S_X(-X) = \underline{S}(-X)$, applying the comparison principle, we have $\underline{S}(\xi)\leq S_X(\xi)$ for $\xi\in[-X, \xi_1)$ and so $\underline{S}(\xi)\leq S_X(\xi)\leq S_0 \ \text{for all}\ \xi\in[-X, X].$

Next, we consider $\mathcal{F}_2[\phi, \varphi](\xi)$. Similarly, we only need to show that $\underline{I}(\xi)\leq I_X(\xi)\leq\overline{I}(\xi)$. First, from the maximum principle, we have $I_X(\xi)\geq0$ for $\xi\in[-X, X]$. Thus, it follows from Lemma 2.4, $\underline{S}(\xi)\leq\phi(\xi), \ \underline{I}(\xi)\leq\widehat{\varphi}(\xi)$, (A1), (A2) and $\underline{I}(\xi) = e^{\lambda_1\xi}(1-Me^{\eta\xi})$ for $\xi\in[-X, \xi_2)$ that

for all $\xi\in[-X, \xi_2)$. Since $I_X(-X) = \underline{I}(-X)$, the comparison principle implies that $\underline{I}(\xi)$ is a sub-solution of (2.18) on $[-X, \xi_2)$. Recalling the fact that $\underline{I}(\xi) = 0$ for $\xi\in[\xi_2, X]$, it is easy to see that

Since $\phi(\xi)\leq S_0$ and $\widehat{\varphi(\xi)}\leq\overline{I}(\xi)$ for all $\xi\in[-X, X]$, from (A1), (A2) and Lemma 2.2, we deduce that

which ensures that $\overline{I}(\xi)$ is a super-solution of (2.18) on $[-X, X]$ by the comparison principle. Combining with (2.20), we know that $\underline{I}(\xi)\leq I_X(\xi)\leq\overline{I}(\xi)\ \text{for}\ \xi\in[-X, X].$ The proof is finished.

Proposition 2.2. The operator $\mathcal{F}:\Gamma_{X, \tau}\rightarrow \Gamma_{X, \tau}$ is completely continuous.

Proof. We first show the compactness of $\mathcal{F} = (\mathcal{F}_1, \mathcal{F}_2)$. We need to prove that, for any bounded subset $B\subset\Gamma_{X, \tau}$, the set $\mathcal{F}(B)$ is precompact. In view of the definition of the operator $\mathcal{F}$, for any $(S_X, I_X)\in\mathcal{F}(B)$, there exists $(\phi, \varphi)\in B$ such that $\mathcal{F}[\phi, \varphi](\xi) = (S_X, I_X)(\xi)$ for $\xi\in[-X, X]$ and $\mathcal{F}[\phi, \varphi](\xi) = (S_X, I_X)(-X)$ for $\xi\in[-X-c\tau, -X]$.

Since $(\phi, \varphi)\in B$, there exists a constant $M_1 > 0$ such that

Moreover, since $(\phi, \varphi)\in B$, from (2.17), (2.18) and the above inequalities, we know that there exists some constant $M_2 > 0$ such that

It follows that $\mathcal{F}(B)$ is a family of the uniformly bounded and equicontinuous functions. The compactness of $\mathcal{F}(B)$ then follows from the Arzelà-Ascoli theorem and the definition of $\Gamma_{X, \tau}$.

Next we prove the continuity of $\mathcal{F} = (\mathcal{F}_1, \mathcal{F}_2)$. By the definition of the operator $\mathcal{F}$, we assume that $(\phi_i(\xi), \varphi_i(\xi))\in\Gamma_{X, \tau}(i = 1, 2)$ and $S_{X, i}(\xi) = \mathcal{F}_1[\phi_i, \varphi_i](\xi), I_{X, i}(\xi) = \mathcal{F}_2[\phi_i, \varphi_i](\xi)$ for $\xi\in[-X, X]$. We will prove the continuity of $\mathcal{F}$ by the following two steps.

Step 1. The continuity of $\mathcal{F}_1$.

It follows from (2.17) that

where $\widehat{\phi}_i(\xi)(i = 1, 2)$ is defined analogously as the $\widehat{\phi}(\xi)$ in (2.15).

In view of

it follows from (2.5), the mean-value theorem, (A1), (A2) and the definition of $\Gamma_{X, \tau}$ that

If $S_{X, 1}(\xi)-S_{X, 2}(\xi) > 0$, then we deduce from (2.21)-(2.23) that

Applying the Gronwall inequality (see p. 90 of ^[31]) to (2.24), we obtain that $\mathcal{F}_1$ is continuous on $\Gamma_{X, \tau}$. If $S_{X, 1}(\xi)-S_{X, 2}(\xi) < 0$, then one can prove the same result. Thus, we show that $\mathcal{F}_1$ is continuous on $\Gamma_{X, \tau}$.

Step 2. The continuity of $\mathcal{F}_2$.

From (2.18), we have

where $\widehat{\varphi}_i(\xi)(i = 1, 2)$ is defined analogously as the $\widehat{\varphi}(\xi)$ in (2.16).

In view of $0\leq I_{X, 1}(\xi)+I_{X, 2}(\xi)\leq 2K_{\varepsilon}$, we then deduce from (2.25) that there exists a nonnegative constant $\widetilde{c}$ such that

Arguing as in the proof of (2.22) and (2.23), we obtain

and

Thus, it follows from (2.26)-(2.28) that

which together with the Gronwall inequality implies that $\mathcal{F}_2$ is continuous on $\Gamma_{X, \tau}$. This completes the proof.

From the definition of $\Gamma_{X, \tau}$, it is easy to see that $\Gamma_{X, \tau}$ is closed and convex. Thus, employing Propositions 2.1, 2.2 and Schauder's fixed point theorem, we can obtain the following result.

Proposition 2.3. There exists $(S_X(\xi), I_X(\xi))\in\Gamma_{X, \tau}$ such that

and

Next, we wish to obtain the existence of traveling wave solutions of (2.1) on $\mathbb{R}$. Before doing this, we need to give some estimates for $S_X(\xi)$ and $I_X(\xi)$ in the following space:

with the norm

Proposition 2.4. Let $(S_X(\xi), I_X(\xi))$ be the fixed point of the operator $\mathcal{F}$ which is guaranteed by Proposition 2.3. Then there exists a positive constant $C_1$ independent of $X$ such that

for all

Proof. First, we know that $(S_X(\xi), I_X(\xi))$ satisfies

and

where

and

By the facts that $S_X(\xi)\leq S_0, \ 0\leq \widehat{S_X}(\xi)\leq S_0, \ 0\leq \widehat{I_X}(\xi)\leq K_{\varepsilon}$ and $I_X(\xi-c\tau)\leq K_{\varepsilon}$ for $\xi\in[-X, X]$, 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 $I'_X(\xi)$, we have

Next, we intend to show that $S_X(\xi), \ I_X(\xi), \ S'_X(\xi)$ and $I'_X(\xi)$ are Lipschitz continuous. For any $\xi, \eta\in[-X, X]$, it follows from (2.31) and (2.32) that

and so $S_X(\xi)$ and $I_X(\xi)$ 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,

and

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

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 $\mathbb{R}$ by a limiting argument.

Theorem 2.1. Let $\mathcal{R}_0 = \frac{f(S_0)g'(0)}{\gamma} > 1.$ Then, for any $c > c^*$, (2.1) admits a solution $(S(\xi), I(\xi))$ such that

Proof. Choose a sequence $\{X_n\}_{n = 1}^\infty$ satisfying

and $\lim_{n\rightarrow +\infty}X_n = +\infty.$ Then, for each $n\in\mathbb{N}$, the solution $(S_{X_n}(\xi), I_{X_n}(\xi))\in\Gamma_{X_n, \tau}$ satisfies Propositions 2.3 and 2.4, Eqs.(2.29) and (2.30) in $\xi\in[-X_n-c\tau, X_n]$ for every $c > c^*$.

According to the estimates in Proposition 2.4, for the sequence $\{(S_{X_n}(\xi), I_{X_n}(\xi))\}$, we can extract a subsequence by a standard diagonal argument, denoted by $\{(S_{X_{n_k}}(\xi), I_{X_{n_k}}(\xi))\}_{k\in\mathbb{N}}$, such that

and

with

and

where $\widehat{S_{X_{n_k}}}(\xi)$ and $\widehat{I_{X_{n_k}}}(\xi)$ are defined analogously as the $\widehat{\phi}(\xi)$ and $\widehat{\varphi}(\xi)$ in (2.15) and (2.16), respectively. Sine $J$ is compactly supported (see(A3)), by the Lebesgue dominated convergence theorem, one has

Similarly, we can show that

Furthermore, in light of the continuity of $f$ and $g$, we obtain

Thus, passing to limits in (2.38), (2.39) and (2.40) as $k\rightarrow +\infty$, we derive from (2.37), (2.41)-(2.43) that $(S(\xi), I(\xi))$ satisfies (2.1) and (2.36). The proof of this theorem is finished.

3.   Existence of traveling wave solutions with critical speed

In this section, we will prove the existence of traveling wave solutions of (1.3) satisfying (1.5).

Theorem 3.1. Let $\mathcal{R}_0 = \frac{f(S_0)g'(0)}{\gamma} > 1.$ Then for any $c\geq c^*$, (1.3) admits a pair of functions $(S(\xi), I(\xi))$ such that

Proof. For $c > c^*$, let $\{\varepsilon_n\}$ be a sequence such that $0 < \varepsilon_{n+1} < \varepsilon_n < 1 (n = 1, 2, 3, \cdots)$ and $\lim_{n\rightarrow +\infty}\varepsilon_n = 0$. By Theorem 2.1 and Proposition 2.4, for each $n\in\mathbb{N}$, there exists a solution $\Phi_n(\xi) = (S_n(\xi), I_n(\xi))$ for $\varepsilon = \varepsilon_n$, such that

and

for all $\xi\in\mathbb{R}$.

Furthermore, we know that

where $C_3$ is a positive constant independent of $\xi$. Then we can assert that $\{\Phi_n(\xi)\}$ and $\{\Phi'_n(\xi)\}$ are equi-continuous and uniformly bounded on $\mathbb{R}$. By the Arzelà-Ascoli theorem, there exists a subsequence of $\{\varepsilon_n\}$, still denoted by $\{\varepsilon_n\}$, such that $\lim_{n\rightarrow \infty}\varepsilon_n = 0$ and

uniformly on every closed bounded interval as $n\rightarrow \infty$, and hence pointwise on $\mathbb{R}$, where $\Phi(\xi) = (S(\xi), I(\xi))$ and $\Phi'(\xi) = (S'(\xi), I'(\xi))$ are bounded. Passing to the limits in (3.2) and (3.3) as $n\rightarrow \infty$ and employing the dominated convergence theorem and the continuity of $f$ and $g$ (see (A1) and (A2)), we obtain that $(S(\xi), I(\xi))$ satisfies (1.3) and (3.1).

For $c = c^*$, one can choose a decreasing sequence $\{c_n\}\in(c^*, c^*+1)$ such that $\lim_{n\rightarrow \infty}c_n = c^*$ and the same reasoning applies to the above case $c > c^*$ and $\varepsilon_n\rightarrow0$. For simplicity, we omit the details. This ends the proof.

Next we aim at the asymptotic behavior of solution $(S(\xi), I(\xi))$ of (1.3), whose existence is guaranteed by Theorem 3.1. For $\xi\in\mathbb{R}$, invoking the Squeeze theorem to (3.1), we deduce the asymptotic behavior of solution $(S(\xi), I(\xi))$ at $-\infty$.

Proposition 3.1. Suppose that $\mathcal{R}_0 = \frac{f(S_0)g'(0)}{\gamma} > 1$ and $c\geq c^*$. Then the solution $(S(\xi), I(\xi))$ of (1.3) satisfies

and

The following proposition shows the asymptotic behavior of $I(\xi)$ at $\infty$.

Proposition 3.2. Assume that $\mathcal{R}_0 = \frac{f(S_0)g'(0)}{\gamma} > 1$ and $c\geq c^*$. Then the solution $(S(\xi), I(\xi))$ of (1.3) satisfies

with $\int_{\mathbb{R}}I(\xi)d\xi < \infty$ and $I(\infty) = 0.$

Proof. Using (3.1), (A1), (A2) and the definitions of $\underline{S}(\xi)$ and $\underline{I}(\xi)$, one has

Note that

Then, by(3.5) and (A3), we get

which implies that, for $x\in\mathbb{R}$,

where we used the fact that $J$ is compactly supported (see(A3)). Taking an integration of the first equation in (1.3) over $(-\infty, x)$ and using (3.5) and (3.7), we get

which implies

Similar to the proof of (3.7), we have

Taking an integration of the second equation in (1.3) over $\mathbb{R}$ gives

where we have used (3.8) and (3.9).

Consequently, it follows from (3.10) that

Upon combining with the fact that $I'(\xi)$ is bounded on $\mathbb{R}$ (see (3.4)), we have

This completes the proof.

The following proposition deals with the asymptotic behavior of $S(\xi)$ at $\infty$.

Proposition 3.3. Assume that $\mathcal{R}_0 = \frac{f(S_0)g'(0)}{\gamma} > 1$ and $c\geq c^*$. Then (1.3) has a solution $(S(\xi), I(\xi))$ such that $\lim_{\xi\rightarrow +\infty}S(\xi)$ exists and

Moreover, there holds

Proof. We prove the existence of $\lim_{\xi\rightarrow +\infty}S(\xi)$ by a contradiction argument. Suppose

for a contrary. Then from Fluctuation Lemma (see Lemma 2.2 in ^[1]), we infer that there exists a sequence $\{\xi_n\}$ satisfying $\xi_n\rightarrow \infty$ as $n\rightarrow \infty$ such that

Meanwhile, there exists another sequence $\{\eta_n\}$ satisfying $\eta_n\rightarrow \infty$ as $n\rightarrow \infty$ such that

Following from the first equation in (1.3), we have

Passing to the limits in (3.14) as $n\rightarrow \infty$, and using (3.11), (3.12) and (A2), we obtain

Set

We will show that $\lim_{n\rightarrow \infty}S_n(y)\rightarrow \sigma_2$ for $y\in supp J: = \Omega$. Take sufficiently small $\varepsilon_1 > 0$ and let

Then from (3.12), (3.15)-(3.17) and (A3) we get

which shows that $m(\Omega_{\varepsilon_1}) = 0$, where $m(\cdot)$ denotes the measure. Therefore, we have $\lim_{n\rightarrow \infty}S_n(y) = \sigma_2$ almost everywhere in $\Omega$.

However, since $\{S_n\}$ is an equi-continuous family, the convergence is everywhere in $\Omega$, that is,

Using the similar arguments, we can prove that

Integrating two sides of the first equation in (1.3) from $\eta_n$ to $\xi_n$, using (3.12), (3.13), (3.18), (3.19) and the fact that

we get

which leads to a contradiction. Thus, $\lim_{\xi\rightarrow \infty}\sup S(\xi) = \lim_{\xi\rightarrow \infty}\inf S(\xi)$ and so $\lim_{\xi\rightarrow \infty}S(\xi): = S_\infty$ exists.

Next, we will prove that $S_\infty < S_0$. Since $S(\xi)\leq S_0$, we have $S_\infty\leq S_0$. Assume that $S_\infty = S_0$. Then it follows from (3.5) that

Taking an integration of the first equation in (1.3) over $\mathbb{R}$ yields

By Fubini's theorem and (A3), one has

From (3.20)-(3.22), we obtain $\int_{\mathbb{R}}f(S(\xi))g(I(\xi-c\tau))d\xi = 0,$ which contradicts (3.6). Thus, we have

and

Moreover, integrating two sides of the second equation in (1.3) on $\mathbb{R}$ and recalling that $I(\pm\infty) = 0$, one has

Using the Fubini theorem again and repeating the above procedures, we can obtain $\int_{\mathbb{R}}[J*I(\xi)-I(\xi)]d\xi = 0$. It follows from (3.23) and (3.24) that

The proof of this proposition is completed.

Finally, combining Theorem 3.1 and Propositions 3.1-3.3, we obtain the existence of traveling wave solutions for system (1.2) satisfying (1.5).

Theorem 3.2. Assume that $\mathcal{R}_0 = \frac{f(S_0)g'(0)}{\gamma} > 1$ and $c\geq c^*$. Then system (1.2) admits a nontrivial and nonnegative traveling wave solution $(S(x+ct), I(x+ct))$ satisfying (1.5).

Remark 3.1

$(i)$ It is worth to point out that, in this paper, we have derived the existence of traveling wave solutions in the absence of assumption (H) and further obtained Theorem 3.2 to confirm that $c^*$ is the minimal wave speed of the existence of traveling wave solutions for (1.2), which improves Theorem 1.1 ([18,Theorem 2.7]).

$(ii)$ Theorem 3.2 states that Theorem 1.2 ([19,Theorem 2.3]) still holds if we take the influences of delays into consideration.

$(iii)$ In $(1.1)$, the choice of $f(S) = \beta S$ and $g(I) = I$ leads to the model investigated by Cheng and Yuan^[17]. Thus, Theorem 3.2 includes Theorem 3.2 of ^[17] as a special case.

4.   Application and conclusion

In this section, we will give a typical example to demonstrate the abstract results presented in Section 3. The choice of $f(S) = S$ and $g(I) = \frac{\beta I}{1+\alpha I}$ $(\alpha, \beta > 0$ are two coefficients) in (1.2) leads to

Obviously, it is easy to verify that $f(S)$ and $g(I)$ satisfy assumptions (A1)-(A2). Applying Theorem 3.2, we obtain the following result.

Theorem 4.1. There exists a positive constant $c^*$ such that if $\mathcal{R}_0 = \frac{\beta S_0}{\gamma} > 1$ and $c\geq c^*$. Then system (4.1) admits a nontrivial and nonnegative traveling wave solution $(S(x+ct), I(x+ct))$ satisfying

We further show that the minimal wave speed $c^*$ is determined by the following system:

where

It is noticed that the minimal wave speed $c^*$ is relevant to the dispersal rate $d_2$ and the delay $\tau$. Due to $\Delta(\lambda^*, c^*) = 0$, by the implicit function theorem, a direct calculation gives

which implies that the geographical movement of infected individuals can increase the speed of the spread of disease. Similarly, we have

That is, the longer the delay $\tau$, 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 $\mathcal{R}_0 > 1$ and $c\geq c^*$, 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 $S(\xi)$ at $+\infty$ is not obtained although the existence of $S(+\infty)$ is established. We leave these problems for future work.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11701460), the Natural Science Foundation of Sichuan Provincial Education Department (Grant No. 18ZB0581), the Meritocracy Research Funds of China West Normal University (17YC373), the research startup foundation of China West Normal University (Grant No. 18Q060) and the Research and Innovation Team of China West Normal University (CXTD2020-5).

Conflict of interest

The authors declare there is no conflict of interest.