Traveling Waves in a Nonlocal Dispersal SIR Model with Standard Incidence Rate and Nonlocal Delayed Transmission

Abstract

In this paper, we study the traveling wave solutions for a nonlocal dispersal SIR epidemic model with standard incidence rate and nonlocal delayed transmission. The existence and nonexistence of traveling wave solutions are determined by the basic reproduction number of the corresponding reaction system and the minimal wave speed. To prove these results, we apply the Schauder’s fixed point theorem and two-sided Laplace transform. The main difficulties are that the complexity of the incidence rate in the epidemic model and the lack of regularity for nonlocal dispersal operator.

1. Introduction

Due to the important significance in modeling the disease transmission, traveling wave solution has been intensively researched in many epidemic models, such as the SIR epidemic models and their various extensions. For instance, Hosono and Ilyas [1] considered the following epidemic model:

$$\left\{\begin{array}{c}\frac{\partial S(x,t)}{\partial t}=d_1\frac{{\partial }^{2}S(x,t)}{\partial x^2}-\beta S(x,t)I(x,t),\hfill \\ \frac{\partial I(x,t)}{\partial t}=d_2\frac{{\partial }^{2}I(x,t)}{\partial x^2}+\beta S(x,t)I(x,t)-\gamma I(x,t),\hfill \\ \frac{\partial R(x,t)}{\partial t}=d_3\frac{{\partial }^{2}R(x,t)}{\partial x^2}+\gamma I(x,t),\hfill \end{array}\right.$$

(1)

where $S,I$ and R denote the densities of the susceptible, infected and removed individuals, respectively. $d_i>0,i=1,2,3$ are the diffusion rates, and the positive constants $\beta ,\gamma$ denote the transmission rate and the recovery rate, respectively. They have proved that system (1) admits a pair of traveling wave solution $\left(S\right(x+ct),I(x+ct\left)\right)$ satisfying $S(-\infty )=S_{-\infty }>S(\infty )=S_{\infty },I(\pm \infty )=0$ if $\frac{\beta S_{-\infty }}{\gamma }>1$ and the wave speed $c\ge c^{*}=2\sqrt{\beta S_{-\infty }{d}_2(1-\frac{\gamma }{\beta S_{-\infty }})}$. Wang and Wu [2] considered the existence and nonexistence of traveling wave solution for a diffusive Kermack–McKendrick epidemic model with nonlocal delayed transmission. The incidence rate in these two papers is bilinear form $\beta SI$. Since then, there has been extensive research with traveling wave solutions for delayed diffusive SIR models with various incidence rates, such as $\frac{\beta SI}{1+\alpha I},\beta I^p{S}^q(p,q>0)$ and general form $f\left(S\right)g\left(I\right)$, see [3,4,5,6,7,8,9,10,11,12,13,14]. There have also been some papers about traveling wave solutions for delayed diffusive SIR models with external supplies [15,16] and delayed diffusive SIRS epidemic models [17,18].

On the other hand, the traveling wave solution for diffusive SIR models with a standard incidence rate has attracted more and more attention. Wang et al. [19] have considered the following SIR disease outbreak model with the standard incidence rate

$$\left\{\begin{array}{c}\frac{\partial S(x,t)}{\partial t}=d_1\frac{{\partial }^{2}S(x,t)}{\partial x^2}-\frac{\beta S(x,t)I(x,t)}{S(x,t)+I(x,t)},\hfill \\ \frac{\partial I(x,t)}{\partial t}=d_2\frac{{\partial }^{2}I(x,t)}{\partial x^2}+\frac{\beta S(x,t)I(x,t)}{S(x,t)+I(x,t)}-\gamma I(x,t),\hfill \\ \frac{\partial R(x,t)}{\partial t}=d_3\frac{{\partial }^{2}R(x,t)}{\partial x^2}+\gamma I(x,t).\hfill \end{array}\right.$$

(2)

They have obtained full information about the existence and nonexistence of traveling wave solutions. Li et al. [20] also have studied diffusive SIR epidemic model with standard incidence rate, which is different from system (2) that they considered the effect of nonlocal delayed transmission. Wang and Wang [21] have studied the following diffusive SIR epidemic model

$$\left\{\begin{array}{c}\frac{\partial S(x,t)}{\partial t}=d_1\frac{{\partial }^{2}S(x,t)}{\partial x^2}-\frac{\beta S(x,t)I(x,t)}{S(x,t)+I(x,t)+R(x,t)},\hfill \\ \frac{\partial I(x,t)}{\partial t}=d_2\frac{{\partial }^{2}I(x,t)}{\partial x^2}+\frac{\beta S(x,t)I(x,t)}{S(x,t)+I(x,t)+R(x,t)}-(\gamma +\delta )I(x,t),\hfill \\ \frac{\partial R(x,t)}{\partial t}=d_3\frac{{\partial }^{2}R(x,t)}{\partial x^2}+\gamma I(x,t),\hfill \end{array}\right.$$

(3)

where $\delta >0$ denotes the death rate due to the disease. They have obtained that system (3) has a traveling wave solution $\left(S\right(x+ct),I(x+ct),R(x+ct\left)\right)$ satisfying $S(-\infty )=S_{-\infty }>S(\infty )=S_{\infty }$, $I(\pm \infty )=0,R(-\infty )=0$, $R(\infty )=\frac{\gamma (S_{-\infty }-S_{\infty })}{\gamma +\delta }$ if $\frac{\beta }{\gamma +\delta }>1,c>c^{*}=2\sqrt{d_2(\beta -\gamma -\delta )}$ and $d_3<2d_2$. More recently, Zhen, Wei, Tian et al. [22] have considered the following diffusive SIR epidemic model with standard incidence rate and spatiotemporal delay

$$\left\{\begin{array}{c}\frac{\partial S(x,t)}{\partial t}=d_1\frac{{\partial }^{2}S(x,t)}{\partial x^2}-\frac{\beta S(x,t)(G\ast I)(x,t)}{S(x,t)+(G\ast I)(x,t)+R(x,t)},\hfill \\ \frac{\partial I(x,t)}{\partial t}=d_2\frac{{\partial }^{2}I(x,t)}{\partial x^2}+\frac{\beta S(x,t)(G\ast I)(x,t)}{S(x,t)+(G\ast I)(x,t)+R(x,t)}-(\gamma +\delta )I(x,t),\hfill \\ \frac{\partial R(x,t)}{\partial t}=d_3\frac{{\partial }^{2}R(x,t)}{\partial x^2}+\gamma I(x,t),\hfill \end{array}\right.$$

(4)

where

$$(G\ast I)(x,t)={\int }_{-\infty }^t{\int }_{-\infty }^{\infty }G(x-y,t-s)I(y,s)dyds.$$

The spatiotemporal kernel $G(x-y,t-s)$ describes the interaction between the infective and the susceptible individuals at location x and the present time t, which occurred at location y and at earlier time s. It should be emphasized that the incidence rate in system (3)–(4) is different from the previous one in model (2), which is $\frac{\beta SI}{S+I}$. The incidence rate $\frac{\beta SI}{S+I+R}$ makes the diffusive systems be totally coupled, and the corresponding traveling wave systems consist of three equations, where few papers have dealt with it, see [23].

The diffusion terms of the above systems are Laplacian operators that account for random motion. However, due to the more frequent interaction with other people, the movements of individuals may be not limited to a small area. Thus, recently, various integral operators have been widely used to model the diffusion phenomena, for example, in [24,25,26]. Yang et al. [27] considered the following nonlocal dispersal epidemic model

$$\left\{\begin{array}{c}\frac{\partial }{\partial t}S(x,t)=d_1[J\ast S(x,t)-S(x,t)]-\beta S(x,t)I(x,t),\hfill \\ \frac{\partial }{\partial t}I(x,t)=d_2[J\ast I(x,t)-I(x,t)]+\beta S(x,t)I(x,t)-\gamma I(x,t),\hfill \\ \frac{\partial }{\partial t}R(x,t)=d_3[J\ast R(x,t)-R(x,t)]+\gamma I(x,t),\hfill \end{array}\right.$$

where $J(·)$ denotes the probability distribution of rates of dispersal and $J\ast u-u$ can describe the net rate of increase due to the dispersal of subpopulation u, where $J\ast u(x,t)$ is the standard convolution with space invariable x and u can be either $S,I$ or R. Under some assumptions about the dispersal kernel function $J(·)$, they obtained the existence and nonexistence of traveling wave solutions. Since then, many researchers pay more attention to the study of traveling wave solutions of nonlocal dispersal SIR epidemic models, for instance, in [28,29,30,31,32,33]. Therefore, in this paper, we will consider the corresponding nonlocal dispersal model of system (4) that is the following nonlocal dispersal SIR epidemic model with nonlocal delayed transmission:

$$\left\{\begin{array}{c}\frac{\partial S(x,t)}{\partial t}=d_1(J\ast S(x,t)-S(x,t))-\frac{\beta S(x,t)(G\ast I)(x,t)}{S(x,t)+(G\ast I)(x,t)+R(x,t)},\hfill \\ \frac{\partial I(x,t)}{\partial t}=d_2(J\ast I(x,t)-I(x,t))+\frac{\beta S(x,t)(G\ast I)(x,t)}{S(x,t)+(G\ast I)(x,t)+R(x,t)}-(\gamma +\delta )I(x,t),\hfill \\ \frac{\partial R(x,t)}{\partial t}=d_3(J\ast R(x,t)-R(x,t))+\gamma I(x,t).\hfill \end{array}\right.$$

(5)

Throughout this paper, we give the following assumptions on the kernel functions J and G:

(J)$J\in C^1\left(\mathbb{R}\right),J\left(y\right)=J(-y)\ge 0,{\int }_{-\infty }^{\infty }J\left(y\right)dy=1,J$ is compactly supported. Furthermore, for any $v\in [0,\tilde{v})$,

$${\int }_{-\infty }^{\infty }J\left(y\right)e^{-vy}dy<+\infty ,$$

and ${\int }_{-\infty }^{\infty }J\left(y\right)e^{-vy}dy\to +\infty$ as $v\to {\tilde{v}}^{-}$ where $\tilde{v}$ may be $+\infty$;

(G)$G(y,s)=G(-y,s)\ge 0,{\int }_0^{\infty }{\int }_{-\infty }^{\infty }G(y,s)dyds=1,{\int }_0^{\infty }{\int }_{-\infty }^{\infty }sG(y,s)dyds<\infty$ and $G(y,s)$ is Lipschitz continuous with the space variable y. Moreover, for each $c\ge 0$,

$${\int }_0^{\infty }{\int }_{-\infty }^{\infty }G(y,s)e^{-\lambda (y+cs)}dyds<\infty ,\lambda \in [0,\infty ),$$

$${\int }_0^{\infty }{\int }_{-\infty }^{\infty }G(y,s)e^{-\lambda (y+cs)}dyds\to \infty ,as\lambda \to \infty .$$

The remainder of this paper is organized as follows. In Section 2, we introduce some preliminaries. In Section 3, we prove the existence of traveling waves by the Schauder’s fixed point theorem and the method of upper-lower solution. We will discuss the nonexistence of the traveling waves in Section 4. Finally, we will provide the conclusions and give a discussion about the effect of the nonlocal delayed transmission on the propagation of the disease in Section 5.

2. Some Preliminaries

In this section, we will consider the traveling wave solutions for system (5). Upon substituting $S(I,R)(x,t)=S(I,R)(x+ct)$ into (5), and denoting $\xi =x+ct$, we derive the following wave profile system for system (5):

$$\left\{\begin{array}{c}c{S}^{\prime }\left(\xi \right)=d_1(J\ast S\left(\xi \right)-S\left(\xi \right))-\frac{\beta S\left(\xi \right)(G\ast I)\left(\xi \right)}{S\left(\xi \right)+(G\ast I)\left(\xi \right)+R\left(\xi \right)},\hfill \\ c{I}^{\prime }\left(\xi \right)=d_2(J\ast I\left(\xi \right)-I\left(\xi \right))+\frac{\beta S\left(\xi \right)(G\ast I)\left(\xi \right)}{S\left(\xi \right)+(G\ast I)\left(\xi \right)+R\left(\xi \right)}-(\gamma +\delta )I\left(\xi \right),\hfill \\ c{R}^{\prime }\left(\xi \right)=d_3(J\ast R\left(\xi \right)-R\left(\xi \right))+\gamma I\left(\xi \right),\hfill \end{array}\right.$$

(6)

where

$$(G\ast I)\left(\xi \right)={\int }_0^{\infty }{\int }_{-\infty }^{\infty }G(y,s)I(\xi -y-cs)dyds.$$

We assume that system (6) has a disease free equilibrium $(S_0,0,0)$, where $S_0>0$ is a constant. According to the meaning of mathematical epidemiology, we intend to find solution $\left(S\right(\xi ),I(\xi ),R(\xi \left)\right)$ for system (6) which is nonnegative and satisfies the following asymptotic boundary conditions:

$$S(-\infty )=S_0,\underset{\xi \to +\infty }{lim}S\left(\xi \right):=S_{\infty }<S_0,I(\pm \infty )=0,R(-\infty )=0.$$

Moreover, if R is bounded, then $\underset{\xi \to +\infty }{lim}R\left(\xi \right)$ exists and

$$R(\infty )=\frac{\gamma (S_0-S_{\infty })}{\gamma +\delta }.$$

For any $\lambda ,c>0$, by linearizing the second equation of system (6) at $(S_0,0,0)$ and letting $I\left(\xi \right)=e^{\lambda \xi }$, we obtain the characteristic equation

$$\begin{array}{cc}\hfill \Delta (\lambda ,c)& =d_2\left({\int }_{-\infty }^{\infty }J\left(y\right)e^{-\lambda y}dy-1\right)-c\lambda +\beta {\int }_0^{\infty }{\int }_{-\infty }^{\infty }G(y,s)e^{-\lambda (y+cs)}dyds-\gamma -\delta .\hfill \end{array}$$

For the convenience, we denote

$$\overline{G}(\lambda ,c)={\int }_0^{\infty }{\int }_{-\infty }^{\infty }G(y,s)e^{-\lambda (y+cs)}dyds,$$

then

$$\Delta (\lambda ,c)=d_2\left({\int }_{-\infty }^{\infty }J\left(y\right)e^{-\lambda y}dy-1\right)-c\lambda +\beta \overline{G}(\lambda ,c)-\gamma -\delta .$$

By a direct calculation, and using (J) and (G), we have

$$\begin{array}{c}\Delta (0,c)=\beta -\gamma -\delta ,\Delta (\lambda ,+\infty )=-\infty ,\hfill \\ \frac{\partial \Delta (0,c)}{\partial \lambda }=-c\left(1+\beta {\int }_0^{\infty }{\int }_{-\infty }^{\infty }sG(y,s)dyds\right)<0,\hfill \\ \frac{\partial \Delta (\lambda ,c)}{\partial c}=-\lambda -\lambda \beta {\int }_0^{\infty }{\int }_{-\infty }^{\infty }sG(y,s)e^{-\lambda (y+cs)}dyds<0,\hfill \\ \frac{{\partial }^2\Delta (\lambda ,c)}{\partial {\lambda }^2}=d_2{\int }_{-\infty }^{\infty }{y}^{2}J\left(y\right)e^{-\lambda y}dy+\beta {\int }_0^{\infty }{\int }_{-\infty }^{\infty }{(y-cs)}^{2}G(y,s)e^{-\lambda (y+cs)}dyds>0.\hfill \end{array}$$

Therefore, we have the following lemma.

Lemma 1.

Assume that ${\mathfrak{R}}_0=\frac{\beta }{\gamma +\delta }>1$, then there exist $c^{*}>0,{\lambda }^{*}>0$ such that

$$\Delta ({\lambda }^{*},c^{*})=0,\frac{\partial \Delta }{\partial \lambda }(\lambda ,c){|}_{({\lambda }^{*},c^{*})}=0.$$

Furthermore,

(i) 

if $0<c<c^{*}$, then $\Delta (\lambda ,c)>0$ for all $\lambda >0$;

(ii) 

if $c>c^{*}$, then $\Delta (\lambda ,c)=0$ has two different positive roots ${\lambda }_1\left(c\right),{\lambda }_2\left(c\right)$ with $0<{\lambda }_1\left(c\right)<{\lambda }^{*}<{\lambda }_2\left(c\right)$ and

$$\Delta (\lambda ,c)\{\begin{array}{c}>0\lambda \in [0,{\lambda }_1\left(c\right))\cup ({\lambda }_2\left(c\right),\infty ),\hfill \\ <0\lambda \in ({\lambda }_1\left(c\right),{\lambda }_2\left(c\right)).\hfill \end{array}$$

When $c>c^{*},i=1,2$, we denote ${\lambda }_i\left(c\right)$ as ${\lambda }_i$.

For any $c>0$, we also define function ${\Delta }_1(\lambda ,c)$ as follows:

$${\Delta }_1(\lambda ,c)=c\lambda -d_3\left({\int }_{-\infty }^{\infty }J\left(x\right)e^{-\lambda x}dx-1\right).$$

Then,

$$\frac{\partial {\Delta }_1(0,c)}{\partial \lambda }=c>0,\frac{{\partial }^2{\Delta }_1(\lambda ,c)}{\partial {\lambda }^2}<0.$$

Thus, there exists ${\lambda }_0>0$ such that ${\Delta }_1(\lambda ,c)>0$ for any $\lambda \in (0,{\lambda }_0)$.

3. Existence of Traveling Waves

3.1. Upper-Lower Solution of System (6)

In this section, we will prove the existence of traveling wave solutions for system (5). In the remainder of this section, we fix ${\mathfrak{R}}_0=\frac{\beta }{\gamma +\delta }>1,c>c^{*}$. At first, we define six nonnegative continuous functions as follows:

$$\begin{array}{cccc}\hfill S_{+}\left(\xi \right)& =S_0,\hfill & \hfill S_{-}\left(\xi \right)& =\left\{\begin{array}{cc}{S}_0(1-\sigma e^{\alpha \xi }),\hfill & \xi <{\xi }_1,\hfill \\ 0,\hfill & \xi \ge {\xi }_1,\hfill \end{array}\right.\hfill \\ \hfill I_{+}\left(\xi \right)& =\left\{\begin{array}{cc}{e}^{{\lambda }_1\xi },\hfill & \xi <{\xi }_2,\hfill \\ \frac{(\beta -\gamma -\delta )S_0}{\gamma +\delta },\hfill & \xi \ge {\xi }_2,\hfill \end{array}\right.\hfill & \hfill I_{-}\left(\xi \right)& =\left\{\begin{array}{cc}{e}^{{\lambda }_1\xi }(1-M{e}^{\eta \xi }),\hfill & \xi <{\xi }_3,\hfill \\ 0,\hfill & \xi \ge {\xi }_3,\hfill \end{array}\right.\hfill \\ \hfill R_{+}\left(\xi \right)& =M_1{e}^{\eta \xi },R_{-}\left(\xi \right)=0,\hfill & & \end{array}$$

where $\sigma ,\alpha ,M,M_1$ are all positive constants, $\sigma =\frac{1}{\alpha },\eta \in (0,{\lambda }_0)$ is sufficiently small satisfying $\eta <min\{\alpha ,{\lambda }_1,{\lambda }_2-{\lambda }_1\}$.

Now, we have the following lemmas.

Lemma 2.

For sufficiently large $M_1$, the following inequalities hold:

$$\begin{array}{cc}\hfill c{S}_{+}^{\prime }\left(\xi \right)& \ge d_1(J\ast S_{+}\left(\xi \right)-S_{+}\left(\xi \right))-\frac{\beta S_{+}\left(\xi \right)(G\ast I_{-})\left(\xi \right)}{S_{+}\left(\xi \right)+(G\ast I_{-})\left(\xi \right)+R_{+}\left(\xi \right)},\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill c{I}_{+}^{\prime }\left(\xi \right)& \ge d_2(J\ast I_{+}\left(\xi \right)-I_{+}\left(\xi \right))+\frac{\beta S_{+}\left(\xi \right)(G\ast I_{+})\left(\xi \right)}{S_{+}\left(\xi \right)+(G\ast I_{+})\left(\xi \right)+R_{-}\left(\xi \right)}\hfill \\ & -(\gamma +\delta )I_{+}\left(\xi \right),\xi \ne {\xi }_2,\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill c{R}_{+}^{\prime }\left(\xi \right)& \ge d_3(J\ast R_{+}\left(\xi \right)-R_{+}\left(\xi \right))+\gamma I_{+}\left(\xi \right).\hfill \end{array}$$

(9)

Proof. 

Since $G\ast I_{-}\left(\xi \right)\ge 0$ and $S_{+}\left(\xi \right)=S_0$ is a positive constant, then inequality (7) holds naturally.

Next, we consider inequality (8). Due to (J), (G) and the definition of $I_{+}\left(\xi \right)$, we have

$$\begin{array}{cc}\hfill J\ast I_{+}\left(\xi \right)& \le min\left\{e^{{\lambda }_1\xi }{\int }_{-\infty }^{\infty }J\left(y\right)e^{-{\lambda }_{1}y}dy,\frac{(\beta -\gamma -\delta )S_0}{\gamma +\delta }\right\},\hfill \\ \hfill G\ast I_{+}\left(\xi \right)& \le min\left\{e^{{\lambda }_1\xi }\overline{G}({\lambda }_1,c),\frac{(\beta -\gamma -\delta )S_0}{\gamma +\delta }\right\}.\hfill \end{array}$$

(10)

If $\xi <{\xi }_2$, by Lemma 1, (10) and $\frac{S_{+}\left(\xi \right)(G\ast I_{+})\left(\xi \right)}{S_{+}\left(\xi \right)+(G\ast I_{+})\left(\xi \right)+R_{-}\left(\xi \right)}\le G\ast I_{+}\left(\xi \right)$, we conclude that $I_{+}\left(\xi \right)=e^{{\lambda }_1\xi }$ satisfies

$$\begin{array}{cc}\hfill c{I}_{+}^{\prime }\left(\xi \right)& =d_2\left({\int }_{-\infty }^{\infty }J\left(y\right)e^{-{\lambda }_{1}y}dy-1\right)I_{+}\left(\xi \right)+\beta I_{+}\left(\xi \right)\overline{G}({\lambda }_1,c)-(\gamma +\delta )I_{+}\left(\xi \right)\hfill \\ & \ge d_2(J\ast I_{+}\left(\xi \right)-I_{+}\left(\xi \right))+\beta (G\ast I_{+})\left(\xi \right)-(\gamma +\delta )I_{+}\left(\xi \right)\hfill \\ & \ge d_2(J\ast I_{+}\left(\xi \right)-I_{+}\left(\xi \right))+\frac{\beta S_{+}\left(\xi \right)(G\ast I_{+})\left(\xi \right)}{S_{+}\left(\xi \right)+(G\ast I_{+})\left(\xi \right)+R_{-}\left(\xi \right)}-(\gamma +\delta )I_{+}\left(\xi \right).\hfill \end{array}$$

If $\xi >{\xi }_2$, then $S_{+}\left(\xi \right)=S_0,I_{+}\left(\xi \right)=\frac{(\beta -\gamma -\delta )S_0}{\gamma +\delta }$ and $R_{-}\left(\xi \right)=0$. Since $\frac{\beta xy}{x+y}$ is nondecreasing with respect with $x,y$, we have that

$$\begin{array}{c}\frac{\beta S_{+}\left(\xi \right)(G\ast I_{+})\left(\xi \right)}{S_{+}\left(\xi \right)+(G\ast I_{+})\left(\xi \right)+R_{-}\left(\xi \right)}-(\gamma +\delta )I_{+}\left(\xi \right)\hfill \\ \le \frac{\beta S_0{I}_{+}\left(\xi \right)}{S_0+I_{+}\left(\xi \right)}-(\gamma +\delta )I_{+}\left(\xi \right)\hfill \\ =\left[\frac{\beta S_0(\gamma +\delta )}{(\gamma +\delta )S_0+(\beta -\gamma -\delta )S_0}-(\gamma +\delta )\right]I_{+}\left(\xi \right)=0.\hfill \end{array}$$

Thus, formula (8) is true.

Finally, we consider inequality (9). When $\xi <{\xi }_2$, then $I_{+}\left(\xi \right)=e^{{\lambda }_1\xi }$ and $R_{+}\left(\xi \right)=M_1{e}^{\eta \xi }$. It suffices to prove

$$c\eta M_1{e}^{\eta \xi }\ge d_3{M}_1{e}^{\eta \xi }\left({\int }_{-\infty }^{\infty }J\left(x\right)e^{-\eta x}dx-1\right)+\gamma e^{{\lambda }_1\xi },$$

that is

$$M_1{e}^{\eta \xi }\left[c\eta -d_3\left({\int }_{-\infty }^{\infty }J\left(x\right)e^{-\eta x}dx-1\right)\right]\ge \gamma e^{{\lambda }_1\xi }.$$

(11)

Due to $\eta \in (0,{\lambda }_0)$, we have ${\Delta }_1(\eta ,c)>0$, and inequality (11) is equivalent to

$$M_1\ge \frac{\gamma e^{({\lambda }_1-\eta )\xi }}{{\Delta }_1(\eta ,c)}.$$

Furthermore, since $\eta <{\lambda }_1$ and $\xi <{\xi }_2$, then formula (9) holds if we take

$$M_1>\frac{\gamma e^{({\lambda }_1-\eta ){\xi }_2}}{{\Delta }_1(\eta ,c)}.$$

Similarly, for $\xi >{\xi }_2$, formula (9) is true if we take

$$M_1>\frac{\gamma (\beta -\gamma -\delta )S_0{e}^{-\eta {\xi }_2}}{(\gamma +\delta ){\Delta }_1(\eta ,c)}.$$

Therefore, we take

$$M_1>max\left\{\frac{\gamma e^{({\lambda }_1-\eta ){\xi }_2}}{{\Delta }_1(\eta ,c)},\frac{\gamma (\beta -\gamma -\delta )S_0{e}^{-\eta {\xi }_2}}{(\gamma +\delta ){\Delta }_1(\eta ,c)}\right\},$$

then inequality (9) is true. The proof is complete.

Lemma 3.

Suppose that $\alpha <{\lambda }_1$ and σ are large enough. Then, the function $S_{-}\left(\xi \right)$ satisfies

$$c{S}_{-}^{\prime }\left(\xi \right)\le d_1(J\ast S_{-}\left(\xi \right)-S_{-}\left(\xi \right))-\frac{\beta S_{-}\left(\xi \right)(G\ast I_{+})\left(\xi \right)}{S_{-}\left(\xi \right)+(G\ast I_{+})\left(\xi \right)+R_{-}\left(\xi \right)}$$

(12)

for any $\xi \ne {\xi }_1:=\frac{1}{\alpha }ln\frac{1}{\sigma }.$

Proof. 

According to the definition of $S_{-}\left(\xi \right)$ and (J), we can obtain that

$$J\ast S_{-}\left(\xi \right)\ge max\left\{S_0-S_0\sigma e^{\alpha \xi }{\int }_{-\infty }^{\infty }J\left(y\right)e^{-\alpha y}dy,0\right\}.$$

(13)

If $\xi >{\xi }_1$, $S_{-}\left(\xi \right)=0$, then inequality (12) holds.

By taking $\sigma$ large enough such that $\sigma >e^{-\alpha {\xi }_2}$, then $\frac{1}{\alpha }ln\frac{1}{\sigma }={\xi }_1<{\xi }_2$. Therefore, if $\xi <{\xi }_1$, $S_{-}\left(\xi \right)=S_0(1-\sigma e^{\alpha \xi })$, $I_{+}\left(\xi \right)=e^{{\lambda }_1\xi }$. Due to $\frac{S_{-}\left(\xi \right)(G\ast I_{+})\left(\xi \right)}{S_{-}\left(\xi \right)+(G\ast I_{+})\left(\xi \right)+R_{-}\left(\xi \right)}\le G\ast I_{+}\left(\xi \right)$, (10) and (13), in order to prove inequality (12), we only need to prove

$$\begin{array}{cc}\hfill c{S}_{-}^{\prime }\left(\xi \right)& \le d_1\left[S_0-S_0\sigma e^{\alpha \xi }{\int }_{-\infty }^{\infty }J\left(y\right)e^{-\alpha y}dy-S_{-}\left(\xi \right)\right]-\beta e^{{\lambda }_1\xi }\overline{G}({\lambda }_1,c).\hfill \end{array}$$

(14)

Through a simple calculation, we know that formula (14) is equivalent to

$$-S_0\sigma \left[c\alpha -d_1\left({\int }_{-\infty }^{\infty }J\left(y\right)e^{-\alpha y}dy-1\right)\right]+\beta e^{({\lambda }_1-\alpha )\xi }\overline{G}({\lambda }_1,c)\le 0.$$

Since $\alpha <{\lambda }_1$ and $\xi <{\xi }_1$, it suffices to show that

$$-S_0\sigma \left[c\alpha -d_1\left({\int }_{-\infty }^{\infty }J\left(y\right)e^{-\alpha y}dy-1\right)\right]+\beta e^{({\lambda }_1-\alpha ){\xi }_1}\overline{G}({\lambda }_1,c)\le 0,$$

that is

$$-S_0\sigma \left[c\alpha -d_1\left({\int }_{-\infty }^{\infty }J\left(y\right)e^{-\alpha y}dy-1\right)\right]+\beta {\sigma }^{\frac{\alpha -{\lambda }_1}{\alpha }}\overline{G}({\lambda }_1,c)\le 0.$$

By taking sufficiently large $\sigma$, the above is true, and the proof is complete. □

Lemma 4.

Supposing that M is large enough, the function $I_{-}\left(\xi \right)$ satisfies the inequality

$$c{I}_{-}^{\prime }\left(\xi \right)\le d_2(J\ast I_{-}\left(\xi \right)-I_{-}\left(\xi \right))+\frac{\beta S_{-}\left(\xi \right)(G\ast I_{-})\left(\xi \right)}{S_{-}\left(\xi \right)+(G\ast I_{-})\left(\xi \right)+R_{+}\left(\xi \right)}-(\gamma +\delta )I_{-}\left(\xi \right)$$

(15)

for $\xi \ne {\xi }_3$.

Proof. 

First, by the definition of $I_{-}\left(\xi \right)$ and (J), (G), we have

$$\begin{array}{cc}\hfill J\ast I_{-}\left(\xi \right)& \ge max\left\{e^{{\lambda }_1\xi }{\int }_{-\infty }^{\infty }J\left(y\right)e^{-{\lambda }_{1}y}dy-M{e}^{({\lambda }_1+\eta )\xi }{\int }_{-\infty }^{\infty }J\left(y\right)e^{-({\lambda }_1+\eta )y}dy,0\right\},\hfill \\ \hfill G\ast I_{-}\left(\xi \right)& \ge max\left\{e^{{\lambda }_1\xi }\overline{G}({\lambda }_1,c)-M{e}^{({\lambda }_1+\eta )\xi }\overline{G}({\lambda }_1+\eta ,c),0\right\}.\hfill \end{array}$$

If $\xi >{\xi }_3$, then $I_{-}\left(\xi \right)=0$. Thus, inequality (15) is obviously true.

If $\xi <{\xi }_3$, we choose an M large enough such that ${\xi }_3=\frac{1}{\eta }ln\frac{1}{M}<\frac{1}{\alpha }ln\frac{1}{\sigma }={\xi }_1$, then $S_{-}\left(\xi \right)=S_0(1-\sigma e^{\alpha \xi })\le S_0$, $I_{-}\left(\xi \right)=e^{{\lambda }_1\xi }(1-M{e}^{\eta \xi })$. In order to prove inequality (15), it is only necessary to prove

$$\begin{array}{cc}\hfill c{I}_{-}^{\prime }\left(\xi \right)& \le d_2(J\ast I_{-}\left(\xi \right)-I_{-}\left(\xi \right))+\beta G\ast I_{-}\left(\xi \right)-(\gamma +\delta )I_{-}\left(\xi \right)\hfill \\ & -\frac{\beta [S_0-S_{-}\left(\xi \right)+(G\ast I_{-})\left(\xi \right)+R_{+}\left(\xi \right)]}{S_0+(G\ast I_{-})\left(\xi \right)+R_{+}\left(\xi \right)}G\ast I_{-}\left(\xi \right).\hfill \end{array}$$

By the estimate of $G\ast I_{-}\left(\xi \right)$ and the definition of $\Delta (\lambda ,c)$, it suffices to prove that

$$\beta [S_0-S_{-}\left(\xi \right)+(G\ast I_{-})\left(\xi \right)+R_{+}\left(\xi \right)](G\ast I_{-})\left(\xi \right)\le -M\Delta ({\lambda }_1+\eta ,c)e^{({\lambda }_1+\eta )\xi }·S_0.$$

(16)

Since $G\ast I_{-}\left(\xi \right)\le e^{{\lambda }_1\xi }\overline{G}({\lambda }_1,c)$, formula (16) is true if

$$\begin{array}{c}\hfill \beta (S_0\sigma e^{\alpha \xi }+e^{{\lambda }_1\xi }\overline{G}({\lambda }_1,c)+M_1{e}^{\eta \xi })e^{{\lambda }_1\xi }\overline{G}({\lambda }_1,c)\le -S_{0}M\Delta ({\lambda }_1+\eta ,c)e^{({\lambda }_1+\eta )\xi }.\end{array}$$

(17)

Due to $\xi <{\xi }_1,\eta <\alpha ,\eta <{\lambda }_1,\Delta ({\lambda }_1+\eta ,c)<0$, by taking M be a constant number which satisfies

$$M\ge \frac{\beta \overline{G}({\lambda }_1,c)\left[S_0{\sigma }^{\frac{\eta }{\alpha }}+{\sigma }^{\frac{\eta -{\lambda }_1}{\alpha }}\overline{G}({\lambda }_1,c)+M_1\right]}{-S_0\Delta ({\lambda }_1+\eta ,c)},$$

formula (17) holds.

In summary, there exists sufficiently large M, such that $I_{-}\left(\xi \right)$ satisfies inequality (15). The proof is complete. □

By Lemmas 2–4, we know that the continuous function $(S_{+}\left(\xi \right),I_{+}\left(\xi \right),$ $R_{+}\left(\xi \right))$ and $(S_{-}\left(\xi \right),I_{-}\left(\xi \right),R_{-}\left(\xi \right))$ is a pair of upper-lower solutions for system (6).

From now on, we will establish the existence of traveling wave solutions of (4). Due to the lack of regularity of nonlocal dispersal operator, we first consider the traveling wave system (6) on a bounded interval. For this purpose, we take $X>\frac{1}{\eta }lnM$, and define the following set:

$${\Gamma }_X=\left\{\left(\varphi \right(·),\chi (·),\psi (·\left)\right)\in C([-X,X],{\mathbb{R}}^3)\left|\begin{array}{c}\varphi (-X)=S_{-}(-X),\chi (-X)=I_{-}(-X),\hfill \\ \psi (-X)=R_{-}(-X),\hfill \\ S_{-}\left(\xi \right)\le \varphi \left(\xi \right)\le S_0,\hfill \\ I_{-}\left(\xi \right)\le \chi \left(\xi \right)\le I_{+}\left(\xi \right),\hfill \\ R_{-}\left(\xi \right)\le \psi \left(\xi \right)\le R_{+}\left(\xi \right),\forall \xi \in [-X,X]\hfill \end{array}\right.\right\}.$$

It is easy to know that ${\Gamma }_X$ is a closed, convex subset of $C([-X,X],{\mathbb{R}}^3)$. For any $(\varphi (·),\chi (·),\psi (·))\in {\Gamma }_X$, we define

$$\begin{array}{c}\hfill \tilde{\varphi }\left(\xi \right)=\left\{\begin{array}{cc}\varphi \left(X\right),\hfill & \xi >X,\hfill \\ \varphi \left(\xi \right),\hfill & \left|\xi \right|\le X,\hfill \\ S_{-}\left(\xi \right),\hfill & \xi <-X,\hfill \end{array}\right.\tilde{\chi }\left(\xi \right)=\left\{\begin{array}{cc}\chi \left(X\right),\hfill & \xi >X,\hfill \\ \chi \left(\xi \right),\hfill & \left|\xi \right|\le X,\hfill \\ I_{-}\left(\xi \right),\hfill & \xi <-X,\hfill \end{array}\right.\end{array}$$

and

$$\begin{array}{c}\hfill \tilde{\psi }\left(\xi \right)=\left\{\begin{array}{cc}\psi \left(X\right),\hfill & \xi >X,\hfill \\ \psi \left(\xi \right),\hfill & \left|\xi \right|\le X,\hfill \\ R_{-}\left(\xi \right),\hfill & \xi <-X.\hfill \end{array}\right.\end{array}$$

We consider the following initial value problem:

$$\begin{array}{cc}\hfill c{S}^{\prime }\left(\xi \right)& =d_1{\int }_{-\infty }^{\infty }J\left(y\right)\tilde{\varphi }(\xi -y)dy-d_{1}S\left(\xi \right)-\frac{\beta \varphi \left(\xi \right)(G\ast \tilde{\chi })\left(\xi \right)}{\varphi \left(\xi \right)+(G\ast \tilde{\chi })\left(\xi \right)+\psi \left(\xi \right)},\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill c{I}^{\prime }\left(\xi \right)& =d_2{\int }_{-\infty }^{\infty }J\left(y\right)\tilde{\chi }(\xi -y)dy-(d_2+\gamma +\delta )I\left(\xi \right)+\frac{\beta \varphi \left(\xi \right)(G\ast \tilde{\chi })\left(\xi \right)}{\varphi \left(\xi \right)+(G\ast \tilde{\chi })\left(\xi \right)+\psi \left(\xi \right)},\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill c{R}^{\prime }\left(\xi \right)& =d_3{\int }_{-\infty }^{\infty }J\left(y\right)\tilde{\psi }(\xi -y)dy-d_{3}R\left(\xi \right)+\gamma \chi \left(\xi \right),\hfill \end{array}$$

(20)

with

$$S(-X)=S_{-}(-X),I(-X)=I_{-}(-X),R(-X)=R_{-}(-X).$$

(21)

By the theory for ODE [34], problems (18)–(21) admit a unique solution $(S_X\left(\xi \right),I_X\left(\xi \right),R_X\left(\xi \right))$ satisfying $S_X(·),I_X(·),R_X(·)\in C^1\left([-X,X]\right)$. Furthermore, we define an operator $F=(F_1,F_2,F_3)$: ${\Gamma }_X\to C\left([-X,X]\right)$ such that, for any $\xi \in [-X,X],$

$$F_1(\varphi ,\chi ,\psi )\left(\xi \right)=S_X\left(\xi \right),F_2(\varphi ,\chi ,\psi )\left(\xi \right)=I_X\left(\xi \right),F_3(\varphi ,\chi ,\psi )\left(\xi \right)=R_X\left(\xi \right).$$

Lemma 5.

The operator F maps ${\Gamma }_X$ into ${\Gamma }_X$, and is completely continuous.

Proof. 

First, by Lemmas 2–4 and using the similar method as that of Theorem 2.4 in [28], we can obtain that F maps ${\Gamma }_X$ into ${\Gamma }_X$. We leave the proof in the Appendix A. Next, we only give the proof of continuity and compactness of F.

By a direct computation, we can obtain the solutions of the initial value problem (18)–(20) as follows:

$$\begin{array}{cc}\hfill S_X\left(\xi \right)& =S_{-}(-X)e^{-\frac{d_1}{c}(\xi +X)}+\frac{1}{c}{\int }_{-X}^{\xi }{e}^{\frac{d_1}{c}(\eta -\xi )}\left(d_1(J\ast \tilde{\varphi })\left(\eta \right)-\frac{\beta \varphi \left(\eta \right)(G\ast \tilde{\chi })\left(\eta \right)}{\varphi \left(\eta \right)+(G\ast \tilde{\chi })\left(\eta \right)+\psi \left(\eta \right)}\right)d\eta ,\hfill \\ \hfill I_X\left(\xi \right)& =I_{-}(-X)e^{-\frac{d_2+\gamma +\delta }{c}(\xi +X)}+\frac{1}{c}{\int }_{-X}^{\xi }{e}^{\frac{d_2+\gamma +\delta }{c}(\eta -\xi )}\left(d_2(J\ast \tilde{\chi })\left(\eta \right)+\frac{\beta \varphi \left(\eta \right)(G\ast \tilde{\chi })\left(\eta \right)}{\varphi \left(\eta \right)+(G\ast \tilde{\chi })\left(\eta \right)+\psi \left(\eta \right)}\right)d\eta ,\hfill \\ \hfill R_X\left(\xi \right)& =\frac{1}{c}{\int }_{-X}^{\xi }{e}^{\frac{d_3}{c}(\eta -\xi )}(d_3(J\ast \tilde{\psi })\left(\eta \right)+\gamma \chi \left(\eta \right))d\eta .\hfill \end{array}$$

For any $({\varphi }_i,{\chi }_i,{\psi }_i)\in {\Gamma }_X,i=1,2$, we denote

$$F_1({\varphi }_i,{\chi }_i,{\psi }_i)\left(\xi \right)=S_{X,i}\left(\xi \right),F_2({\varphi }_i,{\chi }_i,{\psi }_i)\left(\xi \right)=I_{X,i}\left(\xi \right),F_3({\varphi }_i,{\chi }_i,{\psi }_i)\left(\xi \right)=R_{X,i}\left(\xi \right).$$

Since

$$J\ast \tilde{\varphi }\left(\eta \right)={\int }_{-\infty }^{-X}J(\eta -y)S_{-}\left(y\right)dy+{\int }_{-X}^{X}J(\eta -y)\varphi \left(y\right)dy+{\int }_X^{\infty }J(\eta -y)\varphi \left(X\right)dy,$$

we have

$$|J\ast {\tilde{\varphi }}_1\left(\eta \right)-J\ast {\tilde{\varphi }}_2\left(\eta \right)|\le 2\underset{y\in [-X,X]}{max}|{\varphi }_1\left(y\right)-{\varphi }_2\left(y\right)|.$$

Similarly, there hold

$$|J\ast {\tilde{\chi }}_1\left(\eta \right)-J\ast {\tilde{\chi }}_2\left(\eta \right)|\le 2\underset{y\in [-X,X]}{max}|{\chi }_1\left(y\right)-{\chi }_2\left(y\right)|,$$

$$|J\ast {\tilde{\psi }}_1\left(\eta \right)-J\ast {\tilde{\psi }}_2\left(\eta \right)|\le 2\underset{y\in [-X,X]}{max}|{\psi }_1\left(y\right)-{\psi }_2\left(y\right)|$$

and

$$|G\ast {\tilde{\chi }}_1\left(\eta \right)-G\ast {\tilde{\chi }}_2\left(\eta \right)|\le 2\underset{y\in [-X,X]}{max}|{\chi }_1\left(y\right)-{\chi }_2\left(y\right)|.$$

By proposition 2.5 in [33], we have that $\frac{\beta \varphi \left(\eta \right)(G\ast \tilde{\chi })\left(\eta \right)}{\varphi \left(\eta \right)+(G\ast \tilde{\chi })\left(\eta \right)+\psi \left(\eta \right)}$ is Lipschitz continuous. Then, by the definitions of $S_{X,i}\left(\xi \right),I_{X,i}\left(\xi \right),R_{X,i}\left(\xi \right)$ and the operator F, we can conclude that F is continuous.

Finally, we show that F is compact. Since $S_X(·)$, $I_X(·),R_X(·)\in C^1\left([-X,X]\right)$ and satisfy (18)–(20), we obtain that $S_X^{\prime },I_X^{\prime }$ and $R_X^{\prime }$ are uniformly bounded. Thus, the operator F is compact, and we obtain that F is completely continuous. □

Based on the above discussion, by using Schauder’s fixed point theorem, we obtain the following result.

Theorem 1.

There exists $(S_X(·),I_X(·),R_X(·))\in {\Gamma }_X$ such that

$$(S_X(·),I_X(·),R_X(·))=F(S_X,I_X,R_X)\left(\xi \right)forany\xi \in (-X,X).$$

3.2. Traveling Wave Solution for (6) on $\mathbb{R}$

Next, we want to obtain the existence of solutions for traveling wave system (6) on $\mathbb{R}$. To this end, we will give some priori estimates for $S_X,I_X,R_X$ in the space $C^{1,1}\left([-X,X]\right)$, where

$$C^{1,1}\left([-X,X]\right)=\{u\in C^1\left([-X,X]\right)\mid u,u^{\prime }\mathrm{are}\mathrm{Lipschitz}\mathrm{continuous}\}$$

with the norm

$${\parallel u\parallel }_{C^{1,1}\left([-X,X]\right)}=\underset{x\in [-X,X]}{max}\left|u\right|+\underset{x\in [-X,X]}{max}\left|u^{\prime }\right|+\underset{x\ne y\in [-X,X]}{sup}\frac{|u^{\prime }\left(x\right)-u^{\prime }\left(y\right)|}{|x-y|}.$$

Lemma 6.

For any $X>\frac{1}{\eta }lnM$, there exists $Y>0$, such that $Y+r<X$, and

$$\parallel S_X{\parallel }_{C^{1,1}\left([-Y,Y]\right)}\le C\left(Y\right),\parallel I_X{\parallel }_{C^{1,1}\left([-Y,Y]\right)}\le C\left(Y\right),{\parallel R_X\parallel }_{C^{1,1}\left([-Y,Y]\right)}\le C\left(Y\right),$$

where r is the radius of $suppJ$, $C\left(Y\right)$ is a constant which is independent from X.

Proof. 

Clearly, $(S_X,I_X,R_X)$ satisfies

$$\begin{array}{cc}\hfill c{S}_X^{\prime }\left(\xi \right)& =d_1{\int }_{-\infty }^{\infty }J(\xi -y){\tilde{S}}_X\left(y\right)dy-d_1{S}_X\left(\xi \right)-\frac{\beta S_X\left(\xi \right)(G\ast {\tilde{I}}_X)\left(\xi \right)}{S_X\left(\xi \right)+(G\ast {\tilde{I}}_X)\left(\xi \right)+R_X\left(\xi \right)},\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill c{I}_X^{\prime }\left(\xi \right)& =d_2{\int }_{-\infty }^{\infty }J(\xi -y){\tilde{I}}_X\left(y\right)dy-(d_2+\gamma +\delta )I_X\left(\xi \right)\hfill \\ & +\frac{\beta S_X\left(\xi \right)(G\ast {\tilde{I}}_X)\left(\xi \right)}{S_X\left(\xi \right)+(G\ast {\tilde{I}}_X)\left(\xi \right)+R_X\left(\xi \right)},\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill c{R}_X^{\prime }\left(\xi \right)& =d_3{\int }_{-\infty }^{\infty }J(\xi -y){\tilde{R}}_X\left(y\right)dy-d_3{R}_X\left(\xi \right)+\gamma I_X\left(\xi \right)\hfill \end{array}$$

(24)

for $\xi \in [-X,X]$, where

$$\begin{array}{c}\hfill {\tilde{S}}_X\left(\xi \right)=\left\{\begin{array}{cc}{S}_X\left(X\right),\hfill & \xi >X,\hfill \\ S_X\left(\xi \right),\hfill & \left|\xi \right|\le X,\hfill \\ S_{-}\left(\xi \right),\hfill & \xi <-X,\hfill \end{array}\right.{\tilde{I}}_X\left(\xi \right)=\left\{\begin{array}{cc}{I}_X\left(X\right),\hfill & \xi >X,\hfill \\ I_X\left(\xi \right),\hfill & \left|\xi \right|\le X,\hfill \\ I_{-}\left(\xi \right),\hfill & \xi <-X\hfill \end{array}\right.\end{array}$$

and

$$\begin{array}{c}\hfill {\tilde{R}}_X\left(\xi \right)=\left\{\begin{array}{cc}{R}_X\left(X\right),\hfill & \xi >X,\hfill \\ R_X\left(\xi \right),\hfill & \left|\xi \right|\le X,\hfill \\ R_{-}\left(\xi \right),\hfill & \xi <-X.\hfill \end{array}\right.\end{array}$$

Since $S_X\left(\xi \right)\le S_0$, we can deduce from equation (22) that

$$\begin{array}{cc}\hfill |S_X^{\prime }\left(\xi \right)|& \le \frac{1}{c}\left(d_1|J\ast {\tilde{S}}_X\left(\xi \right)|+d_1\left|S_X\left(\xi \right)\right|+\beta \left|\frac{S_X(G\ast {\tilde{I}}_X)}{S_X+(G\ast {\tilde{I}}_X)+R_X}\right|\right)\hfill \\ & \le \frac{2d_1+\beta }{c}{S}_0.\hfill \end{array}$$

(25)

Similarly, we have that

$$|I_X^{\prime }\left(\xi \right)|\le \frac{2d_2+\beta +\gamma +\delta }{c}\left(\frac{\beta }{\gamma +\delta }-1\right)S_0.$$

(26)

Since $R_X\left(\xi \right)\in {\Gamma }_X$, combined with the definitions of ${\Gamma }_X$ and $R_{+}\left(\xi \right)$, we know that $R_X\left(\xi \right)\le M_1{e}^{\eta Y}$ for any $\xi \in [-Y,Y]$, where $0<Y<X-r$. By formula (24) and using the analogous argument as inequality (25), we can obtain that

$$|R_X^{\prime }\left(\xi \right)|\le \frac{1}{c}\left(2d_3{M}_1{e}^{\eta Y}+\frac{\gamma (\beta -\gamma -\delta )}{\gamma +\delta }{S}_0\right),\xi \in [-Y,Y].$$

Consequently, we can conclude that, for any $\xi ,\eta \in [-Y,Y]$, there exist positive constants $L_1,L\left(Y\right)$ such that,

$$|S_X\left(\xi \right)-S_X\left(\eta \right)|\le L_1|\xi -\eta |,|I_X\left(\xi \right)-I_X\left(\eta \right)|\le L_1|\xi -\eta |,|R_X\left(\xi \right)-R_X\left(\eta \right)|\le L\left(Y\right)|\xi -\eta |.$$

Finally, we will prove that $S_X^{\prime }\left(\xi \right),I_X^{\prime }\left(\xi \right),R_X^{\prime }\left(\xi \right)$ are Lipschitz continuous. For any $\xi ,\eta \in [-Y,Y]$, it can be inferred from equation (22) that

$$\begin{array}{cc}\hfill |S_X^{\prime }\left(\xi \right)-S_X^{\prime }\left(\eta \right)|& \le \frac{d_1}{c}\left|{\int }_{-\infty }^{\infty }[J(\xi -y)-J(\eta -y)]{\tilde{S}}_X\left(y\right)dy\right|+\frac{d_1}{c}|S_X\left(\xi \right)-S_X\left(\eta \right)|\hfill \\ & +\frac{\beta }{c}|\frac{S_X\left(\xi \right)(G\ast {\tilde{I}}_X)\left(\xi \right)}{S_X\left(\xi \right)+(G\ast {\tilde{I}}_X)\left(\xi \right)+R_X\left(\xi \right)}-\frac{S_X\left(\eta \right)(G\ast {\tilde{I}}_X)\left(\eta \right)}{S_X\left(\eta \right)+(G\ast {\tilde{I}}_X)\left(\eta \right)+R_X\left(\eta \right)}|\hfill \\ & :={\Lambda }_1+{\Lambda }_2+{\Lambda }_3.\hfill \end{array}$$

Since

$$\begin{array}{cc}\hfill \left|{\int }_{-X}^X[J(\xi -y)-J(\eta -y)]S_X\left(y\right)dy\right|& =|{\int }_{\xi -X}^{\xi +X}J\left(y\right)S_X(\xi -y)dy-{\int }_{\eta -X}^{\eta +X}J\left(y\right)S_X(\eta -y)dy|\hfill \\ & =|{\int }_{\xi -X}^{\eta -X}J\left(y\right)S_X(\xi -y)dy+{\int }_{\eta -X}^{\xi +X}J\left(y\right)S_X(\xi -y)dy\hfill \\ & -{\int }_{\eta -X}^{\xi +X}J\left(y\right)S_X(\eta -y)dy-{\int }_{\xi +X}^{\eta +X}J\left(y\right)S_X(\eta -y)dy|\hfill \\ & \le (2S_0\parallel J{\parallel }_{L^{\infty }}+L_1)|\xi -\eta |,\hfill \end{array}$$

then we can obtain the following estimation about ${\Lambda }_1$:

$$\begin{array}{c}\left|{\int }_{-\infty }^{\infty }[J(\xi -y)-J(\eta -y)]{\tilde{S}}_X\left(y\right)dy\right|\hfill \\ =|{\int }_{-\infty }^{-X}(J(\xi -y)-J(\eta -y))S_{-}\left(y\right)dy+{\int }_{-X}^X(J(\xi -y)-J(\eta -y))S_X\left(y\right)dy\hfill \\ +{\int }_X^{\infty }(J(\xi -y)-J(\eta -y))S_X\left(X\right)dy|\hfill \\ \le S_0|{\int }_{-\infty }^{-X}(J(\xi -y)-J(\eta -y))dy|+(2S_0{\parallel J\parallel }_{L^{\infty }}+L_1\left)\right|\xi -\eta |+S_0|{\int }_X^{\infty }(J(\xi -y)-J(\eta -y))dy|\hfill \\ =S_0\left(\left|{\int }_{\xi +X}^{\eta +X}J\left(y\right)dy\right|+\left|{\int }_{\eta -X}^{\xi -X}J\left(y\right)dy\right|\right)+(2S_0{\parallel J\parallel }_{L^{\infty }}+L_1\left)\right|\xi -\eta |\hfill \\ \le (4S_0\parallel J{\parallel }_{L^{\infty }}+L_1)|\xi -\eta |.\hfill \end{array}$$

Moreover, a simple calculation implies that

$${\Lambda }_3\le \frac{\beta }{c}\left[\right|S_X\left(\xi \right)-S_X\left(\eta \right)|+|G\ast {\tilde{I}}_X\left(\xi \right)-G\ast {\tilde{I}}_X\left(\eta \right)|+|R_X\left(\xi \right)-R_X\left(\eta \right)\left|\right].$$

By a direct calculation, we have

$$\begin{array}{c}|G\ast {\tilde{I}}_X\left(\xi \right)-G\ast {\tilde{I}}_X\left(\eta \right)|\hfill \\ =\left|{\int }_0^T{\int }_{-\infty }^{\infty }[G(\xi -y-cs,s)-G(\eta -y-cs,s)]{\tilde{I}}_X\left(y\right)dyds\right|\hfill \\ =|{\int }_0^T{\int }_{-\infty }^{-X}[G(\xi -y-cs,s)-G(\eta -y-cs,s)]I_{-}\left(y\right)dyds\hfill \\ +{\int }_0^T{\int }_{-X}^X[G(\xi -y-cs,s)-G(\eta -y-cs,s)]I_X\left(y\right)dyds\hfill \\ +{\int }_0^T{\int }_X^{\infty }[G(\xi -y-cs,s)-G(\eta -y-cs,s)]I_X\left(X\right)dyds|\hfill \\ \le T{L}_G{\int }_{-\infty }^{\frac{1}{\eta }ln\frac{1}{M}}{I}_{-}\left(y\right)dy|\xi -\eta |+\left(\frac{\beta }{\gamma +\delta }-1\right)S_0{\int }_0^T{\int }_{\xi -X-cs}^{\eta -X-cs}\left|G(y,s)\right|dyds\hfill \\ +\left|{\int }_0^T{\int }_{-X}^X[G(\xi -y-cs,s)-G(\eta -y-cs,s)]I_X\left(y\right)dyds\right|,\hfill \end{array}$$

where $L_G$ is the Lipschitz constant of kernel $G(y,s)$ with invariable y. Since

$$\begin{array}{c}\left|{\int }_0^T{\int }_{-X}^X[G(\xi -y-cs,s)-G(\eta -y-cs,s)]I_X\left(y\right)dyds\right|\hfill \\ \le \left|{\int }_0^T{\int }_{\eta +X-cs}^{\xi +X-cs}G(y,s)I_X(\xi -y-cs)dyds\right|+\left|{\int }_0^T{\int }_{\eta -X-cs}^{\xi -X-cs}G(y,s)I_X(\eta -y-cs)dyds\right|\hfill \\ +\left|{\int }_0^T{\int }_{\xi -X-cs}^{\eta +X-cs}G(y,s)[I_X(\xi -y-cs)-I_X(\eta -y-cs)]dyds\right|\hfill \\ \le \left[2\left(\frac{\beta }{\delta +\gamma }-1\right)S_{0}T{\parallel G\parallel }_{L^{\infty }(\mathbb{R}\times [0,T])}+L_1\right]|\xi -\eta |,\hfill \end{array}$$

we have

$${\Lambda }_3\le \frac{\beta }{c}\left[2L_1+T{L}_G{\int }_{-\infty }^{\frac{1}{\eta }ln\frac{1}{M}}{I}_{-}\left(y\right)dy+3\left(\frac{\beta }{\delta +\gamma }-1\right)S_{0}T{\parallel G\parallel }_{L^{\infty }(\mathbb{R}\times [0,T])}+L\left(Y\right)\right]|\xi -\eta |.$$

Thus, there exists a constant $C_1\left(Y\right)$ such that

$$|S_X^{\prime }\left(\xi \right)-S_X^{\prime }\left(\eta \right)|\le C_1\left(Y\right)|\xi -\eta |.$$

Similarly, there exist two constants $C_2\left(Y\right),C_3\left(Y\right)$ such that

$$|I_X^{\prime }\left(\xi \right)-I_X^{\prime }\left(\eta \right)|\le C_2\left(Y\right)|\xi -\eta |,|R_X^{\prime }\left(\xi \right)-R_X^{\prime }\left(\eta \right)|\le C_3\left(Y\right)|\xi -\eta |.$$

Therefore, there exists some constant $C\left(Y\right)>0$, such that

$$\parallel S_X{\parallel }_{C^{1,1}\left([-Y,Y]\right)}\le C\left(Y\right),\parallel I_X{\parallel }_{C^{1,1}\left([-Y,Y]\right)}\le C\left(Y\right),{\parallel R_X\parallel }_{C^{1,1}\left([-Y,Y]\right)}\le C\left(Y\right).$$

□

Now, we derive the existence of solutions for system (6) on $\mathbb{R}$ by a limiting argument. Choose a sequence ${\left\{X_n\right\}}_{n=1}^{+\infty }$ such that $X_n>\frac{1}{\eta }lnM,X_n>Y+r$ and $\underset{n\to +\infty }{lim}{X}_n=+\infty$. For every n, we know that there exists $(S_{X_n},I_{X_n},R_{X_n})\in {\Gamma }_{X_n}$ satisfying the conclusion in Lemma 6. Therefore, there exists a subsequence $\left\{X_{n_k}\right\}$ by diagonal extraction argument, such that $\underset{k\to +\infty }{lim}{X}_{n_k}=+\infty$ and $S_{n_k}\to S,I_{n_k}\to I,R_{n_k}\to R$ when $k\to +\infty$.

By (J), (G) and Lebesgue dominated convergence theorem, we obtain

$$\begin{array}{cc}\hfill \underset{k\to +\infty }{lim}{\int }_{-\infty }^{\infty }J(\xi -y){\tilde{S}}_{n_k}\left(y\right)dy& ={\int }_{-\infty }^{\infty }J(\xi -y)S\left(y\right)dy=J\ast S\left(\xi \right),\hfill \\ \hfill \underset{k\to +\infty }{lim}{\int }_{-\infty }^{\infty }J(\xi -y){\tilde{I}}_{n_k}\left(y\right)dy& ={\int }_{-\infty }^{\infty }J(\xi -y)I\left(y\right)dy=J\ast I\left(\xi \right),\hfill \\ \hfill \underset{k\to +\infty }{lim}{\int }_{-\infty }^{\infty }J(\xi -y){\tilde{R}}_{n_k}\left(y\right)dy& ={\int }_{-\infty }^{\infty }J(\xi -y)R\left(y\right)dy=J\ast R\left(\xi \right)\hfill \end{array}$$

and

$$\underset{k\to +\infty }{lim}{\int }_0^{\infty }{\int }_{-\infty }^{\infty }G(\xi -y-cs){\tilde{I}}_{n_k}\left(y\right)dy=G\ast I\left(\xi \right).$$

Hence, $(S,I,R)$ satisfies the traveling wave system (6) with

$$S_{-}\left(\xi \right)<S\left(\xi \right)\le S_0,I_{-}\left(\xi \right)\le I\left(\xi \right)\le I_{+}\left(\xi \right),0<R\left(\xi \right)\le R_{+}\left(\xi \right).$$

By the definition of $(S_{-}\left(\xi \right),I_{-}\left(\xi \right),R_{-}\left(\xi \right)$ and $(S_{+}\left(\xi \right),I_{+}\left(\xi \right),R_{+}\left(\xi \right)$ and utilizing squeeze theorem, we have the following existence theorem.

Theorem 2.

Supposing that ${\mathfrak{R}}_0=\frac{\beta }{\gamma +\delta }>1$. For any $c>c^{*}$, there exists $\left(S\right(\xi ),I(\xi ),R(\xi \left)\right)$, which satisfies the traveling wave system (6) with

$$S_{-}\left(\xi \right)<S\left(\xi \right)\le S_0,I_{-}\left(\xi \right)\le I\left(\xi \right)\le I_{+}\left(\xi \right),0<R\left(\xi \right)\le R_{+}\left(\xi \right)$$

and

$$S(-\infty )=S_0,I(-\infty )=0,R(-\infty )=0.$$

3.3. Asymptotic Behavior

In the following, we will consider the asymptotic behavior of traveling waves $\left(S\right(\xi ),I(\xi ),R(\xi \left)\right)$ at $+\infty$. For this purpose, we give some estimations in advance.

Lemma 7.

Suppose that ${\mathfrak{R}}_0=\frac{\beta }{\gamma +\delta }>1$ and $c>c^{*}$, then the solution $\left(S\right(\xi ),I(\xi ),R(\xi \left)\right)$ of system (6) satisfies

$$0<{\int }_{-\infty }^{\infty }\frac{\beta S\left(\xi \right)(G\ast I)\left(\xi \right)}{S\left(\xi \right)+(G\ast I)\left(\xi \right)+R\left(\xi \right)}<+\infty$$

and

$${\int }_{-\infty }^{\infty }I\left(\xi \right)d\xi <+\infty ,I(\infty )=0.$$

Proof. 

First, due to the positive of $\frac{\beta S\left(\xi \right)(G\ast I)\left(\xi \right)}{S\left(\xi \right)+(G\ast I)\left(\xi \right)+R\left(\xi \right)}$, we get

$${\int }_{-\infty }^{\infty }\frac{\beta S\left(\xi \right)(G\ast I)\left(\xi \right)}{S\left(\xi \right)+(G\ast I)\left(\xi \right)+R\left(\xi \right)}>0.$$

By using the Fubini theorem, we have that

$$\begin{array}{cc}\hfill {\int }_z^x(J\ast S\left(\xi \right)-S\left(\xi \right))d\xi & ={\int }_z^x{\int }_{-\infty }^{\infty }J\left(y\right)(S(\xi -y)-S\left(\xi \right))dyd\xi \hfill \\ & =-{\int }_z^x{\int }_{-\infty }^{\infty }yJ\left(y\right){\int }_0^1{S}^{\prime }(\xi -\theta y)d\theta dyd\xi \hfill \\ & ={\int }_{-\infty }^{\infty }yJ\left(y\right){\int }_0^1(S(z-\theta y)-S(x-\theta y))d\theta dy.\hfill \end{array}$$

By assumption (J) and passing a limit above, we have

$$\begin{array}{cc}\hfill \underset{z\to -\infty }{lim}{\int }_z^x(J\ast S\left(\xi \right)-S\left(\xi \right))d\xi & ={\int }_{-\infty }^{\infty }yJ\left(y\right){\int }_0^1(S_0-S(x-\theta y))d\theta dy\hfill \\ & =-{\int }_{-\infty }^{\infty }yJ\left(y\right){\int }_0^{1}S(x-\theta y))d\theta dy,\hfill \end{array}$$

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

$$\left|{\int }_{-\infty }^x(J\ast S\left(\xi \right)-S\left(\xi \right))d\xi \right|\le S_0{\int }_{-\infty }^{\infty }\left|y\right|J\left(y\right)dy:={\varrho }_1.$$

(27)

Integrating the first equation of system (6) from $-\infty$ to x and using formula (27) yields

$$\begin{array}{cc}\hfill {\int }_{-\infty }^x\frac{\beta S\left(\xi \right)(G\ast \tilde{I})\left(\xi \right)}{S\left(\xi \right)+(G\ast \tilde{I})\left(\xi \right)+R\left(\xi \right)}d\xi & =d_1{\int }_{-\infty }^x(J\ast S\left(\xi \right)-S\left(\xi \right)d\xi +c[S_0-S\left(x\right)]\hfill \\ & \le d_1{\varrho }_1+c{S}_0.\hfill \end{array}$$

Then, we conclude that

$${\int }_{-\infty }^{\infty }\frac{\beta S\left(\xi \right)(G\ast I)\left(\xi \right)}{S\left(\xi \right)+(G\ast I)\left(\xi \right)+R\left(\xi \right)}<\infty .$$

(28)

Through a similar calculation as inequality (27), we have

$$\left|{\int }_{-\infty }^x(J\ast I\left(\xi \right)-I\left(\xi \right))d\xi \right|\le \left(\frac{\beta -\gamma -\delta }{\gamma +\delta }\right)S_0{\int }_{-\infty }^{\infty }\left|y\right|J\left(y\right)dy:={\varrho }_2.$$

(29)

Integrating the second equation of system (6) from $-\infty$ to ∞ and using Formulas (28) and (29), we have

$$\begin{array}{cc}\hfill cI(\infty )+(\gamma +\delta ){\int }_{\mathbb{R}}I\left(\xi \right)d\xi & =d_2{\int }_{\mathbb{R}}(J\ast I\left(\xi \right)-I\left(\xi \right))d\xi \hfill \\ & +{\int }_{\mathbb{R}}\frac{\beta S\left(\xi \right)(G*\tilde{I})\left(\xi \right)}{S\left(\xi \right)+(G*\tilde{I})\left(\xi \right)+R\left(\xi \right)}d\xi \hfill \\ & <\infty .\hfill \end{array}$$

Hence, we conclude that ${\int }_{-\infty }^{\infty }I\left(\xi \right)d\xi <\infty$. Combined with the claim obtained before that $I^{\prime }\left(\xi \right)$ is bounded on $\mathbb{R}$, we have that $I(\infty )=0$. □

Theorem 3.

Assuming that ${\mathfrak{R}}_0=\frac{\beta }{\gamma +\delta }>1$ and $c>c^{*}$, then

(1) 

$\underset{\xi \to +\infty }{lim}S\left(\xi \right)$ exists and $S_{\infty }:=\underset{\xi \to +\infty }{lim}S\left(\xi \right)<S_0$;

(2) 

If $\underset{\xi \to +\infty }{lim}supR\left(\xi \right)<\infty$, then $\underset{\xi \to +\infty }{lim}R\left(\xi \right)=\frac{\gamma (S_0-S_{\infty })}{\gamma +\delta }$.

Proof. 

First, we prove the existence of $\underset{\xi \to +\infty }{lim}S\left(\xi \right)$. On the contrary, we assume that

$$\underset{\xi \to +\infty }{lim}infS\left(\xi \right)=m_1<\underset{\xi \to +\infty }{lim}supS\left(\xi \right)=m_2.$$

Thus, we can find two point sequences $\left\{{\xi }_n\right\}$ and $\left\{{\eta }_n\right\}$ such that

$$\begin{array}{c}\underset{n\to +\infty }{lim}S\left({\xi }_n\right)=lim\underset{\xi \to +\infty }{inf}S\left(\xi \right)=m_1,S^{\prime }\left({\xi }_n\right)=0,\hfill \\ \underset{n\to +\infty }{lim}S\left({\eta }_n\right)=lim\underset{\xi \to +\infty }{sup}S\left(\xi \right)=m_2,S^{\prime }\left({\eta }_n\right)=0,\hfill \\ \underset{n\to +\infty }{lim}J*S\left({\xi }_n\right)\ge m_1,\underset{n\to +\infty }{lim}J\ast S\left({\eta }_n\right)\le m_2.\hfill \end{array}$$

Following the first equation of system (6), we have

$$0=c{S}^{\prime }\left({\xi }_n\right)=d_1(J\ast S\left({\xi }_n\right)-S\left({\xi }_n\right))-\frac{\beta S\left({\xi }_n\right)(G\ast I)\left({\xi }_n\right)}{S\left({\xi }_n\right)+(G\ast I)\left({\xi }_n\right)+R\left({\xi }_n\right)}.$$

Letting $n\to +\infty$, we can obtain that $\underset{n\to +\infty }{lim}J\ast S\left({\xi }_n\right)=\underset{n\to +\infty }{lim}S\left({\xi }_n\right)=m_1$. We prove that $S({\xi }_n-z)\to m_1$ as $n\to +\infty$ for any $z\in suppJ:=\Omega$. Choose a sufficiently small $ϵ>0$, let ${\tilde{S}}_n\left(z\right)=S({\xi }_n-z)$ and ${\Omega }_{ϵ}=\Omega \cap \left\{z\right|\underset{n\to +\infty }{lim}\tilde{S}\left(z\right)>m_1+ϵ\}$. Hence, we have

$$\begin{array}{cc}\hfill m_1& =\underset{n\to +\infty }{lim}J\ast S\left({\xi }_n\right)=\underset{n\to +\infty }{lim}{\int }_{\Omega }J\left(z\right){\tilde{S}}_n\left(z\right)dz\hfill \\ & \ge \underset{n\to +\infty }{lim\; inf}{\int }_{\Omega \setminus {\Omega }_{ϵ}}J\left(z\right){\tilde{S}}_n\left(z\right)dz+\underset{n\to +\infty }{lim\; inf}{\int }_{{\Omega }_{ϵ}}J\left(z\right){\tilde{S}}_n\left(z\right)dz\hfill \\ & \ge m_1{\int }_{\Omega \setminus {\Omega }_{ϵ}}J\left(z\right)dz+(m_1+ϵ){\int }_{{\Omega }_{ϵ}}J\left(z\right)dz\hfill \\ & =m_1+ϵ{\int }_{{\Omega }_{ϵ}}J\left(z\right)dz,\hfill \end{array}$$

which yields that $\mu \left(I_{ϵ}\right)=0$ where $\mu (·)$ denotes the measure. Thus, we have for any $z\in \Omega$,

$$S({\xi }_n-z)\to m_1,a.e.n\to +\infty .$$

(30)

On the other hand, we have

$$0=c{S}^{\prime }\left({\eta }_n\right)=d_1(J\ast S\left({\eta }_n\right)-S\left({\eta }_n\right))-\frac{\beta S\left({\eta }_n\right)(G\ast I)\left({\eta }_n\right)}{S\left({\eta }_n\right)+(G\ast I)\left({\eta }_n\right)+R\left({\eta }_n\right)};$$

then, letting $n\to +\infty$, by $I(+\infty )=0$, we can obtain that

$$\underset{n\to +\infty }{lim}J\ast S\left({\eta }_n\right)=\underset{n\to +\infty }{lim}S\left({\xi }_n\right)=m_2>m_1.$$

Then, by a similar discussion to formula (30), we can get that, for any $z\in \Omega$,

$$S({\eta }_n-z)\to m_2,a.e.n\to +\infty .$$

Note that, when $n\to +\infty$,

$${\int }_{{\xi }_n}^{{\eta }_n}\frac{\beta S\left(\xi \right)G\ast I\left(\xi \right)}{S\left(\xi \right)+G\ast I\left(\xi \right)+R\left(\xi \right)}d\xi \to 0.$$

Integrating the first equation of system (6) from ${\xi }_n$ to ${\eta }_n$, we have

$$\begin{array}{cc}\hfill 0& <c(m_2-m_1)\hfill \\ & =d_1\underset{n\to +\infty }{lim}{\int }_{{\xi }_n}^{{\eta }_n}(J\ast S\left(\xi \right)-S\left(\xi \right))d\xi -\underset{n\to +\infty }{lim}{\int }_{{\xi }_n}^{{\eta }_n}\frac{\beta S\left(\xi \right)(G\ast I)\left(\xi \right)}{S\left(\xi \right)+(G\ast I)\left(\xi \right)+R\left(\xi \right)}d\xi \hfill \\ & =d_1\underset{n\to +\infty }{lim}{\int }_{-\infty }^{\infty }yJ\left(y\right){\int }_0^1[S({\xi }_n-ty)-S({\eta }_n-ty)]dtdy\hfill \\ & =d_1(m_1-m_2){\int }_{-\infty }^{\infty }yJ\left(y\right)dy=0,\hfill \end{array}$$

which leads to a contradiction. Therefore,

$$\underset{\xi \to +\infty }{lim}infS\left(\xi \right)=\underset{\xi \to +\infty }{lim}supS\left(\xi \right),$$

which implies that $\underset{\xi \to +\infty }{lim}S\left(\xi \right)=:S_{\infty }$ exists.

Next, we will derive that $S_{\infty }<S_0$. Since $S\left(\xi \right)\le S_0$, then $S_{\infty }\le S_0$. We assume, on the contrary, that $S_{\infty }=S_0$. Integrating the first equation of system (6) from $-x$ to x yields

$$\begin{array}{cc}\hfill c\left[S\right(x)-S(-x\left)\right]& =d_1{\int }_{-\infty }^{\infty }yJ\left(y\right){\int }_0^1(S(-x-ty)-S(x-ty))dtdy\hfill \\ & -{\int }_{-x}^x\frac{\beta S\left(\xi \right)G\ast I\left)\right(\xi )}{S\left(\xi \right)+(G\ast I)\left(\xi \right)+R\left(\xi \right)}d\xi .\hfill \end{array}$$

Letting $x\to +\infty$, we have

$${\int }_{-\infty }^{\infty }\frac{\beta S\left(\xi \right)(G\ast I)\left(\xi \right)}{S\left(\xi \right)+(G\ast I)\left(\xi \right)+R\left(\xi \right)}d\xi =0,$$

which leads to a contradiction. Thus, we have $S_{\infty }<S_0$.

Using the similar method above, we can obtain that, when $\underset{\xi \to +\infty }{lim}supR\left(\xi \right)<+\infty$, $\underset{\xi \to +\infty }{lim}R\left(\xi \right):=R_{\infty }$. Now, integrating the first equation of system (6) on $\mathbb{R}$ yields that

$${\int }_{-\infty }^{\infty }\frac{\beta S\left(\xi \right)(G\ast I)\left(\xi \right)}{S\left(\xi \right)+(G\ast I)\left(\xi \right)+R\left(\xi \right)}d\xi =c(S_0-S_{\infty }).$$

By integrating the second and third equation of system (6) on $\mathbb{R}$, we can obtain that

$$\begin{array}{c}(\gamma +\delta ){\int }_{-\infty }^{\infty }I\left(\xi \right)d\xi ={\int }_{-\infty }^{\infty }\frac{\beta S\left(\xi \right)(G\ast I)\left(\xi \right)}{S\left(\xi \right)+(G\ast I)\left(\xi \right)+R\left(\xi \right)}d\xi ,\hfill \\ c{R}_{\infty }=\gamma {\int }_{-\infty }^{\infty }I\left(\xi \right)d\xi .\hfill \end{array}$$

Therefore, $R_{\infty }=\frac{\gamma (S_0-S_{\infty })}{\gamma +\delta }.$ □

4. Nonexistence of Traveling Waves

In this section, we will study the nonexistence of traveling wave solutions for system (5).

Theorem 4.

Suppose that ${\mathfrak{R}}_0>1$ and $0<c<c^{*}$, then system (5) has no nontrivial positive solution $(S,I,R)$ that satisfies the following asymptotic boundary conditions:

$$S(-\infty )=S_0,\underset{\mathbb{R}}{sup}S\left(x\right)\le S_0,I(\pm \infty )=0,R(-\infty )=0,\underset{\mathbb{R}}{sup}R\left(x\right)<+\infty .$$

(31)

Proof. 

Assume that $\left(S\right(\xi ),I(\xi ),R(\xi \left)\right)$ is a nontrivial positive solution of system (6) satisfying formula (31). By formula (31), we have $\frac{\beta S\left(\xi \right)}{S\left(\xi \right)+(G\ast I)\left(\xi \right)+R\left(\xi \right)}\to \beta ,\xi \to -\infty$. By using the continuity and ${\mathfrak{R}}_0>1$, we have that there exists ${\xi }^{*}$ such that, for any $\xi <{\xi }^{*}$,

$$\frac{\beta S\left(x\right)}{S\left(x\right)+(G\ast I)\left(x\right)+R\left(x\right)}>\frac{\beta +\gamma +\delta }{2}.$$

Then, for $\xi <{\xi }^{*}$, it follows from the second equation of (6) that

$$\begin{array}{cc}\hfill c{I}^{\prime }\left(\xi \right)& \ge d_2[J\ast I\left(\xi \right)-I\left(\xi \right)]+\frac{\beta +\gamma +\delta }{2}(G\ast I)\left(\xi \right)-(\gamma +\delta )I\left(\xi \right)\hfill \\ & =d_2[J\ast I\left(\xi \right)-I\left(\xi \right)]+\frac{\beta +\gamma +\delta }{2}[G\ast I\left(\xi \right)-I\left(\xi \right)]+\frac{\beta -\gamma -\delta }{2}I\left(\xi \right).\hfill \end{array}$$

(32)

Integrating equation (32) from $-\infty$ to $\xi <{\xi }^{*}$, we get

$$\begin{array}{cc}\hfill c\left[I\right(\xi )-I(-\infty \left)\right]& \ge d_2{\int }_{-\infty }^{\xi }[J\ast I\left(\tau \right)-I\left(\tau \right)]d\tau +\frac{\beta -\gamma -\delta }{2}{\int }_{-\infty }^{\xi }I\left(\tau \right)d\tau \hfill \\ & +\frac{\beta +\gamma +\delta }{2}{\int }_{-\infty }^{\xi }[G\ast I\left(\tau \right)-I\left(\tau \right)]d\tau .\hfill \end{array}$$

(33)

Denote $K\left(\xi \right)={\int }_{-\infty }^{\xi }I\left(\tau \right)d\tau ,\xi \in \mathbb{R}$. It is obvious that $K(-\infty )=0$ and $K\left(\xi \right)$ is bounded for any $\xi \in \mathbb{R}$. By making use of the Fubini theorem, we have

$$\begin{array}{cc}\hfill {\int }_{-\infty }^{\xi }J\ast I\left(\tau \right)d\tau & ={\int }_{-\infty }^{\xi }{\int }_{-\infty }^{\infty }J\left(y\right)I(\tau -y)dyd\tau ={\int }_{-\infty }^{\infty }J\left(y\right){\int }_{-\infty }^{\xi -y}I\left(z\right)dzdy\hfill \\ & ={\int }_{-\infty }^{\infty }J\left(y\right)K(\xi -y)dy=J\ast K\left(\xi \right),\hfill \\ \hfill {\int }_{-\infty }^{\xi }G\ast I\left(\tau \right)d\tau & ={\int }_{-\infty }^{\xi }{\int }_0^{\infty }{\int }_{-\infty }^{\infty }G(y,s)I(\tau -y-cs)dydsd\tau \hfill \\ & ={\int }_0^{\infty }{\int }_{-\infty }^{\infty }G(y,s){\int }_{-\infty }^{\xi -y-cs}I\left(z\right)dzdyds\hfill \\ & ={\int }_0^{\infty }{\int }_{-\infty }^{\infty }G(y,s)K(\xi -y-cs)dyds=G*K\left(\xi \right),\hfill \end{array}$$

then by formula (33) and $I(-\infty )=0$, we can obtain that

$$\frac{\beta -\gamma -\delta }{2}K\left(\xi \right)\le cI\left(\xi \right)-d_2[J\ast K\left(\xi \right)-K\left(\xi \right)]-\frac{\beta +\gamma +\delta }{2}[G\ast K\left(\xi \right)-K\left(\xi \right)].$$

(34)

Integrating Equation (34) from $-\infty$ to $\xi <{\xi }^{*}$, we have

$$\begin{array}{cc}\hfill \frac{\beta -\gamma -\delta }{2}{\int }_{-\infty }^{\xi }K\left(\tau \right)d\tau & \le c{\int }_{-\infty }^{\xi }I\left(\tau \right)d\tau -d_2{\int }_{-\infty }^{\xi }[J\ast K\left(\tau \right)-K\left(\tau \right)]d\tau \hfill \\ & -\frac{\beta +\gamma +\delta }{2}{\int }_{-\infty }^{\xi }[G\ast K\left(\tau \right)-K\left(\tau \right)]d\tau .\hfill \end{array}$$

(35)

By calculation, we get

$$\begin{array}{cc}\hfill {\int }_{-\infty }^{\xi }[J\ast K\left(\tau \right)-K\left(\tau \right)]d\tau & ={\int }_{-\infty }^{\xi }{\int }_{-\infty }^{\infty }J\left(y\right)[K(\tau -y)-K\left(\tau \right)]dyd\tau \hfill \\ & ={\int }_{-\infty }^{\infty }J\left(y\right){\int }_{-\infty }^{\xi }[K(\tau -y)-K\left(\tau \right)]d\tau dy\hfill \\ & =-{\int }_{-\infty }^{\infty }{\int }_0^{1}yJ\left(y\right)K(\xi -\theta y)d\theta dy.\hfill \end{array}$$

Similarly,

$${\int }_{-\infty }^{\xi }[G\ast K\left(\tau \right)-K\left(\tau \right)]d\tau =-{\int }_0^{\infty }{\int }_{-\infty }^{\infty }{\int }_0^1(y+cs)G(y,s)K(\xi -\theta (y+cs))d\theta dyds.$$

Thus, formula (35) is equivalent to

$$\begin{array}{cc}\hfill \frac{\beta -\gamma -\delta }{2}{\int }_{-\infty }^{\xi }K\left(\tau \right)d\tau & \le cK\left(\xi \right)+d_2{\int }_{-\infty }^{\infty }{\int }_0^{1}yJ\left(y\right)K(\xi -\theta y)d\theta dy\hfill \\ & +\frac{\beta +\gamma +\delta }{2}{\int }_0^{\infty }{\int }_{-\infty }^{\infty }{\int }_0^1(y+cs)G(y,s)K(\xi -\theta (y+cs))d\theta dyds.\hfill \end{array}$$

Since $yK(\xi -\theta y)$ is monotone decreasing with respect to $\theta \in [0,1]$, we have $yK(\xi -\theta y)\le yK\left(\xi \right)$ and $(y+cs)K(\xi -\theta (y+cs\left)\right)\le (y+cs)K\left(\xi \right)$. Then, we obtain

$$\begin{array}{cc}\hfill \frac{\beta -\gamma -\delta }{2}{\int }_{-\infty }^{\xi }K\left(\tau \right)d\tau & \le cK\left(\xi \right)+d_2{\int }_{-\infty }^{\infty }yJ\left(y\right)K\left(\xi \right)dy\hfill \\ & +\frac{\beta +\gamma +\delta }{2}{\int }_0^{\infty }{\int }_{-\infty }^{\infty }(y+cs)G(y,s)K\left(\xi \right)dyds\hfill \\ & =c\left[1+\frac{\beta +\gamma +\delta }{2}{\int }_0^{\infty }{\int }_{-\infty }^{\infty }sG(y,s)dyds\right]K\left(\xi \right):=\tilde{L}K\left(\xi \right).\hfill \end{array}$$

(36)

Furthermore, since $K(·)$ is nondecreasing, and by using formula (36), we can obtain that there exists some $\omega >0$ such that $K(\xi -\omega )<1/2K\left(\xi \right)$.

For ${\mu }_0\in (0,{\lambda }_1)$, define $P\left(\xi \right)=K\left(\xi \right)e^{-{\mu }_0\xi }$. Then, there exists ${\mu }_0$ such that $P(\xi -\omega )<P\left(\xi \right)$. Thus, $P\left(\xi \right)$ is bounded as $\xi \to -\infty$, which infers that there exists a constant $K_0>0$ such that $K\left(\xi \right)\le K_0{e}^{{\mu }_0\xi }$ for all $\xi \in \mathbb{R}$. By the definition of $K\left(\xi \right)$, we have

$$\underset{\xi \in \mathbb{R}}{sup}\left\{I\left(\xi \right)e^{-{\mu }_0\xi }\right\}<\infty .$$

Similarly, from the assumptions (J) and (G), we can obtain that

$$J\ast I\left(\xi \right)e^{-{\mu }_0\xi }<\infty ,G\ast I\left(\xi \right)e^{-{\mu }_0\xi }<\infty .$$

Moreover, by the second equation of system (6), we can obtain that $I^{\prime }\left(\xi \right)e^{-{\mu }_0\xi }<+\infty$ for any $\xi \in \mathbb{R}$. That is, we have

$$\underset{\xi \in \mathbb{R}}{sup}\{J\ast I\left(\xi \right)e^{-{\mu }_0\xi }\}<\infty ,\underset{\xi \in \mathbb{R}}{sup}\{G\ast I\left(\xi \right)e^{-{\mu }_0\xi }\}<\infty ,\underset{\xi \in \mathbb{R}}{sup}\left\{I^{\prime }\left(\xi \right)e^{-{\mu }_0\xi }\right\}<\infty .$$

For any $\xi \in \mathbb{R}$, it follows from the second equation of system (6) that

$$\begin{array}{c}{d}_2[J\ast I\left(\xi \right)-I\left(\xi \right)]-c{I}^{\prime }\left(\xi \right)+\beta (G\ast I)\left(\xi \right)-(\gamma +\delta )I\left(\xi \right)=\beta \left[G\ast I\left(\xi \right)-\frac{S\left(x\right)(G\ast I)\left(\xi \right)}{S\left(\xi \right)+(G\ast I)\left(\xi \right)+R\left(\xi \right)}\right].\end{array}$$

(37)

For any $\lambda \in \mathbb{C}$ with $0<Re\lambda <{\mu }_0$, taking a two-sided Laplace transform of $I\left(\xi \right)$ on Equation (37), we have

$$\Delta (\lambda ,c)\mathfrak{L}\left(\lambda \right)=\beta {\int }_{-\infty }^{\infty }\frac{e^{-\lambda \xi }[{(G\ast I)}^2\left(\xi \right)+(G\ast I)\left(\xi \right)R\left(\xi \right)]}{S\left(\xi \right)+(G\ast I)\left(\xi \right)+R\left(\xi \right)}d\xi ,$$

(38)

where $\mathfrak{L}\left(\lambda \right)={\int }_{-\infty }^{\infty }{e}^{-\lambda \xi }I\left(\xi \right)d\xi$. Using the property of Laplace transform, we know that either there exists positive constant ${\lambda }_0$ such that $\mathfrak{L}\left(\lambda \right)$ is analytic for $\lambda \in \mathbb{C}$ with $0<Re\lambda <{\lambda }_0$ and has singularity at $\lambda ={\lambda }_0$ or for $\lambda \in \mathbb{C}$ with $Re\lambda >0,\mathfrak{L}\left(\lambda \right)$ is well defined. According to the previous discussion, we know that the integral term on the right-hand side of formula (38) is uniformly bounded on the real line. Then, the two-sided Laplace integrals can be analytically continued to the whole right half plane. By Lemma 1, $\Delta (\lambda ,c)>0$ for all $\lambda >0$ when $0<c<c^{*}$, thus $\mathfrak{L}\left(\lambda \right)$ is analytic in the right half plane. According to the definition of $\Delta (\lambda ,c)$, we know that $\Delta (\lambda ,c)\to +\infty$ as $\lambda \to +\infty$, which leads to a contradiction from formula (38). Thus, the conclusion follows.

Theorem 5.

Assume that ${\mathfrak{R}}_0\le 1$. For any $c>0$, system (5) has no nontrivial positive traveling wave solution $(S,I,R)$ satisfying formula (31).

Proof. 

On the contrary, we suppose that system (5) has a traveling wave solution $(S,I,R)$ satisfying formula (31). For the case ${\mathfrak{R}}_0<1$, integrating the second equation of system (6) on $\mathbb{R}$, we have

$$(\gamma +\delta ){\int }_{-\infty }^{\infty }I\left(\xi \right)d\xi \le d_2{\int }_{-\infty }^{\infty }(J\ast I\left(\xi \right)-I\left(\xi \right))d\xi +\beta {\int }_{-\infty }^{\infty }I\left(\xi \right)d\xi ,$$

that is

$${\int }_{-\infty }^{\infty }I\left(\xi \right)d\xi \le \frac{d_2}{\gamma +\delta +d_2-\beta }{\int }_{-\infty }^{\infty }J\ast I\left(\xi \right)d\xi <{\int }_{-\infty }^{\infty }I\left(\xi \right)d\xi ,$$

which is a contradiction and the assumption does not hold.

When ${\mathfrak{R}}_0=1$ that is $\beta =\gamma +\delta$,

$$c{I}^{\prime }\left(\xi \right)=d_2(J\ast I\left(\xi \right)-I\left(\xi \right))+\beta \left[\frac{S\left(\xi \right)(G\ast I)\left(\xi \right)}{S\left(\xi \right)+(G\ast I)\left(\xi \right)+R\left(\xi \right)}-I\left(\xi \right)\right].$$

Then, integrating the above equation, we have

$${\int }_{-\infty }^{\infty }\left[\frac{S\left(\xi \right)(G\ast I)\left(\xi \right)}{S\left(\xi \right)+(G\ast I)\left(\xi \right)+R\left(\xi \right)}-I\left(\xi \right)\right]d\xi =0.$$

Due to the continuity and nonnegativity of $S,I,R$, we can obtain that

$$\begin{array}{cc}\hfill 0& ={\int }_{-\infty }^{\infty }\left[\frac{S\left(\xi \right)(G\ast I)\left(\xi \right)}{S\left(\xi \right)+(G\ast I)\left(\xi \right)+R\left(\xi \right)}-I\left(\xi \right)\right]d\xi \hfill \\ & <{\int }_{-\infty }^{\infty }G\ast I\left(\xi \right)d\xi -{\int }_{-\infty }^{\infty }I\left(\xi \right)d\xi \hfill \\ & ={\int }_0^{\infty }{\int }_{-\infty }^{\infty }G(y,s){\int }_{-\infty }^{\infty }I(\xi -y-cs)d\xi dyds-{\int }_{-\infty }^{\infty }I\left(\xi \right)d\xi \hfill \\ & =0,\hfill \end{array}$$

which is also a contradiction and completes the proof.

5. Conclusions and Discussion

In this paper, we have studied the existence and nonexistence of nontrivial traveling wave solutions for system (5). Combined with Theorems 2, 4 and 5, we obtain the threshold condition for the existence and nonexistence of traveling wave solutions, which is determined by the basic reproduction number ${\mathfrak{R}}_0$ of the corresponding reaction system and the minimal wave speed $c^{*}$. From Lemma 1, we know that the minimal wave speed $c^{*}$ is the unique root of the algebraic equations

$$\Delta (\lambda ,c)=0,\frac{\partial \Delta (\lambda ,c)}{\partial \lambda }=0\mathrm{for}\lambda >0,c>0,$$

where

$$\Delta (\lambda ,c)=d_2\left({\int }_{-\infty }^{\infty }J\left(y\right)e^{-\lambda y}dy-1\right)-c\lambda +\beta {\int }_0^{\infty }{\int }_{-\infty }^{+\infty }G(y,s)e^{-\lambda (y+cs)}dyds-\gamma -\delta .$$

It is obvious that the minimal wave speed $c^{*}$ is dependent on the dispersal rate $d_2$, the pattern of nonlocal interaction between the infected and the susceptible individuals, and the latent period of disease. In order to see the quantitative effect of nonlocal interaction and time delay on the minimum wave speed, we let $G(y,s)=\delta (s-\tau )\frac{1}{\sqrt{4\pi \varrho }}{e}^{-\frac{y^2}{4\varrho }}$ with $\tau >0,\varrho >0$ and $\delta (·)$ be the Dirac function. By simple calculation, we have

$$\Delta (\lambda ,c)=d_2\left({\int }_{-\infty }^{\infty }J\left(y\right)e^{-\lambda y}dy-1\right)-c\lambda +\beta e^{\varrho {\lambda }^2-c\lambda \tau }-\gamma -\delta .$$

Utilizing the implicit function theorem, we have

$$\begin{array}{cc}\hfill \frac{d{c}^{*}}{d{d}_2}& =\frac{{\int }_{\mathbb{R}}J\left(y\right)e^{-{\lambda }^{*}y}dy-1}{{\lambda }^{*}+\beta {\lambda }^{*}\tau e^{\varrho {\lambda }^{*2}-c^{*}{\lambda }^{*}\tau }}>0,\hfill \\ \hfill \frac{d{c}^{*}}{d\tau }& =\frac{-\beta c^{*}{\lambda }^{*}{e}^{\varrho {\lambda }^{*2}-c^{*}{\lambda }^{*}\tau }}{{\lambda }^{*}+\beta {\lambda }^{*}\tau e^{\varrho {\lambda }^{*2}-c^{*}{\lambda }^{*}\tau }}<0,\hfill \\ \hfill \frac{d{c}^{*}}{d\varrho }& =\frac{\beta {\lambda }^{*2}{e}^{\varrho {\lambda }^{*2}-c^{*}{\lambda }^{*}\tau }}{{\lambda }^{*}+\beta {\lambda }^{*}\tau e^{\varrho {\lambda }^{*2}-c^{*}{\lambda }^{*}\tau }}>0.\hfill \end{array}$$

Hence, we conclude that the dispersal rate $d_2$ and the nonlocal interaction can increase the minimal wave speed, while time delay can reduce the minimal wave speed.