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Research article Special Issues

Traveling waves for SVIR epidemic model with nonlocal dispersal

  • Received: 09 December 2018 Accepted: 30 January 2019 Published: 27 February 2019
  • In this paper, we studied an SVIR epidemic model with nonlocal dispersal and delay, and we find that the existence of traveling wave is determined by the basic reproduction number 0 and minimal wave speed c. By applying Schauder's fixed point theorem and Lyapunov functional, the existence and boundary asymptotic behaviour of traveling wave solutions is investigated for 0>1 and c>c. The existence of traveling waves is obtained for 0>1 and c=c by employing a limiting argument. We also show that the nonexistence of traveling wave solutions by Laplace transform. Our results imply that (ⅰ) the diffusion and infection ability of infected individuals can accelerate the wave speed; (ⅱ) the latent period and successful rate of vaccination can slow down the wave speed.

    Citation: Ran Zhang, Shengqiang Liu. Traveling waves for SVIR epidemic model with nonlocal dispersal[J]. Mathematical Biosciences and Engineering, 2019, 16(3): 1654-1682. doi: 10.3934/mbe.2019079

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  • In this paper, we studied an SVIR epidemic model with nonlocal dispersal and delay, and we find that the existence of traveling wave is determined by the basic reproduction number 0 and minimal wave speed c. By applying Schauder's fixed point theorem and Lyapunov functional, the existence and boundary asymptotic behaviour of traveling wave solutions is investigated for 0>1 and c>c. The existence of traveling waves is obtained for 0>1 and c=c by employing a limiting argument. We also show that the nonexistence of traveling wave solutions by Laplace transform. Our results imply that (ⅰ) the diffusion and infection ability of infected individuals can accelerate the wave speed; (ⅱ) the latent period and successful rate of vaccination can slow down the wave speed.


    As one of the most basic models in modeling infectious diseases, the SIR epidemiological model was introduced by Kermack and McKendrick [1] in 1927. Since then, a lot of differential equations have been studied as models for the spread of infectious diseases. Considering a continuous vaccination strategy, let V be a new group of vaccinated individuals, Liu et al. [2] formulated the following system of ordinary differential equations:

    {dS(t)dt=Λβ1S(t)I(t)αS(t)μ1S(t),dV(t)dt=αS(t)β2V(t)I(t)(γ1+μ1)V(x,t),dI(t)dt=β1S(t)I(t)+β2V(t)I(t)γI(x,t)μ3I(x,t),dR(t)dt=γ1V(t)+γI(t)μ1R(t), (1.1)

    where S(t), V(t), I(t) and R(t) denote the densities of susceptible, vaccinated, infective and removed individuals at time t, respectively. Λ denote the recruitment rate of susceptible individuals, μ1 denote the natural death rate. β1 is the rate of disease transmission between susceptible and infectious individuals, and β2 is the rate of disease transmission between vaccinated and infected individuals. γ denote the recovery rate, α is the vaccination rate and γ1 is the rate at which a vaccinated individual obtains immunity. In [2], the authors shown that the global dynamics of model (1.1) is completely determined by the basic reproduction number: that is, if the number is less than unity, then the disease-free equilibrium is globally asymptotically stable, while if the number is greater than unity, then a positive endemic equilibrium exists and it is globally asymptotically stable. Moreover, it was observed in Liu et al. [2] that vaccination has an effect of decreasing the basic reproduction number. By using the classical method of Lyapunov and graph-theoretic approach, Kuniya [3] studied the global stability of a multi-group SVIR epidemic model. Xu et.al [4] formulated a multi-group epidemic model with distributed delay and vaccination age, the authors established the global stability of the model, furthermore, the stochastic perturbation of the model is studied and it is proved that the endemic equilibrium of the stochastic model is stochastically asymptotically stable in the large under certain conditions. In [5,6,7], the global stability of different SVIR models with age structure are investigated.

    On the other hand, in order to understand the geographic spread of infectious disease, the spatial effect would give insights into disease spread and control. Due to this fact, many literatures have studied the spatial effects on epidemics by using reaction-diffusion equations (see, for instance, [8,9,10,11,12,13,14,15,16,17] and the references therein). In the study of reaction-diffusion models, the Laplacian operator describes the random diffusion of each individual, but it can not describe the long range diffusion. Therefore, a nonlocal dispersal term has been established, which is by a convolution operator:

    Jϕ(x)ϕ(x)=IRJ(xy)ϕ(y)dyϕ(x), (1.2)

    where ϕ(x) denote the densities of individuals at position x, J(xy) is interpreted as the probability of jumping from position y to position x, the convolution IRJ(xy)ϕ(y)dy is the rate at which individuals arrive at position x from all other positions, while IRJ(xy)ϕ(x)dy=ϕ(x) is the rate at which they leave position x to reach any other position. Problems involving such operators are called nonlocal diffusion problems and have appeared in various references [18,19,20,21,22,23,24,25,26,27,28,29,30,31].

    Recently, Li et al. [23] proposed a nonlocal dispersal SIR model with delay:

    {S(x,t)t=d1(JS(x,t)S(x,t))+Λβ1S(x,t)I(x,tτ)1+θI(x,tτ)μ1S(x,t),I(x,t)t=d2(JI(x,t)I(x,t))+β1S(x,t)I(x,tτ)1+θI(x,tτ)γI(x,t)μ3I(x,t),R(x,t)t=d3(JR(x,t)R(x,t))+γI(x,t)μ1R(x,t), (1.3)

    where S(x,t), I(x,t) and R(x,t) denote the densities of susceptible, infective and removed individuals at position x and time t, respectively. θ measures the saturation level. di(i=1,2,3) describes the spatial motility of each compartments. The biological meaning of other parameters are the same as in model (1.1). The authors find that there exists traveling wave solution if the basic reproduction number 0>1 and the wave speed cc, where c is the minimal wave speed. They also obtain the nonexistence of traveling wave solution for 0>1 and any 0<c<c or 0<1.

    Motivated by [2] and [23], in this paper, we consider a nonlocal dispersal epidemic model with vaccination and delay. Precisely, we study the following model.

    {S(x,t)t=d1(JS(x,t)S(x,t))+Λβ1S(x,t)I(x,tτ)αS(x,t)μ1S(x,t),V(x,t)t=d2(JV(x,t)V(x,t))β2V(x,t)I(x,tτ)+αS(x,t)(γ1+μ1)V(x,t),I(x,t)t=d3(JI(x,t)I(x,t))+β1S(x,t)I(x,tτ)+β2V(x,t)I(x,tτ)γI(x,t)μ3I(x,t),R(x,t)t=d4(JR(x,t)R(x,t))+β1S(x,t)I(x,tτ)+γ1V(x,t)+γI(x,t)μ4R(x,t), (1.4)

    where S(x,t), V(x,t), I(x,t) and R(x,t) denote the densities of susceptible, vaccinated, infective and removed individuals at position x and time t, respectively. di(i=1,2,3,4) describes the spatial motility of each compartments. The biological meaning of other parameters are the same as in model (1.1). J is the standard convolution operator satisfying the following assumptions.

    Assumption 1.1. [23,24] The kernel function J satisfies

    (J1) JC1(IR), J(x)0,  J(x)=J(x),  IRJ(x)dx=1 and J is compactly supported.

    (J2) There exists a constant λM(0,+) such that

    IRJ(x)eλxdx<+,  for  any  λ[0,λM)

    and

    limλλM0IRJ(x)eλxdx+.

    The organization of this paper is as follows. In section 2, we proved the existence of traveling wave solutions of (1.4) for c>c by applying Schauder's fixed point theorem and Lyapunov method. In section 3, we show that the existence of traveling wave solutions of (1.4) for c=c. Furthermore, we investigate the nonexistence of traveling wave solutions under some conditions in section 4. At last, there is a brief discussion.

    In this section, we study the existence of traveling wave solutions of system (1.4). Since we have assumed that the recovered have gained permanent immunity and R(x,t) is decoupled from other equations, we indeed need to study the following subsystem of (1.4)

    {S(x,t)t=d1(JS(x,t)S(x,t))+Λβ1S(x,t)I(x,tτ)αS(x,t)μ1S(x,t),V(x,t)t=d2(JV(x,t)V(x,t))β2V(x,t)I(x,tτ)+αS(x,t)μ2V(x,t),I(x,t)t=d3(JI(x,t)I(x,t))+(β1S(x,t)+β2V(x,t))I(x,tτ)γI(x,t)μ3I(x,t). (2.1)

    where μ2=γ1+μ1. Obviously, system (2.1) always has a disease-free equilibrium E0=(S0,V0,0)=(Λμ1+α,Λαμ2(μ1+α),0). Denote the basic reproduction number as following:

    0=β1S0+β2V0μ3+γ. (2.2)

    Furthermore, there exists another equilibrium E=(S,V,I) satisfying

    {Λβ1SIαSμ1S=0,β2VI+αSμ2V=0,(β1S+β2V)IγIμ3I=0. (2.3)

    From [2,Theorem 2.1], system (2.1) has a unique positive equilibrium E if 0>1.

    Let ξ=x+ct and substituting ξ into system (2.1), then we obtain the wave form equations as

    {cS(ξ)=d1(JS(ξ)S(ξ))+Λβ1S(ξ)I(ξcτ)αS(ξ)μ1S(ξ),cV(ξ)=d2(JV(ξ)V(ξ))+αS(ξ)β2V(ξ)I(ξcτ)μ2V(ξ),cI(ξ)=d3(JI(ξ)I(ξ))+β1S(ξ)I(ξcτ)+β2V(ξ)I(ξcτ)γI(ξ)μ3I(ξ). (2.4)

    We want to find traveling wave solutions with the following asymptotic boundary conditions:

    limξ(S(ξ),V(ξ),I(ξ))=(S0,V0,0) (2.5)

    and

    limξ+(S(ξ),V(ξ),I(ξ))=(S,V,I). (2.6)

    Consider the following linear system of system (2.4) at infection-free equilibrium (S0,V0,0),

    cI(ξ)=d3(JI(ξ)I(ξ))+β1S0I(ξcτ)+β2V0I(ξcτ)(γ+μ3)I(ξ). (2.7)

    Let I(ξ)=eλξ, we have

    Δ(λ,c) (2.8)

    By some calculations, we obtain

    \begin{align*} &\Delta(0, c) = \beta_1S_0+\beta_2V_0-\gamma-\mu_3, \ \ \ \lim\limits_{c\rightarrow+\infty}\Delta(\lambda, c) = -\infty\ \ \textrm{for}\ \ \lambda \gt 0, \\ &\frac{\partial \Delta(\lambda, c)}{\partial\lambda}\bigg|_{(0, c)} = -c-c\tau(\beta_1S_0+\beta_2V_0) \lt 0\ \ \textrm{for}\ \ c \gt 0, \\ &\frac{\partial \Delta(\lambda, c)}{\partial c} = -\lambda-\tau\lambda e^{-c\tau\lambda}(\beta_1S_0+\beta_2V_0) \lt 0\ \ \textrm{for}\ \ \lambda \gt 0, \\ &\frac{\partial^2 \Delta(\lambda, c)}{\partial\lambda^2} = d_3\int_{{\rm IR}}J(x)x^2e^{-\lambda x}\rm{d} x+(c\tau)^2e^{-c\tau\lambda}(\beta_1S_0+\beta_2V_0) \gt 0. \end{align*}

    For any c\in{\rm IR}, \Delta(0, c) = \Re_0 -1, gives us \Delta(0, c)>0 if \Re_0>1. Then there exist c^*>0 and \lambda^*>0 such that \frac{\partial \Delta(\lambda, c)}{\partial\lambda}\big|_{(\lambda^*, c^*)} = 0, we have the following lemma.

    Lemma 2.1. Let \Re_{0}>1, we have

    (ⅰ) If c = c^*, then \Delta(\lambda, c) = 0 has two same positive real roots \lambda^*;

    (ⅱ) If 0<c<c^*, then \Delta(\lambda, c)>0 for all \lambda\in(0, \lambda_{c, \tau}), where \lambda_{c, \tau}\in(0, +\infty];

    (ⅲ) If c>c^*, then \Delta(\lambda, c) = 0 has two positive real roots \lambda_1(c), \lambda_2(c).

    Denote \lambda_c = \lambda_1(c), from Lemma 2.1, we have

    0 \lt \lambda_c \lt \lambda^* \lt \lambda_2(c) \lt \lambda_{c, \tau}.

    For the followings in this section, we always fix c>c^* and \Re_0>1. Define the following functions:

    \begin{equation*} \label{UpLowSolution}\left\{ \begin{array}{l} \displaystyle \overline{S}(\xi) = S_0, \\ \displaystyle \overline{V}(\xi) = V_0, \\ \displaystyle \overline{I}(\xi) = e^{\lambda_c\xi}, \end{array}\right. \ \ \left\{ \begin{array}{l} \displaystyle \underline{S}(\xi) = \max\{S_0-M_1 e^{\varepsilon_1 \xi}, 0\}, \\ \displaystyle \underline{V}(\xi) = \max\{V_0-M_2 e^{\varepsilon_2 \xi}, 0\}, \\ \displaystyle \underline{I}(\xi) = \max\{e^{\lambda_c\xi}(1-M_3e^{\varepsilon_3 \xi}), 0\}, \end{array}\right. \end{equation*}

    where M_i and \varepsilon_i(i = 1, 2, 3) are some positive constants to be determined in the following lemmas.

    Lemma 2.2. The function \overline{I}(\xi) = e^{\lambda_c\xi} satisfies

    \begin{equation}\label{upI} cI'(\xi) \geq d_3(J*I(\xi)-I(\xi)) + \beta_1 S_0I(\xi-c\tau) + \beta_2 V_0I(\xi-c\tau) - \gamma I(\xi) - \mu_3 I(\xi). \end{equation} (2.9)

    Lemma 2.3. The functions \overline{S}(\xi) = S_0 and \overline{V}(\xi) = V_0 satisfy

    \begin{equation} \left\{ \begin{array}{l} \displaystyle cS'(\xi)\geq d_1(J*S(\xi)-S(\xi))+\Lambda - \beta_1 S(\xi)\underline{I}(\xi-c\tau) - \alpha S(\xi) - \mu_1 S(\xi), \\ \displaystyle cV'(\xi)\geq d_2(J*V(\xi)-V(\xi)) + \alpha S(\xi)- \beta_2 V(\xi)\underline{I}(\xi-c\tau) - \mu_2 V(\xi). \end{array}\right. \end{equation} (2.10)

    The proof is trivial, so we omitted the above two lemmas.

    Lemma 2.4. For each 0<\varepsilon_1<\lambda_c sufficiently small and M_1 large enough, the function \underline{S}(\xi) = \max\{S_0-M_1 e^{\varepsilon_1 \xi}, 0\} satisfies

    \begin{equation}\label{lowS} cS'(\xi) \leq d_1(J*S(\xi)-S(\xi))+\Lambda - \beta_1S(\xi)\overline{I}(\xi-c\tau)- (\mu_1+\alpha) S(\xi), \end{equation} (2.11)

    with \xi\neq\mathfrak{X}_1\triangleq\frac{1}{\varepsilon_1}\ln\frac{S_0}{M_1}.

    Proof. See Appendix A.

    Lemma 2.5. For each 0<\varepsilon_2<\lambda_c sufficiently small and M_2 large enough, the function \underline{V}(\xi) = \max\{V_0-M_2 e^{\varepsilon_2 \xi}, 0\} satisfies

    \begin{equation}\label{lowV} cV'(\xi) \leq d_2(J*V(\xi)-V(\xi))+\alpha \underline{S}(\xi) - \beta_2V(\xi)\overline{I}(\xi-c\tau)- \mu_2 V(\xi), \end{equation} (2.12)

    with \xi\neq\mathfrak{X}_2\triangleq\frac{1}{\varepsilon_2}\ln\frac{V_0}{M_2}.

    The proof is similar with Lemma 2.4.

    Lemma 2.6. Let 0<\varepsilon_3<\min\{\varepsilon_1/2, \varepsilon_2/2\} and M_3>\max\{S_0, V_0\} is large enough, then the function \underline{I}(\xi) = \max\{e^{\lambda_c\xi}(1-M_3e^{\varepsilon_3 \xi}), 0\} satisfies

    \begin{equation}\label{lowIEqu} cI'(\xi)\leq d_3(J*I(\xi)-I(\xi)) + \beta_1 \underline{S}(\xi)I(\xi-c\tau) + \beta_2 \underline{V}(\xi)I(\xi-c\tau) - \gamma I(\xi) - \mu_3 I(\xi), \end{equation} (2.13)

    with \xi\neq\mathfrak{X}_3\triangleq\frac{1}{\varepsilon_3}\ln\frac{1}{M_3}.

    Proof. See Appendix B.

    Let X>\max\{\mathfrak{X}_1, \mathfrak{X}_2, \mathfrak{X}_3\}, define

    \begin{equation*} \label{Gamma} \Gamma_X = \left\{\left( \begin{array}{c} \phi \\ \varphi \\ \psi \\ \end{array} \right) \in C([-X, X], {\rm IR}^3)\left| \begin{array}{l} \displaystyle \underline{S}(\xi)\leq \phi(\xi)\leq S_0, \ \ \phi(-X) = \underline{S}(-X), \ \ \mathrm{for}\ \xi\in [-X, X];\\ \displaystyle \underline{V}(\xi)\leq \varphi(\xi)\leq V_0, \ \ \varphi(-X) = \underline{V}(X), \ \ \mathrm{for}\ \xi\in [-X, X];\\ \displaystyle \underline{I}(\xi)\leq \psi(\xi)\leq \overline{I}(\xi), \ \ \psi(-X) = \underline{I}(X), \ \ \mathrm{for}\ \xi\in [-X, X].\\ \end{array}\right.\right\}. \end{equation*}

    For given (\phi(\xi), \varphi(\xi), \psi(\xi))\in \Gamma_X, define

    \begin{equation*} \label{hat1} \hat{\phi}(\xi) = \left\{ \begin{array}{l} \displaystyle \phi(X), \ \ \mathrm{for}\ \xi \gt X, \\ \displaystyle \phi(\xi), \ \ \ \mathrm{for}\ \xi\in[-X-c\tau, X], \\ \displaystyle \underline{S}(\xi), \ \ \ \mathrm{for}\ \xi\leq -X-c\tau, \\ \end{array}\right.\ \ \ \hat{\varphi}(\xi) = \left\{ \begin{array}{l} \displaystyle \varphi(X), \ \ \mathrm{for}\ \xi \gt X, \\ \displaystyle \varphi(\xi), \ \ \ \mathrm{for}\ \xi\in[-X-c\tau, X], \\ \displaystyle \underline{V}(\xi), \ \ \ \mathrm{for}\ \xi\leq -X-c\tau, \\ \end{array}\right. \end{equation*}

    and

    \begin{equation*} \label{hat2} \hat{\psi}(\xi) = \left\{ \begin{array}{l} \displaystyle \psi(X), \ \ \mathrm{for}\ \xi \gt X, \\ \displaystyle \psi(\xi), \ \ \ \mathrm{for}\ \xi\in[-X-c\tau, X], \\ \displaystyle \underline{I}(\xi), \ \ \ \ \mathrm{for}\ \xi\leq -X-c\tau.\\ \end{array}\right. \end{equation*}

    We have

    \begin{equation*} \label{hat3} \left\{ \begin{array}{l} \displaystyle \underline{S}(\xi)\leq\hat{\phi}(\xi)\leq S_0, \\ \displaystyle \underline{V}(\xi)\leq\hat{\varphi}(\xi)\leq V_0, \\ \displaystyle \underline{I}(\xi)\leq\hat{\psi}(\xi)\leq \overline{I}(\xi). \end{array}\right. \end{equation*}

    For any \xi\in[-X, X], consider the following initial value problem

    \begin{equation} \label{Initial}\left\{ \begin{array}{l} \displaystyle cS'(\xi) = d_1\int_{{\rm IR}}J(y)\hat{\phi}(\xi-y)\rm{d} y + \Lambda - \beta_1S(\xi)\psi(\xi-c\tau) - (d_1+\mu_1+\alpha) S(\xi), \\ \displaystyle cV'(\xi) = d_2\int_{{\rm IR}}J(y)\hat{\varphi}(\xi-y)\rm{d} y + \alpha\phi(\xi)- \beta_2V(\xi)\psi(\xi-c\tau) - (d_2+\mu_2) V(\xi), \\ \displaystyle cI'(\xi) = d_3\int_{{\rm IR}}J(y)\hat{\psi}(\xi-y)\rm{d} y + \beta_1\phi(\xi)\psi(\xi-c\tau) +\beta_2\varphi(\xi)\psi(\xi-c\tau) - (d_3+\gamma+\mu_3) I(\xi), \\ \displaystyle S(-X) = \underline{S}(-X), \ \ \ V(-X) = \underline{V}(-X), \ \ \ I(-X) = \underline{I}(-X). \end{array}\right. \end{equation} (2.14)

    From the standard theory of functional differential equations (see [32]), the initial value problem (2.14) admits a unique solution (S_X(\xi), V_X(\xi), I_X(\xi)) satisfying

    (S_X, V_X, I_X)\in C^1([-X, X]),

    this defines an operator \mathcal{A} = (\mathcal{A}_1, \mathcal{A}_2, \mathcal{A}_3):\Gamma_X\rightarrow C^1([-X, X]) as

    S_X = \mathcal{A}_1(\phi, \varphi, \psi), \ V_X = \mathcal{A}_2(\phi, \varphi, \psi), \ I_X = \mathcal{A}_3(\phi, \varphi, \psi).

    Next we show the operator \mathcal{A} = (\mathcal{A}_1, \mathcal{A}_2, \mathcal{A}_3) has a fixed point in \Gamma_X.

    Lemma 2.7. The operator \mathcal{A} = (\mathcal{A}_1, \mathcal{A}_2, \mathcal{A}_3) maps \Gamma_X into itself.

    Proof. Firstly, we show that \underline{S}(\xi)\leq S_X(\xi) for any \xi\in[-X, X]. If \xi\in(\mathfrak{X}_1, X), \underline{S}(\xi) = 0 is a lower solution of the first equation of (2.14). If \xi\in(-X, \mathfrak{X}_1), \underline{S}(\xi) = S_0-M_1 e^{\varepsilon_1 \xi}, by Lemma 2.4, we have

    \begin{align*} &c\underline{S}'(\xi) - d_1\int_{{\rm IR}}J(y)\hat{\phi}(\xi-y)\rm{d} y - \Lambda + \beta_1\underline{S}(\xi)\psi(\xi-c\tau) - (d_1+\mu_1+\alpha) \underline{S}(\xi)\\ \nonumber \leq&c\underline{S}'(\xi) - d_1\int_{{\rm IR}}J(y)\underline{S}(\xi-y)\rm{d} y - \Lambda + \beta_1\underline{S}(\xi)\overline{I}(\xi-c\tau) - (d_1+\mu_1+\alpha) \underline{S}(\xi))\\ \leq&0, \nonumber \end{align*}

    which implies that \underline{S}(\xi) = S_0-M_1 e^{\varepsilon_1 \xi} is a lower solution of the first equation of (2.14). Thus \underline{S}(\xi)\leq S_X(\xi) for any \xi\in[-X, X].

    Secondly, we show that S_X(\xi)\leq \overline{S}(\xi) = S_0 for any \xi\in[-X, X]. In fact,

    \begin{align*} &d_1\int_{{\rm IR}}J(y)\hat{\phi}(\xi-y)\rm{d} y + \Lambda - \beta_1S_0\psi(\xi-c\tau) - (d_1+\mu_1+\alpha) S_0\\ \nonumber \leq& d_1\int_{{\rm IR}}J(y)S_0\rm{d} y + \Lambda - \beta_1S_0\underline{I}(\xi-c\tau) - (d_1+\mu_1+\alpha) S_0\\ \leq&0, \nonumber \end{align*}

    thus \overline{S}(\xi) = S_0 is an upper solution to the first equation of (2.14), which gives us S_X(\xi)\leq S_0 for any \xi\in[-X, X].

    Similarly, \underline{V}(\xi)\leq V_X(\xi)\leq \overline{V}(\xi) and \underline{I}(\xi)\leq I_X(\xi)\leq \overline{I}(\xi) for any \xi\in[-X, X].

    Lemma 2.8. The operator \mathcal{A} is completely continuous.

    Proof. Suppose (\phi_i(\xi), \varphi_i(\xi), \psi_i(\xi))\in\Gamma_X, \ i = 1, 2.

    \begin{align*} S_{X, i}(\xi)& = \mathcal{A}_1(\phi_i(\xi), \varphi_i(\xi), \psi_i(\xi)), \\ V_{X, i}(\xi)& = \mathcal{A}_2(\phi_i(\xi), \varphi_i(\xi), \psi_i(\xi)), \\ I_{X, i}(\xi)& = \mathcal{A}_3(\phi_i(\xi), \varphi_i(\xi), \psi_i(\xi)), \end{align*}

    We show the operator \mathcal{A} is continuous. By direct calculation, we have

    \begin{align*} S_X(\xi) = &\underline{S}(-X) \exp \left\{-\frac{1}{c}\int_{-X}^\xi(d_1 +\mu_1+\alpha+\beta_1\psi(s-c\tau))\rm{d} s\right\}\\ &+ \frac{1}{c}\int_{-X}^\xi\exp \left\{-\frac{1}{c}\int_{\eta}^\xi(d_1 +\mu_1+\alpha+\beta_1\psi(s-c\tau))\rm{d} s\right\}f_{\phi}(\eta)\rm{d} \eta, \end{align*}
    \begin{align*} V_X(\xi) = &\underline{V}(-X) \exp \left\{-\frac{1}{c}\int_{-X}^\xi(d_2 +\mu_2+\beta_2\psi(s-c\tau))\rm{d} s\right\}\\ &+ \frac{1}{c}\int_{-X}^\xi\exp \left\{-\frac{1}{c}\int_{\eta}^\xi(d_2 +\mu_2+\beta_2\psi(s-c\tau))\rm{d} s\right\}f_{\varphi}(\eta)\rm{d} \eta, \end{align*}

    and

    \begin{align*} I_X(\xi) = &\underline{I}(-X) \exp \left\{-\frac{(d_3+\gamma+\mu_3)(\xi+X)}{c}\right\}\\ &+ \frac{1}{c}\int_{-X}^\xi\exp \left\{-\frac{(d_3+\gamma+\mu_3)(\xi-\eta)}{c}\right\}f_{\psi}(\eta)\rm{d} \eta. \end{align*}

    where

    \begin{align*} f_{\phi}(\eta) & = d_1\int_{{\rm IR}}J(\eta-y)\hat{\phi}(y)\rm{d} y + \Lambda, \\ f_{\varphi}(\eta) & = d_2\int_{{\rm IR}}J(\eta-y)\hat{\varphi}(y)\rm{d} y + \alpha\phi(\eta), \\ f_{\psi}(\eta) & = d_3\int_{{\rm IR}}J(\eta-y)\hat{\psi}(y)\rm{d} y + (\beta_1\phi(\eta)+\beta_2\varphi(\eta))\psi(\eta-c\tau). \end{align*}

    For any (\phi_i, \varphi_i, \psi_i)\in\Gamma_X, i = 1, 2, we have

    \begin{align*} |f_{\phi_1}(\eta) - f_{\phi_2}(\eta)| = &d_1\left|\int_{{\rm IR}}J(\eta-y)[\hat{\phi}_1(y) - \hat{\phi}_2(y)]\rm{d} y\right|\\ \leq &d_1\left|\int_{-X}^X J(\xi-y)(\phi_1(y)-\phi_2(y))\rm{d} y\right|+d_1\left|\int_X^\infty J(\xi-y)(\phi_1(X)-\phi_2(X))\rm{d} y\right|\\ \leq &2d_1\max\limits_{y\in[-X, X]}|\phi_1(y)-\phi_2(y)|, \end{align*}
    \begin{align*} |f_{\varphi_1}(\eta) - f_{\varphi_2}(\eta)| = &d_2\left|\int_{{\rm IR}}J(\eta-y)[\hat{\varphi}_1(y) - \hat{\varphi}_2(y)]\rm{d} y + \alpha(\phi_1(\eta) - \phi_2(\eta))\right|\\ \leq &2d_2\max\limits_{y\in[-X, X]}|\varphi_1(y)-\varphi_2(y)| + \alpha \max\limits_{y\in[-X, X]}|\phi_1(y)-\phi_2(y)|, \end{align*}
    \begin{align*} |f_{\psi_1}(\eta) - f_{\psi_2}(\eta)|\leq &(2d_2+\beta_1S_0+\beta_2V_0)\max\limits_{y\in[-X, X]}|\psi_1(y)-\psi_2(y)|\\ &+\beta_1e^{\lambda_c\xi}\max\limits_{y\in[-X, X]}|\phi_1(y)-\phi_2(y)|+\beta_2e^{\lambda_c\xi}\max\limits_{y\in[-X, X]}|\varphi_1(y)-\varphi_2(y)|. \end{align*}

    Here we use

    \begin{align*} &\left|\beta_1\phi_2(\xi)\psi_2(\xi-c\tau)-\beta_1\phi_1(\xi)\psi_1(\xi-c\tau)\right|\\ \leq &\left|\beta_1\phi_2(\xi)\psi_2(\xi-c\tau)-\beta_1\phi_2(\xi)\psi_1(\xi-c\tau)\right| +\left|\beta_1\phi_2(\xi)\psi_1(\xi-c\tau)-\beta_1\phi_1(\xi)\psi_1(\xi-c\tau)\right|\\ \leq &\beta_1S_0\max\limits_{y\in[-X, X]}|\psi_1(y)-\psi_2(y)|+\beta_1e^{\lambda_c\xi}\max\limits_{y\in[-X, X]}|\phi_1(y)-\phi_2(y)|. \end{align*}

    and

    \begin{align*} &\left|\beta_2\varphi_2(\xi)\psi_2(\xi-c\tau)-\beta_2\varphi_1(\xi)\psi_1(\xi-c\tau)\right|\\ \leq &\beta_2V_0\max\limits_{y\in[-X, X]}|\psi_1(y)-\psi_2(y)|+\beta_2e^{\lambda_c\xi}\max\limits_{y\in[-X, X]}|\varphi_1(y)-\varphi_2(y)|. \end{align*}

    Thus, we obtain that the operator \mathcal{A} is continuous. Next, we show \mathcal{A} is compact. Indeed, since S_X, V_X and I_X are class of C^1([-X, X]), note that

    \begin{align*} &c(S_{X, 1}'(\xi)-S_{X, 2}'(\xi))+(d_1+\mu_1+\alpha)(S_{X, 1}(\xi)-S_{X, 2}(\xi))\\ = &d_1\int_{{\rm IR}}J(\xi-y)(\hat{\phi}_1(y)-\hat{\phi}_2(y))\rm{d} y+ \beta_1\phi_2(\xi)\psi_2(\xi-c\tau)-\beta_1\phi_1(\xi)\psi_1(\xi-c\tau)\\ \leq&\left(2d_1+\beta_1e^{\lambda_c\xi}\right)\max\limits_{y\in[-X, X]}|\phi_1(y)-\phi_2(y)|+\beta_1S_0\max\limits_{y\in[-X, X]}|\psi_1(y)-\psi_2(y)|. \end{align*}

    Same arguments with V_X' and I_X', give us S_X', V_X' and I_X' are bounded. Then \mathcal{A} is compact and the operator \mathcal{A} is completely continuous. This ends the proof.

    Obviously, \Gamma_X is a bounded closed convex set, applying the Schauder's fixed point theorem ([33] Corollary 2.3.10), we have the following theorem.

    Theorem 2.1. There exists (S_X, V_X, I_X)\in \Gamma_X such that

    (S_X(\xi), V_X(\xi), I_X(\xi)) = \mathcal{A}(S_X, V_X, I_X)(\xi)

    for \xi\in[-X, X].

    Now we are in position to show the existence of traveling wave solutions, before that we do some estimates for S_X(\cdot), \ V_X(\cdot) and I_X(\cdot).

    Define

    C^{1, 1}([-X, X]) = \{u\in C^1([-X, X])|u, u' \textrm{are Lipschitz continuous}\}

    with norm

    \begin{gather*} \|u\|_{C^{1, 1}([-X, X])} = \max\limits_{x\in[-X, X]}|u|+\max\limits_{x\in[-X, X]}|u'|+ \sup\limits_{\begin{subarray}{c} x, y\in [-X, X] \\ x\neq y \end{subarray}}\frac{|u'(x)-u'(y)|}{|x-y|}. \end{gather*}

    Lemma 2.9. There exists a constant C(Y)>0 such that

    \begin{equation*} \|S_X\|_{C^{1, 1}([-Y, Y])}\leq C(Y), \ \ \|V_X\|_{C^{1, 1}([-Y, Y])}\leq C(Y), \ \ \|I_X\|_{C^{1, 1}([-Y, Y])}\leq C(Y) \end{equation*}

    for Y<X and X>\max\{\mathfrak{X}_1, \mathfrak{X}_2, \mathfrak{X}_3\}.

    Proof. Recall that (S_X, E_X, I_X) is the fixed point of the operator \mathcal{A}, then

    cS_X'(\xi) = d_1\int_{-\infty}^{+\infty}J(y)\hat{S}_X(\xi-y)\rm{d} y + \Lambda - \beta_1S_X(\xi)I_X(\xi-c\tau) - (d_1+\mu_1+\alpha) S_X(\xi), (2.15)
    cV_X'(\xi) = d_2\int_{-\infty}^{+\infty}J(y)\hat{V}_X(\xi-y)\rm{d} y + \alpha S_X(\xi)- \beta_2V_X(\xi)I_X(\xi-c\tau) - (d_2+\mu_2)V_X(\xi), (2.16)
    cI_X'(\xi) = d_3\int_{-\infty}^{+\infty}J(y)\hat{I}_X(\xi-y)\rm{d} y + \beta_1S_X(\xi)I_X(\xi-c\tau) + \beta_2V_X(\xi)I_X(\xi-c\tau) - (d_3+\mu_3) I_X(\xi), (2.17)

    where

    \begin{equation*} (\hat{S}_X(\xi), \hat{V}_X(\xi), \hat{I}_X(\xi)) = \left\{ \begin{array}{l} \displaystyle (S_X(X), V_X(X), I_X(X)), \ \ \mathrm{for}\ \xi \gt X, \\ \displaystyle (S_X(\xi), V_X(\xi), I_X(\xi)), \ \ \ \ \ \ \mathrm{for}\ \xi\in [-X-c\tau, X], \\ \displaystyle (\underline{S}(\xi), \underline{V}(\xi), \underline{I}(\xi)), \ \ \ \ \ \ \ \ \ \ \ \ \mathrm{for}\ \xi\leq -X-c\tau, \\ \end{array}\right. \end{equation*}

    following that S_X(\xi)\leq S_0, \ \ V_X(\xi)\leq V_0, \ \ I_X(\xi)\leq e^{\lambda_c Y} for any \xi\in[-Y, Y]. Then

    \begin{align*} |S'_X(\xi)|\leq &\frac{d_1}{c}\left|\int_{-\infty}^{+\infty}J(y)\hat{S}_X(\xi-y)\rm{d} y\right|+\frac{\Lambda}{c}+\frac{d_1+\mu_1+\alpha}{c}|S_X(\xi)|+\frac{\beta_1}{c}|S_X(\xi)||I_X(\xi-c\tau)|\\ \leq&\frac{2d_1+\mu_1+\alpha}{c} S_0+\frac{\Lambda}{c}+\frac{\beta_1S_0}{c}e^{\lambda_c Y}, \\ |V'_X(\xi)|\leq &\frac{2d_2+\mu_2}{c}V_0+\frac{\alpha S_0}{c}+\frac{\beta_2V_0}{c}e^{\lambda_c Y}, \\ |I'_X(\xi)|\leq &\left(\frac{d_3+\mu_3}{c}+\frac{\beta_1S_0}{c}+\frac{\beta_2V_0}{c}\right)e^{\lambda_c Y}. \end{align*}

    Thus, there exists some constant C_1(Y)>0 such that

    \begin{equation*} \|S_X\|_{C^{1}([-Y, Y])}\leq C_1(Y), \ \ \|V_X\|_{C^{1}([-Y, Y])}\leq C_1(Y), \ \ \|I_X\|_{C^{1}([-Y, Y])}\leq C_1(Y). \end{equation*}

    Then for any \xi_1, \xi_2\in[-Y, Y] such that

    \begin{equation*} |S_X(\xi_1)-S_X(\xi_2)|\leq C_1(Y)|\xi_1-\xi_2|, \ \ |V_X(\xi_1)-V_X(\xi_2)|\leq C_1(Y)|\xi_1-\xi_2|, \ \ |I_X(x_1)-I_X(x_2)|\leq C_1(Y)|\xi_1-\xi_2|. \end{equation*}

    From (2.15), we have

    \begin{align*} c|S'_X(\xi_1)-S'_X(\xi_2)|\leq&d_1\left|\int_{-\infty}^{+\infty}J(y)(\hat{S}_X(\xi_1-y)-\hat{S}_X(\xi_2-y))\rm{d} y\right|\\ &+ (d_1+\mu_1+\alpha)|S_X(\xi_1)-S_X(\xi_2)| + S_0|I_X(\xi_1)-I_X(\xi_2)|. \end{align*}

    Recall (J1) of Assumption 1.1, we know J is Lipschitz continuous and compactly supported on {\rm IR}, let L be the Lipschitz constant for J and R be the radius of suppJ. Then

    \begin{align*} &d_1\left|\int_{-\infty}^{+\infty}J(y)(\hat{S}_X(\xi_1-y)-\hat{S}_X(\xi_2-y))\rm{d} y\right|\\ = &d_1\left|\int_{-R}^{R}J(y)\hat{S}_X(\xi_1-y)\rm{d} y-\int_{-R}^{R}J(y)\hat{S}_X(\xi_2-y)\rm{d} y\right|\\ = &d_1\left|\int_{\xi_1-R}^{\xi_1+R}J(\xi_1-y)\hat{S}_X(y)\rm{d} y-\int_{\xi_2-R}^{\xi_2+R}J(y)\hat{S}_X(y)\rm{d} y\right|\\ = &d_1\left|\left(\int_{\xi_1-R}^{\xi_2-R}+\int_{\xi_2-R}^{\xi_2+R}+\int_{\xi_2+R}^{\xi_1+R}\right)J(\xi_1-y)\hat{S}_X(y)\rm{d} y-\int_{\xi_2-R}^{\xi_2+R}J(y)\hat{S}_X(y)\rm{d} y\right|\\ \leq& d_1\left|\int_{\xi_2+R}^{\xi_1+R}J(\xi_1-y)\hat{S}_X(y)\rm{d} y\right|+d_1\left|\int_{\xi_1-R}^{\xi_2-R}J(\xi_1-y)\hat{S}_X(y)\rm{d} y\right|\\ &+d_1\left|\int_{\xi_2-R}^{\xi_2+R}(J(\xi_1-y)-J(\xi_2-y))\hat{S}_X(y)\rm{d} y\right|\\ \leq& d_1(2S_0\|J\|_{L^\infty}+2RLS_0)|\xi_1-\xi_2|. \end{align*}

    Thus there exists some constant C_2(Y)>0 such that

    |S'_X(\xi_1)-S'_X(\xi_2)|\leq C_2(Y)|\xi_1-\xi_2|.

    Similarly

    |V'_X(\xi_1)-V'_X(\xi_2)|\leq C_2(Y)|\xi_1-\xi_2|, \ \ |I'_X(\xi_1)-I'_X(\xi_2)|\leq C_2(Y)|\xi_1-\xi_2|.

    From the above discussion, there exists some constant C(Y)>0 for any Y<X that is independent of X such that

    \begin{equation*} \|S_X\|_{C^{1, 1}([-Y, Y])}\leq C(Y), \ \ \|V_X\|_{C^{1, 1}([-Y, Y])}\leq C(Y), \ \ \|I_X\|_{C^{1, 1}([-Y, Y])}\leq C(Y). \end{equation*}

    Now let \{X_n\}_{n = 1}^{+\infty} be an increasing sequence such that X_n\geq\{\mathfrak{X}_1, \mathfrak{X}_2, \mathfrak{X}_3\}, X_n>Y+R for each n and \lim\limits X_n = +\infty, where R is the radius of suppJ. For every c>c^*, we have (S_{X_n}, V_{X_n}, I_{X_n})\in\Gamma_{X_n} satisfying Lemma 2.9 and Equations (2.15)-(2.17).For the sequence (S_{X_n}, V_{X_n}, I_{X_n}), we can extract a subsequence denoted by \{S_{X_{n_k}}\}_{k\in{\rm IN}}, \{V_{X_{n_k}}\}_{k\in{\rm IN}} and \{I_{X_{n_k}}\}_{k\in{\rm IN}} tending to functions (S, V, I)\in C^1({\rm IR}) in the following topologies

    S_{X_{n_k}}\rightarrow S, \ \ V_{X_{n_k}}\rightarrow V\ \ \textrm{and}\ \ I_{X_{n_k}}\rightarrow I\ \ \textrm{in}\ \ C^1_{\textrm{loc}}({\rm IR})\ \ \textrm{as}\ \ k\rightarrow+\infty.

    Since J is compactly supported, applying the dominated convergence theorem, thus

    \lim\limits_{k\rightarrow+\infty}\int_{{\rm IR}}J(y)\hat{S}_{X_{n_k}}(\xi-y)\rm{d} y = \int_{{\rm IR}}J(y)S(\xi-y)\rm{d} y = J*S(\xi),
    \lim\limits_{k\rightarrow+\infty}\int_{{\rm IR}}J(y)\hat{V}_{X_{n_k}}(\xi-y)\rm{d} y = \int_{{\rm IR}}J(y)V(\xi-y)\rm{d} y = J*V(\xi)

    and

    \lim\limits_{k\rightarrow+\infty}\int_{{\rm IR}}J(y)\hat{I}_{X_{n_k}}(\xi-y)\rm{d} y = \int_{{\rm IR}}J(y)I(\xi-y)\rm{d} y = J*I(\xi).

    Moreover, (S, V, I) satisfies system (2.4) and

    \underline{S}(\xi)\leq S(\xi)\leq S_0, \ \ \underline{V}(\xi)\leq V(\xi)\leq V_0, \ \ \underline{I}(\xi)\leq I(\xi)\leq e^{\lambda_c \xi}.

    Next, we show that I(\xi) is bounded in {\rm IR} by the method in [34] (see also [35,36]).

    Lemma 2.10. There exists some positive constant C such that

    \int_{{\rm IR}} J(y)\frac{I(\xi-y)}{I(\xi)}\rm{d} y \lt C, \ \ \ \frac{I(\xi-c\tau)}{I(\xi)} \lt C \ \ \ \mathit{and}\ \ \ \bigg|\frac{I'(\xi)}{I(\xi)}\bigg| \lt C.

    Proof. Let \theta(\xi) = \frac{I'(\xi)}{I(\xi)}, from the third equation of (2.4), we have

    \begin{align*} \theta(\xi) &\geq \frac{d_3}{c}\left(\int_{{\rm IR}} J(y)\frac{I(\xi-y)}{I(\xi)}\rm{d} y-1\right) - \frac{\gamma + \mu_3}{c}\\ & = \frac{d_3}{c} \int_{{\rm IR}} J(y) e^{\int_{\xi}^{\xi-y}\theta(s)\rm{d} s}\rm{d} y - \left(\frac{d_3+\gamma+\mu_3}{c}\right). \end{align*}

    Set \varpi = \left(\frac{d_3+\gamma+\mu_3}{c}\right) and W(\xi) = \exp\left\{\varpi\xi + \int_0^{\xi}\theta(s)\rm{d} s\right\}, thus

    W'(\xi) = (\varpi + \theta(\xi))W(\xi) \geq \frac{d_3}{c} \int_{{\rm IR}} J(y) e^{\int_{\xi}^{\xi-y}\theta(s)\rm{d} s}\rm{d} y W(\xi),

    that is W(\xi) is non-decreasing. We can take some R_0>0 with 2R_0<R, where R is the radius of suppJ. Then by the same argument in [34,Lemma 2.2], we have

    W(\xi)\geq \frac{d_3}{c}R_0\int_{{\rm IR}}J(y)e^{\varpi y}W(\xi-R_0-y)\rm{d} y

    and

    W(\xi+R_0)\leq \sigma_0 W(\xi)\ \ \ \textrm{for}\ \ \textrm{all}\ \ \xi\in{\rm IR},

    where

    \sigma_0 \triangleq \frac{d_3}{cR_0\int_{-\infty}^{-2R_0}J(y)e^{\varpi y}\rm{d} y}.

    Thus

    \begin{align*} \int_{{\rm IR}} J(y)\frac{I(\xi-y)}{I(\xi)}\rm{d} y = &\int_{-\infty}^0 J(y)\frac{I(\xi-y)}{I(\xi)}\rm{d} y + \int_0^{+\infty} J(y)\frac{I(\xi-y)}{I(\xi)}\rm{d} y\\ = &\int_{-\infty}^0 J(y)e^{\varpi y}\frac{W(\xi-y)}{W(\xi)}\rm{d} y + \int_0^{+\infty} J(y)e^{\varpi y}\frac{W(\xi-y)}{W(\xi)}\rm{d} y\\ \leq&\sigma_0\int_{-\infty}^0 J(y)e^{\varpi y}\frac{W(\xi-y-R_0)}{W(\xi)}\rm{d} y + \int_0^{+\infty} J(y)e^{\varpi y}\rm{d} y\\ \leq&\frac{c\sigma_0}{d_3R_0} + \int_0^{+\infty} J(y)e^{\varpi y}\rm{d} y. \end{align*}

    Again with the third equation of (2.4), we have

    I'(\xi) + \varpi I(\xi) = d_3J*I(\xi) + \beta_1S(\xi)I(\xi-c\tau) + \beta_2V(\xi)I(\xi-c\tau) \gt 0\ \ \ \textrm{for}\ \ \textrm{all}\ \ \xi\in{\rm IR}.

    Let U(\xi) = e^{\varpi\xi} I(\xi), then U'(\xi)\geq0, it follows that

    \frac{I(\xi-c\tau)}{I(\xi)}\leq e^{\varpi c\tau}.

    Furthermore,

    \bigg|\frac{I'(\xi)}{I(\xi)}\bigg|\leq \frac{d_3}{c}\int_{{\rm IR}} J(y)\frac{I(\xi-y)}{I(\xi)}\rm{d} y + (\beta_1S_0+\beta_2V_0)\frac{I(\xi-c\tau)}{I(\xi)} + \varpi.

    This completes the proof.

    Lemma 2.11. Choose c_k\in(c^*, c^*+1) and let \{c_k, S_k, V_k, I_k\} be a sequence of traveling waves of (2.1) with speeds \{c_k\}. If there is a sequence \{\xi_k\} such that I_k(\xi_k)\rightarrow+\infty as k\rightarrow+\infty, then S_k(\xi_k)\rightarrow0 and V_k(\xi_k)\rightarrow0 as k\rightarrow+\infty.

    Proof. Assume that there is a subsequence of \{\xi_k\}_{k\in{\rm IN}} again denoted by \xi_k, such that I_k(\xi_k)\rightarrow+\infty as k\rightarrow+\infty and S_k(\xi_k)\geq\varepsilon in {\rm IR} for all k\in{\rm IN} with some positive constant \varepsilon. From the first equation of (2.4), we have

    S'_k(\xi)\leq \frac{2d_1S_0+\Lambda}{c^*} \triangleq \rm{d}elta_0\ \ \textrm{in}\ \ {\rm IR}.

    It follows that

    S_k(\xi)\geq\frac{\varepsilon}{2}, \ \ \ \forall\xi\in[\xi_k-\rm{d}elta, \xi_k],

    for all k\in{\rm IN}, where \rm{d}elta = \frac{\varepsilon}{\rm{d}elta_0}. By Lemma 2.10, we have \bigg|\frac{I'_k}{I_k}\bigg|<C_0 for some C_0>0. Then

    \frac{I_k(\xi_k)}{I_k(\xi-c\tau)} = \exp\left\{\int_{\xi-c\tau}^{\xi_k}\frac{I'_k(s)}{I_k(s)}\rm{d} s\right\}\leq e^{C_0(c\tau+\rm{d}elta)}, \ \ \forall\xi\in[\xi_k-\rm{d}elta, \xi_k]

    for all k\in{\rm IN}. Thus

    \min\limits_{\xi\in[\xi_k-\rm{d}elta, \xi_k]}I_k(\xi-c\tau)\geq e^{-C_0(c\tau+\rm{d}elta)}I_k(\xi_k),

    which give us

    \min\limits_{\xi\in[\xi_k-\rm{d}elta, \xi_k]}I_k(\xi-c\tau) \rightarrow +\infty\ \ \textrm{as}\ \ k\rightarrow+\infty

    since I_k(\xi_k)\rightarrow+\infty as k\rightarrow+\infty. Recalling the first equation of (2.4), one can have

    \max\limits_{\xi\in[\xi_k-\rm{d}elta, \xi_k]}S'_k(\xi)\leq \rm{d}elta_0 - \frac{\beta_1\varepsilon}{2}\min\limits_{\xi\in[\xi_k-\rm{d}elta, \xi_k]}I_k(\xi-c\tau)\rightarrow-\infty\ \ \textrm{as}\ \ k\rightarrow+\infty.

    Moreover, there exists some K>0 such that

    S'_k(\xi)\leq - \frac{2S_0}{\rm{d}elta}, \ \ \forall k\geq K\ \ \textrm{and}\ \ \xi\in[\xi_k-\rm{d}elta, \xi_k].

    Note that S_k<S_0 in {\rm IR} for each k\in{\rm IN}. Hence S_k(\xi_k)\leq-S_0 for all k\geq K, which reduces to a contradiction since S_k(\xi_k)\geq\varepsilon in {\rm IR} for all k\in{\rm IN} with some positive constant \varepsilon. Similarly, we can show that V_k(\xi_k)\rightarrow0 as k\rightarrow+\infty. This completes the proof.

    Lemma 2.12. If \limsup\limits_{\xi\rightarrow\infty}I(\xi) = \infty, then \lim\limits_{\xi\rightarrow\infty}I(\xi) = \infty.

    The proof is similar to that of [34,Lemma 2.4], so we omit the details. With the previous lemmas, we can show that I(\xi) is bounded in {\rm IR}.

    Theorem 2.2.I(\xi) is bounded in {\rm IR}.

    Proof. Assume that \limsup\limits_{\xi\rightarrow\infty}I(\xi) = \infty, then we have \lim\limits_{\xi\rightarrow\infty}S(\xi) = 0 and \lim\limits_{\xi\rightarrow\infty}V(\xi) = 0 from Lemma 2.11 and Lemma 2.12. Set \theta(\xi) = \frac{I'(\xi)}{I(\xi)}, from the third equation of (2.4), we have

    c\theta(\xi) = d_3\int_{{\rm IR}}J(y)e^{\int_\xi^{\xi-y}\theta(s)\rm{d} s} - (d_3+\gamma+\mu_3) + B(\xi),

    where

    B(\xi) = [\beta_1S(\xi)+\beta_2V(\xi)]\frac{I(\xi-c\tau)}{I(\xi)}.

    Since \frac{I(\xi-c\tau)}{I(\xi)}<C for some positive constant C from Lemma 2.10. By using [34,Lemma 2.5], we can get that \lim\limits_{\xi\rightarrow+\infty}\theta(\xi) exists and satisfies the following equation

    f(\lambda, c) \triangleq d_3\left(\int_{\rm IR}J(y)e^{-\lambda y}-1\right)-c\lambda-(\gamma+\mu_3).

    By some calculations, we obtain

    f(0, c) \lt 0, \ \ \frac{\partial f(\lambda, c)}{\partial \lambda}\bigg|_{\lambda = 0} \lt 0, \ \ \frac{\partial^2 f(\lambda, c)}{\partial \lambda^2} \gt 0\ \ \textrm{and}\ \ \lim\limits_{\lambda\rightarrow+\infty}f(\lambda, c) = -\infty.

    Thus, I(\xi) is bounded by using the same arguments in [34,Theorem 2.6]. This ends the proof.

    Since I(\xi) is bounded in {\rm IR}, we assume that there exists a positive constant \rho<\infty such that I(\xi)<\rho. Furthermore, it can be verified \frac{\Lambda}{\mu_1+\alpha+\beta_1\rho} is a lower solution of S and \frac{\alpha\Lambda}{(\mu_1+\alpha+\beta_1\rho)(\mu_2+\beta_2\rho)} is a lower solution of V. Then we have the following proposition.

    Proposition 2.1.S(\xi), V(\xi) and I(\xi) satisfy

    \begin{align*} \frac{\Lambda}{\mu_1+\alpha+\beta_1\rho}\leq S(\xi)\leq S_0, \ \ \frac{\alpha\Lambda}{(\mu_1+\alpha+\beta_1\rho)(\mu_2+\beta_2\rho)}\leq V(\xi)\leq V_0, \ \ \underline{I}(\xi)\leq I(\xi)\leq\rho \end{align*}

    for \xi\in {\rm IR}.

    The following lemma is to show that I(\xi) cannot approach 0.

    Lemma 2.13. Assume that \Re_0>1, then for each c>c^*, we have

    \liminf\limits_{\xi\rightarrow\infty}I(\xi) \gt 0.

    Proof. We only need to show that if I(\xi)\leq\varepsilon_0 for some small enough constant \varepsilon_0>0, then I'(\xi)>0 for all \xi\in {\rm IR}. Assume by way of contradiction that there is no such \varepsilon_0, that is there exist some sequence \{\xi_k\}_{k\in{\rm IN}} such that I(\xi_k)\rightarrow0 as k\rightarrow+\infty and I'(\xi_k)\leq0. Denote

    S_k(\xi)\triangleq S(\xi_k+\xi), \ \ V_k(\xi)\triangleq V(\xi_k+\xi)\ \ \textrm{and}\ \ I_k(\xi)\triangleq I(\xi_k+\xi).

    Thus we have I_k(0)\rightarrow0 as k\rightarrow+\infty and I_k(\xi)\rightarrow0 locally uniformly in {\rm IR} as k\rightarrow+\infty. As a consequence, there also holds that I_k'(\xi)\rightarrow0 locally uniformly in {\rm IR} as k\rightarrow+\infty by the third equation of (2.4). From the argument in [25,Theorem 2.9], we can obtain that S_\infty = S_0 and V_\infty = V_0.

    Let \psi_k(\xi)\triangleq\frac{I_k(\xi)}{I_k(0)}. By Lemma 2.10, and in the view of

    \psi_k'(\xi) = \frac{I_k'(\xi)}{I_k(0)} = \frac{I_k'(\xi)}{I_k(\xi)}\psi_k(\xi),

    we have \psi_k(\xi) and \psi_k'(\xi) are also locally uniformly in {\rm IR} as k\rightarrow+\infty. Letting k\rightarrow+\infty, thus

    c\psi_\infty'(\xi) = d_3\int_{\rm IR}J(y)\psi_\infty(\xi-y)\rm{d} y + (\beta_1 S_0+\beta_2 V_0)\psi_\infty(\xi-c\tau) - (d_3+\gamma+\mu_3) \psi_\infty(\xi).

    One can have \psi_\infty(\xi)>0 in {\rm IR}. In fact, if there exist some \xi_0 such that \psi_\infty(\xi_0) = 0 and \psi_\infty(\xi)>0 for all \xi<\xi_0, then

    0 = d_3\int_{\rm IR}J(y)\psi_\infty(\xi_0-y)\rm{d} y + (\beta_1 S_0+\beta_2 V_0)\psi_\infty(\xi_0-c\tau) \gt 0,

    which is a contradiction.

    Denote Z(\xi)\triangleq\frac{\psi_\infty'(\xi)}{\psi_\infty(\xi)}, it is easy to verify Z(\xi) satisfies

    \begin{equation}\label{Z} cZ(\xi) = d_3\int_{\rm IR}J(y)e^{\int^{\xi-y}_\xi Z(s)\rm{d} s}\rm{d} y + (\beta_1 S_0+\beta_2 V_0)e^{\int^{\xi-c\tau}_\xi Z(s)\rm{d} s} - (d_3+\gamma+\mu_3). \end{equation} (2.18)

    Then by similar discussion in [25,Theorem 2.9], for \Re_0>1 and c>c^*, we have

    0 \lt \psi_\infty'(0) = \lim\limits_{k\rightarrow+\infty}\psi_n'(0) = \lim\limits_{k\rightarrow+\infty}\frac{I_n'(0)}{I_n(0)}.

    Thus, I'(\xi_k) = I_n'(0)>0, which is a contradiction. This completes the proof.

    Remark 2.1. In the proof of Lemma 2.13, we need to show that Z(\pm\infty) exist in Equation (2.18). In [25], the authors applying [37,Lemma 3.4] to show that Z(\pm\infty) exist. There is a time delay term in Equation (2.18) which is different from [37,Lemma 3.4], but we can still using the method in [37,Lemma 3.4] to proof Z(\pm\infty) exist. The proof is trivial, so we omitted it.

    Now, we can give the main result in this section.

    Theorem 2.3. Suppose \Re_0>1, then for every c>c^*, system (2.1) admits a nontrivial traveling wave solution (S(x+ct), V(x+ct), I(x+ct)) satisfying the asymptotic boundary condition (2.5) and (2.6).

    Proof. First, it is easy to verify that S(-\infty) = S_0, V(-\infty) = V_0, I(-\infty) = 0 by Lemmas 2.4, 2.5 and 2.6.

    Next, we will show (S(\xi), V(\xi), I(\xi)) = (S^*, V^*, I^*) as \xi\rightarrow +\infty by using Lyapunov function. From Proposition 2.1 and Lemma 2.13, we have S(\xi)>0, \ V(\xi)>0 and I(\xi)>0.

    Let g(x) = x-1-\ln x, \alpha^+(y) = \int_y^{+\infty}J(x)\rm{d} x, \alpha^-(y) = \int_{-\infty}^y J(x)\rm{d} x. Since J is compactly supported, and recall that R is the radius of suppJ, hence

    \begin{equation}\label{suppJ} \alpha^+(y) \equiv 0\ \ \textrm{and}\ \ \alpha^-(y) \equiv 0\ \ \textrm{for}\ \ \ |y|\geq R. \end{equation} (2.19)

    Define the following Lyapunov functional

    L(S, V, I)(\xi) = cS^*L_1(\xi)+cV^*L_2(\xi)+cI^*L_3(\xi)+d_1S^*U_1(\xi)+d_2V^*U_2(\xi)+d_3I^*U_3(\xi)

    where

    \begin{align*} L_1(\xi) = &g\left(\frac{S(\xi)}{S^*}\right);\ \ \ L_2(\xi) = g\left(\frac{V(\xi)}{V^*}\right);\\ L_3(\xi) = &g\left(\frac{I(\xi)}{I^*}\right) + (\mu_3+\gamma)I^*\int_0^{c\tau}g\left(\frac{I(\xi-\theta)}{I^*}\right)\rm{d} \theta;\\ U_1(\xi) = &\int_0^{+\infty}\alpha^+(y)g\left(\frac{S(\xi-y)}{S^*}\right)\rm{d} y-\int_{-\infty}^0\alpha^-(y)g\left(\frac{S(\xi-y)}{S^*}\right)\rm{d} y;\\ U_2(\xi) = &\int_0^{+\infty}\alpha^+(y)g\left(\frac{V(\xi-y)}{V^*}\right)\rm{d} y-\int_{-\infty}^0\alpha^-(y)g\left(\frac{V(\xi-y)}{V^*}\right)\rm{d} y;\\ U_3(\xi) = &\int_0^{+\infty}\alpha^+(y)g\left(\frac{I(\xi-y)}{I^*}\right)\rm{d} y-\int_{-\infty}^0\alpha^-(y)g\left(\frac{I(\xi-y)}{I^*}\right)\rm{d} y. \end{align*}

    Thanks to [38,Theorem 1] and S(\xi)>0, V(\xi)>0, I(\xi)>0, we can get that L_1(\xi), L_2(\xi) and L_3(\xi) are bounded from below. Furthermore, by using (2.19), Proposition 2.1 and Lemma 2.13, we can claim that U_1(\xi), U_2(\xi) and U_3(\xi) is bounded from below. Thus L(S, V, I)(\xi) is well defined and bounded from below. Note that \alpha^{\pm} = \frac{1}{2}, \frac{\rm{d} \alpha^{+}(y)}{\rm{d} y} = J(y) and \frac{\rm{d} \alpha^{-}(y)}{\rm{d} y} = -J(y), we have

    \begin{align*} \frac{\rm{d} U_1(\xi)}{\rm{d} \xi} = &\frac{\rm{d}}{\rm{d} \xi}\int_0^{+\infty}\alpha^+(y)g\left(\frac{S(\xi-y)}{S^*}\right)\rm{d} y-\frac{\rm{d}}{\rm{d} \xi}\int_{-\infty}^0\alpha^-(y)g\left(\frac{S(\xi-y)}{S^*}\right)\rm{d} y\\ = &\int_0^{+\infty}\alpha^+(y)\frac{\rm{d}}{\rm{d} \xi}g\left(\frac{S(\xi-y)}{S^*}\right)\rm{d} y-\int_{-\infty}^0\alpha^-(y)\frac{\rm{d}}{\rm{d} \xi}g\left(\frac{S(\xi-y)}{S^*}\right)\rm{d} y\\ = &-\int_0^{+\infty}\alpha^+(y)\frac{\rm{d}}{\rm{d} y}g\left(\frac{S(\xi-y)}{S^*}\right)\rm{d} y+\int_{-\infty}^0\alpha^-(y)\frac{\rm{d}}{\rm{d} y}g\left(\frac{S(\xi-y)}{S^*}\right)\rm{d} y\\ = &g\left(\frac{S(\xi)}{S^*}\right)-\int_{-\infty}^{+\infty}J(y)g\left(\frac{S(\xi-y)}{S^*}\right)\rm{d} y. \end{align*}

    Similarly,

    \begin{align*} \frac{\rm{d} U_2(\xi)}{\rm{d} \xi} = g\left(\frac{V(\xi)}{V^*}\right)-\int_{-\infty}^{+\infty}J(y)g\left(\frac{V(\xi-y)}{V^*}\right)\rm{d} y;\\ \frac{\rm{d} U_3(\xi)}{\rm{d} \xi} = g\left(\frac{I(\xi)}{I^*}\right)-\int_{-\infty}^{+\infty}J(y)g\left(\frac{I(\xi-y)}{I^*}\right)\rm{d} y. \end{align*}

    By some calculations, it can be shown that

    \begin{align*} \nonumber\frac{\rm{d}}{\rm{d} \xi}\int_0^{c\tau}g\left(\frac{I(\xi-\theta)}{I^*}\right)\rm{d} \theta = &\int_0^{c\tau}\frac{\rm{d}}{\rm{d} \xi}g\left(\frac{I(\xi-\theta)}{I^*}\right)\rm{d} \theta = -\int_0^{c\tau}\frac{\rm{d}}{\rm{d} \theta}g\left(\frac{I(\xi-\theta)}{I^*}\right)\rm{d} \theta\\ = &\frac{I(\xi)}{I^*}-\frac{I(\xi-c\tau)}{I^*}+\ln \frac{I(\xi-c\tau)}{I(\xi)}. \end{align*}

    Thus

    \begin{align*} \frac{\rm{d} L(\xi)}{\rm{d} \xi} = &\left(1-\frac{S^*}{S(\xi)}\right)(d_1(J*S(\xi)-S(\xi))+\Lambda-\beta_1S(\xi)I(\xi-c\tau)-(\alpha+\mu_1)S(\xi))\\ &+\left(1-\frac{V^*}{V(\xi)}\right)(d_2(J*V(\xi)-V(\xi))+\alpha S(\xi) - \beta_2V(\xi)I(\xi-c\tau)- \mu_2E(\xi))\\ &+\left(1-\frac{I^*}{I(\xi)}\right)(d_3(J*I(\xi)-I(\xi))+\beta_1S(\xi)I(\xi-c\tau)+\beta_2V(\xi)I(\xi-c\tau)-(\gamma+\mu_3)I(\xi))\\ &+(\mu_3+\gamma)I^*\left(\frac{I(\xi)}{I^*}-\frac{I(\xi-c\tau)}{I^*}+\ln \frac{I(\xi-c\tau)}{I(\xi)}\right)\\ &+d_1S^*g\left(\frac{S(\xi)}{S^*}\right)-d_1S^*\int_{-\infty}^{+\infty}J(y)g\left(\frac{S(\xi-y)}{S^*}\right)\rm{d} y\\ &+d_2V^*g\left(\frac{V(\xi)}{V^*}\right)-d_2V^*\int_{-\infty}^{+\infty}J(y)g\left(\frac{V(\xi-y)}{V^*}\right)\rm{d} y\\ &+d_3I^*g\left(\frac{I(\xi)}{I^*}\right)-d_3I^*\int_{-\infty}^{+\infty}J(y)g\left(\frac{I(\xi-y)}{I^*}\right)\rm{d} y\\ \triangleq&B_1+B_2, \end{align*}

    where

    \begin{align*} B_1 = &\left(1-\frac{S^*}{S(\xi)}\right)d_1(J*S(\xi)-S(\xi))+d_1S^*g\left(\frac{S(\xi)}{S^*}\right)-d_1S^*\int_{-\infty}^{+\infty}J(y)g\left(\frac{S(\xi-y)}{S^*}\right)\rm{d} y\\ &+\left(1-\frac{V^*}{V(\xi)}\right)d_2(J*V(\xi)-V(\xi))+d_2V^*g\left(\frac{V(\xi)}{V^*}\right)-d_2V^*\int_{-\infty}^{+\infty}J(y)g\left(\frac{V(\xi-y)}{V^*}\right)\rm{d} y\\ &+\left(1-\frac{I^*}{I(\xi)}\right)d_3(J*I(\xi)-I(\xi))+d_3I^*g\left(\frac{I(\xi)}{I^*}\right)-d_3I^*\int_{-\infty}^{+\infty}J(y)g\left(\frac{I(\xi-y)}{I^*}\right)\rm{d} y, \\ \end{align*}

    and

    \begin{align*} B_2 = &\left(1-\frac{S^*}{S(\xi)}\right)(\Lambda-\beta_1S(\xi)I(\xi-c\tau)-(\alpha+\mu_1)S(\xi))\\ &+\left(1-\frac{V^*}{V(\xi)}\right)(\alpha S(\xi) - \beta_2V(\xi)I(\xi-c\tau)- \mu_2E(\xi))\\ &+\left(1-\frac{I^*}{I(\xi)}\right)(\beta_1S(\xi)I(\xi-c\tau)+\beta_2V(\xi)I(\xi-c\tau)-(\gamma+\mu_3)I(\xi))\\ &+(\mu_3+\gamma)I^*\left(\frac{I(\xi)}{I^*}-\frac{I(\xi-c\tau)}{I^*}+\ln \frac{I(\xi-c\tau)}{I(\xi)}\right). \end{align*}

    For B_1, using \ln\frac{S(\xi)}{S^*} = \ln\frac{S(\xi-y)}{S^*}-\ln\frac{S(\xi-y)}{S(\xi)}, thus

    \begin{align*} &\left(1-\frac{S^*}{S(\xi)}\right)d_1(J*S(\xi)-S(\xi))+d_1S^*g\left(\frac{S(\xi)}{S^*}\right)-d_1S^*\int_{-\infty}^{+\infty}J(y)g\left(\frac{S(\xi-y)}{S^*}\right)\rm{d} y\\ = &d_1S^*\int_{-\infty}^{+\infty}J(y)\left[\frac{S(\xi-y)}{S^*}-\frac{S(\xi-y)}{S(\xi)}-\ln \frac{S(\xi)}{S^*}\right]-d_1S^*\int_{-\infty}^{+\infty}J(y)g\left(\frac{S(\xi-y)}{S^*}\right)\rm{d} y\\ = &d_1S^*\int_{-\infty}^{+\infty}J(y)\left[g\left(\frac{S(\xi-y)}{S^*}\right)-g\left(\frac{S(\xi-y)}{S(\xi)}\right)\right]-d_1S^*\int_{-\infty}^{+\infty}J(y)g\left(\frac{S(\xi-y)}{S^*}\right)\rm{d} y\\ = &-d_1S^*\int_{-\infty}^{+\infty}J(y)g\left(\frac{S(\xi-y)}{S(\xi)}\right)\rm{d} y. \end{align*}

    Then

    \begin{align} \nonumber B_1 = &-d_1S^*\int_{-\infty}^{+\infty}J(y)g\left(\frac{S(\xi-y)}{S(\xi)}\right)\rm{d} y-d_2V^*\int_{-\infty}^{+\infty}J(y)g\left(\frac{V(\xi-y)}{V(\xi)}\right)\rm{d} y\\ &-d_3I^*\int_{-\infty}^{+\infty}J(y)g\left(\frac{I(\xi-y)}{I(\xi)}\right)\rm{d} y. \end{align} (2.20)

    For B_2, by some calculation yields

    \begin{align*} B_2 = &\mu_1S^*\left(2-\frac{S(\xi)}{S^*}-\frac{S^*}{S(\xi)}\right)-\beta_1S^*I^*g\left(\frac{S(\xi)I(\xi-c\tau)}{S^*I(\xi)}\right)\\ &-\beta_2V^*I^*\left[g\left(\frac{V(\xi)I(\xi-c\tau)}{V^*I(\xi)}\right)+g\left(\frac{S(\xi)V^*}{S^*V(\xi)}\right)\right]\\ &-\mu_2V^*\left[g\left(\frac{V(\xi)}{V^*}\right)+g\left(\frac{S(\xi)V^*}{S^*V(\xi)}\right)\right]\\ &-(\alpha S^*+\beta_1S^*I^*)g\left(\frac{S^*}{S(\xi)}\right), \end{align*}

    here we use (\mu_3+\gamma)I^* = \beta_1S^*I^*+\beta_2V^*I^* and \alpha S(\xi)\frac{V^*}{V(\xi)} = (\beta_2V^*I^*+\mu_2V^*)\frac{S(\xi)V^*}{S^*V(\xi)}. Combining B_1 and B_2, we obtain L(\xi) is decreasing in \xi.

    Consider an increasing sequence \{\xi_n\}_{n\geq 0} with \xi_n>0 such that \xi_n\rightarrow+\infty when n\rightarrow+\infty and denote

    \{S_n(\xi) = S(\xi+\xi_n)\}_{n\geq 0}, \ \ \{V_n(\xi) = V(\xi+\xi_n)\}_{n\geq 0, }\ \ \textrm{and}\ \ \{I_n(\xi) = I(\xi+\xi_n)\}_{n\geq 0}.

    We can assume that S_n, \ V_n and I_n converge to some nonnegative functions S_\infty, \ V_\infty and I_\infty. Furthermore, since L(S, V, I)(\xi) is non-increasing on \xi, then there exists a constant \hat{C} and large n such that

    \hat{C}\leq L(S_n, V_n, I_n)(\xi) = L(S, V, I)(\xi+\xi_n)\leq L(S, V, I)(\xi).

    Therefore there exists some \tilde{\rm{d}elta}\in {\rm IR} such that \lim_{n\rightarrow\infty} L(S_n, V_n, I_n)(\xi) = \tilde{\rm{d}elta} for any \xi\in {\rm IR}. By Lebegue dominated convergence theorem, gives us

    \lim\limits_{n\rightarrow+\infty}L(S_n, V_n, I_n)(\xi) = L(S_\infty, V_\infty, I_\infty)(\xi), \ \xi\in {\rm IR}.

    Thus

    L(S_\infty, V_\infty, I_\infty)(\xi) = \tilde{\rm{d}elta}.

    Note that \frac{\rm{d} L}{\rm{d} \xi} = 0 if and only if S(\xi)\equiv S^*, V(\xi)\equiv V^* and I(\xi)\equiv I^*, it follows that

    (S_\infty, V_\infty, I_\infty)\equiv (S^*, V^*, I^*).

    This completes the proof.

    In this section, we investigate the existence of traveling wave solutions for the case c = c^* by a limiting argument(see [23,39]).

    Theorem 3.1. Suppose \Re_0>1, then for every c = c^*, system (2.1) admits a nontrivial traveling wave solution (S(x+c^*t), V(x+c^*t), I(x+c^*t)) satisfying

    \lim\limits_{\xi\rightarrow+\infty}(S(\xi), V(\xi), I(\xi)) = (S^*, V^*, I^*).

    Furthermore, if we assume that S(-\infty) and V(-\infty) exist, then (S(x+c^*t), V(x+c^*t), I(x+c^*t)) also satisfying

    \lim\limits_{\xi\rightarrow-\infty}(S(\xi), V(\xi), I(\xi)) = (S_0, V_0, 0).

    Proof. Let \{c_n\}\subset(c^*, c^*+1) be a decreasing sequence such that \lim\limits_{n\rightarrow\infty}c_n = c^*. Then for each c_n, there exists a traveling wave solution (S_n(\cdot), V_n(\cdot), I_n(\cdot)) of system (2.4) with asymptotic boundary condition (2.5) and (2.6). Since (S_n(\cdot+a), V_n(\cdot+a), I_n(\cdot+a)) are also solutions of (2.4) for any a\in{\rm IR}, we can assume that

    I_n(0) = \rm{d}elta^*, \ \ I_n(\xi)\leq\rm{d}elta^*, \ \ \xi \lt 0

    with 0<\rm{d}elta<I^* is small enough.

    Similar to [23,39], we can find a subsequence of (S_n, V_n, I_n), again denoted by (S_n, V_n, I_n), such that (S_n, V_n, I_n) and (S'_n, V'_n, I'_n) converge uniformly on every bounded interval to function (S, V, I) and (S', V', I'), respectively. Applying the Lebesgue dominated convergence theorem, it then follows that

    \lim\limits_{n\rightarrow\infty}J*S_n = J*S, \ \ \lim\limits_{n\rightarrow\infty}J*V_n = J*V, \ \ \textrm{and}\ \ \lim\limits_{n\rightarrow\infty}J*I_n = J*I

    on every bounded interval. Then we get that (S, V, I) satisfies system (2.4). From the proof of Theorem 2.3, the Lyapunov functional is independent of c. By the same argument in the proof of Theorem 2.3, we claim that I(\xi)>0 for any \xi\in{\rm IR}. Hence, we can still get that

    \lim\limits_{\xi\rightarrow+\infty}S(\xi) = S^*, \ \ \lim\limits_{\xi\rightarrow+\infty}V(\xi) = V^*, \ \ \lim\limits_{\xi\rightarrow+\infty}I(\xi) = I^*.

    Moreover, we have

    I(0) = \rm{d}elta^*, \ \ I(\xi)\leq\rm{d}elta^*, \ \ \xi \lt 0.

    Let

    S_{sup} = \limsup\limits_{\xi\rightarrow-\infty}S(\xi), \ \ V_{sup} = \limsup\limits_{\xi\rightarrow-\infty}V(\xi), \ \ I_{sup} = \limsup\limits_{\xi\rightarrow-\infty}I(\xi)

    and

    S_{inf} = \liminf\limits_{\xi\rightarrow-\infty}S(\xi), \ \ V_{inf} = \liminf\limits_{\xi\rightarrow-\infty}V(\xi), \ \ I_{inf} = \liminf\limits_{\xi\rightarrow-\infty}I(\xi).

    Next, we show that I(-\infty) exists. By way of contradiction, assume that I_{inf} < I_{sup}. Then there exist sequences \{x_n\} and \{y_n\} satisfying x_n, \ y_n\rightarrow-\infty as n\rightarrow+\infty such that

    \lim\limits_{n\rightarrow+\infty}I(x_n) = I_{inf}\ \ \textrm{}\ \ \lim\limits_{n\rightarrow+\infty}I(y_n) = I_{sup}.

    Since we assumed that S(-\infty) and V(-\infty) exist, then S_{sup} = S_{inf} = S(-\infty) and V_{sup} = V_{inf} = V(-\infty). From [40,Lemma 2.3], we can obtain that S'(-\infty) = 0 and V'(-\infty) = 0. For any sequence \{\xi_n\}, \xi_n\rightarrow -\infty as n\rightarrow+\infty, using Fatou Lemma, one have that

    S(-\infty)\leq\liminf\limits_{n\rightarrow\infty}J*S(\xi_n)\leq\limsup\limits_{n\rightarrow\infty}J*S(\xi_n)\leq S(-\infty).

    and

    V(-\infty)\leq\liminf\limits_{n\rightarrow\infty}J*V(\xi_n)\leq\limsup\limits_{n\rightarrow\infty}J*V(\xi_n)\leq V(-\infty).

    Thus, we have

    \lim\limits_{n\rightarrow\infty}[J*S(\xi_n)-S(\xi_n)] = 0\ \ \textrm{and}\ \ \lim\limits_{n\rightarrow\infty}[J*V(\xi_n)-V(\xi_n)] = 0

    Taking \xi = x_n and \xi = y_n in the first equation of system 2.4, and letting n\rightarrow\infty, we obtain that I_{inf} = I_{sup}, which is a contradiction. Hence, I(-\infty) exists and I(-\infty)<\rm{d}elta^*. From system (2.4) and [40,Lemma 2.3], we obtain

    \begin{equation} \left\{ \begin{array}{l} \displaystyle \Lambda - \beta_1 S(-\infty)I(-\infty) - \alpha S(-\infty) - \mu_1 S(-\infty = 0), \\ \displaystyle \alpha S(-\infty)- \beta_2 V(-\infty)I(-\infty) - \mu_2 V(-\infty) = 0, \\ \displaystyle \beta_1 S(-\infty)I(-\infty) + \beta_2 V(-\infty)I(-\infty)) - \gamma I(-\infty) - \mu_3 I(-\infty) = 0. \end{array}\right. \end{equation} (3.1)

    In the view of \rm{d}elta^*<I^*, it follows that

    \lim\limits_{\xi\rightarrow-\infty}S(\xi) = S_0, \ \ \lim\limits_{\xi\rightarrow-\infty}V(\xi) = V_0, \ \ \lim\limits_{\xi\rightarrow-\infty}I(\xi) = 0.

    This completes the proof.

    Remark 3.1. For the case c = c^*, there is a priori condition assuming S(-\infty) and V(-\infty) exist. This condition is only necessary for the difficulty in mathematics. In [34], the authors have given some results for the case c = c^* in a nonlocal diffusive SIR model without constant recruitment, but some estimates is much more difficult for our model with constant recruitment and time delay as in [34,Section 3]. Thus, how to extend the methods in [34] to our model, it will be an interesting problem for further investigation.

    In this section, we show the nonexistence of traveling waves when \Re_0>1 with 0<c<c^*.

    Theorem 4.1. If \Re_0>1 and 0<c<c^*, then there exists no nontrivial positive solutions of (2.4) with (2.5) and (2.6).

    Proof. Since \Re_0>1 gives us \beta_1S_0+\beta_2V_0>\mu_3+\gamma. Assume there exists nontrivial positive solution (S, V, I) of (2.4) with (2.5) and (2.6). Then there exists a positive constant K>0 large enough such that, for any \xi<-K, we have

    \begin{equation}\label{Equ1} c I'(\xi) \geq d_3(J*I(\xi) - I(\xi)) +\frac{\beta_1S_0 + \beta_2V_0 - (\gamma + \mu_3)}{2}I(\xi-c\tau) +(\gamma+\mu_3)(I(\xi-c\tau) - I(\xi)) \end{equation} (4.1)

    holds. Let K(\xi) = \int_{-\infty}^\xi I(\eta)\rm{d} \eta. By Fubini theorem, thus

    \begin{align}\label{Equ2} d_3\int_{-\infty}^{\xi}J * I(s)\rm{d} s = &d_3\int_{-\infty}^{\xi} \int_{{\rm IR}} J(y) I(s-y)\rm{d} y\rm{d} s\\ \nonumber = &d_3\int_{{\rm IR}} \int_{-\infty}^{\xi} J(y) I(s-y)\rm{d} s\rm{d} y\\ \nonumber = &d_3\int_{{\rm IR}} J(y) \int_{-\infty}^{\xi} I(s-y)\rm{d} s\rm{d} y\\ \nonumber = & d_3 J * K(\xi).\nonumber \end{align} (4.2)

    Integrating the both sides of (4.1) from -\infty to \xi with \xi\leq-K, we have

    \begin{align}\label{Equ3} \nonumber cI(\xi) \geq &d_3(J*K(\xi) - K(\xi)) + (\gamma + \mu_3)[K(\xi-c\tau) - K(\xi)]\\ & + \frac{\beta_1S_0 + \beta_2V_0 - (\gamma + \mu_3)}{2} K(\xi-c\tau). \end{align} (4.3)

    Furthermore, the following two equations hold.

    \begin{align}\label{Equ4} \nonumber\int_{-\infty}^\xi[K(\eta-c\tau) - K(\eta)]\rm{d} \eta = &\int_{-\infty}^\xi (-c\tau) \int_0^1\frac{\partial K(\eta-c\tau s)}{\partial s}\rm{d} s\rm{d} \eta\\ = &- c\tau \int_0^1 K(\xi-c\tau s)\rm{d} s \end{align} (4.4)

    and

    \begin{align}\label{Equ5} d_3\nonumber\int_{-\infty}^\xi[J*K(\eta) - K(\eta)]\rm{d} \eta = &d_3\int_{-\infty}^\xi \int_{-\infty}^{+\infty}(-x)J(x)\int_0^1\frac{\partial K(\eta-x s)}{\partial s}\rm{d} s\rm{d} x\rm{d} \eta\\ = &d_3\int_{-\infty}^{+\infty} (-x) J(x)\int_0^1 K(\xi-x s)\rm{d} s \rm{d} x. \end{align} (4.5)

    Integrating both sides of inequality (4.3) from -\infty to \xi, and combining Equations (4.4) and (4.5) yield

    \begin{align}\label{Equ6} \nonumber&\frac{\beta_1S_0 + \beta_2V_0 - (\gamma + \mu_3)}{2} \int_{-\infty}^\xi K(\eta-c\tau)\rm{d} \eta\\ \nonumber\leq&c K(\xi) + (\gamma+\mu_3)c\tau \int_0^1 K(\xi-c\tau s)\rm{d} s\\ \nonumber& + d_3\int_{-\infty}^{+\infty} x J(x)\int_0^1 K(\xi-x s)\rm{d} s \rm{d} x\\ \leq&\left(c+d_3\int_{{\rm IR}}xJ(x)\rm{d} x+(\gamma+\mu_3)c\tau\right)K(\xi), \end{align} (4.6)

    Here we use xK(\xi-sx) as a non-increasing function with s\in(0, 1). By (J1) of Assumption 1.1, we have \int_{{\rm IR}} x J(x)\rm{d} x = 0. Then for \xi<-K, we have

    \begin{align}\label{Equ7} \nonumber&\frac{\beta_1S_0 + \beta_2V_0 - (\gamma + \mu_3)}{2} \int_0^{+\infty} K(\xi - \eta - c\tau)\rm{d} \eta\\ \leq&(c+(\gamma+\mu_3)c\tau)K(\xi), \end{align} (4.7)

    For the non-decreasing function K(\xi), there exists some \tilde{\eta} with \tilde{\eta} + c\tau > 0 such that

    \begin{align}\label{Equ8} \nonumber&\frac{\beta_1S_0 + \beta_2V_0 - (\gamma + \mu_3)}{2} (\tilde{\eta} + c\tau) K(\xi - \tilde{\eta} - c\tau)\\ \leq&(c+(\gamma+\mu_3)c\tau)K(\xi), \end{align} (4.8)

    Thus there exists a sufficiently large constant \theta>-c\tau and some constant \varepsilon\in (0, 1), such that

    K(\xi-\theta -c\tau)\leq\varepsilon K(\xi), \ \ \xi\leq-M.

    Let

    p(\xi) = K(\xi)e^{-\nu\xi},

    where

    0 \lt \nu\triangleq\frac{1}{\theta+c\tau}\ln \frac{1}{\varepsilon} \lt \lambda_c,

    By some simple calculation, we have

    p(\xi-\theta-c\tau)\leq p(\xi).

    Using L'Hospital's rule yields

    \lim\limits_{\xi\rightarrow+\infty}p(\xi) = \lim\limits_{\xi\rightarrow+\infty}\frac{K(\xi)}{e^{\nu\xi}} = \lim\limits_{\xi\rightarrow+\infty}\frac{I(\xi)}{\nu e^{\nu\xi}} = 0,

    Note that p(\xi)\geq 0, thus there exists a constant p_0 such that

    \begin{align}\label{Equ9} p(\xi) = K(\xi)e^{-\nu\xi}\leq p_0, \ \ \xi\in {\rm IR}. \end{align} (4.9)

    On the other hand, since S(\xi)\leq S_0 and V(\xi)\leq V_0 for \xi\in{\rm IR}, recall the third equation of (2.4), we have

    \begin{align}\label{Equ91} \nonumber cI'(\xi) = &d_3(J*I(\xi)-I(\xi)) + \beta_1 S(\xi)I(\xi-c\tau) + \beta_2 V(\xi)I(\xi-c\tau) - \gamma I(\xi) - \mu_3 I(\xi)\\ \leq&d_3(J*I(\xi)-I(\xi)) + \beta_1 S_0I(\xi-c\tau) + \beta_2 V_0I(\xi-c\tau) - \gamma I(\xi) - \mu_3 I(\xi). \end{align} (4.10)

    Integrating the both sides of (4.10) from -\infty to \xi yields

    \begin{align}\label{Equ10} cI(\xi)\leq d_3J*K(\xi)-(\gamma+\mu_3+d_3)K(\xi) + (\beta_1S_0+\beta_2V_0)K(\xi-c\tau). \end{align} (4.11)

    From (4.9), using J is compactly supported, for \xi\in{\rm IR}, there exists a positive constant M_1 such that

    \begin{align}\label{Equ11} \nonumber (d_3J*K(\xi))e^{-\nu \xi} = &d_3\int_{{\rm IR}}J(y)e^{-\nu \xi}K(\xi-y)\rm{d} y\\ = & d_3\int_{{\rm IR}}J(y)e^{-\nu y}K(\xi-y) e^{-\nu (\xi-y)}\rm{d} y\\ \nonumber\leq& d_3p_0\int_{{\rm IR}} J(y)e^{-\nu y} \rm{d} y\\ \nonumber\leq&M_1. \end{align} (4.12)

    Thus there exists a constant M_2>0 such that

    \begin{align} I(\xi) e^{-\nu \xi}\leq M_2, \ \ \xi\in{\rm IR}, \end{align} (4.13)

    since (4.9), (4.11) and (4.12) hold. Then

    \begin{align} \sup\limits_{\xi\in{\rm IR}}\{I(\xi) e^{-\nu \xi}\} \lt +\infty. \end{align} (4.14)

    By the same procedure in (4.12), there exists a positive constant M_2 such that

    \begin{align}\label{Equ12} (d_3J*I(\xi))e^{-\nu \xi}\leq&M_2. \end{align} (4.15)

    Hence

    \begin{align} \sup\limits_{\xi\in{\rm IR}}\{I'(\xi) e^{-\nu \xi}\} \lt +\infty. \end{align} (4.16)

    For \lambda\in{\rm IC} with 0<\textrm{Re}\lambda<\nu, define the following two-side Laplace transform of I(\xi),

    \begin{align} \nonumber \mathcal{L}_I(\lambda):& = \int_{{\rm IR}}I(\xi)e^{-\lambda \xi}\rm{d} \xi. \end{align}

    From (2.4), we have

    \begin{align} \nonumber &d_3(J*I(\xi)-I(\xi)) - cI'(\xi) + (\beta_1 S_0 + \beta_2V_0) I(\xi-c\tau) - (\gamma+\mu_3) I(\xi)\\ = &\beta_1 (S_0 - S(\xi))I(\xi-c\tau) + \beta_2 (V_0 - V(\xi))I(\xi-c\tau). \end{align} (4.17)

    Take the two-side Laplace transform to the above equation, thus

    \begin{align}\label{Equ13} \Delta(\lambda, c)\mathcal{L}_I(\lambda) = \int_{\rm IR}e^{-\lambda \xi}[\beta_1 (S_0 - S(\xi))I(\xi-c\tau) + \beta_2 (V_0 - V(\xi))I(\xi-c\tau)]\rm{d} \xi \end{align} (4.18)

    for \lambda \in {\rm IC} with 0<\textrm{Re}\lambda<\nu. Let L(\xi) = S_0 -S(\xi), we have 0\leq L(\xi)\leq S_0 and \lim_{\xi\rightarrow -\infty} L(\xi) = 0. Then from the first equation of (2.4), we have

    c L'(\xi) = d_1 (J*L(\xi)-L(\xi)) + \beta_1 S_(\xi)I(\xi-c\tau) + (\alpha+\mu_1)S(\xi).

    Let \eta\in C^{\infty}({\rm IR}, [0, 1]) be a nonnegative nondecreasing function, \eta(x)\equiv0 in (-\infty, -2] and \eta(x)\equiv1 in [-1, +\infty). For N\in{\rm IN}, set \eta_N = \eta\left(\frac{x}{N}\right). Then, taking 0\leq \nu_0\leq\nu, we have

    c \int_{{\rm IR}}L'(\xi)e^{-\nu_0 \xi}\eta_N\rm{d} \xi = d_1\int_{{\rm IR}}(J*L(\xi)-L(\xi))e^{-\nu_0 \xi}\eta_N\rm{d} \xi + \int_{{\rm IR}}S(\xi)[\beta_1I(\xi-c\tau) + \alpha + \mu_1] e^{-\nu_0 \xi}\eta_N\rm{d} \xi.

    By the argument in [22,Theorem 3.1], there exists a constant \Xi>0 dependent on \nu_0 such that

    \int_{{\rm IR}}L(\xi)e^{-\nu_0 \xi}\rm{d} \xi \leq \Xi.

    Thus,

    \int_{{\rm IR}}\beta_1(S_0 - S(\xi))I(\xi-c\tau)e^{-(\nu+\nu_0)\xi}\rm{d} \xi\leq \beta_1\sup\limits_{\xi\in{\rm IR}}\{I(\xi) e^{-\nu \xi}\}\int_{{\rm IR}}L(\xi)e^{-\nu_0 \xi}\rm{d} \xi \lt \infty.

    Similarly,

    \int_{{\rm IR}}\beta_2(V_0 - V(\xi))I(\xi-c\tau)e^{-(\nu+\nu_0)\xi}\rm{d} \xi \lt \infty.

    From the property of Laplace transform [41], \mathcal{L}_I(\lambda) is well defined with \textrm{Re}\lambda>0. Note that Equation (4.18) can be rewritten as

    \begin{align}\label{Equ16} \int_{\rm IR}e^{-\lambda \xi}\left[\Delta(\lambda, c)I(\xi)+\beta_1 (S_0 - S(\xi))I(\xi-c\tau) + \beta_2 (V_0 - V(\xi))I(\xi-c\tau)\right]\rm{d} \xi = 0. \end{align} (4.19)

    Recall (J2) of Assumption 1.1, then \Delta(\lambda, c)\rightarrow+\infty as \xi\rightarrow+\infty for c\in(0, c^*) which is a contradiction of (4.19). This completes the proof.

    As traveling wave solutions describe the transition from disease-free equilibrium to endemic equilibrium when the wave speed is larger than the minimal wave speed. Now, we focus on how the parameters in system (2.1) can affect the wave speed. Suppose (\hat{\lambda}, \hat{c}) be a zero root of \Delta(\lambda, c), recall that V_0 = \frac{\Lambda\alpha}{\mu_2(\mu_1+\alpha)} and \mu_2 = \mu_1 +\gamma_1, we have

    \Delta(\hat{\lambda}, \hat{c}) = d_3\int_{{\rm IR}}J(x)e^{-\hat{\lambda} x}\rm{d} x-(d_3+\gamma+\mu_3)-\hat{c}\hat{\lambda}+\beta_1S_0e^{-\hat{c}\tau\hat{\lambda}}+\frac{\beta_2\Lambda\alpha}{(\mu_1 +\gamma_1)(\mu_1+\alpha)}e^{-\hat{c}\tau\hat{\lambda}} = 0.

    By some calculations, we obtain

    \frac{\rm{d} \hat{c}}{\rm{d} d_3} = \frac{\int_{{\rm IR}}J(x)[e^{-\hat\lambda x}-1]\rm{d} x}{\hat\lambda(1+ [\beta_1 S_0 + \beta_2 V_0] \tau e^{-\hat{c}\tau\hat{\lambda}})} \gt 0, \ \ \frac{\rm{d} \hat{c}}{\rm{d} \tau} = -\frac{\beta_1S_0 + \beta_2V_0}{e^{\hat{c}\tau\hat{\lambda}} + \beta_1S_0\tau + \beta_2V_0\tau} \lt 0,
    \frac{\rm{d} \hat{c}}{\rm{d} \beta_1} = \frac{S_0e^{-\hat{c}\tau\hat{\lambda}}}{\hat\lambda(1+ [\beta_1 S_0 + \beta_2 V_0] \tau e^{-\hat{c}\tau\hat{\lambda}})} \gt 0, \ \ \frac{\rm{d} \hat{c}}{\rm{d} \beta_2} = \frac{V_0e^{-\hat{c}\tau\hat{\lambda}}}{\hat\lambda(1+ [\beta_1 S_0 + \beta_2 V_0] \tau e^{-\hat{c}\tau\hat{\lambda}})} \gt 0,

    and

    \frac{\rm{d} \hat{c}}{\rm{d} \gamma_1} = -\frac{\beta_2V_0 e^{-\hat{c}\tau\hat{\lambda}}}{(\mu_1+\gamma_1)\hat\lambda(1+ [\beta_1 S_0 + \beta_2 V_0] \tau e^{-\hat{c}\tau\hat{\lambda}})} \lt 0,

    that is, \hat{c} is a decreasing function on \gamma_1 and \tau, while \hat{c} is an increasing function on d_3, \beta_1 and \beta_2. From the biological point of view, this indicates the following four scenarios:

    Ⅰ. The more successful the vaccination, the slower the disease spreads;

    Ⅱ. The longer the latent period, the slower the disease spreads;

    Ⅲ. The faster infected individuals move, the faster the disease spreads;

    Ⅳ. The more effective the infections are, the faster the disease spreads.

    Now, we are in a position to make the following summary:

    Mathematically, we investigated a nonlocal dispersal epidemic model with vaccination and delay; The existence of traveling wave solutions is studied by applying Schauder fixed point theorem with upper-lower solutions, that is there exists traveling wave solutions when \Re_0>1 with c>c^*. Furthermore, the boundary asymptotic behaviour of traveling wave solutions at +\infty was established by the methods of constructing suitable Lyapunov like function. We also showed that there exists traveling wave solutions when \Re_0>1 with c = c^*. Finally, we proved the nonexistence of traveling wave solutions under the assumptions \Re_0>1 and 0<c<c^*.

    Biologically, our results imply that the nonlocal dispersal and infection ability of infected individuals can accelerate the spreading of infectious disease, while the latent period and successful rate of vaccination can slow down the disease spreads.

    The authors are very grateful to the editors and three reviewers for their valuable comments and suggestions that have helped us improving the presentation of this paper. We would also very grateful to Prof.Shigui Ruan, Dr. Sanhong Liu and Dr.Wen-Bing Xu for their valuable comments and helpful advice. This work is supported by Natural Science Foundation of China (No.11871179; No.11771374), and the first author was also partially supported by China Scholarship Council (No.201706120216). R. Zhang acknowledges the kind hospitality received from the Department of Mathematics at the University of Miami, where part of the work was completed.

    All authors declare no conflicts of interest in this paper.

    Proof. If \xi>\mathfrak{X}_1, then \underline{S}(\xi) = 0, equation (2.11) holds. If \xi<\mathfrak{X}_1, then \underline{S}(\xi) = S_0-M_1 e^{\varepsilon_1 \xi}, we have

    \begin{align*} &c{\underline{S}}'(\xi)- d_1(J*\underline{S}(\xi)-\underline{S}(\xi)) - \Lambda + \beta_1\underline{S}(\xi)\overline{I}(\xi-c\tau) +(\mu_1+\alpha) \underline{S}(\xi)\\ = &-c\varepsilon_1M_1e^{\varepsilon_1\xi}+d_1M_1e^{\varepsilon_1\xi}\int_{{\rm IR}}J(x)e^{-\varepsilon_1x}\rm{d} x-d_1M_1e^{\varepsilon_1\xi}-\Lambda\\ &+\beta_1(S_0-M_1e^{\varepsilon_1\xi})e^{\lambda_c(\xi-c\tau)}+(\mu_1+\alpha)(S_0-M_1 e^{\varepsilon_1 \xi})\\ \leq&e^{\varepsilon_1\xi}\left[-c\varepsilon_1M_1e^{\varepsilon_1\xi}+d_1M_1e^{\varepsilon_1\xi}\int_{{\rm IR}}J(x)e^{-\varepsilon_1x}\rm{d} x-d_1M_1e^{\varepsilon_1\xi}+\beta_1S_0\left(\frac{S_0}{M_1}\right)^{\frac{\lambda-\varepsilon_1}{\varepsilon_1}}\right]. \end{align*}

    Here we use

    e^{(\lambda_c-\varepsilon_1)\xi} \lt \left(\frac{S_0}{M_1}\right)^{\frac{\lambda_c-\varepsilon_1}{\varepsilon_1}}\ \ \ \textrm{for}\ \ \ \xi \lt \mathfrak{X}_1.

    Keeping \varepsilon_1M_1 = 1, letting M_1\rightarrow+\infty for some M_1>S_0 large enough and \varepsilon_1 small enough, we have

    \begin{equation*} c{\underline{S}}'(\xi)- d_1(J*\underline{S}(\xi)-\underline{S}(\xi)) - \Lambda + \beta_1\underline{S}(\xi)\overline{I}(\xi-c\tau) +(\mu_1+\alpha) \underline{S}(\xi)\leq0. \end{equation*}

    This completes the proof.

    Proof. If \xi > \frac{1}{\varepsilon_3}\ln \frac{1}{M_3}, the Equation (2.13) holds since \underline{I}(\xi) = 0. If \xi < \frac{1}{\varepsilon_3}\ln \frac{1}{M_3}, then \underline{I}(\xi) = e^{\lambda_c\xi}(1-M_3e^{\varepsilon_3 \xi}), we have the following four cases.

    Case Ⅰ: \xi>\max\{\mathfrak{X}_1, \mathfrak{X}_2\}.

    In this case, \underline{S}(\xi) = \underline{V}(\xi) = 0. Thus, Equation (2.13) is equivalent to

    c{\underline{I}}'(\xi) \leq d_3(J*\underline{I}(\xi)-\underline{I}(\xi)) - \gamma \underline{I}(\xi) - \mu_3 \underline{I}(\xi),

    that is

    \begin{align*} &c\lambda_c-d_3\int_{{\rm IR}}J(y)e^{-\lambda_cy}\rm{d} y+d_3+\gamma+\mu_3\\ \leq& M_3e^{\varepsilon_3\xi}\left[c(\lambda+\varepsilon_3)-d_3\int_{{\rm IR}}J(y)e^{-(\lambda_c+\varepsilon_3)y}\rm{d} y+d_3+\gamma+\mu_3\right]. \end{align*}

    From \Delta(\lambda_c, c) = 0 and \Delta(\lambda_c+\varepsilon_3, c)<0, we have

    \beta_1S_0e^{-c\tau\lambda_c}+\beta_2V_0e^{-c\tau\lambda_c}\\ \leq M_3e^{\varepsilon_3\xi}\left[-\Delta(\lambda_c+\varepsilon_3, c)+\beta_1S_0e^{-c\tau(\lambda_c+\varepsilon_3)}+\beta_2V_0e^{-c\tau(\lambda_c+\varepsilon_3)}\right],

    Because \tau>0, \lambda_c>0, it suffices to prove

    \begin{align*} \beta_1S_0+\beta_2V_0\leq M_3e^{\varepsilon_3\xi}\left[-\Delta(\lambda_c+\varepsilon_3, c)+\beta_1S_0e^{-c\tau(\lambda_c+\varepsilon_3)}+\beta_2V_0e^{-c\tau(\lambda_c+\varepsilon_3)}\right]. \end{align*}

    Since \xi>\max\{\mathfrak{X}_1, \mathfrak{X}_2\}, M_3>\max\{S_0, V_0\} and 0<\varepsilon_3<\min\{\varepsilon_1/2, \varepsilon_2/2\}, note that e^{\varepsilon_3\xi}\geq\left(\frac{S_0}{M_1}\right)^{\frac{1}{2}}\left(\frac{V_0}{M_2}\right)^{\frac{1}{2}}, then we only need to ensure

    \beta_1S_0+\beta_2V_0\leq-\Delta(\lambda_c+\varepsilon_3, c)M_3\left(\frac{S_0}{M_1}\right)^{\frac{1}{2}}\left(\frac{V_0}{M_2}\right)^{\frac{1}{2}}.

    Thus, Equation (2.13) holds for sufficiently large M_3>0 with

    M_3\geq \frac{\beta_1S_0+\beta_2V_0}{-\Delta(\lambda_c+\varepsilon_3, c)}\sqrt{\frac{S_0}{M_1}}\sqrt{\frac{V_0}{M_2}}\triangleq\Pi_1.

    Case Ⅱ: \mathfrak{X}_1>\xi>\mathfrak{X}_2.

    In this case, \underline{S}(\xi) = S_0-M_1e^{\varepsilon_1\xi} and \underline{V}(\xi) = 0. Hence, Equation (2.13) is equivalent to

    c{\underline{I}}'(\xi) \leq d_3(J*\underline{I}(\xi)-\underline{I}(\xi)) - \gamma \underline{I}(\xi) - \mu_3 \underline{I}(\xi) + \beta_1\underline{S}(\xi)\underline{I}(\xi-c\tau),

    that is

    \begin{align*} &c\lambda_c-d_3\int_{{\rm IR}}J(y)e^{-\lambda_cy}\rm{d} y+d_3+\gamma+\mu_3-\beta_1S_0e^{-\lambda_cc\tau}+\beta_1M_1e^{\varepsilon_1\xi-\lambda_cc\tau}\\ \leq& M_3e^{\varepsilon_3\xi}\left[c(\lambda+\varepsilon_3)-d_3\int_{{\rm IR}}J(y)e^{-(\lambda_c+\varepsilon_3)y}\rm{d} y+d_3+\gamma+\mu_3-\beta_1S_0e^{-(\varepsilon_3+\lambda_c)c\tau}+\beta_1M_1e^{\varepsilon_1\xi-(\varepsilon_3+\lambda_c)c\tau}\right], \end{align*}

    we need to prove

    \begin{equation*} \beta V_0\leq -\Delta(\lambda_c+\varepsilon_3, c)M_3e^{\varepsilon_3\xi}. \end{equation*}

    Choose M_3 large enough with

    M_3\geq \frac{\beta_2 \sqrt{V_0M_2}}{-\Delta(\lambda_c+\varepsilon_3, c)}\triangleq\Pi_2.

    Case Ⅲ: \mathfrak{X}_2>\xi>\mathfrak{X}_1.

    In this case, \underline{V}(\xi) = V_0-M_2e^{\varepsilon_2\xi} and \underline{S}(\xi) = 0. Similar to Case Ⅱ, Equation (2.13) holds if we choose

    M_3\geq \frac{\beta_1 \sqrt{S_0M_1}}{-\Delta(\lambda_c+\varepsilon_3, c)}\triangleq\Pi_3

    large enough.

    Case Ⅵ: \xi<\min\{\mathfrak{X}_1, \mathfrak{X}_2\}.

    In this case, \underline{S}(\xi) = S_0-M_1e^{\varepsilon_1\xi} and \underline{V}(\xi) = V_0-M_2e^{\varepsilon_2\xi}, Equation (2.13) is equivalent to

    c{\underline{I}}'(\xi) \leq d_3(J*\underline{I}(\xi)-\underline{I}(\xi)) - \gamma \underline{I}(\xi) - \mu_3 \underline{I}(\xi) + \beta_1\underline{S}(\xi)\underline{I}(\xi-c\tau) + \beta_2\underline{V}(\xi)\underline{I}(\xi-c\tau),

    that is

    \begin{align*} c\lambda_c-d_3&\int_{{\rm IR}}J(y)e^{-\lambda_cy}\rm{d} y+d_3+\gamma+\mu_3-\beta_1S_0e^{-\lambda_cc\tau}-\beta_2V_0e^{-\lambda_cc\tau}+\beta_1M_1e^{\varepsilon_1\xi-\lambda_cc\tau}+\beta_2M_2e^{\varepsilon_2\xi-\lambda_cc\tau}\\ \leq M_3e^{\varepsilon_3\xi}&\left(c(\lambda+\varepsilon_3)-d_3\int_{{\rm IR}}J(y)e^{-(\lambda_c+\varepsilon_3)y}\rm{d} y+d_3+\gamma+\mu_3-\beta_1S_0e^{-(\varepsilon_3+\lambda_c)c\tau}\right.\\ &+\left.\beta_1M_1e^{\varepsilon_1\xi-(\varepsilon_3+\lambda_c)c\tau}-\beta_2V_0e^{-(\varepsilon_3+\lambda_c)c\tau}+\beta_2M_2e^{\varepsilon_1\xi-(\varepsilon_3+\lambda_c)c\tau}\right) \end{align*}

    we only need to ensure

    M_3\geq\frac{\beta_1M_1e^{(\varepsilon_1-\varepsilon_3)\xi-\lambda_cc\tau}+\beta_2M_2e^{(\varepsilon_2-\varepsilon_3)\xi-\lambda_cc\tau}}{-\Delta(\lambda_c+\varepsilon_3, c)+\beta_1M_1e^{\varepsilon_1\xi-(\varepsilon_3+\lambda_c)c\tau}+\beta_2M_2e^{\varepsilon_2\xi-(\varepsilon_3+\lambda_c)c\tau}}

    Since \xi<\min\{\mathfrak{X}_1, \mathfrak{X}_2\}, 0<S_0<M_3, 0<V_0<M_3, \varepsilon_3<\min\{\varepsilon_1/2, \varepsilon_2/2\} and \tau>0, we have

    \frac{\beta_1M_1e^{(\varepsilon_1-\varepsilon_3)\xi-\lambda_cc\tau}+\beta_2M_2e^{(\varepsilon_2-\varepsilon_3)\xi-\lambda_cc\tau}}{-\Delta(\lambda_c+\varepsilon_3, c)+\beta_1M_1e^{\varepsilon_1\xi-(\varepsilon_3+\lambda_c)c\tau}+\beta_2M_2e^{\varepsilon_2\xi-(\varepsilon_3+\lambda_c)c\tau}} \lt \frac{\beta_1\sqrt{S_0M_1}+\beta_2\sqrt{V_0M_2}}{-\Delta(\lambda_c+\varepsilon_3, c)}.

    Then Equation (2.13) holds if we choose M_3 large enough with

    M_3\geq\frac{\beta_1\sqrt{S_0M_1}+\beta_2\sqrt{V_0M_2}}{-\Delta(\lambda_c+\varepsilon_3, c)}\triangleq\Pi_4.

    Through the above discussion, Equation (2.13) holds if we choose M_3\geq\max\{\Pi_1, \Pi_2, \Pi_3, \Pi_4\} large enough for all \xi\in{\rm IR}. Here we completes the proof.



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