Existence of traveling wave solutions with critical speed in a delayed diffusive epidemic model

Abstract

In this paper, we prove the existence of a critical traveling wave solution for a delayed diffusive SIR epidemic model with saturated incidence. Moreover, we establish the nonexistence of traveling wave solutions with nonpositive wave speed for this model. Our results solve some open problems left in the recent paper (Z. Xu in Nonlinear Anal. 111:66–81, 2014).

1 Introduction

In the past few decades, more research has focused on spatial propagation of communicable diseases in mathematical epidemiology and more reaction-diffusion SIR models have been proposed to describe the transmission of communicable diseases [1, 4, 6,7,8,9, 13, 19,20,21,22,23,24,25,26, 29,30,31]. For most epidemic diseases models, the traveling wave solutions can describe the phase transmission from a disease-free state to an infective state. The existence and non-existence of the traveling wave solutions can predict whether or not the epidemic disease transmits in the population and how fast it spreads geographically [2, 3, 10,11,12,13,14, 16,17,18, 25]. Recently, Xu [26] considered the following delayed diffusive SIR model:

$$ \textstyle\begin{cases} \frac{\partial }{\partial t}S(t,x)=d_{1}\frac{\partial ^{2}S(t,x)}{ \partial x^{2}}- \frac{\beta S(t,x)I(t-\tau ,x)}{1+\alpha I(t-\tau ,x)}, \\ \frac{\partial }{\partial t}I(t,x)=d_{2}\frac{\partial ^{2}I(t,x)}{ \partial x^{2}}+ \frac{\beta S(t,x)I(t-\tau ,x)}{1+\alpha I(t-\tau ,x)}-\gamma I(t,x), \\ \frac{\partial }{\partial t}R(t,x)=d_{3}\frac{\partial ^{2}R(t,x)}{ \partial x^{2}}+\gamma I(t,x), \end{cases} $$

(1.1)

where $S(t,x)$, $I(t,x)$, and $R(t,x)$ denote the densities of the susceptible, infected, and recovered individuals at time t and location x, respectively. The constants $d_{i}>0$ ($i=1,2,3$) are the diffusion rates, $\tau >0$ is the incubation period, $\beta >0$ is the transmission coefficient, and $\gamma >0$ represents the recovery rate. The nonlinear incidence $\frac{\beta S I}{1+\alpha I}$ ($\alpha >0$) is called a saturated incidence [5, 15, 30]. Since the third equation in (1.1) is decoupled with the first two, the author studied the subsystem of (1.1) for S and I. In [26] he proved that if $\mathcal{R}_{0}=\beta S_{0}/\gamma >1$ and $c>c^{*}$ ($c^{*}$ is the critical wave speed), then the subsystem of (1.1) admits a traveling wave solution $(S(x+ct),I(x+ct))$ satisfying the wave system

$$ \textstyle\begin{cases} d_{1} S''(\xi )-cS'(\xi )-\frac{\beta S(\xi )I(\xi -c\tau )}{1+\alpha I(\xi -c\tau )}=0, \\ d_{2} I''(\xi )-cI'(\xi )+\frac{\beta S(\xi )I(\xi -c\tau )}{1+\alpha I(\xi -c\tau )}-\gamma I(\xi )=0, \\ S(-\infty )=S_{0}, \qquad S(\infty )\in [0,S_{0}),\qquad I(\pm \infty )=0, \end{cases} $$

(1.2)

where $\xi =x+ct$ is the moving coordinate, $c\in \mathbb{R}$ is the wave speed, and $S_{0}>0$ is a given constant. On the other hand, he showed that if $\mathcal{R}_{0}<1$ and $c\geq 0$ or $\mathcal{R}_{0}>1$ and $c\in (0,c^{*})$, then the subsystem of (1.1) has no nontrivial and nonnegative traveling wave solutions.

Observing his results in [26], one can find that there exist some open problems listed as follows:

• (P1) Does the traveling wave solution of (1.1) exist if (i) $\mathcal{R}_{0}>1$ and $c=c^{*}$; (ii) $\mathcal{R}_{0}=1$ and $c\in \mathbb{R}$; (iii) $\mathcal{R}_{0}>1$ and $c\leq 0$?

• (P2) How does the R-component in (1.1) change?

As was discussed in [19], traveling wave solutions with the critical speed play an important role in the research of epidemic spread. However, it is challenging to investigate the existence of critical traveling wave solutions. There has been some work on the existence of critical traveling wave solutions for diffusive epidemic systems [2, 7, 14, 19, 23, 25, 27, 28, 30]. In this paper, we solve problems (P1) and (P2). For our purpose, we need the following lemma which is established in Lemma 3.1 of [26].

Lemma 1.1

Assume that $\mathcal{R}_{0} =\beta S_{0} /\gamma >1$, and let

$$ \Delta (\lambda ,c):=d_{2}\lambda ^{2}-c \lambda -\gamma +\beta S_{0}e ^{-\lambda c \tau }. $$

(1.3)

Then there exist $c^{*}>0$and $\lambda ^{*}>0$such that

$$ \Delta \bigl(\lambda ^{*},c^{*} \bigr)=d_{2}\bigl(\lambda ^{*}\bigr)^{2}-c^{*} \lambda ^{*}- \gamma +\beta S_{0}e^{-\lambda ^{*} c^{*} \tau }=0 $$

(1.4)

and

$$ \frac{\partial \Delta (\lambda , c)}{\partial \lambda }\bigg|_{( \lambda ^{*},c^{*})}=2d_{2} \lambda ^{*}-c^{*}-\beta S_{0}c^{*} \tau e ^{-\lambda ^{*} c^{*}\tau } =0. $$

(1.5)

Proof

Due to $\mathcal{R}_{0}>1$, for every $c>0$, at $\lambda =0$, it is obvious to get that

$$ \begin{aligned} &\Delta (0, c)=\beta S_{0}-\gamma >0 \\ &\frac{\partial \Delta (\lambda , c)}{\partial \lambda }\bigg|_{ \lambda =0}=2d_{2}\lambda -c-\beta S_{0}c\tau e^{-\lambda c\tau } |_{\lambda =0}< 0, \\ &\frac{\partial ^{2} \Delta (\lambda , c)}{\partial \lambda ^{2}} \bigg|_{\lambda =0}=2d_{2}+\beta S_{0} (c\tau )^{2} e^{-\lambda c \tau }|_{\lambda =0}>0, \\ &\Delta (+\infty ,c)=+\infty . \end{aligned} $$

For every $\lambda >0$, we can get that

$$ \begin{aligned} &\Delta (\lambda ,0)=d_{2}\lambda ^{2}-\gamma +\beta S_{0}>0, \\ &\frac{\partial \Delta (\lambda , c)}{\partial c}=-\lambda -\lambda \beta S_{0}\lambda \tau e^{-\lambda c\tau }< 0, \quad c>0, \\ &\Delta (\lambda ,+\infty )=-\infty . \end{aligned} $$

Making full use of the above computations and the rough graphs of $\Delta (\lambda ,c)$, we obtain the desired results of this lemma. □

Now, we state our strategies and organization as follows. In Sect. 2, we state our results and some remarks. In Sect. 3, we construct a pair of upper and lower solutions of the wave system and apply Schauder’s fixed point theorem to derive the existence of a critical traveling wave solution for (1.1). In addition, employing the subtle analysis and a limiting approach, we obtain the asymptotic boundary conditions of the traveling wave solution and its other properties. In Sect. 4, by contradictory arguments, we establish the non-existence of the traveling wave solutions for the cases $\mathcal{R}_{0}=1$ and $c\in \mathbb{R}$ or $\mathcal{R}_{0}>1$ and $c\leq 0$. In Sect. 5, we make a brief conclusion.

2 Main results

Now we introduce the definition concerning critical traveling wave solutions of (1.1).

Definition 1

A critical traveling solution of (1.1) is a special solution in the form of $(S(\xi ),I(\xi ),R(\xi ))=(S(x+c^{*}t),I(x+c^{*}t),R(x+c ^{*}t))$, where $\xi :=x+c^{*}t$ and $c^{*}$ is the critical wave speed (see Lemma 1.1). Meanwhile, $(S(\xi ),I(\xi ),R(\xi ))\in C^{2}( \mathbb{R},\mathbb{R}^{3})$ is the wave profile that propagates in the one-dimension spatial domain at the constant critical wave speed and connects the initial disease-free equilibrium $(S(-\infty ),I(-\infty ),R(-\infty ))$ to the final disease-free equilibrium $(S(\infty ),I( \infty ),R(\infty ))$.

Next, we mainly consider the following critical wave system:

$$ \textstyle\begin{cases} d_{1} S''(\xi )-c^{*}S'(\xi )-\frac{\beta S(\xi )I(\xi -c^{*}\tau )}{1+ \alpha I(\xi -c^{*}\tau )}=0, \\ d_{2} I''(\xi )-c^{*}I'(\xi )+\frac{\beta S(\xi )I(\xi -c^{*}\tau )}{1+ \alpha I(\xi -c^{*}\tau )}-\gamma I(\xi )=0, \\ d_{3} R''(\xi )-c^{*}R'(\xi )+\gamma I(\xi )=0. \end{cases} $$

(2.1)

For all $\xi \in \mathbb{R}$, we will show the existence and non-existence of the critical traveling wave solution $(S(\xi ),I( \xi ),R(\xi ))$ of (1.1) satisfying the asymptotic boundary conditions

$$ (S,I,R) (-\infty )=(S_{0},0,0),\qquad (S,I,R) ( \infty )=(\varepsilon_{0} ,0,S _{0}-\varepsilon_{0} ), $$

(2.2)

where $\varepsilon_{0}$ is some constant and $0\leq \varepsilon_{0} < S_{0}$. Now we state our results.

Theorem 2.1

If $\mathcal{R}_{0}>1$and $c=c^{*}$, then system (1.1) admits a critical traveling wave solution $(S(\xi ),I(\xi ),R(\xi ))$satisfying (2.2). Moreover,

• (1) $S(\xi )>0$, $I(\xi )>0$, and $R(\xi )>0$on $\mathbb{R}$;

• (2) $S(-\infty )=S_{0}$, $I(-\infty )=0$, $R(-\infty )=0$, and $I(\xi )=O(-\xi e^{\lambda ^{*}\xi })$as $\xi \rightarrow -\infty$;

• (3) $S(\xi )$is strictly decreasing and $R(\xi )$is strictly increasing on $\mathbb{R}$; $S(\infty )=\varepsilon_{0} < S_{0}$, $I(\infty )=0$and $R(\infty )=S_{0}-\varepsilon_{0}$; $S'(\xi )$, $I'(\xi )$, $R'(\xi )$, $S''(\xi )$, $I''(\xi )$, $R''(\xi )\rightarrow 0$as $\xi \rightarrow \pm \infty$; $\gamma \int _{\mathbb{R}}I( \xi )\,d\xi =\beta \int _{\mathbb{R}}\frac{S(\xi )I(\xi -c^{*}\tau )}{1+ \alpha I(\xi -c^{*}\tau )}\,d\xi =c^{*}(S_{0}-\varepsilon_{0} )$;

• (4) $S(\xi )< S_{0}$, $I(\xi )<\frac{1}{\alpha } (\frac{ \beta S_{0}}{\gamma }-1 )$, and $R(\xi )<(S_{0}-\varepsilon_{0} )$on $\mathbb{R}$.

Theorem 2.2

Assume that $\mathcal{R}_{0}=1$and $c\in \mathbb{R}$or $\mathcal{R} _{0}>1$and $c\leq 0$, then system (1.1) admits no positive traveling wave solutions $(S(\xi ),I(\xi ),R(\xi ))$satisfying (2.2).

Remark 1

It is necessary to point out that in Theorems 2.1 and 2.2 we have solved open problems (P1) and (P2). Our method adopted here can be used to improve the corresponding results for super-critical traveling wave solutions in [26].

Remark 2

In order to address the change of the number for R-component in (1.1), we study the three equations in (1) together, please refer to our construction of upper-lower solutions. In Theorem 2.1, we proved the existence of the traveling wave solutions, meanwhile we obtained a lot of nice properties of the traveling wave solutions for (1.1).

3 Proof of Theorem 2.1

3.1 Upper and lower solutions

First, we give the definition of upper and lower solutions of (2.1).

Definition 2

The pair of continuous functions $(\bar{S}(\xi ),\bar{I}(\xi ), \bar{R}(\xi ))$ and $(\underline{S}(\xi ),\underline{I}(\xi ), \underline{R}(\xi ))$ is called a pair of upper and lower solutions for (2.1) if they satisfy

$$d_1{\overline{S}}^{″}(\xi )-c^{\ast }{\overline{S}}^{\prime }(\xi )-\frac{\beta \overline{S}(\xi )\underset{\_}{I}(\xi -c^{\ast }\tau )}{1+\alpha \underset{\_}{I}(\xi -c^{\ast }\tau )}\le 0,$$

(3.1)

$$d_2{\overline{I}}^{″}(\xi )-c^{\ast }{\overline{I}}^{\prime }(\xi )+\frac{\beta \overline{S}(\xi )\overline{I}(\xi -c^{\ast }\tau )}{1+\alpha \overline{I}(\xi -c^{\ast }\tau )}-\gamma \overline{I}(\xi )\le 0,$$

(3.2)

$$d_3{\overline{R}}^{″}(\xi )-c^{\ast }{\overline{R}}^{\prime }(\xi )+\gamma \overline{I}(\xi )\le 0,$$

(3.3)

$$d_1{\underset{\_}{S}}^{″}(\xi )-c^{\ast }{\underset{\_}{S}}^{\prime }(\xi )-\frac{\beta \underset{\_}{S}(\xi )\overline{I}(\xi -c^{\ast }\tau )}{1+\alpha \overline{I}(\xi -c^{\ast }\tau )}\ge 0,$$

(3.4)

$$d_2{\underset{\_}{I}}^{″}(\xi )-c^{\ast }{\underset{\_}{I}}^{\prime }(\xi )+\frac{\beta \underset{\_}{S}(\xi )\underset{\_}{I}(\xi -c^{\ast }\tau )}{1+\alpha \underset{\_}{I}(\xi -c^{\ast }\tau )}-\gamma \underset{\_}{I}(\xi )\ge 0,$$

(3.5)

$$d_3{\underset{\_}{R}}^{″}(\xi )-c^{\ast }{\underset{\_}{R}}^{\prime }(\xi )+\gamma \underset{\_}{I}(\xi )\ge 0$$

(3.6)

except for finitely many points of ξ on $\mathbb{R}$.

For $\xi \in \mathbb{R}$, define the following nonnegative continuous functions:

$$\begin{aligned}& \bar{S}(\xi ):=S_{0}, \\& \underline{S}(\xi ):=\textstyle\begin{cases} S_{0}(1-\sigma _{2}^{-1}e^{\sigma _{2}\xi }), \quad \xi < \xi _{2}, \\ 0, \quad \xi \geq \xi _{2}, \end{cases}\displaystyle \\& \bar{I}(\xi ):=\textstyle\begin{cases} -L_{1} \xi e^{\lambda ^{*} \xi }, \quad \xi < \xi _{1}, \\ \frac{1}{\alpha } (\frac{\beta S_{0}}{\gamma }-1 ),\quad \xi \geq \xi _{1}, \end{cases}\displaystyle \\& \underline{I}(\xi ):=\textstyle\begin{cases} -L_{1} \xi e^{\lambda ^{*} \xi }-L_{2}(-\xi )^{\frac{1}{2}}e^{\lambda ^{*} \xi },\quad \xi < \xi _{3}, \\ 0, \quad \xi \geq \xi _{3}, \end{cases}\displaystyle \\& \bar{R}(\xi ):=L_{3}e^{\sigma _{1} \xi },\qquad \underline{R}(\xi ):=0, \end{aligned}$$

where $\lambda ^{*}$ is defined in Lemma 1.1,

$$ \xi _{1}=-\frac{1}{\lambda ^{*}},\qquad \xi _{2}= \frac{\ln \sigma _{2}}{\sigma _{2}},\qquad \xi _{3}=-\frac{L_{2}^{2}}{L_{1}^{2}},\qquad L_{1}=\frac{e\lambda ^{*}}{\alpha } \biggl(\frac{\beta S_{0}}{\gamma }-1 \biggr), $$

and the constants $\sigma _{1}$, $\sigma _{2}$, $L_{2}$, and $L_{3}\in \mathbb{R^{+}}$ are to be determined later. In the next lemma, we will prove that $(\bar{S}(\xi ),\bar{I}(\xi ),\bar{R}(\xi ))$ and $(\underline{S}(\xi ),\underline{I}(\xi ),\underline{R}(\xi ))$ are a pair of upper and lower solutions of (2.1).

Lemma 3.1

The function $\bar{S}(\xi )$satisfies the inequality

$$ d_{1} \bar{S}''(\xi )-c^{*}\bar{S}'(\xi )-\frac{\beta \bar{S}(\xi ) \underline{I}(\xi -c^{*}\tau )}{1+\alpha \underline{I}(\xi -c^{*} \tau )}\leq 0 $$

and the function $\underline{R}(\xi )$satisfies the inequality

$$ d_{3} \underline{R}''(\xi )-c^{*} \underline{R}'(\xi )+\gamma \underline{I}(\xi )\geq 0 $$

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

Proof

By the definitions of $\bar{S}(\xi )$, $\underline{R}(\xi )$, and $\underline{I}(\xi )$ on $\mathbb{R}$, one can get

$$ d_{1} \bar{S}''(\xi )-c^{*}\bar{S}'(\xi )-\frac{\beta \bar{S}(\xi ) \underline{I}(\xi -c^{*}\tau )}{1+\alpha \underline{I}(\xi -c^{*} \tau )}=- \frac{\beta S_{0}\underline{I}(\xi -c^{*}\tau )}{1+\alpha \underline{I}(\xi -c^{*}\tau )}\leq 0 $$

(3.7)

and

$$ d_{3} \underline{R}''( \xi )-c^{*} \underline{R}'(\xi )+\gamma \underline{I}( \xi )=\gamma \underline{I}(\xi )\geq 0, $$

(3.8)

which completes the proof. □

Lemma 3.2

The function $\bar{I}(\xi )$satisfies the inequality

$$ d_{2} \bar{I}''(\xi )-c^{*}\bar{I}'(\xi )+\frac{\beta \bar{S}(\xi ) \bar{I}(\xi -c^{*}\tau )}{1+\alpha \bar{I}(\xi -c^{*}\tau )}-\gamma \bar{I}(\xi )\leq 0 $$

for all $\xi \neq \xi _{1}$.

Proof

When $\xi <\xi _{1}$, $\bar{I}(\xi )=-L_{1}\xi e^{\lambda ^{*}\xi }$, then it follows that

$$\begin{aligned} &d_{2} \bar{I}''(\xi )-c^{*}\bar{I}'(\xi )+\frac{\beta \bar{S}(\xi ) \bar{I}(\xi -c^{*}\tau )}{1+\alpha \bar{I}(\xi -c^{*}\tau )}-\gamma \bar{I}(\xi ) \\ &\quad \leq d_{2} \bar{I}''(\xi )-c^{*}\bar{I}'(\xi )+\beta S_{0}\bar{I} \bigl( \xi -c^{*}\tau \bigr)-\gamma \bar{I}(\xi ) \\ &\quad =-L_{1} e^{\lambda ^{*} \xi } \biggl[\Delta \bigl(\lambda ^{*},c^{*}\bigr)+\frac{ \partial \Delta }{\partial \lambda }\bigl(\lambda ^{*},c^{*}\bigr) \biggr]=0,\quad \xi < \xi _{1}. \end{aligned}$$

When $\xi >\xi _{1}$, $\bar{I}(\xi -c^{*}\tau )<\bar{I}(\xi )=\frac{1}{ \alpha } (\frac{\beta S_{0}}{\gamma }-1 )$, we have that

$$\begin{aligned} &d_{2} \bar{I}''(\xi )-c^{*}\bar{I}'(\xi )+\frac{\beta \bar{S}(\xi ) \bar{I}(\xi -c^{*}\tau )}{1+\alpha \bar{I}(\xi -c^{*}\tau )}-\gamma \bar{I}(\xi ) \\ &\quad =\frac{\beta \bar{S}(\xi )\bar{I}(\xi -c^{*}\tau )}{1+\alpha \bar{I}(\xi -c^{*}\tau )}-\gamma \bar{I}(\xi ) \\ &\quad \leq \frac{\beta {S}_{0}\bar{I}(\xi )}{1+\alpha \bar{I}(\xi )}- \gamma \bar{I}(\xi )=0. \end{aligned}$$

This is the end of the proof. □

Lemma 3.3

The function $\bar{R}(\xi )$satisfies the inequality

$$ d_{3} \bar{R}''(\xi )-c^{*}\bar{R}'(\xi )+\gamma \bar{I}(\xi )\leq 0 $$

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

Proof

Choose a sufficiently large $L_{3}>0$ and a sufficiently small $\sigma _{1} \in (0,\min \{\lambda ^{*},c^{*}/d_{3}\})$ such that

$$ d_{3} \sigma _{1}^{2}-c^{*} \sigma _{1}-\gamma L_{1} {L_{3}}^{-1} \xi e ^{(\lambda ^{*}-\sigma _{1})\xi }< 0, \quad \xi < \xi _{1} $$

(3.9)

and

$$ d_{3} \sigma _{1}^{2}-c^{*} \sigma _{1}+ \frac{\gamma }{\alpha L_{3}} \biggl(\frac{\beta S_{0}}{\gamma }-1 \biggr)e^{-\sigma _{1}\xi }< 0, \quad \xi \geq \xi _{1}. $$

(3.10)

When $\xi <\xi _{1}$, $\overline{I}(\xi )=-L_{1}\xi e^{\lambda ^{*} \xi }$. Then, by using (3.9), we obtain that

$$\begin{aligned} &d_{3} \bar{R}''(\xi )-c^{*}\bar{R}'(\xi )+\gamma \bar{I}(\xi ) \\ &\quad =d_{3} {\sigma _{1}}^{2} L_{3} e^{\sigma _{1} \xi }-c^{*}L_{3}\sigma _{1} e^{\sigma _{1} \xi }+\gamma (-L_{1})\xi e^{\lambda ^{*}\xi } \\ &\quad =L_{3} e^{\sigma _{1} \xi }\bigl[d_{3} {\sigma _{1}}^{2}-c^{*}\sigma _{1}- \gamma L_{1} {L_{3}}^{-1}\xi e^{(\lambda ^{*}-\sigma _{1})\xi }\bigr] \\ &\quad \leq 0,\quad \xi < \xi _{1}. \end{aligned}$$

When $\xi \geq \xi _{1}$, $\bar{I}(\xi )=\frac{1}{\alpha } (\frac{ \beta S_{0}}{\gamma }-1 )$. Using (3.10), we get that

$$\begin{aligned} &d_{3} \bar{R}''(\xi )-c^{*}\bar{R}'(\xi )+\gamma \bar{I}(\xi ) \\ &\quad =d_{3} {\sigma _{1}}^{2} L_{3} e^{\sigma _{1} \xi }-c^{*}L_{3}\sigma _{1} e^{\sigma _{1} \xi }+\frac{\gamma }{\alpha } \biggl( \frac{\beta S _{0}}{\gamma }-1 \biggr) \\ &\quad =L_{3}e^{\sigma _{1} \xi } \biggl[d_{3} {\sigma _{1}}^{2}-c^{*}\sigma _{1}+ \frac{\gamma }{\alpha L_{3}} \biggl(\frac{\beta S_{0}}{\gamma }-1 \biggr)e^{-\sigma _{1}\xi } \biggr]\leq 0, \quad \xi \geq \xi _{1}. \end{aligned}$$

The proof of this lemma is finished. □

Lemma 3.4

The function $\underline{S}(\xi )$satisfies the inequality

$$ d_{1} \underline{S}''(\xi )-c^{*}\underline{S}'(\xi )-\frac{\beta \underline{S}(\xi )\bar{I}(\xi -c^{*}\tau )}{1+\alpha \bar{I}(\xi -c ^{*}\tau )}\geq 0 $$

for all $\xi \neq \xi _{2}$.

Proof

Select $\sigma _{2}$ to be small enough such that $\sigma _{2}\in (0, \min \{\lambda ^{*},c^{*}/d_{1}\})$, $\xi _{2} <\xi _{1}$, and

$$ -d_{1} \sigma _{2}+c^{*}+ \beta {L_{1}}\bigl(\xi -c^{*}\tau \bigr) \bigl(1-{\sigma _{2}} ^{-1}e^{\sigma _{2} \xi }\bigr)e^{(\lambda ^{*}-\sigma _{2})\xi -\lambda ^{*} c ^{*} \tau }\geq 0,\quad \xi < \xi _{2}. $$

(3.11)

If $\xi <\xi _{2}$, then $\underline{S}(\xi ) =S_{0}(1-\sigma _{2}^{-1}e ^{\sigma _{2}\xi })$ and $\bar{I}(\xi )=-L_{1} \xi e^{\lambda ^{*} \xi }$. Utilizing (3.11), we deduce that

$$\begin{aligned} &d_{1} \underline{S}''(\xi )-c^{*}\underline{S}'(\xi )-\frac{\beta \underline{S}(\xi )\bar{I}(\xi -c^{*}\tau )}{1+\alpha \bar{I}(\xi -c ^{*}\tau )} \\ &\quad \geq d_{1} \underline{S}''(\xi )-c^{*}\underline{S}'(\xi )-\beta \underline{S}(\xi ) \bar{I}\bigl(\xi -c^{*}\tau \bigr) \\ &\quad =-d_{1} S_{0}\sigma _{2} e^{\sigma _{2} \xi }+c^{*}S_{0}e^{\sigma _{2} \xi }+L_{1} \beta S_{0}\bigl(1-\sigma _{2}^{-1}e^{\sigma _{2}\xi } \bigr) \bigl(\xi -c ^{*}\tau \bigr)e^{\lambda ^{*}(\xi -c^{*}\tau )} \\ &\quad =S_{0} e^{\sigma _{2} \xi } \bigl[-d_{1} \sigma _{2}+c^{*}+\beta {L _{1}}\bigl(\xi -c^{*}\tau \bigr) \bigl(1-{\sigma _{2}}^{-1}e^{\sigma _{2} \xi } \bigr)e^{( \lambda ^{*}-\sigma _{2})\xi -\lambda ^{*} c^{*} \tau } \bigr] \\ &\quad \geq 0, \quad \xi < \xi _{2}. \end{aligned}$$

If $\xi >\xi _{2}$, then $\underline{S}(\xi )=0$ and

$$ d_{1} \underline{S}''(\xi )-c^{*}\underline{S}'(\xi )-\frac{\beta \underline{S}(\xi )\bar{I}(\xi -c^{*}\tau )}{1+\alpha \bar{I}(\xi -c ^{*}\tau )}=0 $$

holds naturally. □

Lemma 3.5

The function $\underline{I}(\xi )$satisfies the inequality

$$ d_{2} \underline{I}''(\xi )-c^{*}\underline{I}'(\xi )+\frac{\beta \underline{S}(\xi )\underline{I}(\xi -c^{*}\tau )}{1+\alpha \underline{I}(\xi -c^{*}\tau )}- \gamma \underline{I}(\xi )\geq 0 $$

for all $\xi \neq \xi _{3}$.

Proof

Choosing large enough $L_{2}>0$ such that $\xi _{3} < \min \{\xi _{2},-c ^{*}\tau \}$ and recalling $\xi _{2} <\xi _{1}$, then for $\xi <\xi _{3}$ we have that

$$\begin{aligned}& S_{0}\bigl(1-{\sigma _{2}}^{-1}e^{\sigma _{2}\xi } \bigr)< S_{0}, \end{aligned}$$

(3.12)

$$\begin{aligned}& \underline{I}(\xi )=\overline{I}(\xi )-L_{2} (-\xi )^{\frac{1}{2}}e ^{\lambda ^{*}\xi }, \end{aligned}$$

(3.13)

$$\begin{aligned}& 1+c^{*}\tau \xi ^{-1}>0, \end{aligned}$$

(3.14)

and

$$ \frac{1}{16} L_{2}\bigl(c^{*} \tau \bigr)^{2}-\sigma _{2}^{-1}L_{1}(- \xi )^{ \frac{5}{2}}e^{\sigma _{2}\xi } -\alpha L_{1}^{2}(- \xi )^{\frac{7}{2}}e ^{\lambda ^{*}(\xi -c^{*}\tau )}\geq 0. $$

(3.15)

From (3.13), for $\xi <\xi _{3}$, we can get that

$$\begin{aligned} d_{2}\underline{I}''( \xi ) =&d_{2}\overline{I}''(\xi )+d_{2}L_{2} e ^{\lambda ^{*}\xi } \biggl[\lambda ^{*}(-\xi )^{-\frac{1}{2}}-\bigl(\lambda ^{*} \bigr)^{2}(-\xi )^{\frac{1}{2}}+\frac{1}{4}(-\xi )^{-\frac{3}{2}} \biggr] \\ \geq &d_{2}\overline{I}''(\xi )+d_{2}L_{2} e^{\lambda ^{*}\xi } \bigl[ \lambda ^{*}(-\xi )^{-\frac{1}{2}}-\bigl(\lambda ^{*} \bigr)^{2}(-\xi )^{ \frac{1}{2}} \bigr] \end{aligned}$$

(3.16)

and

$$\begin{aligned} -c^{*}\underline{I}'(\xi )-\gamma \underline{I}(\xi ) = &-c^{*} \overline{I}'( \xi )-c^{*}L_{2} \biggl[\frac{1}{2}(-\xi )^{-\frac{1}{2}}- \lambda ^{*}(-\xi )^{\frac{1}{2}} \biggr]e^{\lambda ^{*}\xi } \\ &{}-\gamma \overline{I}(\xi )+\gamma L_{2} (-\xi )^{\frac{1}{2}}e^{ \lambda ^{*}\xi }. \end{aligned}$$

(3.17)

Using the inequality $\frac{x}{1+\alpha x}\geq x(1-\alpha x)$ for $x\geq 0$ and $\alpha >0$, we obtain from (3.12) that

$$\begin{aligned} &\frac{\beta \underline{S}(\xi )\underline{I}(\xi -c^{*}\tau )}{1+ \alpha \underline{I}(\xi -c^{*}\tau )} \\ &\quad \geq \beta \underline{S}(\xi )\underline{I}\bigl(\xi -c^{*} \tau \bigr)\bigl[1- \alpha \underline{I}\bigl(\xi -c^{*}\tau \bigr) \bigr] \\ &\quad \geq \beta \underline{S}(\xi )\underline{I}\bigl(\xi -c^{*} \tau \bigr)-\alpha \beta \underline{S}(\xi ) \bigl(\bar{I}\bigl(\xi -c^{*}\tau \bigr)\bigr)^{2} \\ &\quad \geq \beta S_{0}\bigl(1-\sigma _{2}^{-1}e^{\sigma _{2}\xi } \bigr)\bigl[\bar{I}\bigl( \xi -c^{*}\tau \bigr)-L_{2} \bigl(-\xi +c^{*}\tau \bigr)^{\frac{1}{2}}e^{\lambda ^{*}( \xi -c^{*}\tau )}\bigr] \\ & \qquad {} -\alpha \beta S_{0}L_{1}^{2} \xi ^{2}e^{2\lambda ^{*}(\xi -c^{*} \tau )} \\ &\quad \geq \beta S_{0}\bigl[\bar{I}\bigl(\xi -c^{*} \tau \bigr)-L_{2}\bigl(-\xi +c^{*}\tau \bigr)^{ \frac{1}{2}}e^{\lambda ^{*}(\xi -c^{*}\tau )} \\ &\qquad {} +\sigma _{2}^{-1}L_{1}\bigl(\xi -c^{*}\tau \bigr)e^{\sigma _{2}\xi + \lambda ^{*}(\xi -c^{*}\tau )}-\alpha L_{1}^{2} \xi ^{2}e^{2\lambda ^{*}( \xi -c^{*}\tau )}\bigr] \end{aligned}$$

(3.18)

for $\xi <\xi _{3}$. By Taylor’s formula, for $\xi <\xi _{3}$, we have that

$$ \bigl(-\xi +c^{*}\tau \bigr)^{\frac{1}{2}}\leq (-\xi )^{\frac{1}{2}}+ \frac{1}{2}(-\xi )^{-\frac{1}{2}}c^{*} \tau -\frac{1}{8}(-\xi )^{- \frac{3}{2}}\bigl(c^{*}\tau \bigr)^{2}+\frac{1}{16}(-\xi )^{-\frac{5}{2}} \bigl(c^{*} \tau \bigr)^{3}. $$

(3.19)

From (3.12)–(3.19), (1.4), and (1.5), we deduce that

$$\begin{aligned} & d_{2} \underline{I}''(\xi )-c^{*}\underline{I}'(\xi )+\frac{\beta \underline{S}(\xi )\underline{I}(\xi -c^{*}\tau )}{1+\alpha \underline{I}(\xi -c^{*}\tau )}- \gamma \underline{I}(\xi ) \\ &\quad \geq d_{2}\overline{I}''(\xi )+d_{2}L_{2} e^{\lambda ^{*}\xi }\bigl[ \lambda ^{*}(-\xi )^{-\frac{1}{2}}-\bigl(\lambda ^{*} \bigr)^{2}(-\xi )^{ \frac{1}{2}}\bigr] \\ &\qquad {} -c^{*}\overline{I}'(\xi )-c^{*}L_{2} \biggl[\frac{1}{2}(- \xi )^{-\frac{1}{2}}-\lambda ^{*}(-\xi )^{\frac{1}{2}} \biggr]e^{\lambda ^{*}\xi } \\ & \qquad {} -\gamma \overline{I}(\xi )+\gamma L_{2} (-\xi )^{\frac{1}{2}}e ^{\lambda ^{*}\xi }+\beta S_{0}\bigl[\bar{I}\bigl( \xi -c^{*}\tau \bigr)-L_{2}\bigl(-\xi +c ^{*} \tau \bigr)^{\frac{1}{2}}e^{\lambda ^{*}(\xi -c^{*}\tau )} \\ &\qquad {} +\sigma _{2}^{-1}L_{1}\bigl(\xi -c^{*}\tau \bigr)e^{\sigma _{2}\xi + \lambda ^{*}(\xi -c^{*}\tau )}-\alpha L_{1}^{2} \xi ^{2}e^{2\lambda ^{*}( \xi -c^{*}\tau )} \bigr] \\ &\quad \geq -L_{1} e^{\lambda ^{*}\xi }\bigl[\xi \bigl(d_{2}\bigl(\lambda ^{*}\bigr)^{2}-c^{*} \lambda ^{*}-\gamma +\beta S_{0}e^{-\lambda ^{*}c^{*}\tau } \bigr)+\bigl(2d_{2} \lambda ^{*}-c^{*}-\beta S_{0} c^{*}\tau e^{-\lambda ^{*}c^{*}\tau }\bigr)\bigr] \\ &\qquad {} +L_{1}\beta S_{0}\bigl(\xi -c^{*}\tau \bigr)e^{\lambda ^{*}(\xi -c^{*} \tau )}-L_{2} e^{\lambda ^{*}\xi }(-\xi )^{\frac{1}{2}}\bigl[d_{2}\bigl(\lambda ^{*}\bigr)^{2}-c^{*}\lambda ^{*}- \gamma +\beta S_{0}e^{-\lambda ^{*}c^{*} \tau }\bigr] \\ &\qquad {} +L_{2} \beta S_{0}(-\xi )^{\frac{1}{2}}e^{\lambda ^{*}(\xi -c ^{*}\tau )}+ \frac{1}{2}L_{2} e^{\lambda ^{*}\xi }(-\xi )^{-\frac{1}{2}} \bigl[2d_{2}\lambda ^{*}-c^{*}-\beta S_{0} c^{*}\tau e^{-\lambda ^{*}c ^{*}\tau } \bigr] \\ &\qquad {} +\frac{1}{2}L_{2} \beta S_{0}c^{*} \tau (-\xi )^{-\frac{1}{2}}e ^{\lambda ^{*}(\xi -c^{*}\tau )}+\beta S_{0} \bigl[ \bar{I}\bigl(\xi -c^{*} \tau \bigr)-L_{2}\bigl(-\xi +c^{*}\tau \bigr)^{\frac{1}{2}}e^{\lambda ^{*}(\xi -c^{*} \tau )} \\ &\qquad {} +\sigma _{2}^{-1}L_{1}\bigl(\xi -c^{*}\tau \bigr)e^{\sigma _{2}\xi + \lambda ^{*}(\xi -c^{*}\tau )}-\alpha L_{1}^{2} \xi ^{2}e^{2\lambda ^{*}( \xi -c^{*}\tau )} \bigr] \\ &\quad \geq -L_{1} e^{\lambda ^{*}\xi } \biggl[\Delta \bigl(\lambda ^{*},c^{*}\bigr)+\frac{ \partial \Delta }{\partial \lambda }\bigl(\lambda ^{*},c^{*}\bigr) \biggr]-L_{2} e ^{\lambda ^{*}\xi }(-\xi )^{\frac{1}{2}}\bigl[\Delta \bigl(\lambda ^{*},c^{*}\bigr)\bigr] \\ &\qquad {} +\frac{1}{2}L_{2} e^{\lambda ^{*}\xi }(-\xi )^{\frac{1}{2}} \biggl[\frac{\partial \Delta }{\partial \lambda }\bigl(\lambda ^{*},c^{*} \bigr) \biggr] \\ &\qquad {}+\beta S_{0} \biggl\{ L_{2}e^{\lambda ^{*}(\xi -c^{*}\tau )} \biggl[ \frac{1}{8}(-\xi )^{-\frac{3}{2}}\bigl(c^{*}\tau \bigr)^{2}-\frac{1}{16}(- \xi )^{-\frac{5}{2}} \bigl(c^{*}\tau \bigr)^{3} \biggr] \\ & \qquad {} +\sigma _{2}^{-1}L_{1}\xi e^{\sigma _{2}\xi +\lambda ^{*}(\xi -c ^{*}\tau )}-\alpha L_{1}^{2}\xi ^{2}e^{2\lambda ^{*}(\xi -c^{*}\tau )} \biggr\} \\ &\quad =\beta S_{0}(-\xi )^{-\frac{3}{2}}e^{\lambda ^{*}(\xi -c^{*}\tau )} \biggl[\frac{1}{16} L_{2}\bigl(c^{*}\tau \bigr)^{2}-\sigma _{2}^{-1}L_{1}(- \xi )^{ \frac{5}{2}}e^{\sigma _{2}\xi } -\alpha L_{1}^{2}(- \xi )^{\frac{7}{2}}e ^{\lambda ^{*}(\xi -c^{*}\tau )} \biggr] \\ &\qquad {} +\frac{1}{16}\beta S_{0}L_{2} \bigl(c^{*}\tau \bigr)^{2}(-\xi )^{- \frac{3}{2}}e^{\lambda ^{*}(\xi -c^{*}\tau )} \bigl(1+c^{*}\tau \xi ^{-1}\bigr) \\ &\quad \geq 0. \end{aligned}$$

If $\xi >\xi _{3}$, then $\underline{I}(\xi )=0$, and the inequality

$$ d_{2} \underline{I}''(\xi )-c^{*}\underline{I}'(\xi )+\frac{\beta \underline{S}(\xi )\underline{I}(\xi -c^{*}\tau )}{1+\alpha \underline{I}(\xi -c^{*}\tau )}- \gamma \underline{I}(\xi )=\frac{ \beta \underline{S}(\xi )\underline{I}(\xi -c^{*}\tau )}{1+\alpha \underline{I}(\xi -c^{*}\tau )}\geq 0 $$

holds trivially. The proof of this lemma is finished. □

3.2 Application of Schauder’s fixed point theorem

Introduce a functional space

$$ B_{\mu }\bigl(\mathbb{R},\mathbb{R}^{3}\bigr):=\Bigl\{ \varphi (\xi )=\bigl(\varphi _{1}( \xi ),\varphi _{2}(\xi ),\varphi _{3}(\xi )\bigr)\in C\bigl(\mathbb{R},\mathbb{R} ^{3}\bigr):\sup_{\xi \in \mathbb{R}} \bigl\vert \varphi _{i}(\xi ) \bigr\vert e^{-\mu \vert \xi \vert }< \infty ,i=1,2,3\Bigr\} $$

equipped with the norm $| \varphi | _{\mu }:=\max \{ \sup_{\xi \in \mathbb{R}}| \varphi _{i}(\xi )| e^{-\mu | \xi | },i=1,2,3\}$, where $\mu \in (\sigma _{1},\mu _{0})$ is a constant and $\mu _{0}$ is also a constant that will be specified later. Define a cone by

$$ \mathcal{S} := \left\{ \bigl(S(\xi ),I(\xi ),R(\xi )\bigr)\in B_{\mu }\bigl( \mathbb{R},\mathbb{R}^{3}\bigr)\left|\vphantom{\textstyle\begin{array}{l} \underline{S}(\xi )\leq S(\xi )\leq \bar{S}(\xi ), \\ \underline{I}(\xi )\leq I(\xi )\leq \bar{I}(\xi ), \\ \underline{R}(\xi )\leq R(\xi )\leq \bar{R}(\xi ) \end{array}\displaystyle }\right. \textstyle\begin{array}{l} \underline{S}(\xi )\leq S(\xi )\leq \bar{S}(\xi ), \\ \underline{I}(\xi )\leq I(\xi )\leq \bar{I}(\xi ), \\ \underline{R}(\xi )\leq R(\xi )\leq \bar{R}(\xi ) \end{array}\displaystyle \right\} . $$

It is easy to see that $\mathcal{S}$ is nonempty, bounded, closed, and convex in $B_{\mu }(\mathbb{R},\mathbb{R}^{3})$. Choosing a constant m to be satisfied $m > \max \{\alpha ^{-1}\beta ,\beta \}$ and noting that $\beta >\gamma$, one can get that:

$$ H_{1}[S,I,R](\xi ):=m S(\xi )-\frac{\beta S(\xi )I(\xi -c^{*}\tau )}{1+ \alpha I(\xi -c^{*}\tau )} $$

is increasing with respect to S and decreasing with respect to I;

$$ H_{2}[S,I,R](\xi ):=\frac{\beta S(\xi )I(\xi -c^{*}\tau )}{1+\alpha I( \xi -c^{*}\tau )}+(m-\gamma )I(z) $$

is increasing in both S and I;

$$ H_{3}[S,I,R](\xi ):=m R(\xi )+\gamma I(\xi ) $$

is increasing in both I and R. For any $(S,I,R)\in \mathcal{S}$, define a nonlinear operator $\mathcal{M}:=(\mathcal{M}_{1}, \mathcal{M}_{2},\mathcal{M}_{3})$ on the space $B_{\mu }(\mathbb{R}, \mathbb{R}^{3})$ by

$$ \mathcal{M}_{i}[S,I,R](\xi ):=\frac{1}{\varLambda _{i}} \biggl\{ \int _{- \infty }^{\xi }e^{r_{i1}(\xi -\eta )}H_{i}[S,I,R]( \eta )\,d\eta + \int _{\xi }^{\infty }e^{r_{i2}(\xi -\eta )}H_{i}[S,I,R]( \eta )\,d\eta \biggr\} , $$

where

$$ r_{i1}=\frac{c^{*}-\sqrt{(c^{*})^{2}+4md_{i}}}{2d_{i}},\qquad r_{i2}= \frac{c ^{*}+\sqrt{(c^{*})^{2}+4md_{i}}}{2d_{i}}, \qquad \varLambda _{i}=d_{i}(r _{i2}-r_{i1}) $$

for $i=1,2,3$. Note that any fixed point of $\mathcal{M}$ is a solution of (2.1).

Lemma 3.6

$\mathcal{M}(\mathcal{S})\subset \mathcal{S}$.

Proof

Clearly, $(\mathcal{M}_{1}[S,I,R](\xi ),\mathcal{M}_{2}[S,I,R](\xi ), \mathcal{M}_{3}[S,I,R](\xi ))\in B_{\mu }(\mathbb{R},\mathbb{R}^{3})$ for any $(S,I,R)\in \mathcal{S}$. Then, by the monotonicity of $H_{i}$ ($i=1,2,3$), we need to prove that

$$\underset{\_}{S}(\xi )\le {\mathcal{M}}_1[\underset{\_}{S},\overline{I},R](\xi )\le {\mathcal{M}}_1[S,I,R](\xi )\le {\mathcal{M}}_1[\overline{S},\underset{\_}{I},R](\xi )\le \overline{S}(\xi ),$$

(3.20)

$$\underset{\_}{I}(\xi )\le {\mathcal{M}}_2[\underset{\_}{S},\underset{\_}{I},R](\xi )\le {\mathcal{M}}_2[S,I,R](\xi )\le {\mathcal{M}}_2[\overline{S},\overline{I},R](\xi )\le \overline{I}(\xi ),$$

(3.21)

$$\underset{\_}{R}(\xi )\le {\mathcal{M}}_3[S,\underset{\_}{I},\underset{\_}{R}](\xi )\le {\mathcal{M}}_3[S,I,R](\xi )\le {\mathcal{M}}_3[S,\overline{I},\overline{R}](\xi )\le \overline{R}(\xi )$$

(3.22)

for any $(S,I,R)\in \mathcal{S}$.

Proof of (3.20). Using (3.1) and $\bar{S}(\xi )=S_{0}$, we derive that

$$\begin{aligned} &\mathcal{M}_{1}[\bar{S},\underline{I},R](\xi ) \\ &\quad =\frac{1}{\varLambda _{1}} \biggl\{ \int _{-\infty }^{\xi }e^{r_{11}( \xi -\eta )}H_{1}[ \bar{S},\underline{I},R](\eta )\,d\eta + \int _{\xi } ^{\infty }e^{r_{12}(\xi -\eta )}H_{1}[ \bar{S},\underline{I},R](\eta )\,d\eta \biggr\} \\ &\quad \leq \frac{1}{\varLambda _{1}} \biggl\{ \int _{-\infty }^{\xi }e^{r_{11}( \xi -\eta )}\bigl[m \bar{S}(\eta )+c^{*}\bar{S}'(\eta )-d_{1} \bar{S}''( \eta )\bigr]\,d\eta \\ &\qquad {} + \int _{\xi }^{\infty }e^{r_{12}(\xi -\eta )}\bigl[m \bar{S}( \eta )+c ^{*}\bar{S}'(\eta )-d_{1} \bar{S}''(\eta )\bigr]\,d\eta \biggr\} \\ &\quad = \frac{m S_{0}}{\varLambda _{1}} \biggl[ \int _{-\infty }^{\xi }e^{r_{11}( \xi -\eta )}\,d\eta + \int _{\xi }^{\infty }e^{r_{12}(\xi -\eta )}\,d\eta \biggr] \\ &\quad =S_{0}, \quad \xi \in \mathbb{R}. \end{aligned}$$

It follows from (3.4) that

$$\begin{aligned} &\mathcal{M}_{1}[\underline{S},\bar{I},R](\xi ) \\ &\quad =\frac{1}{\varLambda _{1}} \biggl\{ \int _{-\infty }^{\xi }e^{r_{11}( \xi -\eta )}H_{1}[ \underline{S},\bar{I},R](\eta )\,d\eta + \int _{\xi } ^{\infty }e^{r_{12}(\xi -\eta )}H_{1}[ \underline{S},\bar{I},R](\eta )\,d\eta \biggr\} \\ &\quad \geq \frac{1}{\varLambda _{1}} \biggl\{ \int _{-\infty }^{\xi }e^{r_{11}( \xi -\eta )}\bigl[m \underline{S}(\eta )+c^{*}\underline{S}'(\eta )-d_{1} \underline{S}''(\eta )\bigr]\,d\eta \\ & \qquad {} + \int _{\xi }^{\infty }e^{r_{12}(\xi -\eta )}\bigl[m \underline{S}( \eta )+c^{*}\underline{S}'(\eta )-d_{1} \underline{S}-''(\eta )\bigr]\,d\eta \biggr\} \\ &\quad = \underline{S}(\xi )+\frac{e^{r_{11}(\xi -\xi _{2})}[\underline{S}'( \xi _{2}+0)-\underline{S}'(\xi _{2}-0)]}{r_{12}-r_{11}} \\ &\quad \geq \underline{S}(\xi ), \quad \xi \neq \xi _{2}. \end{aligned}$$

By the continuity of both $\mathcal{M}_{1}[\underline{S},\bar{I},R]( \xi )$ and $\underline{S}(\xi )$ on $\mathbb{R}$, we get that

$$ \mathcal{M}_{1}[\underline{S},\bar{I},R](\xi )\geq \underline{S}( \xi ), \quad \xi \in \mathbb{R}. $$

Proof of (3.21). From (3.2) and (3.5), we get that

$$\begin{aligned} &\mathcal{M}_{2}[\bar{S},\bar{I},R](\xi ) \\ &\quad =\frac{1}{\varLambda _{2}} \biggl\{ \int _{-\infty }^{\xi }e^{r_{21}( \xi -\eta )}H_{2}[ \bar{S},\bar{I},R](\eta )\,d\eta + \int _{\xi }^{\infty }e^{r_{22}(\xi -\eta )}H_{2}[ \bar{S},\bar{I},R](\eta )\,d\eta \biggr\} \\ &\quad \leq \frac{1}{\varLambda _{2}} \biggl\{ \int _{-\infty }^{\xi }e^{r_{21}( \xi -\eta )}\bigl[m \bar{I}(\eta )+c^{*}\bar{I}'(\eta )-d_{2} \bar{I}''( \eta )\bigr]\,d\eta \\ &\qquad {} + \int _{\xi }^{\infty }e^{r_{22}(\xi -\eta )}\bigl[m \bar{I}( \eta )+c ^{*}\bar{I}'(\eta )-d_{2} \bar{I}''(\eta )\bigr]\,d\eta \biggr\} \\ &\quad = \bar{I}(\xi )+\frac{e^{r_{21}(\xi -\xi _{1})}[\bar{I}'(\xi _{1}+0)- \bar{I}'(\xi _{1}-0)]}{r_{22}-r_{21}} \\ &\quad \leq \bar{I}(\xi ), \quad \xi \neq \xi _{1}, \end{aligned}$$

and

$$\begin{aligned} &\mathcal{M}_{2}[\underline{S},\underline{I},R](\xi ) \\ &\quad =\frac{1}{\varLambda _{2}} \biggl\{ \int _{-\infty }^{\xi }e^{r_{21}( \xi -\eta )}H_{2}[ \underline{S},\underline{I},R](\eta )\,d\eta + \int _{\xi }^{\infty }e^{r_{22}(\xi -\eta )}H_{2}[ \underline{S}, \underline{I},R](\eta )\,d\eta \biggr\} \\ &\quad \geq \frac{1}{\varLambda _{2}} \biggl\{ \int _{-\infty }^{\xi }e^{r_{21}( \xi -\eta )}\bigl[m \underline{I}(\eta )+c^{*}\underline{I}'(\eta )-d_{2} \underline{I}''(\eta )\bigr]\,d\eta \\ & \qquad {} + \int _{\xi }^{\xi _{3}}e^{r_{22}(\xi -\eta )}\bigl[m \underline{I}( \eta )+c^{*}\underline{I}'(\eta )-d_{2}\underline{I}''(\eta )\bigr]\,d\eta \biggr\} \\ &\quad =\bar{I}(\xi )+\frac{e^{r_{21}(\xi -\xi _{3})}[\bar{I}'(\xi _{3}+0)- \bar{I}'(\xi _{3}-0)]}{r_{22}-r_{21}} \\ &\quad \geq \underline{I}(\xi ), \quad \xi \neq \xi _{3}. \end{aligned}$$

Using the continuity of both $\mathcal{M}_{2}[\bar{S},\bar{I},R]( \xi )$, $\mathcal{M}_{2}[\underline{S},\underline{I},R](\xi )$, $\bar{I}( \xi )$ and $\underline{I}(\xi )$ on $\mathbb{R}$, we obtain

$$ \mathcal{M}_{2}[\bar{S},\bar{I},R](\xi )\leq \bar{I}(\xi ),\qquad \mathcal{M}_{2}[\underline{S},\underline{I},R](\xi )\geq \underline{I}(\xi ), \quad \xi \in \mathbb{R}. $$

Proof of (3.22). From (3.3), (3.6), and the expressions of $\bar{R}(\xi )$ and $\underline{R}(\xi )$, we deduce that

$$\begin{aligned} &\mathcal{M}_{3}[S,\bar{I},\bar{R}](\xi ) \\ &\quad =\frac{1}{\varLambda _{3}} \biggl\{ \int _{-\infty }^{\xi }e^{r_{31}( \xi -\eta )}H_{3}[S, \bar{I},\bar{R}](\eta )\,d\eta + \int _{\xi }^{\infty }e^{r_{32}(\xi -\eta )}H_{3}[S, \bar{I},\bar{R}](\eta )\,d\eta \biggr\} \\ &\quad \leq \frac{1}{\varLambda _{3}} \biggl\{ \int _{-\infty }^{\xi }e^{r_{31}( \xi -\eta )}\bigl[m \bar{R}(\eta )+c^{*}\bar{R}'(\eta )-d_{3} \bar{R}''( \eta )\bigr]\,d\eta \\ & \qquad {} + \int _{\xi }^{\infty }e^{r_{32}(\xi -\eta )}\bigl[m \bar{R}( \eta )+c ^{*}\bar{R}'(\eta )-d_{3} \bar{R}''(\eta )\bigr]\,d\eta \biggr\} \\ &\quad = \bar{R}(\xi ),\quad \xi \in \mathbb{R}, \end{aligned}$$

and

$$\begin{aligned} &\mathcal{M}_{3}[S,\underline{I},\underline{R}](\xi ) \\ &\quad =\frac{1}{\varLambda _{3}} \biggl\{ \int _{-\infty }^{\xi }e^{r_{31}( \xi -\eta )}H_{3}[S, \underline{I},\underline{R}](\eta )\,d\eta + \int _{\xi }^{\infty }e^{r_{32}(\xi -\eta )}H_{3}[S, \underline{I}, \underline{R}](\eta )\,d\eta \biggr\} \\ &\quad \geq \frac{1}{\varLambda _{3}} \biggl\{ \int _{-\infty }^{\xi }e^{r_{31}( \xi -\eta )}\bigl[m \underline{R}(\eta )+c^{*}\underline{R}'(\eta )-d_{3} \underline{R}''(\eta )\bigr]\,d\eta \\ & \qquad {} + \int _{\xi }^{\infty }e^{r_{32}(\xi -\eta )}\bigl[m \underline{R}( \eta )+c^{*}\underline{R}'(\eta )-d_{3}\underline{R}''(\eta )\bigr]\,d\eta \biggr\} \\ &\quad = \underline{R}(\xi ), \quad \xi \in \mathbb{R}. \end{aligned}$$

The proof of this lemma is finished. □

Lemma 3.7

The operator $\mathcal{M}:=(\mathcal{M}_{1},\mathcal{M}_{2}, \mathcal{M}_{3})$is completely continuous with respect to the norm $|\cdot |_{\mu }$in $B_{\mu }(\mathbb{R},\mathbb{R}^{3})$.

Proof

First, we show that $\mathcal{M}$ is continuous with respect to the norm $|\cdot |_{\mu }$ in $B_{\mu }(\mathbb{R},\mathbb{R}^{3})$. For any $\varPsi _{1}=(S_{1},I_{1},R_{1})\in \mathcal{S}$ and $\varPsi _{2}=(S_{2},I _{2},R_{2})\in \mathcal{S}$, we derive that

$$\begin{aligned} & \bigl\vert H_{1}(S_{1},I_{1},R_{1}) (\xi )-H_{1}(S_{2},I_{2},R_{2}) (\xi ) \bigr\vert e^{- \mu \vert \xi \vert } \\ &\quad \leq \biggl(m+\frac{\beta }{\alpha } \biggr) \bigl\vert S_{1}(\xi )-S_{2}(\xi ) \bigr\vert e ^{-\mu \vert \xi \vert }+\beta S_{0} \bigl\vert I_{1}\bigl( \xi -c^{*}\tau \bigr)-I_{2}\bigl(\xi -c^{*} \tau \bigr) \bigr\vert e^{-\mu \vert \xi \vert } \\ &\quad \leq \biggl(m+\frac{\beta }{\alpha } \biggr) \vert S_{1}-S_{2} \vert _{\mu }+ \beta S_{0} e^{\mu c^{*}\tau } \vert I_{1}-I_{2} \vert _{\mu } \\ &\quad \leq l \vert \varPsi _{1}-\varPsi _{2} \vert _{\mu }, \\ & \bigl\vert H_{2}(S_{1},I_{1},R_{1}) (\xi )-H_{2}(S_{2},I_{2},R_{2}) ( \xi ) \bigr\vert e^{- \mu \vert \xi \vert } \\ &\quad \leq \frac{\beta }{\alpha } \vert S_{1}-S_{2} \vert _{\mu }+\bigl(m+\beta S_{0}e^{\mu c^{*}\tau }- \gamma \bigr) \vert I_{1}-I_{2} \vert _{\mu } \\ &\quad \leq l \vert \varPsi _{1}-\varPsi _{2} \vert _{\mu }, \end{aligned}$$

and

$$\begin{aligned} & \bigl\vert H_{3}(S_{1},I_{1},R_{1}) (\xi )-H_{3}(S_{2},I_{2},R_{2}) ( \xi ) \bigr\vert e^{- \mu \vert \xi \vert } \\ &\quad \leq m \vert R_{1}-R_{2} \vert _{\mu }+\gamma \vert I_{1}-I_{2} \vert _{\mu } \\ &\quad \leq l \vert \varPsi _{1}-\varPsi _{2} \vert _{\mu }, \end{aligned}$$

where $l=m+\frac{\beta }{\alpha }+\gamma +\beta S_{0} e^{\mu c^{*} \tau }$. Then, choosing $\mu \in (\sigma _{2},-r_{i1})$, we have that

$$\begin{aligned} & \bigl\vert \mathcal{M}_{i}[S_{1},I_{1},R_{1}]( \xi )-\mathcal{M}_{i}[S_{2},I _{2},R_{2}]( \xi ) \bigr\vert e^{-\mu \vert \xi \vert } \\ &\quad \leq \frac{1}{\varLambda _{i}} \bigl\vert H_{i}(S_{1},I_{1},R_{1})-H_{i}(S_{2},I _{2},R_{2}) \bigr\vert _{\mu } \biggl[ \int _{-\infty }^{\xi }e^{r_{i1}(\xi -\eta )}e ^{\mu \vert \eta \vert -\mu \vert \xi \vert }\,d\eta \\ & \qquad {} + \int _{\xi }^{\infty }e^{r_{i2}(\xi -\eta )}e^{\mu \vert \eta \vert - \mu \vert \xi \vert }\,d\eta \biggr] \\ &\quad \leq \frac{l}{\varLambda _{i}} \vert \varPsi _{1}-\varPsi _{2} \vert _{\mu } \biggl[ \int _{-\infty }^{\xi }e^{r_{i1}(\xi -\eta )}e^{\mu \vert \eta -\xi \vert }\,d\eta + \int _{\xi }^{\infty }e^{r_{i2}(\xi -\eta )}e^{\mu \vert \eta -\xi \vert }\,d\eta \biggr] \\ &\quad =\frac{l(2\mu +r_{i1}-r_{i2})}{d_{i}(r_{i2}-r_{i1})(r_{i2}-\mu )(r _{i1}+\mu )} \vert \varPsi _{1}-\varPsi _{2} \vert _{\mu }, \quad i=1,2,3, \end{aligned}$$

which implies that $\mathcal{M}$ is continuous with respect to the norm $|\cdot |_{\mu }$ in $B_{\mu }(\mathbb{R},\mathbb{R}^{3})$.

Now we turn to proving that $\mathcal{M}$ is compact with respect to the norm $|\cdot |_{\mu }$ in $B_{\mu }(\mathbb{R},\mathbb{R}^{3})$. For any $(S,I,R)\in \mathcal{S}$, we deduce for $\xi \in \mathbb{R}$ that

$$\begin{aligned} & \biggl\vert \frac{d\mathcal{M}_{1}[S,I,R](\xi )}{d\xi } \biggr\vert \\ &\quad = \biggl\vert \frac{r _{11}}{\varLambda _{1}} \int _{-\infty }^{\xi }e^{r _{11}(\xi -\eta )}H_{1}[S,I,R]( \eta )\,d\eta + \frac{r_{12}}{\varLambda _{1}} \int _{\xi }^{\infty }e^{r_{12}(\xi -\eta )}H _{1}[S,I,R](\eta )\,d\eta \biggr\vert \\ &\quad \leq -\frac{r_{11}}{\varLambda _{1}} \int _{-\infty }^{\xi }e^{r_{11}( \xi -\eta )}H_{1}[S,I,R]( \eta )\,d\eta +\frac{r_{12}}{\varLambda _{1}} \int _{\xi }^{\infty }e^{r_{12}(\xi -\eta )}H_{1}[S,I,R]( \eta )\,d\eta \\ &\quad \leq -\frac{r_{11}m S_{0}}{\varLambda _{1}} \int _{-\infty }^{\xi }e^{r _{11}(\xi -\eta )}\,d\eta + \frac{r_{12}m S_{0}}{\varLambda _{1}} \int _{ \xi }^{\infty }e^{r_{12}(\xi -\eta )}\,d\eta \\ &\quad =\frac{2m S_{0}}{\varLambda _{1}}, \end{aligned}$$

(3.23)

$$\begin{aligned} & \biggl\vert \frac{d\mathcal{M}_{2}[S,I,R](\xi )}{d\xi } \biggr\vert = \biggl\vert \frac{r _{21}}{\varLambda _{2}} \int _{-\infty }^{\xi }e^{r_{21}(\xi -\eta )}H_{2}[S,I,R]( \eta )\,d\eta \\ &\hphantom{ |\frac{d\mathcal{M}_{2}[S,I,R](\xi )}{d\xi } |=}{} +\frac{r_{22}}{\varLambda _{2}} \int _{\xi }^{\infty }e^{r_{22}( \xi -\eta )}H_{2}[S,I,R]( \eta )\,d\eta \biggr\vert \\ &\hphantom{ |\frac{d\mathcal{M}_{2}[S,I,R](\xi )}{d\xi } |}\leq -\frac{r_{21}}{\varLambda _{2}} \int _{-\infty }^{\xi }e^{r_{21}( \xi -\eta )}H_{2}[S,I,R]( \eta )\,d\eta \\ &\hphantom{ |\frac{d\mathcal{M}_{2}[S,I,R](\xi )}{d\xi } |=}{} +\frac{r_{22}}{\varLambda _{2}} \int _{\xi }^{\infty }e^{r_{22}( \xi -\eta )}H_{2}[S,I,R]( \eta )\,d\eta \\ &\hphantom{ |\frac{d\mathcal{M}_{2}[S,I,R](\xi )}{d\xi } |}\leq -\frac{r_{21}(m+\beta S_{0}-\gamma )\bar{I}}{\varLambda _{2}} \int _{-\infty }^{\xi }e^{r_{21}(\xi -\eta )}\,d\eta \\ &\hphantom{ |\frac{d\mathcal{M}_{2}[S,I,R](\xi )}{d\xi } |=}{} +\frac{r_{22}(m+\beta S_{0}-\gamma )\bar{I}}{\varLambda _{2}} \int _{\xi }^{\infty }e^{r_{22}(\xi -\eta )}\,d\eta \\ &\hphantom{ |\frac{d\mathcal{M}_{2}[S,I,R](\xi )}{d\xi } |}=\frac{2(m+\beta S_{0}-\gamma )\bar{I}}{\varLambda _{2}}, \end{aligned}$$

(3.24)

and

$$\begin{aligned} \biggl\vert \frac{d\mathcal{M}_{3}[S,I,R](\xi )}{d\xi } \biggr\vert =& \biggl\vert \frac{r _{31}}{\varLambda _{3}} \int _{-\infty }^{\xi }e^{r_{31}(\xi -\eta )}H_{3}[S,I,R]( \eta )\,d\eta \\ &{} +\frac{r_{32}}{\varLambda _{3}} \int _{\xi }^{\infty }e^{r_{32}( \xi -\eta )}H_{3}[S,I,R]( \eta )\,d\eta \biggr\vert \\ \leq& -\frac{r_{31}}{\varLambda _{3}} \int _{-\infty }^{\xi }e^{r_{31}( \xi -\eta )}\bigl(m L_{3} e^{\sigma _{1}\eta }+\gamma \bar{I}\bigr)\,d\eta \\ &{} +\frac{r_{32}}{\varLambda _{3}} \int _{\xi }^{\infty }e^{r_{32}( \xi -\eta )}\bigl(m L_{3} e^{\sigma _{1}\eta }+\gamma \bar{I}\bigr)\,d\eta \\ \leq& \frac{m L_{3} \vert 2r_{31}r_{32}-\sigma _{1}(r_{31}+r_{32}) \vert }{\varLambda _{3} \vert (\sigma _{1}-r_{31})(\sigma _{1}-r_{32}) \vert }e ^{\sigma _{1} \xi }+\frac{2\gamma \bar{I}}{\varLambda _{3}}. \end{aligned}$$

(3.25)

It follows from Lemma 3.6 that $|\mathcal{M}_{1}[S,I,R](\xi ) |+ |\mathcal{M}_{2}[S,I,R](\xi ) |+ |\mathcal{M}_{3}[S,I,R]( \xi ) |\leq S_{0}+\bar{I}+L_{3} e^{\sigma _{1}\xi }$ on $\mathbb{R}$. Recall that $\mu >\sigma _{1}$. Then, for any $\varepsilon >0$, there is a sufficiently large number $N>0$ such that

$$\begin{aligned} & \bigl\{ \bigl\vert \mathcal{M}_{1}[S,I,R]( \xi ) \bigr\vert + \bigl\vert \mathcal{M}_{2}[S,I,R]( \xi ) \bigr\vert + \bigl\vert \mathcal{M}_{3}[S,I,R](\xi ) \bigr\vert \bigr\} e^{-\mu \vert \xi \vert } \\ & \quad \leq \bigl(S_{0}+\bar{I}+L_{3} e^{\sigma _{1}\xi }\bigr)e^{-\mu \vert \xi \vert } \\ &\quad < (S_{0}+\bar{I})e^{-\mu N}+L_{3} e^{(\sigma _{1}-\mu )N} \\ & \quad < \varepsilon , \quad \vert \xi \vert >N. \end{aligned}$$

(3.26)

Utilizing (3.23)–(3.25) and Arzerà–Ascoli theorem, we can select finite elements in $\mathcal{M}(\mathcal{S})$ such that they are a finite ε-net of $\mathcal{M}(\mathcal{S})( \xi )$ on $[-N,N]$ with the supremum norm, a finite ε-net of $\mathcal{M}(\mathcal{S})(\xi )$ on $\mathbb{R}$ with the norm $|\cdot |_{\mu }$ (see (3.26)). Thus $\mathcal{M}$ is compact with respect to the norm $|\cdot |_{\mu }$ in $B_{\mu }(\mathbb{R}, \mathbb{R}^{3})$. The proof of this lemma is completed. □

By Lemma 3.6, Lemma 3.7, and Schauder’s fixed point theorem, we deduce that the operator $\mathcal{M}$ has a fixed point $(S(\xi ),I(\xi ),R( \xi ))\in \mathcal{S}$, which is a solution of the system

$$ \textstyle\begin{cases} d_{1} S''(\xi )-c^{*}S'(\xi )-\frac{\beta S(\xi )I(\xi -c^{*}\tau )}{1+ \alpha I(\xi -c^{*}\tau )}=0, \\ d_{2} I''(\xi )-c^{*}I'(\xi )+\frac{\beta S(\xi )I(\xi -c^{*}\tau )}{1+ \alpha I(\xi -c^{*}\tau )}-\gamma I(\xi )=0, \\ d_{3} R''(\xi )-c^{*}R'(\xi )+\gamma I(\xi )=0. \end{cases} $$

(3.27)

Based on the above analysis, we have the following results.

Proposition 3.1

If $\mathcal{R}_{0} >1$and $c=c^{*}$, then system (1.1) admits a traveling wave solution $(S(\xi ),I(\xi ),R(\xi ))$such that

$$ \underline{S}(\xi )\leq S(\xi )\leq \bar{S}(\xi ),\qquad \underline{I}(\xi )\leq I(\xi )\leq \bar{I}(\xi ),\qquad \underline{R}(\xi )\leq R(\xi )\leq \bar{R}(\xi ), \quad \xi \in \mathbb{R}. $$

(3.28)

3.3 Properties of the critical traveling wave solutions

In this section, we focus on some properties of the critical traveling wave solution of (2.1), that is, the proof of the four properties in Theorem 2.1.

Proof

(1) By contradiction, suppose that $I(\hat{\xi })=0$ for some $\hat{\xi }\in \mathbb{R}$. Then there are two constants $a,b\in \mathbb{R}$ such that $a<\xi _{3}\leq b$ and $a<\hat{\xi }<b$, which implies that $I(\xi )$ attains its minimum in $(a,b)$. It follows from the second equation in (3.27) that $-d_{2} I''(\xi )+c^{*}I'( \xi )+\gamma I(\xi )\geq 0$ for $\xi \in [a,b]$. By the strong maximum principle, we deduce that $I(\xi )\equiv 0$ for $\xi \in [a,b]$, which contradicts the fact that $I(\xi )\geq I_{-}(\xi )>0$ for $\xi \in [a, \xi _{3})$. Thus, $I(\xi )>0$ on $\mathbb{R}$. Similarly, one can obtain $S(\xi )>0$ on $\mathbb{R}$. Assume that $R(\tilde{\xi })=0$ for some $\tilde{\xi }\in \mathbb{R}$, then $R'(\tilde{\xi })=0$ and $R''(\tilde{\xi })\geq 0$. We infer from the third equation in (3.28) that $I(\tilde{\xi })\leq 0$, which contradicts the positiveness of $I(\xi )$ on $\mathbb{R}$. This implies that $R(\xi )>0$ on $\mathbb{R}$.

(2) From (3.28), we get

$$ \begin{aligned} &S_{0}\bigl(1-\sigma _{2}^{-1}e^{\sigma _{2}\xi } \bigr)\leq S(\xi )\leq S_{0}, \\ &\bigl[-L_{1} \xi -L_{2}(-\xi )^{\frac{1}{2}} \bigr]e^{\lambda ^{*} \xi }\leq I( \xi )\leq -L_{1} \xi e^{\lambda ^{*}\xi }, \\ &0\leq R(\xi )\leq L_{3}e^{\sigma _{1} \xi },\quad \xi \in \mathbb{R}. \end{aligned} $$

Then using the squeeze rule yields that

$$ S(-\infty )=S_{0}, \qquad I(-\infty )=0, \qquad R(-\infty )=0\quad \text{and} \quad I(\xi )=O\bigl(-\xi e^{\lambda ^{*}\xi } \bigr) $$

(3.29)

as $\xi \rightarrow -\infty$.

(3) Since $S(\xi )$ and $I(\xi )$ are uniformly bounded on $\mathbb{R}$, we have from the first two equations in (3.27) that

$$ \textstyle\begin{cases} S(\xi )=\frac{1}{\varLambda _{1}} \{\int _{-\infty }^{\xi }e^{r_{11}( \xi -\eta )}H_{1}[S,I,R](\eta )\,d\eta +\int _{\xi }^{\infty }e^{r_{12}( \xi -\eta )}H_{1}[S,I,R](\eta )\,d\eta \}, \\ I(\xi )=\frac{1}{\varLambda _{2}} \{\int _{-\infty }^{\xi }e^{r_{21}( \xi -\eta )}H_{2}[S,I,R](\eta )\,d\eta +\int _{\xi }^{\infty }e^{r_{22}( \xi -\eta )}H_{2}[S,I,R](\eta )\,d\eta \}, \end{cases} $$

(3.30)

where

$$ H_{1}[S,I,R](\eta )=m S(\eta )-\frac{\beta S(\eta )I(\eta -c^{*} \tau )}{1+\alpha I(\eta -c^{*}\tau )} $$

and

$$ H_{2}[S,I,R](\eta )=\frac{\beta S(\eta )I(\eta -c^{*}\tau )}{1+\alpha I(\eta -c^{*}\tau )}+(m-\gamma )I(\eta ). $$

Using L’Hôpital rule in (3.30) gives

$$ S'(\pm \infty )=0 \quad \text{and}\quad I'(\pm \infty )=0. $$

(3.31)

Integrating the first equation in (3.27) from −∞ to ξ and using (3.29) and (3.31), we have that

$$ \beta \int _{-\infty }^{\xi }\frac{ S(\eta )I(\eta -c^{*}\tau )}{1+ \alpha I(\eta -c^{*}\tau )}\,d\eta =c^{*}\bigl[S(\xi )-S_{0}\bigr]-d_{1} S'( \xi ) \leq c^{*}S_{0}-d_{1} S'(\xi ), \quad \xi \in \mathbb{R}. $$

(3.32)

Again, integrating the second equation in (3.27) from −∞ to ξ and utilizing (3.29), (3.31), and (3.32), we get that

$$\begin{aligned} \gamma \int _{-\infty }^{\xi }I(\eta )\,d\eta =&d_{2} I'(\xi )-c^{*} I( \xi )+\beta \int _{-\infty }^{\xi }\frac{ S(\eta )I(\eta -c^{*}\tau )}{1+ \alpha I(\eta -c^{*}\tau )}\,d\eta \\ \leq& d_{2} I'(\xi )+\beta \int _{-\infty }^{\xi }\frac{ S(\eta )I( \eta -c^{*}\tau )}{1+\alpha I(\eta -c^{*}\tau )}\,d\eta \\ \leq& d_{2} I'(\xi )+c^{*}S_{0}-d_{1} S'(\xi ),\quad \xi \in \mathbb{R}. \end{aligned}$$

Then, by the virtue of (3.31), we further obtain $\int _{\mathbb{R}}I(\xi )\,d\xi <\infty$, which together with the boundedness of $I'(\xi )$ on $\mathbb{R}$ (see (3.31)) implies that

$$ I(\infty )=0. $$

(3.33)

It follows from the first equation in (3.27) that

$$ \bigl[e^{-\frac{c^{*}}{d_{1}}\xi }S'(\xi ) \bigr]'= \frac{\beta }{d_{1}}e^{-\frac{c^{*}}{d_{1}}\xi } \frac{S(\xi )I(\xi -c ^{*}\tau )}{1+\alpha I(\xi -c^{*}\tau )}. $$

(3.34)

Integrating (3.34) from ξ to ∞, utilizing $S'(\infty )=0$ and $S(\xi ),I(\xi )>0$ on $\mathbb{R}$, we deduce

$$ S'(\xi )=-\frac{\beta }{d_{1}} \int _{\xi }^{\infty }e^{\frac{c^{*}}{d _{1}}(\xi -\eta )} \frac{S(\eta )I(\eta -c^{*}\tau )}{1+\alpha I( \eta -c^{*}\tau )}\,d\eta < 0, $$

(3.35)

which means that $S(\xi )$ is strictly decreasing on $\mathbb{R}$. This together with $S(\xi )>0$ on $\mathbb{R}$ gives that the limit $S(\infty )$ exists and $S(\infty ):=\varepsilon_{0} < S_{0}$. Moreover, an integration of the first equation in (3.27) over $\mathbb{R}$ gives

$$ \beta \int _{-\infty }^{\infty }\frac{ S(\xi )I(\xi -c^{*}\tau )}{1+ \alpha I(\xi -c^{*}\tau )}\,d\xi =c^{*}( S_{0}-\varepsilon_{0} ), $$

(3.36)

where we have used (3.29) and (3.31). Another integration of the second equation in (3.27) over $\mathbb{R}$ yields

$$ \gamma \int _{-\infty }^{\infty }I(\xi )\,d\xi =\beta \int _{-\infty } ^{\infty }\frac{ S(\xi )I(\xi -c^{*}\tau )}{1+\alpha I(\xi -c^{*} \tau )}\,d\xi , $$

(3.37)

since $I(\pm \infty )=I'(\pm \infty )=0$. Solving the third equation in (3.27) and using $R(-\infty )=0$ lead to

$$ R(\xi )=Ce^{\frac{c^{*}}{d_{3}}\xi }+\frac{\gamma }{c^{*}} \int _{- \infty }^{\xi }I(\eta )\,d\eta + \frac{\gamma }{c^{*}} \int _{\xi }^{+ \infty }e^{\frac{c^{*}}{d_{3}}(\xi -\eta )}I(\eta )\,d\eta , $$

where C is a constant of integration. Since $R(\xi )\leq L_{3} e ^{\sigma _{1}\xi }$ and $\sigma _{1}< c^{*}/d_{3}$ (see the proof of Lemma 3.4), we obtain

$$ R(\xi )=\frac{\gamma }{c^{*}} \int _{-\infty }^{\xi }I(\eta )\,d\eta + \frac{ \gamma }{c^{*}} \int _{\xi }^{\infty }e^{\frac{c^{*}}{d_{3}}(\xi - \eta )}I(\eta )\,d\eta . $$

(3.38)

We infer from (3.36)–(3.38) and L’Hôpital’s rule that

$$ R(\infty )=\frac{\gamma }{c^{*}} \int _{-\infty }^{\infty }I(\xi )\,d\xi =S_{0}- \varepsilon_{0} . $$

(3.39)

Differentiating (3.38) with respect to ξ and using $I(\xi )>0$ on $\mathbb{R}$, we have

$$ R'(\xi )=\frac{\gamma }{d_{3}} \int _{\xi }^{\infty }e^{\frac{c^{*}}{d _{3}}(\xi -\eta )}I(\eta )\,d\eta >0, $$

(3.40)

which means that $R(\xi )$ is strictly increasing on $\mathbb{R}$. Combining (3.40), $I(\pm \infty )=0$, and L’Hôpital’s rule yields

$$ R'(\pm \infty )=0. $$

(3.41)

Note from (3.29), (3.31), (3.32), (3.39), and (3.41) that

$$ S''(\pm \infty )=0, \qquad I''(\pm \infty )=0,\quad \text{and}\quad R''(\pm \infty )=0. $$

(3.42)

(4) Since $S(\xi )$ is strictly decreasing and $R(\xi )$ is strictly increasing on $\mathbb{R}$, we obtain $S(\xi )< S_{0}$ and $R(\xi )< S _{0}$ for $\xi \in \mathbb{R}$. Now we claim that $I(\xi )<\frac{1}{ \alpha } (\frac{\beta S_{0}}{\gamma }-1 )$ on $\mathbb{R}$. For contradiction, we assume that $I(\acute{\xi })=\frac{1}{\alpha } (\frac{\beta S_{0}}{\gamma }-1 )$ for some $\acute{\xi } \in \mathbb{R}$, which results in $I'(\acute{\xi })=0$ and $I''( \acute{\xi })\leq 0$. By the second equation in (3.27) and $S(\acute{\xi })< S_{0}$, we deduce that

$$\begin{aligned} 0 =&d_{2} I''(\acute{\xi })-c^{*}I'(\acute{\xi })+\frac{\beta S( \acute{\xi })I(\acute{\xi }-c^{*}\tau )}{1+\alpha I(\acute{\xi }-c ^{*}\tau )}-\gamma I(\acute{\xi }) \\ \leq& \frac{\beta S(\acute{\xi })I(\acute{\xi }-c^{*}\tau )}{1+ \alpha I(\acute{\xi }-c^{*}\tau )}-\gamma I(\acute{\xi }) \\ < &\frac{\frac{\beta S_{0}}{\alpha } (\frac{\beta S_{0}}{\gamma }-1 )}{1+ (\frac{\beta S_{0}}{\gamma }-1 )}- \frac{ \gamma }{\alpha } \biggl(\frac{\beta S_{0}}{\gamma }-1 \biggr) \\ =&0, \end{aligned}$$

leading to a contradiction. Thus $I(\xi )<\frac{1}{\alpha } (\frac{ \beta S_{0}}{\gamma }-1 )$ on $\mathbb{R}$. The proof is completed. □

4 Proof of Theorem 2.2

This proof is based on the contradictory argument. Suppose that the pair of continuous positive functions $(S(\xi ),I(\xi ),R(\xi ))$ ($\xi \in \mathbb{R}$) is a solution of the wave system of (1.1)

$$ \textstyle\begin{cases} d_{1} S''(\xi )-cS'(\xi )-\frac{\beta S(\xi )I(\xi -c\tau )}{1+\alpha I(\xi -c\tau )}=0, \\ d_{2} I''(\xi )-cI'(\xi )+\frac{\beta S(\xi )I(\xi -c\tau )}{1+\alpha I(\xi -c\tau )}-\gamma I(\xi )=0, \\ d_{3} R''(\xi )-cR'(\xi )+\gamma I(\xi )=0, \end{cases} $$

(4.1)

satisfying the asymptotic boundary conditions

$$ (S,I,R) (-\infty )=(S_{0},0,0),\qquad (S,I,R) ( \infty )=(\varepsilon ,0,S _{0}-\varepsilon ), $$

(4.2)

where $c\in \mathbb{R}$ is the wave speed. The proof of Theorem 2.2 is divided into two cases: the one is $\mathcal{R}_{0}=1$ and $c\in \mathbb{R}$; the other one is $\mathcal{R}_{0}>1$ and $c\leq 0$.

4.1 Case 1: $\mathcal{R}_{0}=1$ and $c\in \mathbb{R}$

From the second equation of (4.1), we get

$$\begin{aligned} I(\xi ) =&\frac{\beta }{d_{2}(\lambda ^{+}-\lambda ^{-})} \biggl[ \int _{-\infty }^{\xi }e^{\lambda ^{-}(\xi -\eta )} \frac{S(\eta )I( \eta -c\tau )}{1+\alpha I(\eta -c\tau )}\,d\eta \\ &{} + \int _{\xi }^{\infty }e^{\lambda ^{+}(\xi -\eta )} \frac{S( \eta )I(\eta -c\tau )}{1+\alpha I(\eta -c\tau )}\,d\eta \biggr], \end{aligned}$$

(4.3)

where

$$ \lambda ^{-}=\frac{c-\sqrt{c^{2}+4d_{2}\gamma }}{2d_{2}} \quad \text{and}\quad \lambda ^{+}=\frac{c+\sqrt{c^{2}+4d_{2}\gamma }}{2d _{2}}. $$

Applying L’Hôpital rule in (4.3) yields $I'(\pm \infty )=0$. Then, integrating the second equation in (4.1) over $\mathbb{R}$ and using $\mathcal{R}_{0} =1$, that is, $\beta S_{0}= \gamma$, we obtain

$$ \gamma \int _{-\infty }^{\infty }I(\xi )\,d\xi =\beta \int _{-\infty } ^{\infty }\frac{S(\xi )I(\xi -c\tau )}{1+\alpha I(\xi -c\tau )}\,d\xi . $$

(4.4)

By $\int _{\mathbb{R}}I(\xi )\,d\xi =\int _{\mathbb{R}}I(\xi -c\tau )\,d\xi$ and $\sup_{\xi \in \mathbb{R}}S(\xi )\leq S_{0}$ and (4.4), we have

$$\begin{aligned} 0 =&\gamma \int _{-\infty }^{\infty }I(\xi )\,d\xi -\beta \int _{-\infty } ^{\infty }\frac{S(\xi )I(\xi -c\tau )}{1+\alpha I(\xi -c\tau )}\,d\xi \\ =&\gamma \int _{-\infty }^{\infty }I(\xi )\,d\xi -\beta \int _{-\infty }^{\infty }\frac{S(\xi +c\tau )I(\xi )}{1+\alpha I(\xi )}\,d\xi \\ \geq& \beta \int _{-\infty }^{\infty }\bigl(S_{0}-S(\xi +c \tau )\bigr)I(\xi )\,d\xi \\ \geq& 0, \end{aligned}$$

which leads to

$$ \beta \int _{-\infty }^{\infty }\bigl(S_{0}-S(\xi +c \tau )\bigr)I(\xi )\,d\xi =0. $$

(4.5)

By a similar argument as that in (3.35) and the fact $\sup_{\xi \in \mathbb{R}}S(\xi )\leq S_{0}$, we get from (4.5) that

$$ \bigl(S_{0}-S(\xi +c\tau )\bigr)I(\xi )=0, \quad \xi \in \mathbb{R}, $$

(4.6)

which together with $I(\xi )>0$ implies that

$$ S(\xi )=S_{0}, \quad \xi \in \mathbb{R}. $$

(4.7)

A contradiction appears. The proof is completed.

4.2 Case 2: $\mathcal{R}_{0}>1$ and $c\leq 0$

Due to $S(-\infty )=S_{0}$ and $I(-\infty )=0$, we have

$$ \lim_{\xi \rightarrow -\infty }\frac{\beta S(\xi )}{1+\alpha I(\xi -c \tau )}=\beta S_{0}. $$

Then there exists a sufficiently small constant $\xi ^{*}<0$ such that

$$ \frac{\beta S(\xi )}{1+\alpha I(\xi -c\tau )}>\frac{\beta S_{0}+ \gamma }{2}, \quad \xi < \xi ^{*}. $$

Thus, from the second equation in (3.27), we obtain

$$\begin{aligned} cI'(\xi ) =&d_{2} I''(\xi )+\frac{\beta S(\xi )I(\xi -c\tau )}{1+ \alpha I(\xi -c\tau )}-\gamma I(\xi ) \\ \geq& d_{2}I''(\xi )+ \frac{\beta S_{0}+\gamma }{2}\bigl(I(\xi -c\tau )-I( \xi )\bigr)+\frac{\beta S_{0}-\gamma }{2}I( \xi ),\quad \xi < \xi ^{*}. \end{aligned}$$

(4.8)

By the integrability of $I(\xi )$ on $\mathbb{R}$, we can define

$$ Q(\xi ):= \int _{-\infty }^{\xi }I(x)\,dx, \quad \xi \in \mathbb{R}. $$

(4.9)

Since $I(\xi )>0$ in $\mathbb{R}$, one can see that $Q(\xi )$ is strictly increasing on $\mathbb{R}$. Integrating (4.8) from −∞ to ξ ($\xi <\xi ^{*}$) and using $I(-\infty )=0$ and $I'(-\infty )=0$ yield that

$$ cI(\xi )\geq d_{2} I'(\xi )+ \frac{\beta S_{0}+\gamma }{2}\bigl(Q(\xi -c \tau )-Q(\xi )\bigr)+\frac{\beta S_{0}-\gamma }{2}Q( \xi ),\quad \xi < \xi ^{*}. $$

(4.10)

Integrating (4.10) from −∞ to ξ, for $\xi <\xi ^{*}$, we get that

$$ cQ(\xi )\geq d_{2} I(\xi )+\frac{\beta S_{0}+\gamma }{2} \int _{-\infty }^{\xi }\bigl(Q(x-c\tau )-Q(x)\bigr)\,dx+ \frac{\beta S_{0}-\gamma }{2} \int _{- \infty }^{\xi }Q(x)\,dx. $$

(4.11)

Noting that $c\leq 0$, $\tau >0$, and $Q(\xi )$ is strictly increasing in $\mathbb{R}$, we obtain from (4.11) that

$$ \begin{aligned}[b] 0 &\geq cQ(\xi ) \\ & \geq d_{2} I(\xi )+\frac{\beta S_{0}+\gamma }{2} \int _{-\infty } ^{\xi }\bigl(Q(x-c\tau )-Q(x)\bigr)\,dx+ \frac{\beta S_{0}-\gamma }{2} \int _{- \infty }^{\xi }Q(x)\,dx \\ &>0, \xi < \xi ^{*}. \end{aligned} $$

(4.12)

A contradiction occurs. The proof is finished.

5 Conclusion

In this paper we have solved the open problems raised in the introduction, which are different from those in [26]. In the proof of the existence of critical traveling waves, we constructed a new pair of upper and lower solutions, which was an innovation of the paper. Then we mainly used the contradictory arguments and subtle analysis to establish the non-existence of traveling wave solutions for the cases: (i) $\mathcal{R}_{0}=1$ and $c\in \mathbb{R}$; (ii) $\mathcal{R}_{0}>1$ and $c\leq 0$. In order to address the change of the number for R-component in (1.1), we study the three equations together, which is helpful to describe the whole transmission behavior of the epidemic model. In Theorem 2.1, we obtained a lot of nice properties of the traveling wave solutions for (1.1). Our method adopted here can be used to improve the corresponding results for super-critical traveling wave solutions in [26] and also be helpful to the study of critical traveling wave solutions.