Impulsive vaccination and dispersal on dynamics of an SIR epidemic model with restricting infected individuals boarding transports

Abstract

To understand the effect of impulsive vaccination and restricting infected individuals boarding transports on disease spread, we establish an SIR model with impulsive vaccination, impulsive dispersal and restricting infected individuals boarding transports. This SIR epidemic model for two regions, which are connected by transportation of non-infected individuals, portrays the evolvement of diseases. We prove that all solutions of the investigated system are uniformly ultimately bounded. We also prove that there exists globally asymptotically stable infection-free boundary periodic solution. The condition for permanence is discussed. It is concluded that the approach of impulsive vaccination and restricting infected individuals boarding transports provides reliable tactic basis for preventing disease spread.

Highlights

• •An SIR epidemic model with restricting infected individuals boarding transports was investigated.
• •Impulsive vaccination and dispersal are considered, which are more reasonable.
• •Two thresholds are obtained for Infection-free of the system.

1. Introduction

The work of Kermack and McKendick  [1] was the fundamental study of epidemic models described by nonlinear differential equations. In this field, epidemic models have recently attracted much attention of mathematical epidemiologists and are perceived as significant  [2], [3], [4], [5], [6], [7]. Wang, Takeuchi and Liu  [8] studied a multi-group SVEIR epidemic model with distributed delay and vaccination. Sun and Shi  [9] considered a multigroup SEIR model with nonlinear incidence of infection and nonlinear removal functions between compartments. To understand the effect of transport-related infection on disease spread, Cui, Takeuchi and Saito  [10] proposed the spreading disease with transport-related infection. Takeuchi, Liu and Cui  [11] investigated the global dynamics of SIS models with transport-related infection, their conclusions implied that transport-related infection on disease can make the disease endemic even if all the isolated regions are disease free. Yan and Zou  [12] considered two control variables representing the quarantine and isolation strategies for SARS epidemics, they gave a theoretical interpretation to the practical experiences that the early quarantine and isolation strategies are critically important to control the outbreaks of epidemic. Chowell and Castillo-Chavez  [13] used the uncertainly and sensitivity analysis of the basic reproductive number $R_0$ to assess the role that the model parameters play in outbreak control. Quarantine and isolation measures have been widely used to control the spread of diseases such as yellow fever, smallpox, measles, ebola, pandemic influenza, diphtheria, plague, cholera, and, more recently, severe acute respiratory syndrome (SARS)  [14], [15], [16], [17], [18], [19]. Xie, et al.  [20] simultaneously use two kinds of measures: expand the treatment ranges of suspected case and limit population flows freely to suppress the diffusion of SARS effectively. Gong et al.  [21] showed that the SARS may fluctuate with import of SARS infectiousness from outside Beijing, weakness of quarantine, more social activities and so on.

Different types of vaccination policies and strategies combining pulse vaccination policy, treatment, pre-outbreak vaccination or isolation have already been introduced by many referees  [22], [23], [24], [25], [26], [27], [28], [29], [30], [31]. The pulse vaccination strategy (PVS) consists of repeated application of vaccine at discrete time with equal interval in a population in contrast to the traditional constant vaccination  [22], [23]. At each vaccination time a constant fraction of the susceptible population is vaccinated successfully. Since 1993, attempts have been made to develop mathematical theory to control infectious diseases using pulse vaccination  [22]. Compared to the proportional vaccination models, the study of pulse vaccination models is in its infancy  [23]. The control of childhood viral infections by pulse vaccination strategy is discussed by Nokes and Swinton  [24], [25]. Stone et al.  [26] presented a theoretical examination of the pulse vaccination strategy in the SIR epidemic model. d’Onofrio  [27] investigated the application of the pulse vaccination policy to eradicate infectious disease for SIR and SEIR epidemic models.

Theories of impulsive differential equations have been introduced into population dynamics lately  [32], [33], [34], [35], [36], [37], [38]. Impulsive equations are found in almost every domain of applied science and have been studied in many investigation  [38], [39], [40], [41], they generally describe phenomena which are subject to steep or instantaneous changes. The theories of population dynamical system and its application have been achieved many good results. However, the oasis vegetation degradation combining with dynamical system has been considered very little. In this paper, we will investigate an impulsive dispersal on SIR model with restricting infected individuals boarding transports. We expect to obtain some dynamical properties of the investigated system. We also expect that impulsive dispersal will provides reliable tactic for controlling epidemic.

The organization of this paper is as follows. In Section  2, we introduce the model and background concepts. In Section  3, some important lemmas are presented. We give the globally asymptotically stable conditions of the infection-free boundary periodic solution of System (2.2), and the permanent condition of System (2.2). In Section  4, a brief discussion is given to conclude this work.

2. The model

Inspired by the above discussion, we establish an SIR model with impulsive vaccination, impulsive dispersal and restricting infected individuals boarding transports.

$$\{\begin{array}{c}\begin{array}{c}\frac{d{S}_1\left(t\right)}{dt}={\lambda }_1-d_1{S}_1\left(t\right)-\frac{{\beta }_1{S}_1\left(t\right)I_1\left(t\right)}{1+{\alpha }_1{I}_1\left(t\right)}\text{,}\hfill \\ \frac{d{I}_1\left(t\right)}{dt}=\frac{{\beta }_1{S}_1\left(t\right)I_1\left(t\right)}{1+{\alpha }_1{I}_1\left(t\right)}-(r_1+d_1+b_1)I_1\left(t\right)\text{,}\hfill \\ \frac{d{R}_1\left(t\right)}{dt}=r_1{I}_1\left(t\right)-d_1{R}_1\left(t\right)\text{,}\hfill \\ \frac{d{S}_2\left(t\right)}{dt}={\lambda }_2-d_2{S}_2\left(t\right)-\frac{{\beta }_2{S}_2\left(t\right)I_2\left(t\right)}{1+{\alpha }_2{I}_2\left(t\right)}\text{,}\hfill \\ \frac{d{I}_2\left(t\right)}{dt}=\frac{{\beta }_2{S}_2\left(t\right)I_2\left(t\right)}{1+{\alpha }_2{I}_2\left(t\right)}-(r_2+d_2+b_2)I_2\left(t\right)\text{,}\hfill \\ \frac{d{R}_2\left(t\right)}{dt}=r_2{I}_2\left(t\right)-d_2{R}_2\left(t\right)\text{,}\hfill \end{array}\}t\ne (n+l)\tau ,t\ne (n+1)\tau \text{,}\hfill \\ \begin{array}{c}△S_1\left(t\right)=D(S_2\left(t\right)-S_1\left(t\right))\text{,}\hfill \\ △I_1\left(t\right)=0\text{,}\hfill \\ △R_1\left(t\right)=D(R_2\left(t\right)-R_1\left(t\right))\text{,}\hfill \\ △S_2\left(t\right)=D(S_1\left(t\right)-S_2\left(t\right))\text{,}\hfill \\ △I_2\left(t\right)=0\text{,}\hfill \\ △R_2\left(t\right)=D(R_1\left(t\right)-R_2\left(t\right))\text{,}\hfill \end{array}\}t=(n+l)\tau ,n\in Z^{+}\text{,}\hfill \\ \begin{array}{c}△S_1\left(t\right)=-{\mu }_1{S}_1\left(t\right)\text{,}\hfill \\ △I_1\left(t\right)=0\text{,}\hfill \\ △R_1\left(t\right)={\mu }_1{S}_1\left(t\right)\text{,}\hfill \\ △S_2\left(t\right)=-{\mu }_2{S}_2\left(t\right)\text{,}\hfill \\ △I_2\left(t\right)=0\text{,}\hfill \\ △R_2\left(t\right)={\mu }_2{S}_2\left(t\right)\text{,}\hfill \end{array}\}t=(n+1)\tau ,n\in Z^{+}\text{,}\hfill \end{array}$$ (2.1)

here $S_i\left(t\right)$, $I_i\left(t\right)$ and $R_i\left(t\right)$ represent the number of susceptible,infected, recovered individuals in city $i(i=1,2)$ at time $t$. It is assumed that we adopt the fixed number of offspring, denoted by ${\lambda }_i(i=1,2)$, joins into the susceptible class per unit time in city $i(i=1,2)$. The natural death rate is assumed as the same constant $d_i(i=1,2)$ for the susceptible,infected, recovered individuals in city $i(i=1,2)$. Disease is transmitted with the incidence rate, that is, the number of new cases of infection per unit time

$$\frac{{\beta }_i{S}_i{I}_i}{1+{\alpha }_i{I}_i}\text{,}i=1,2\text{,}$$

with city $i=1,2$. The transmission rate with city $i$ is a constant ${\beta }_i(i=1,2)$. ${\alpha }_i$ is a nonnegative constant representing the half saturation constant with city $i(i=1,2)$. The infected individuals in city $i(i=1,2)$ suffer an extra disease-related death with constant rate $b_i(i=1,2)$. $r_i(i=1,2)$ is the recovery rate of the infected individuals in city $i(i=1,2)$. By boarding transports, the susceptible and recovered individuals of city $i$ leave to city $j(i\ne j,i,j=1,2)$ with a dispersal rate $0<D<1$ at moment $t=(n+l)\tau ,n\in Z_{+}$. The susceptible is successfully vaccinated with ${\mu }_i(i=1,2)$ in city $i(i=1,2)$ at moment $t=(n+1)\tau ,n\in Z_{+}$.

Because $R_i\left(t\right)(i=1,2)$ do not affect the other equations of (2.1), we can simplify system (2.1) and restrict our attention to the following system

$$\{\begin{array}{c}\begin{array}{c}\frac{d{S}_1\left(t\right)}{dt}={\lambda }_1-d_1{S}_1\left(t\right)-\frac{{\beta }_1{S}_1\left(t\right)I_1\left(t\right)}{1+{\alpha }_1{I}_1\left(t\right)}\text{,}\hfill \\ \frac{d{I}_1\left(t\right)}{dt}=\frac{{\beta }_1{S}_1\left(t\right)I_1\left(t\right)}{1+{\alpha }_1{I}_1\left(t\right)}-(r_1+d_1+b_1)I_1\left(t\right)\text{,}\hfill \\ \frac{d{S}_2\left(t\right)}{dt}={\lambda }_2-d_2{S}_2\left(t\right)-\frac{{\beta }_2{S}_2\left(t\right)I_2\left(t\right)}{1+{\alpha }_2{I}_2\left(t\right)}\text{,}\hfill \\ \frac{d{I}_2\left(t\right)}{dt}=\frac{{\beta }_2{S}_2\left(t\right)I_2\left(t\right)}{1+{\alpha }_2{I}_2\left(t\right)}-(r_2+d_2+b_2)I_2\left(t\right)\text{,}\hfill \end{array}\}t\ne (n+l)\tau ,t\ne (n+1)\tau \text{,}\hfill \\ \begin{array}{c}△S_1\left(t\right)=D(S_2\left(t\right)-S_1\left(t\right))\text{,}\hfill \\ △I_1\left(t\right)=0\text{,}\hfill \\ △S_2\left(t\right)=D(S_1\left(t\right)-S_2\left(t\right))\text{,}\hfill \\ △I_2\left(t\right)=0\text{,}\hfill \end{array}\}t=(n+l)\tau ,n\in Z^{+}\text{,}\hfill \\ \begin{array}{c}△S_1\left(t\right)=-{\mu }_1{S}_1\left(t\right)\text{,}\hfill \\ △I_1\left(t\right)=0\text{,}\hfill \\ △S_2\left(t\right)=-{\mu }_2{S}_2\left(t\right)\text{,}\hfill \\ △I_2\left(t\right)=0\text{,}\hfill \end{array}\}t=(n+1)\tau ,n\in Z^{+}\text{.}\hfill \end{array}$$ (2.2)

3. The lemmas

The solution of (2.1), denote by $X\left(t\right)={(S_1\left(t\right),I_1\left(t\right),R_1\left(t\right),S_2\left(t\right),I_2\left(t\right),R_2\left(t\right))}^T$, is a piecewise continuous function $X:R_{+}\to R_{+}^6$, $X\left(t\right)$ is continuous on $(n\tau ,(n+l)\tau ]$ and $((n+l)\tau ,(n+1)\tau ]$, $n\in Z_{+}$ and $X\left(n{\tau }^{+}\right)={lim}_{t\to n{\tau }^{+}}X\left(t\right)$, $X\left((n+l){\tau }^{+}\right)={lim}_{t\to (n+l){\tau }^{+}}X\left(t\right)$ exist. Obviously the global existence and uniqueness of solutions of (2.1) is guaranteed by the smoothness properties of $f$, which denotes the mapping defined by right-side of system (2.1) (see Lakshmikantham  [39]).

Let $V:R_{+}\times R_{+}^6\to R_{+}$, then $V$ is said to belong to class $V_0$, if

• (i) $V$ is continuous in $(n\tau ,(n+l)\tau ]\times R_{+}^6$ and $((n+l)\tau ,(n+1)\tau ]\times R_{+}^6$, for each $z\in R_{+}^6,n\in Z_{+}$, $V(n{\tau }^{+},z)={lim}_{(t,y)\to (n{\tau }^{+},z)}V(t,y)$, $V((n+l){\tau }^{+},z)={lim}_{(t,y)\to ((n+l){\tau }^{+},y)}V(t,y)$ exist.
• (ii) $V$ is locally Lipschitzian in $z$.

Definition 3.1

$V\in V_0$, then, for $(t,z)\in (n\tau ,(n+l)\tau ]\times R_{+}^6$ and $((n+l)\tau ,(n+1)\tau ]\times R_{+}^6$, the upper right derivative of $V(t,z)$ with respect to the impulsive differential system (2.1) is defined as

$$D^{+}V(t,z)=\underset{h\to 0}{lim\; sup}\frac{1}{h}[V(t+h,z+hf(t,z))-V(t,z)]\text{.}$$

It can easily be obtained from the following lemma.

Lemma 3.2

Suppose $X\left(t\right)$ is a solution of (2.1)   with $X\left(0^{+}\right)\ge 0$ , then $X\left(t\right)\ge 0$ for $t\ge 0$ and further $X\left(t\right)>0(t\ge 0)$ for $X\left(0^{+}\right)>0$ .

Lemma 3.3 [39] —

Let the function $m\in P{C}^{\prime }[R^{+},R]$ ​satisfy the inequalities

$$\{\begin{array}{c}{m}^{\prime }\left(t\right)\le p\left(t\right)m\left(t\right)+q\left(t\right)\text{,}\hfill \\ t\ge t_0\text{,}t\ne t_k\text{,}k=1,2,\dots \text{,}\hfill \\ m\left(t_k^{+}\right)\le d_{k}m\left(t_k\right)+b_k\text{,}t=t_k\text{,}\hfill \end{array}$$ (3.1)

where $p,q\in PC[R^{+},R]$ and $d_k\ge 0,b_k$ are constants, then

$$m\left(t\right)\le m\left(t_0\right)\prod _{t_0<t_k<t}{d}_{k}exp\left({\int }_{t_0}^{t}p\left(s\right)ds\right)+\sum _{t_0<t_k<t}(\prod _{t_k<t_j<t}{d}_{j}exp\left({\int }_{t_0}^{t}p\left(s\right)ds\right))b_k+{\int }_{t_0}^t\prod _{s<t_k<t}{d}_{k}exp\left({\int }_s^{t}p\left(\sigma \right)d\sigma \right)q\left(s\right)ds\text{,}t\ge t_0\text{.}$$

Now, we show that all solutions of (2.1) are uniformly ultimately bounded.

Lemma 3.4

There exists a constant $M>0$ such that $S_i\left(t\right)\le M,I_i\left(t\right)\le M,R_i\left(t\right)\le M(i=1,2)$ for each solution $(S_1\left(t\right),I_1\left(t\right),R_1\left(t\right),S_2\left(t\right),I_2\left(t\right),R_2\left(t\right))$ of   (2.1)   with all $t$ large enough.

Proof

Define

$$V\left(t\right)=S_1\left(t\right)+I_1\left(t\right)+R_1\left(t\right)+S_2\left(t\right)+I_2\left(t\right)+R_2\left(t\right)\text{,}$$

and $d=min\{d_1,d_2\}$. Then $t\ne n\tau ,t\ne (n+l)\tau$, we have

$$D^{+}V\left(t\right)+dV\left(t\right)={\lambda }_1+{\lambda }_2-\sum _{i=1}^2[(d_i-d)S_i\left(t\right)+(d_i-d)I_i\left(t\right)+(d_i-d)R_i\left(t\right)]-\sum _{i=1}^2{b}_i{I}_i\left(t\right)\le {\lambda }_1+{\lambda }_2\text{.}$$

When $t=n\tau ,V\left(n{\tau }^{+}\right)=S_1\left(n{\tau }^{+}\right)+I_1\left(n{\tau }^{+}\right)+R_1\left(n{\tau }^{+}\right)+S_2\left(n{\tau }^{+}\right)+I_2\left(n{\tau }^{+}\right)+R_2\left(n{\tau }^{+}\right)=S_1\left(n\tau \right)+I_1\left(n\tau \right)+R_1\left(n\tau \right)+S_2\left(n\tau \right)+I_2\left(n\tau \right)+R_2\left(n\tau \right)=V\left(n\tau \right)$. When $t=(n+l)\tau ,V\left((n+l){\tau }^{+}\right)=S_1\left((n+l){\tau }^{+}\right)+I_1\left((n+l){\tau }^{+}\right)+R_1\left((n+l){\tau }^{+}\right)+S_2\left((n+l){\tau }^{+}\right)+I_2\left((n+l){\tau }^{+}\right)+R_2\left((n+l){\tau }^{+}\right)=S_1\left((n+l)\tau \right)+I_1\left((n+l)\tau \right)+R_1\left((n+l)\tau \right)+S_2\left((n+l)\tau \right)+I_2\left((n+l)\tau \right)+R_2\left((n+l)\tau \right)=V\left((n+l)\tau \right)$. By Lemma 3.3, for $t\in (n\tau ,(n+1)\tau ]$, we have

$$V\left(t\right)\le V\left(0\right)exp(-dt)+{\int }_0^t({\lambda }_1+{\lambda }_2)exp(-d(t-s))ds=V\left(0\right)exp(-dt)+\frac{{\lambda }_1+{\lambda }_2}{d}(1-exp(-dt))\to \frac{{\lambda }_1+{\lambda }_2}{d}\text{,}\text{as }t\to \infty \text{.}$$

So $V\left(t\right)$ is uniformly ultimately bounded. Hence, by the definition of $V\left(t\right)$, we have there exists a constant $M>0$ such that $S_i\left(t\right)\le M,I_i\left(t\right)\le M,R_i\left(t\right)\le M(i=1,2)$ for $t$ large enough. The proof is complete.

If $I_i\left(t\right)=0(i=1,2)$, we have the following subsystem of (2.2)

$$\{\begin{array}{c}\begin{array}{c}\frac{d{S}_1\left(t\right)}{dt}={\lambda }_1-d_1{S}_1\left(t\right)\text{,}\hfill \\ \frac{d{S}_2\left(t\right)}{dt}={\lambda }_2-d_2{S}_2\left(t\right)\text{,}\hfill \end{array}\}t\ne (n+l)\tau ,t\ne (n+1)\tau \text{,}\hfill \\ \begin{array}{c}△S_1\left(t\right)=D(S_2\left(t\right)-S_1\left(t\right))\text{,}\hfill \\ △S_2\left(t\right)=D(S_1\left(t\right)-S_2\left(t\right))\text{,}\hfill \end{array}\}t=(n+l)\tau \text{,}\hfill \\ \begin{array}{c}△S_1\left(t\right)=-{\mu }_1{S}_1\left(t\right)\text{,}\hfill \\ △S_2\left(t\right)=-{\mu }_2{S}_2\left(t\right)\text{,}\hfill \end{array}\}t=(n+1)\tau \text{,}n=1,2\dots \text{.}\hfill \end{array}$$ (3.2)

We can easily obtain the analytic solution of (3.2) between pulses as follows

$$\{\begin{array}{c}{S}_1\left(t\right)=\{\begin{array}{c}\frac{1}{d_1}[{\lambda }_1-({\lambda }_1-d_1{S}_1\left(n{\tau }^{+}\right))e^{-d_1(t-n\tau )}]\text{,}t\in [n\tau ,(n+l)\tau )\text{,}\hfill \\ \frac{1}{d_1}[{\lambda }_1-({\lambda }_1-d_1{S}_1\left((n+l){\tau }^{+}\right))e^{-d_1(t-(n+l)\tau )}]\text{,}t\in [(n+l)\tau ,(n+1)\tau )\text{,}\hfill \end{array}\hfill \\ S_2\left(t\right)=\{\begin{array}{c}\frac{1}{d_2}[{\lambda }_2-({\lambda }_2-d_2{S}_2\left(n{\tau }^{+}\right))e^{-d_2(t-n\tau )}]\text{,}t\in [n\tau ,(n+l)\tau )\text{.}\hfill \\ \frac{1}{d_2}[{\lambda }_2-({\lambda }_2-d_2{S}_2\left((n+l){\tau }^{+}\right))e^{-d_2(t-(n+l)\tau )}]\text{,}t\in [(n+l)\tau ,(n+1)\tau )\text{.}\hfill \end{array}\hfill \end{array}$$ (3.3)

Considering the third and fourth equations of (3.2), we have

$$\{\begin{array}{c}{S}_1\left((n+l){\tau }^{+}\right)=\frac{1-D}{d_1}[{\lambda }_1-({\lambda }_1-d_1{S}_1\left(n{\tau }^{+}\right))e^{-d_{1}l\tau }]+\frac{D}{d_2}[{\lambda }_2-({\lambda }_2-d_2{S}_2\left(n{\tau }^{+}\right))e^{-d_{2}l\tau }]\text{,}\hfill \\ S_2\left((n+l){\tau }^{+}\right)=\frac{D}{d_1}[{\lambda }_1-({\lambda }_1-d_1{S}_1\left(n{\tau }^{+}\right))e^{-d_{1}l\tau }]+\frac{1-D}{d_2}[{\lambda }_2-({\lambda }_2-d_2{S}_2\left(n{\tau }^{+}\right))e^{-d_{2}l\tau }]\text{.}\hfill \end{array}$$ (3.4)

Considering the fifth and sixth equations of (3.2), we also have

$$\{\begin{array}{c}{S}_1\left((n+1){\tau }^{+}\right)=\frac{1-{\mu }_1}{d_1}[{\lambda }_1-({\lambda }_1-d_1{S}_1\left((n+l){\tau }^{+}\right))e^{-d_1(1-l)\tau }]\text{,}\hfill \\ S_2\left((n+1){\tau }^{+}\right)=\frac{1-{\mu }_2}{d_2}[{\lambda }_2-({\lambda }_2-d_2{S}_2\left((n+l){\tau }^{+}\right))e^{-d_2(1-l)\tau }]\text{.}\hfill \end{array}$$ (3.5)

Substituting (3.4) into (3.5), we have the stroboscopic map of (3.2)

$$\{\begin{array}{c}{S}_1\left((n+1){\tau }^{+}\right)=(1-{\mu }_1)(1-D)e^{-d_1\tau }{S}_1\left(n{\tau }^{+}\right)+(1-{\mu }_1)D{e}^{-[d_1(1-l)+d_{2}l]\tau }{S}_2\left(n{\tau }^{+}\right)+(1-{\mu }_1)\times [\frac{{\lambda }_1(1-e^{-d_{1}l\tau })(1-(1-D)e^{-d_1(1-l)\tau })}{d_1}+\frac{D{\lambda }_2(1-e^{-d_{2}l\tau })e^{-d_1(1-l)\tau }}{d_2}]\text{,}\hfill \\ S_2\left((n+1){\tau }^{+}\right)=(1-{\mu }_2)D{e}^{-[d_{1}l+d_2(1-l)]\tau }{S}_1\left(n{\tau }^{+}\right)+(1-{\mu }_2)(1-D)e^{-d_2\tau }{S}_2\left(n{\tau }^{+}\right)+(1-{\mu }_2)\times [\frac{D{\lambda }_1(1-e^{-d_{1}l\tau })e^{-d_2(1-l)\tau }}{d_1}+\frac{{\lambda }_2(1-e^{-d_{2}l\tau })(1-(1-D)e^{-d_2(1-l)\tau })}{d_2}]\text{.}\hfill \end{array}$$ (3.6)

(3.5) has one fixed point as

$$\{\begin{array}{c}{S}_1^{\ast }=\frac{(1-A_1)B-A{A}_2}{(1-A_1)(1-B_2)-A_2{B}_1}>0\text{,}\hfill \\ S_2^{\ast }=\frac{B_{1}B-A(1-B_2)}{(1-A_1)(1-B_2)-A_2{B}_1}>0\text{,}\hfill \end{array}$$ (3.7)

where

$$A_1=(1-{\mu }_1)(1-D)e^{-d_1\tau }<1\text{,}$$

$$B_1=(1-{\mu }_1)D{e}^{-[d_1(1-l)+d_{2}l]\tau }<1\text{,}$$

$$A_2=(1-{\mu }_2)D{e}^{-[d_{1}l+d_2(1-l)]\tau }<1\text{,}$$

$$B_2=(1-{\mu }_2)(1-D)e^{-d_2\tau }<1\text{,}$$

$$A=(1-{\mu }_1)\times [\frac{{\lambda }_1(1-e^{-d_{1}l\tau })(1-(1-D)e^{-d_1(1-l)\tau })}{d_1}+\frac{D{\lambda }_2(1-e^{-d_{2}l\tau })e^{-d_1(1-l)\tau }}{d_2}]>0\text{,}$$

$$B=(1-{\mu }_2)\times [\frac{D{\lambda }_1(1-e^{-d_{1}l\tau })e^{-d_2(1-l)\tau }}{d_1}+\frac{{\lambda }_2(1-e^{-d_{2}l\tau })(1-(1-D)e^{-d_2(1-l)\tau })}{d_2}]>0\text{.}$$

Lemma 3.5

The unique fixed point $(S_1^{\ast },S_2^{\ast })$ of   (3.6)   is globally asymptotically stable.

Proof

For convenience, we make a notation as $(S_1^n,S_2^n)=(S_1\left(n{\tau }^{+}\right),S_2\left(n{\tau }^{+}\right))$. The linear form of (3.6) can be written as

$$\left(\begin{array}{c}\hfill S_1^{n+1}\hfill \\ \hfill S_2^{n+1}\hfill \end{array}\right)=M\left(\begin{array}{c}\hfill S_1^n\hfill \\ \hfill S_2^n\hfill \end{array}\right)\text{.}$$ (3.8)

Obviously, the near dynamics of $(S_1^{\ast },S_2^{\ast })$ is determined by linear system (3.6). The stabilities of $(S_1^{\ast },S_2^{\ast })$ ​are determined by the eigenvalue of $M$ less than 1. If $M$ satisfies the Jury criteria  [22], we can know the eigenvalue of $M$ less than $1$,

$$1-\mathrm{tr}M+detM>0\text{.}$$ (3.9)

We can easily know that $(S_1^{\ast },S_2^{\ast })$ is unique fixed point of (3.6), and

$$M=\left(\begin{array}{cc}\hfill A_1\hfill & \hfill B_1\hfill \\ \hfill A_2\hfill & \hfill B_2\hfill \end{array}\right)\text{.}$$ (3.10)

For

$$1-\mathrm{tr}M+detM=1-(A_1+B_2)+(A_1{B}_2-A_2{B}_1)=(1-A_1)(1-B_2)-A_2{B}_1=[(1-(1-{\mu }_1)e^{-d_1\tau })+(1-{\mu }_1)D{e}^{-d_1\tau }][(1-(1-{\mu }_2)e^{-d_2\tau })+(1-{\mu }_2)D{e}^{-d_2\tau }]-(1-{\mu }_1)(1-{\mu }_2)D^2{e}^{-(d_1+d_2)\tau }=[1-(1-{\mu }_1)e^{-d_1\tau }][1-(1-{\mu }_2)e^{-d_2\tau }]+[1-(1-{\mu }_1)e^{-d_1\tau }](1-{\mu }_2)D{e}^{-d_2\tau }+[1-(1-{\mu }_2)e^{-d_2\tau }](1-{\mu }_1)D{e}^{-d_1\tau }>0\text{.}$$

From Jury criteria, $(S_1^{\ast },S_2^{\ast })$ is locally stable. Because the fixed point $(S_1^{\ast },S_2^{\ast })$ of (3.6) is unique, then, it is globally asymptotically stable. This completes the proof.

Lemma 3.6

The periodic solution $(\widetilde{S_1\left(t\right)},\widetilde{S_2\left(t\right)})$ of System   (3.2)   is globally asymptotically stable, where

$$\{\begin{array}{c}\widetilde{S_1\left(t\right)}=\{\begin{array}{c}\frac{1}{d_1}[{\lambda }_1-({\lambda }_1-d_1{S}_1^{\ast })e^{-d_1(t-n\tau )}]\text{,}t\in [n\tau ,(n+l)\tau )\text{,}\hfill \\ \frac{1}{d_1}[{\lambda }_1-({\lambda }_1-d_1{S}_1^{\ast \ast })e^{-d_1(t-(n+l)\tau )}]\text{,}t\in [(n+l)\tau ,(n+1)\tau )\text{,}\hfill \end{array}\hfill \\ \widetilde{S_2\left(t\right)}=\{\begin{array}{c}\frac{1}{d_2}[{\lambda }_2-({\lambda }_2-d_2{S}_2^{\ast })e^{-d_2(t-n\tau )}]\text{,}t\in [n\tau ,(n+l)\tau )\text{.}\hfill \\ \frac{1}{d_2}[{\lambda }_2-({\lambda }_2-d_2{S}_2^{\ast \ast })e^{-d_2(t-(n+l)\tau )}]\text{,}t\in [(n+l)\tau ,(n+1)\tau )\text{,}\hfill \end{array}\hfill \end{array}$$ (3.11)

here $S_1^{\ast }$ and $S_2^{\ast }$ are determined as   (3.7) , $S_1^{\ast \ast }$ and $S_2^{\ast \ast }$ are defined as

$$\{\begin{array}{c}{S}_1^{\ast \ast }=\frac{1-D}{d_1}[{\lambda }_1-({\lambda }_1-d_1{S}_1^{\ast })e^{-d_{1}l\tau }]+\frac{D}{d_2}[{\lambda }_2-({\lambda }_2-d_2{S}_2^{\ast })e^{-d_{2}l\tau }]\text{,}\hfill \\ S_2^{\ast \ast }=\frac{D}{d_1}[{\lambda }_1-({\lambda }_1-d_1{S}_1^{\ast })e^{-d_{1}l\tau }]+\frac{1-D}{d_2}[{\lambda }_2-({\lambda }_2-d_2{S}_2^{\ast })e^{-d_{2}l\tau }]\text{.}\hfill \end{array}$$ (3.12)

4. The dynamics

Theorem 4.1

If

$$D<\frac{1}{2}\text{,}$$ (4.1)

and

$$\underset{i=1,2}{max}\frac{{\beta }_i}{d_i}[{\lambda }_i\tau +\frac{{\lambda }_i-d_i\underset{i}{\overset{\ast }{S}}}{d_i}(e^{-d_{i}l\tau }-1)+\frac{{\lambda }_i-d_i\underset{i}{\overset{\ast \ast }{S}}}{d_i}(e^{-d_i\tau }-e^{-d_{i}l\tau })]-(r_i+d_i+b_i)\tau <0\text{,}$$ (4.2)

hold, the infection-free boundary periodic solution $(\widetilde{S_1\left(t\right)},0,\widetilde{S_2\left(t\right)},0)$ of   (2.2)   is globally asymptotically stable, where $S_i^{\ast }(i=1,2)$ and $S_i^{\ast \ast }(i=1,2)$ are defined as   (3.7), (3.12) .

Proof

First, we prove the local stability of the infection-free boundary periodic solution $(\widetilde{S_1\left(t\right)},0,\widetilde{S_2\left(t\right)},0)$ of (2.2). Defining $S_{11}\left(t\right)=S_1\left(t\right)-\widetilde{S_1\left(t\right)},S_{12}\left(t\right)=S_2\left(t\right)-\widetilde{S_2\left(t\right)},I_1\left(t\right)=I_1\left(t\right),I_2\left(t\right)=I_2\left(t\right)$, then we have the following linearly similar system for (2.2) which is concerning one periodic solution $(\widetilde{S_1\left(t\right)},0,\widetilde{S_2\left(t\right)},0)$

$$\left(\begin{array}{c}\hfill \frac{d{S}_{11}\left(t\right)}{dt}\hfill \\ \hfill \frac{d{I}_1\left(t\right)}{dt}\hfill \\ \hfill \frac{d{S}_{12}\left(t\right)}{dt}\hfill \\ \hfill \frac{d{I}_2\left(t\right)}{dt}\hfill \end{array}\right)=\left(\begin{array}{cccc}\hfill -d_1\hfill & \hfill -{\beta }_1\widetilde{S_1\left(t\right)}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\beta }_1\widetilde{S_1\left(t\right)}-(r_1+d_1+b_1)\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -d_2\hfill & \hfill -{\beta }_2\widetilde{S_2\left(t\right)}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\beta }_2\widetilde{S_2\left(t\right)}-(r_2+d_2+b_2)\hfill \end{array}\right)\left(\begin{array}{c}\hfill S_{11}\left(t\right)\hfill \\ \hfill I_1\left(t\right)\hfill \\ \hfill S_{12}\left(t\right)\hfill \\ \hfill I_2\left(t\right)\hfill \end{array}\right)\text{.}$$

It is easy to obtain the fundamental solution matrix

$$\Phi \left(t\right)=\left(\begin{array}{cccc}\hfill exp(-d_{1}t)\hfill & \hfill {\ast }_1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill exp\left({\int }_0^t({\beta }_1\widetilde{S_1\left(s\right)}-(r_1+d_1+b_1))ds\right)\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill exp(-d_{2}t)\hfill & \hfill {\ast }_2\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill exp\left({\int }_0^t({\beta }_2\widetilde{S_2\left(s\right)}-(r_2+d_2+b_2))ds\right)\hfill \end{array}\right)\text{.}$$

There is no need to calculate the exact form of ${\ast }_i(i=1,2)$, as they are not required in the analysis that follows. The linearization of the fifth, sixth, seventh and eighth equations of (2.2) is

$$\left(\begin{array}{c}\hfill S_{11}\left(n{\tau }^{+}\right)\hfill \\ \hfill I_1\left(n{\tau }^{+}\right)\hfill \\ \hfill S_{12}\left(n{\tau }^{+}\right)\hfill \\ \hfill I_2\left(n{\tau }^{+}\right)\hfill \end{array}\right)=\left(\begin{array}{cccc}\hfill 1-D\hfill & \hfill 0\hfill & \hfill D\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill D\hfill & \hfill 0\hfill & \hfill 1-D\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right)\left(\begin{array}{c}\hfill S_{11}\left(n\tau \right)\hfill \\ \hfill I_1\left(n\tau \right)\hfill \\ \hfill S_{12}\left(n\tau \right)\hfill \\ \hfill I_2\left(n\tau \right)\hfill \end{array}\right)\text{.}$$

The linearization of the ninth, tenth, eleventh and twelfth equations of (2.2) is

$$\left(\begin{array}{c}\hfill S_{11}\left(n{\tau }^{+}\right)\hfill \\ \hfill I_1\left(n{\tau }^{+}\right)\hfill \\ \hfill S_{12}\left(n{\tau }^{+}\right)\hfill \\ \hfill I_2\left(n{\tau }^{+}\right)\hfill \end{array}\right)=\left(\begin{array}{cccc}\hfill 1-{\mu }_1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1-{\mu }_2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right)\left(\begin{array}{c}\hfill S_{11}\left(n\tau \right)\hfill \\ \hfill I_1\left(n\tau \right)\hfill \\ \hfill S_{12}\left(n\tau \right)\hfill \\ \hfill I_2\left(n\tau \right)\hfill \end{array}\right)\text{.}$$

The stability of the periodic solution $(\widetilde{S_1\left(t\right)},0,\widetilde{S_2\left(t\right)},0)$ is determined by the eigenvalues of

$$M=\left(\begin{array}{cccc}\hfill 1-D\hfill & \hfill 0\hfill & \hfill D\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill D\hfill & \hfill 0\hfill & \hfill 1-D\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right)\left(\begin{array}{cccc}\hfill 1-{\mu }_1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1-{\mu }_2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right)\Phi \left(\tau \right)\text{,}$$

where are

$${\lambda }_1=(1-{\mu }_1)(1-D)e^{-d_1\tau }<1\text{,}$$

and

$${\lambda }_2=\left|\frac{(1-D)(K_1+K_3)+\sqrt{{(1-D)}^2{(K_1+K_3)}^2-4(1-2D)K_1{K}_3}}{2}\right|\le \left|\frac{(1-D)(K_1+K_3)+\sqrt{{(1+D)}^2{(K_1+K_3)}^2}}{2}\right|\le \left|\frac{(K_1+K_3)}{2}\right|\text{,}$$

and

$${\lambda }_3=(1-{\mu }_2)(1-D)e^{-d_2\tau }<1\text{,}$$

and

$${\lambda }_4=\left|\frac{(1-D)(K_1+K_3)-\sqrt{{(1-D)}^2{(K_1+K_3)}^2-4(1-2D)K_1{K}_3}}{2}\right|\le \left|\frac{(1-D)(K_1+K_3)-\sqrt{{(1-D)}^2{(K_1-K_3)}^2}}{2}\right|=\left|\frac{(1-D)(K_1+K_3)-\sqrt{{(1-D)}^2{(K_1-K_3)}^2}}{2}\right|\le (1-D)max\{K_1,K_3\}\text{,}$$

where $K_1=e^{{\int }_0^{\tau }[{\beta }_1\widetilde{S_1\left(t\right)}-(r_1+d_1+b_1)]dt}$ and $K_3=e^{{\int }_0^{\tau }[{\beta }_2\widetilde{S_2\left(t\right)}-(r_2+d_2+b_2)]dt}$. According to condition (4.1), (4.2) and the Floquet theory  [39], i.e. 

$$exp\left[{\int }_0^{\tau }({\beta }_i\widetilde{S_i\left(t\right)}-(r_i+d_i+b_i))dt\right]<1(i=1,2)\text{,}$$

then,

$${\lambda }_2\le \left|\frac{(K_1+K_3)}{2}\right|<1\text{,}$$

and

$${\lambda }_4\le (1-D)max\{K_1,K_3\}<1\text{,}$$

hold, the infection-free boundary periodic solution $(\widetilde{S_1\left(t\right)},0,\widetilde{S_2\left(t\right)},0)$ of (2.2) is locally stable.

In the following, we will prove the global attraction. Choose a $\epsilon >0$ such that

$${\rho }_i=exp\left[{\int }_0^{\tau }({\beta }_i(\widetilde{S_i\left(t\right)}+\epsilon )-(r_i+d_i+b_i))dt\right]<1(i=1,2)\text{.}$$

From the first and third equations of (2.2), we notice that $\frac{d{S}_i\left(t\right)}{dt}\le {\lambda }_i-d_i{S}_i\left(t\right)(i=1,2)$. Then, we consider following impulsive comparative differential equation

$$\{\begin{array}{c}\begin{array}{c}\frac{d{S}_{21}\left(t\right)}{dt}={\lambda }_1-d_1{S}_{21}\left(t\right)\text{,}\hfill \\ \frac{d{S}_{22}\left(t\right)}{dt}={\lambda }_2-d_2{S}_{22}\left(t\right)\text{,}\hfill \end{array}\}t\ne (n+l)\tau ,t\ne (n+1)\tau \text{,}\hfill \\ \begin{array}{c}△S_{21}\left(t\right)=D(S_{22}\left(t\right)-S_{21}\left(t\right))\text{,}\hfill \\ △S_{22}\left(t\right)=D(S_{21}\left(t\right)-S_{22}\left(t\right))\text{,}\hfill \end{array}\}t=(n+l)\tau \text{,}\hfill \\ \begin{array}{c}△S_{21}\left(t\right)=-{\mu }_1{S}_{21}\left(t\right)\text{,}\hfill \\ △S_{22}\left(t\right)=-{\mu }_2{S}_{22}\left(t\right)\text{,}\hfill \end{array}\}t=(n+1)\tau \text{.}\hfill \end{array}$$ (4.3)

From Lemma 3.6 and comparison theorem of impulsive equation (see theorem 3.1.1 in Ref.  [39]), we have $S_1\left(t\right)\le S_{21}\left(t\right)$, $S_2\left(t\right)\le S_{22}\left(t\right)$ and $S_{21}\left(t\right)\to \widetilde{S_1\left(t\right)}$, $S_{22}\left(t\right)\to \widetilde{S_2\left(t\right)}$, as $t\to \infty$. Then

$$\{\begin{array}{c}{S}_1\left(t\right)\le S_{21}\left(t\right)\le \widetilde{S_1\left(t\right)}+\epsilon \text{,}\hfill \\ S_2\left(t\right)\le S_{22}\left(t\right)\le \widetilde{S_2\left(t\right)}+\epsilon \text{,}\hfill \end{array}$$ (4.4)

for all $t$ large enough. For convenience, we may assume (4.4) holds for all $t\ge 0$. From (2.2) and (4.4), we get

$$\frac{d{I}_i\left(t\right)}{dt}\le [{\beta }_i(\widetilde{S_i\left(t\right)}+\epsilon )-(r_i+d_i+b_i)]I_i\left(t\right)(i=1,2)\text{.}$$ (4.5)

So $I_i\left((n+1)\tau \right)\le I_i\left(n{\tau }^{+}\right)exp\left[{\int }_{n\tau }^{(n+1)\tau }({\beta }_i(\widetilde{S_i\left(S\right)}+\epsilon )-(r_i+d_i+b_i))ds\right](i=1,2)$. Hence $I_i\left(n\tau \right)\le I_i\left(0^{+}\right){\rho }_i^n(i=1,2)$ and $I_i\left(n\tau \right)\to 0(i=1,2)$ as $n\to \infty$, therefore $I_i\left(t\right)\to 0(i=1,2)$ as $t\to \infty$.

Next, we will prove that $S_i\left(t\right)\to \widetilde{S_i\left(t\right)}(i=1,2)$ as $t\to \infty$. For ${\epsilon }_1>0$, there must exist a $t_0>0$ such that $0<I_i\left(t\right)<{\epsilon }_1(i=1,2)$ for all $t\ge t_0$. Without loss of generality, we may assume that $0<I_i\left(t\right)<{\epsilon }_1$ for all $t\ge 0$. For system (2.2) we have

$${\lambda }_i-(d_i+\frac{{\beta }_i{\epsilon }_1}{1+{\alpha }_i{\epsilon }_1})S_i\left(t\right)\le \frac{d{S}_i\left(t\right)}{dt}\le {\lambda }_i-d_i{x}_i\left(t\right)(i=1,2)\text{,}$$ (4.6)

then we have $S_{31}\left(t\right)\le S_1\left(t\right)\le S_{21}\left(t\right)$, $S_{32}\left(t\right)\le S_2\left(t\right)\le S_{22}\left(t\right)$ and $S_{31}\left(t\right)\to \widetilde{S_{31}\left(t\right)},S_{32}\left(t\right)\to \widetilde{S_{32}\left(t\right)}$, $S_{21}\left(t\right)\to \widetilde{S_1\left(t\right)},S_{22}\left(t\right)\to \widetilde{S_2\left(t\right)}$ as $t\to \infty$. While $(S_{31}\left(t\right),S_{32}\left(t\right))$ and $(S_{21}\left(t\right),S_{22}\left(t\right))$ are the solutions of

$$\{\begin{array}{c}\begin{array}{c}\frac{d{S}_{31}\left(t\right)}{dt}={\lambda }_1-(d_1+\frac{{\beta }_1{\epsilon }_1}{1+{\alpha }_1{\epsilon }_1})S_{31}\left(t\right)\text{,}\hfill \\ \frac{d{S}_{32}\left(t\right)}{dt}={\lambda }_2-(d_2+\frac{{\beta }_2{\epsilon }_1}{1+{\alpha }_2{\epsilon }_1})S_{32}\left(t\right)\text{,}\hfill \end{array}\}t\ne (n+l)\tau ,t\ne (n+1)\tau \text{,}\hfill \\ \begin{array}{c}△S_{31}\left(t\right)=D(S_{32}\left(t\right)-S_{31}\left(t\right))\text{,}\hfill \\ △S_{32}\left(t\right)=D(S_{31}\left(t\right)-S_{32}\left(t\right))\text{,}\hfill \end{array}\}t=(n+l)\tau \text{,}\hfill \\ \begin{array}{c}△S_{31}\left(t\right)=-{\mu }_1{S}_{31}\left(t\right)\text{,}\hfill \\ △S_{32}\left(t\right)=-{\mu }_2{S}_{32}\left(t\right)\text{,}\hfill \end{array}\}t=(n+1)\tau \text{,}\hfill \end{array}$$ (4.7)

and

$$\{\begin{array}{c}\begin{array}{c}\frac{d{S}_{21}\left(t\right)}{dt}={\lambda }_1-d_1{S}_{21}\left(t\right)\text{,}\hfill \\ \frac{d{S}_{22}\left(t\right)}{dt}={\lambda }_2-d_2{S}_{22}\left(t\right)\text{,}\hfill \end{array}\}t\ne (n+l)\tau ,t\ne (n+1)\tau \text{,}\hfill \\ \begin{array}{c}△S_{21}\left(t\right)=D(S_{22}\left(t\right)-S_{21}\left(t\right))\text{,}\hfill \\ △S_{22}\left(t\right)=D(S_{21}\left(t\right)-S_{22}\left(t\right))\text{,}\hfill \end{array}\}t=(n+l)\tau \text{,}\hfill \\ \begin{array}{c}△S_{21}\left(t\right)=-{\mu }_1{S}_{21}\left(t\right)\text{,}\hfill \\ △S_{22}\left(t\right)=-{\mu }_2{S}_{22}\left(t\right)\text{,}\hfill \end{array}\}t=(n+1)\tau \hfill \end{array}$$ (4.8)

respectively.

$$\{\begin{array}{c}\widetilde{S_{31}\left(t\right)}=\{\begin{array}{c}\frac{1}{(d_1+\frac{{\beta }_1{\epsilon }_1}{1+{\alpha }_1{\epsilon }_1})}[{\lambda }_1-({\lambda }_1-(d_1+\frac{{\beta }_1{\epsilon }_1}{1+{\alpha }_1{\epsilon }_1})S_{31}^{\ast })e^{-(d_1+\frac{{\beta }_1{\epsilon }_1}{1+{\alpha }_1{\epsilon }_1})(t-n\tau )}]\text{,}t\in [n\tau ,(n+l)\tau )\text{,}\hfill \\ \frac{1}{(d_1+\frac{{\beta }_1{\epsilon }_1}{1+{\alpha }_1{\epsilon }_1})}[{\lambda }_1-({\lambda }_1-(d_1+\frac{{\beta }_1{\epsilon }_1}{1+{\alpha }_1{\epsilon }_1})S_{31}^{\ast \ast })e^{-(d_1+\frac{{\beta }_1{\epsilon }_1}{1+{\alpha }_1{\epsilon }_1})(t-(n+l)\tau )}],t\in [(n+l)\tau ,(n+1)\tau )\text{,}\hfill \end{array}\hfill \\ \widetilde{S_{32}\left(t\right)}=\{\begin{array}{c}\frac{1}{(d_2+\frac{{\beta }_2{\epsilon }_1}{1+{\alpha }_2{\epsilon }_1})}[{\lambda }_2-({\lambda }_2-(d_2+\frac{{\beta }_2{\epsilon }_1}{1+{\alpha }_2{\epsilon }_1})S_{32}^{\ast })e^{-(d_2+\frac{{\beta }_2{\epsilon }_1}{1+{\alpha }_2{\epsilon }_1})(t-n\tau )}]\text{,}t\in [n\tau ,(n+l)\tau )\text{.}\hfill \\ \frac{1}{(d_2+\frac{{\beta }_2{\epsilon }_1}{1+{\alpha }_2{\epsilon }_1})}[{\lambda }_2-({\lambda }_2-(d_2+\frac{{\beta }_2{\epsilon }_1}{1+{\alpha }_2{\epsilon }_1})S_{32}^{\ast \ast })e^{-(d_2+\frac{{\beta }_2{\epsilon }_1}{1+{\alpha }_2{\epsilon }_1})(t-(n+l)\tau )}],t\in [(n+l)\tau ,(n+1)\tau )\text{,}\hfill \end{array}\hfill \end{array}$$ (4.9)

here $S_{31}^{\ast }$ and $S_{32}^{\ast }$ are determined as

$$\{\begin{array}{c}{S}_{31}^{\ast }=\frac{(1-A_{31})B_3-A_3{A}_{32}}{(1-A_{31})(1-B_{32})-A_{32}{B}_{31}}>0\text{,}\hfill \\ S_{32}^{\ast }=\frac{B_{31}{B}_3-A_3(1-B_{32})}{(1-A_{31})(1-B_{32})-A_{32}{B}_{31}}>0\text{,}\hfill \end{array}$$ (4.10)

and $S_{31}^{\ast \ast }$ and $S_{32}^{\ast \ast }$ are defined as

$$\{\begin{array}{c}{S}_{31}^{\ast \ast }=\frac{1-D}{(d_1+\frac{{\beta }_1{\epsilon }_1}{1+{\alpha }_1{\epsilon }_1})}[{\lambda }_1-({\lambda }_1-(d_1+\frac{{\beta }_1{\epsilon }_1}{1+{\alpha }_1{\epsilon }_1})S_{31}^{\ast })e^{-(d_1+\frac{{\beta }_1{\epsilon }_1}{1+{\alpha }_1{\epsilon }_1})l\tau }]+\frac{D}{(d_2+\frac{{\beta }_2{\epsilon }_1}{1+{\alpha }_2{\epsilon }_1})}[{\lambda }_2-({\lambda }_2-(d_2+\frac{{\beta }_2{\epsilon }_1}{1+{\alpha }_2{\epsilon }_1})S_{32}^{\ast })e^{-(d_2+\frac{{\beta }_2{\epsilon }_1}{1+{\alpha }_2{\epsilon }_1})l\tau }]\text{,}\hfill \\ S_{32}^{\ast \ast }=\frac{D}{(d_1+\frac{{\beta }_1{\epsilon }_1}{1+{\alpha }_1{\epsilon }_1})}[{\lambda }_1-({\lambda }_1-(d_1+\frac{{\beta }_1{\epsilon }_1}{1+{\alpha }_1{\epsilon }_1})S_{31}^{\ast })e^{-(d_1+\frac{{\beta }_1{\epsilon }_1}{1+{\alpha }_1{\epsilon }_1})l\tau }]+\frac{1-D}{(d_2+\frac{{\beta }_2{\epsilon }_1}{1+{\alpha }_2{\epsilon }_1})}[{\lambda }_2-({\lambda }_2-(d_2+\frac{{\beta }_2{\epsilon }_1}{1+{\alpha }_2{\epsilon }_1})S_{32}^{\ast })e^{-(d_2+\frac{{\beta }_2{\epsilon }_1}{1+{\alpha }_2{\epsilon }_1})l\tau }]\text{,}\hfill \end{array}$$ (4.11)

where

$$A_{31}=(1-{\mu }_1)(1-D)e^{-(d_1+\frac{{\beta }_1{\epsilon }_1}{1+{\alpha }_1{\epsilon }_1})\tau }<1\text{,}$$

$$B_{31}=(1-{\mu }_1)D{e}^{-[(d_1+\frac{{\beta }_1{\epsilon }_1}{1+{\alpha }_1{\epsilon }_1})(1-l)+(d_2+\frac{{\beta }_2{\epsilon }_1}{1+{\alpha }_2{\epsilon }_1})l]\tau }<1\text{,}$$

$$A_{32}=(1-{\mu }_2)D{e}^{-[(d_1+\frac{{\beta }_1{\epsilon }_1}{1+{\alpha }_1{\epsilon }_1})l+(d_2+\frac{{\beta }_2{\epsilon }_1}{1+{\alpha }_2{\epsilon }_1})(1-l)]\tau }<1\text{,}$$

$$B_{32}=(1-{\mu }_2)(1-D)e^{-(d_2+\frac{{\beta }_2{\epsilon }_1}{1+{\alpha }_2{\epsilon }_1})\tau }<1\text{,}$$

$$A_3=(1-{\mu }_1)\times [\frac{{\lambda }_1(1-e^{-(d_1+\frac{{\beta }_1{\epsilon }_1}{1+{\alpha }_1{\epsilon }_1})l\tau })(1-(1-D)e^{-(d_1+\frac{{\beta }_1{\epsilon }_1}{1+{\alpha }_1{\epsilon }_1})(1-l)\tau })}{(d_1+\frac{{\beta }_1{\epsilon }_1}{1+{\alpha }_1{\epsilon }_1})}+\frac{D{\lambda }_2(1-e^{-(d_2+\frac{{\beta }_2{\epsilon }_1}{1+{\alpha }_2{\epsilon }_1})l\tau })e^{-(d_1+\frac{{\beta }_1{\epsilon }_1}{1+{\alpha }_1{\epsilon }_1})(1-l)\tau }}{(d_2+\frac{{\beta }_2{\epsilon }_1}{1+{\alpha }_2{\epsilon }_1})}]>0\text{,}$$

$$B_3=(1-{\mu }_2)\times [\frac{D{\lambda }_1(1-e^{-(d_1+\frac{{\beta }_1{\epsilon }_1}{1+{\alpha }_1{\epsilon }_1})l\tau })e^{-(d_2+\frac{{\beta }_2{\epsilon }_1}{1+{\alpha }_2{\epsilon }_1})(1-l)\tau }}{(d_1+\frac{{\beta }_1{\epsilon }_1}{1+{\alpha }_1{\epsilon }_1})}+\frac{{\lambda }_2(1-e^{-(d_2+\frac{{\beta }_2{\epsilon }_1}{1+{\alpha }_2{\epsilon }_1})l\tau })(1-(1-D)e^{-(d_2+\frac{{\beta }_2{\epsilon }_1}{1+{\alpha }_2{\epsilon }_1})(1-l)\tau })}{(d_2+\frac{{\beta }_2{\epsilon }_1}{1+{\alpha }_2{\epsilon }_1})}]>0\text{.}$$

For any ${\epsilon }_2>0$, there exists a $t_1,t>t_1$ such that

$$\widetilde{S_{31}\left(t\right)}-{\epsilon }_2<S_1\left(t\right)<\widetilde{S_1\left(t\right)}+{\epsilon }_2\text{,}$$

and

$$\widetilde{S_{32}\left(t\right)}-{\epsilon }_2<S_2\left(t\right)<\widetilde{S_2\left(t\right)}+{\epsilon }_2\text{.}$$

Let ${\epsilon }_1\to 0$, so we have

$$\widetilde{S_1\left(t\right)}-{\epsilon }_2<S_1\left(t\right)<\widetilde{S_1\left(t\right)}+{\epsilon }_2\text{,}$$

and

$$\widetilde{S_2\left(t\right)}-{\epsilon }_2<S_2\left(t\right)<\widetilde{S_2\left(t\right)}+{\epsilon }_2\text{,}$$

for $t$ large enough, which implies $S_1\left(t\right)\to \widetilde{S_1\left(t\right)}$ and $S_2\left(t\right)\to \widetilde{S_2\left(t\right)}$ as $t\to \infty$. This completes the proof.

The next work is to investigate the permanence of system (2.2).

Definition 4.2

System (2.2) is said to be permanent if there are constants $m,M>0$ (independent of initial value) and a finite time $T_0$ such that for all solutions $(S_1\left(t\right),I_1\left(t\right),S_2\left(t\right),I_2\left(t\right))$ with all initial values $S_1\left(0^{+}\right)>0,I_1\left(0^{+}\right)>0,S_2\left(0^{+}\right)>0,I_2\left(0^{+}\right)>0$, $m\le S_1\left(t\right)\le M,m\le I_1\left(t\right)\le M,m\le S_2\left(t\right)\le M,m\le I_2\left(t\right)\le M$, hold for all $t\ge T_0$. Here $T_0$ may depend on the initial values $(S_1\left(0^{+}\right),I_1\left(0^{+}\right),S_2\left(0^{+}\right),I_2\left(0^{+}\right))$.

Theorem 4.3

If

$$\underset{i=1,2}{min}\frac{{\beta }_i}{d_i}[{\lambda }_i\tau +\frac{{\lambda }_i-d_i\underset{i}{\overset{\ast }{S}}}{d_i}(e^{-d_{i}l\tau }-1)+\frac{{\lambda }_i-d_i\underset{i}{\overset{\ast \ast }{S}}}{d_i}(e^{-d_i\tau }-e^{-d_{i}l\tau })]-(r_i+d_i+b_i)\tau >0\text{,}$$ (4.12)

holds, system (2.2)   is permanent. Where $S_i^{\ast }(i=1,2)$ and $S_i^{\ast \ast }(i=1,2)$ are defined as   (3.7), (3.12) .

Proof

Suppose $(S_1\left(t\right),I_1\left(t\right),S_2\left(t\right),I_2\left(t\right))$ is a solution of (2.2) with $S_1\left(0\right)>0,I_1\left(0\right)>0,S_2\left(0\right)>0,I_2\left(0\right)>0$. By Lemma 3.4, we have proved there exists a constant $M>0$ such that $S_1\left(t\right)\le M,I_1\left(t\right)\le M,S_2\left(t\right)\le M,I_2\left(t\right)\le M$, for $t$ large enough. From system (2.2), we know $I_i\left(t\right)>I_i\left(0^{+}\right)e^{-(r_i+d_i+b_i)t}(i=1,2)$ for all $t$ large enough. Thus, we only need to find $m_1>0$ and ${\epsilon }_3$ such that $I_i\left(t\right)\ge m_1(i=1,2)$ for $t$ large enough. Otherwise, we can select $m_2>0$ small enough, and prove $I_i\left(t\right)<m_2(i=1,2)$ cannot hold for $t\ge 0$. By the condition (4.12), we can obtain

$${\sigma }_i=\frac{{\beta }_i}{(d_i+\frac{{\beta }_i{m}_2}{1+{\alpha }_i{m}_2})}[{\lambda }_i\tau +\frac{{\lambda }_i-(d_i+\frac{{\beta }_i{m}_2}{1+{\alpha }_i{m}_2})S_{4i}^{\ast }}{(d_i+\frac{{\beta }_i{m}_2}{1+{\alpha }_i{m}_2})}(e^{-(d_i+\frac{{\beta }_i{m}_2}{1+{\alpha }_i{m}_2})l\tau }-1)+\frac{{\lambda }_i-(d_i+\frac{{\beta }_i{m}_2}{1+{\alpha }_i{m}_2})S_{4i}^{\ast \ast }}{(d_i+\frac{{\beta }_i{m}_2}{1+{\alpha }_i{m}_2})}(e^{-(d_i+\frac{{\beta }_i{m}_2}{1+{\alpha }_i{m}_2})\tau }-e^{-(d_i+\frac{{\beta }_i{m}_2}{1+{\alpha }_i{m}_2})l\tau })]-(r_i+(d_i+\frac{{\beta }_i{m}_2}{1+{\alpha }_i{m}_2})+b_i)\tau -{\beta }_i{\epsilon }_3\tau >0\text{,}$$

with $S_{4i}^{\ast }(i=1,2)$ and $S_{4i}^{\ast \ast }(i=1,2)$ are defined as (4.16), (4.17). Then,

$$\{\begin{array}{c}\begin{array}{c}\frac{d{S}_1\left(t\right)}{dt}>{\lambda }_1-(d_1+\frac{{\beta }_1{m}_2}{1+{\alpha }_1{m}_2})S_1\left(t\right)\text{,}\hfill \\ \frac{d{S}_2\left(t\right)}{dt}>{\lambda }_2-(d_1+\frac{{\beta }_2{m}_2}{1+{\alpha }_2{m}_2})S_2\left(t\right)\text{,}\hfill \end{array}\}t\ne (n+l)\tau ,t\ne (n+1)\tau \text{,}\hfill \\ \begin{array}{c}△S_1\left(t\right)=D(S_2\left(t\right)-S_1\left(t\right))\text{,}\hfill \\ △S_2\left(t\right)=D(S_1\left(t\right)-S_2\left(t\right))\text{,}\hfill \end{array}\}t=(n+l)\tau \text{,}\hfill \\ \begin{array}{c}△S_1\left(t\right)=-{\mu }_1{S}_1\left(t\right)\text{,}\hfill \\ △S_2\left(t\right)=-{\mu }_2{S}_2\left(t\right)\text{,}\hfill \end{array}\}t=(n+1)\tau \text{.}\hfill \end{array}$$ (4.13)

By Lemmas 3.6, we have $S_1\left(t\right)\ge S_{41}\left(t\right)$, $S_2\left(t\right)\ge S_{42}\left(t\right)$ and $S_{41}\left(t\right)\to \overline{S_{41}\left(t\right)},S_{42}\left(t\right)\to \overline{S_{42}\left(t\right)},t\to \infty$, where $(S_{41}\left(t\right),S_{42}\left(t\right))$ is the solution of

$$\{\begin{array}{c}\begin{array}{c}\frac{d{S}_{41}\left(t\right)}{dt}={\lambda }_1-(d_1+\frac{{\beta }_1{m}_2}{1+{\alpha }_1{m}_2})S_{41}\left(t\right)\text{,}\hfill \\ \frac{d{S}_{42}\left(t\right)}{dt}={\lambda }_2-(d_2+\frac{{\beta }_2{m}_2}{1+{\alpha }_2{m}_2})S_{42}\left(t\right)\text{,}\hfill \end{array}\}t\ne (n+l)\tau ,t\ne (n+1)\tau \text{,}\hfill \\ \begin{array}{c}△S_{41}\left(t\right)=D(S_{42}\left(t\right)-S_{41}\left(t\right))\text{,}\hfill \\ △S_{42}\left(t\right)=D(S_{41}\left(t\right)-S_{42}\left(t\right))\text{,}\hfill \end{array}\}t=(n+l)\tau \text{,}\hfill \\ \begin{array}{c}△S_{41}\left(t\right)=-{\mu }_1{S}_{41}\left(t\right)\text{,}\hfill \\ △S_{42}\left(t\right)=-{\mu }_2{S}_{42}\left(t\right)\text{,}\hfill \end{array}\}t=(n+1)\tau \text{,}\hfill \end{array}$$ (4.14)

with

$$\{\begin{array}{c}\overline{S_{41}\left(t\right)}=\{\begin{array}{c}\frac{1}{(d_1+\frac{{\beta }_1{m}_2}{1+{\alpha }_1{m}_2})}[{\lambda }_1-({\lambda }_1-(d_1+\frac{{\beta }_1{m}_2}{1+{\alpha }_1{m}_2})S_{31}^{\ast })e^{-(d_1+\frac{{\beta }_1{m}_2}{1+{\alpha }_1{m}_2})(t-n\tau )}]\text{,}t\in [n\tau ,(n+l)\tau )\text{,}\hfill \\ \frac{1}{(d_1+\frac{{\beta }_1{m}_2}{1+{\alpha }_1{m}_2})}[{\lambda }_1-({\lambda }_1-(d_1+\frac{{\beta }_1{m}_2}{1+{\alpha }_1{m}_2})S_{31}^{\ast \ast })e^{-(d_1+\frac{{\beta }_1{m}_2}{1+{\alpha }_1{m}_2})(t-(n+l)\tau )}]\text{,}t\in [(n+l)\tau ,(n+1)\tau )\text{,}\hfill \end{array}\hfill \\ \overline{S_{42}\left(t\right)}=\{\begin{array}{c}\frac{1}{(d_2+\frac{{\beta }_2{m}_2}{1+{\alpha }_2{m}_2})}[{\lambda }_2-({\lambda }_2-(d_2+\frac{{\beta }_2{m}_2}{1+{\alpha }_2{m}_2})S_{32}^{\ast })e^{-(d_2+\frac{{\beta }_2{m}_2}{1+{\alpha }_2{m}_2})(t-n\tau )}]\text{,}t\in [n\tau ,(n+l)\tau )\text{.}\hfill \\ \frac{1}{(d_2+\frac{{\beta }_2{m}_2}{1+{\alpha }_2{m}_2})}[{\lambda }_2-({\lambda }_2-(d_2+\frac{{\beta }_2{m}_2}{1+{\alpha }_2{m}_2})S_{32}^{\ast \ast })e^{-(d_2+\frac{{\beta }_2{m}_2}{1+{\alpha }_2{m}_2})(t-(n+l)\tau )}]\text{,}t\in [(n+l)\tau ,(n+1)\tau )\text{,}\hfill \end{array}\hfill \end{array}$$ (4.15)

here $S_{41}^{\ast }$ and $S_{42}^{\ast }$ are determined as

$$\{\begin{array}{c}{S}_{41}^{\ast }=\frac{(1-A_{41})B_4-A_4{A}_{42}}{(1-A_{41})(1-B_{42})-A_{42}{B}_{41}}>0\text{,}\hfill \\ S_{42}^{\ast }=\frac{B_{41}{B}_4-A_4(1-B_{42})}{(1-A_{41})(1-B_{42})-A_{42}{B}_{41}}>0\text{,}\hfill \end{array}$$ (4.16)

and $S_{41}^{\ast \ast }$ and $S_{42}^{\ast \ast }$ are defined as

$$\{\begin{array}{c}{S}_{41}^{\ast \ast }=\frac{1-D}{(d_1+\frac{{\beta }_1{m}_2}{1+{\alpha }_1{m}_2})}[{\lambda }_1-({\lambda }_1-(d_1+\frac{{\beta }_1{m}_2}{1+{\alpha }_1{m}_2})S_{41}^{\ast })e^{-(d_1+\frac{{\beta }_1{m}_2}{1+{\alpha }_1{m}_2})l\tau }]+\frac{D}{(d_2+\frac{{\beta }_2{m}_2}{1+{\alpha }_2{m}_2})}[{\lambda }_2-({\lambda }_2-(d_2+\frac{{\beta }_2{m}_2}{1+{\alpha }_2{m}_2})S_{32}^{\ast })e^{-(d_2+\frac{{\beta }_2{m}_2}{1+{\alpha }_2{m}_2})l\tau }]\text{,}\hfill \\ S_{42}^{\ast \ast }=\frac{D}{(d_1+\frac{{\beta }_1{m}_2}{1+{\alpha }_1{m}_2})}[{\lambda }_1-({\lambda }_1-(d_1+\frac{{\beta }_1{m}_2}{1+{\alpha }_1{m}_2})S_{31}^{\ast })e^{-(d_1+\frac{{\beta }_1{m}_2}{1+{\alpha }_1{m}_2})l\tau }]+\frac{1-D}{(d_2+\frac{{\beta }_2{m}_2}{1+{\alpha }_2{m}_2})}[{\lambda }_2-({\lambda }_2-(d_2+\frac{{\beta }_2{m}_2}{1+{\alpha }_2{m}_2})S_{32}^{\ast })e^{-(d_2+\frac{{\beta }_2{m}_2}{1+{\alpha }_2{m}_2})l\tau }]\text{,}\hfill \end{array}$$ (4.17)

where

$$A_{41}=(1-{\mu }_1)(1-D)e^{-(d_1+\frac{{\beta }_1{m}_2}{1+{\alpha }_1{m}_2})\tau }<1\text{,}$$

$$B_{41}=(1-{\mu }_1)D{e}^{-[(d_1+\frac{{\beta }_1{m}_2}{1+{\alpha }_1{m}_2})(1-l)+(d_2+\frac{{\beta }_2{m}_2}{1+{\alpha }_2{m}_2})l]\tau }<1\text{,}$$

$$A_{42}=(1-{\mu }_2)D{e}^{-[(d_1+\frac{{\beta }_1{m}_2}{1+{\alpha }_1{m}_2})l+(d_2+\frac{{\beta }_2{m}_2}{1+{\alpha }_2{m}_2})(1-l)]\tau }<1\text{,}$$

$$B_{42}=(1-{\mu }_2)(1-D)e^{-(d_2+\frac{{\beta }_2{m}_2}{1+{\alpha }_2{m}_2})\tau }<1\text{,}$$

$$A_4=(1-{\mu }_1)\times [\frac{{\lambda }_1(1-e^{-(d_1+\frac{{\beta }_1{m}_2}{1+{\alpha }_1{m}_2})l\tau })(1-(1-D)e^{-(d_1+\frac{{\beta }_1{m}_2}{1+{\alpha }_1{m}_2})(1-l)\tau })}{(d_1+\frac{{\beta }_1{m}_2}{1+{\alpha }_1{m}_2})}+\frac{D{\lambda }_2(1-e^{-(d_2+\frac{{\beta }_2{m}_2}{1+{\alpha }_2{m}_2})l\tau })e^{-(d_1+\frac{{\beta }_1{m}_2}{1+{\alpha }_1{m}_2})(1-l)\tau }}{(d_2+\frac{{\beta }_2{m}_2}{1+{\alpha }_2{m}_2})}]>0\text{,}$$

$$B_4=(1-{\mu }_2)\times [\frac{D{\lambda }_1(1-e^{-(d_1+\frac{{\beta }_1{m}_2}{1+{\alpha }_1{m}_2})l\tau })e^{-(d_2+\frac{{\beta }_2{m}_2}{1+{\alpha }_2{m}_2})(1-l)\tau }}{(d_1+\frac{{\beta }_1{m}_2}{1+{\alpha }_1{m}_2})}+\frac{{\lambda }_2(1-e^{-(d_2+\frac{{\beta }_2{m}_2}{1+{\alpha }_2{m}_2})l\tau })(1-(1-D)e^{-(d_2+\frac{{\beta }_2{m}_2}{1+{\alpha }_2{m}_2})(1-l)\tau })}{(d_2+\frac{{\beta }_2{m}_2}{1+{\alpha }_2{m}_2})}]>0\text{.}$$

Therefore, there exist $T_1>0$ and ${\epsilon }_3>0$ such that

$$S_1\left(t\right)\ge S_{41}\left(t\right)\ge \overline{S_{41}\left(t\right)}-{\epsilon }_3\text{,}$$

and

$$S_2\left(t\right)\ge S_{42}\left(t\right)\ge \overline{S_{42}\left(t\right)}-{\epsilon }_3\text{.}$$

Then,

$$\frac{d{I}_i\left(t\right)}{dt}\ge [{\beta }_i(\overline{S_{4i}\left(t\right)}-{\epsilon }_3)-(r_i+d_i+b_i)]I_i\left(t\right)(i=1,2)\text{,}$$ (4.18)

for $t\ge T_1$. Let $N_1\in N$ and $N_1\tau >T_1$. Integrating (4.18) on $(n\tau ,(n+1)\tau ),n\ge N_1$, we have

$$I_i\left((n+1)\tau \right)\ge I_i\left(n{\tau }^{+}\right)exp\left({\int }_{n\tau }^{(n+1)\tau }[{\beta }_i(\overline{S_{4i}\left(t\right)}-{\epsilon }_3)-(r_i+d_i+b_i)]dt\right)=I_i\left(n\tau \right)e^{{\sigma }_i}(i=1,2)\text{,}$$

then $I_i\left((N_1+k)\tau \right)\ge I_i\left(N_1{\tau }^{+}\right)e^{k{\sigma }_i}\to \infty$, as $k\to \infty$, which is a contradiction to the boundedness of $I_i\left(t\right)$. Hence, there exists a $t_1>0$ such that $I_i\left(t\right)\ge m_1(i=1,2)$.

Thus, we can obtain $\frac{d{S}_i\left(t\right)}{dt}>{\lambda }_i-(d_i+\frac{{\beta }_{i}M}{1+{\alpha }_{i}M})S_i\left(t\right)(i=1,2)$. Then, the following comparatively impulsive differential equation is

$$\{\begin{array}{c}\begin{array}{c}\frac{d{S}_{51}\left(t\right)}{dt}={\lambda }_1-(d_1+\frac{{\beta }_{1}M}{1+{\alpha }_{1}M})S_{51}\left(t\right)\text{,}\hfill \\ \frac{d{S}_{52}\left(t\right)}{dt}={\lambda }_2-(d_2+\frac{{\beta }_{2}M}{1+{\alpha }_{1}M})S_{51}\left(t\right)\text{,}\hfill \end{array}\}t\ne (n+l)\tau ,t\ne (n+1)\tau \text{,}\hfill \\ \begin{array}{c}△S_{51}\left(t\right)=D(S_{52}\left(t\right)-S_{51}\left(t\right))\text{,}\hfill \\ △S_{52}\left(t\right)=D(S_{51}\left(t\right)-S_{52}\left(t\right))\text{.}\hfill \end{array}\}t=(n+l)\tau \text{,}\hfill \\ \begin{array}{c}△S_{51}\left(t\right)=-{\mu }_1{S}_{51}\left(t\right)\text{,}\hfill \\ △S_{52}\left(t\right)=-{\mu }_1{S}_{52}\left(t\right)\text{.}\hfill \end{array}\}t=(n+1)\tau \text{.}\hfill \end{array}$$ (4.19)

Similar to Lemma 3.6, we have

$$\{\begin{array}{c}\widetilde{S_{51}\left(t\right)}=\{\begin{array}{c}\frac{1}{(d_1+\frac{{\beta }_{1}M}{1+{\alpha }_{1}M})}[{\lambda }_1-({\lambda }_1-(d_1+\frac{{\beta }_{1}M}{1+{\alpha }_{1}M})S_{51}^{\ast })e^{-(d_1+\frac{{\beta }_{1}M}{1+{\alpha }_{1}M})(t-n\tau )}]\text{,}t\in [n\tau ,(n+l)\tau )\text{,}\hfill \\ \frac{1}{(d_1+\frac{{\beta }_{1}M}{1+{\alpha }_{1}M})}[{\lambda }_1-({\lambda }_1-(d_1+\frac{{\beta }_{1}M}{1+{\alpha }_{1}M})S_{51}^{\ast \ast })e^{-(d_1+\frac{{\beta }_{1}M}{1+{\alpha }_{1}M})(t-(n+l)\tau )}]\text{,}t\in [(n+l)\tau ,(n+1)\tau )\text{,}\hfill \end{array}\hfill \\ \widetilde{S_{52}\left(t\right)}=\{\begin{array}{c}\frac{1}{(d_2+\frac{{\beta }_{2}M}{1+{\alpha }_{2}M})}[{\lambda }_2-({\lambda }_2-(d_2+\frac{{\beta }_{2}M}{1+{\alpha }_{2}M})S_{32}^{\ast })e^{-(d_2+\frac{{\beta }_{2}M}{1+{\alpha }_{2}M})(t-n\tau )}]\text{,}t\in [n\tau ,(n+l)\tau )\text{.}\hfill \\ \frac{1}{(d_2+\frac{{\beta }_{2}M}{1+{\alpha }_{2}M})}[{\lambda }_2-({\lambda }_2-(d_2+\frac{{\beta }_{2}M}{1+{\alpha }_{2}M})S_{32}^{\ast \ast })e^{-(d_2+\frac{{\beta }_{2}M}{1+{\alpha }_{2}M})(t-(n+l)\tau )}]\text{,}t\in [(n+l)\tau ,(n+1)\tau )\text{,}\hfill \end{array}\hfill \end{array}$$ (4.20)

here $S_{51}^{\ast }$ and $S_{52}^{\ast }$ are determined as

$$\{\begin{array}{c}{S}_{51}^{\ast }=\frac{(1-A_{51})B_5-A_5{A}_{52}}{(1-A_{51})(1-B_{52})-A_{52}{B}_{51}}>0\text{,}\hfill \\ S_{52}^{\ast }=\frac{B_{51}{B}_4-A_5(1-B_{52})}{(1-A_{51})(1-B_{52})-A_{52}{B}_{51}}>0\text{,}\hfill \end{array}$$ (4.21)

and $S_{51}^{\ast \ast }$ and $S_{52}^{\ast \ast }$ are defined as

$$\{\begin{array}{c}{S}_{51}^{\ast \ast }=\frac{1-D}{(d_1+\frac{{\beta }_{1}M}{1+{\alpha }_{1}M})}[{\lambda }_1-({\lambda }_1-(d_1+\frac{{\beta }_{1}M}{1+{\alpha }_{1}M})S_{51}^{\ast })e^{-(d_1+\frac{{\beta }_{1}M}{1+{\alpha }_{1}M})l\tau }]+\frac{D}{(d_2+\frac{{\beta }_{2}M}{1+{\alpha }_{2}M})}[{\lambda }_2-({\lambda }_2-(d_2+\frac{{\beta }_{2}M}{1+{\alpha }_{2}M})S_{52}^{\ast })e^{-(d_2+\frac{{\beta }_{2}M}{1+{\alpha }_{2}M})l\tau }]\text{,}\hfill \\ S_{52}^{\ast \ast }=\frac{D}{(d_1+\frac{{\beta }_{1}M}{1+{\alpha }_{1}M})}[{\lambda }_1-({\lambda }_1-(d_1+\frac{{\beta }_{1}M}{1+{\alpha }_{1}M})S_{51}^{\ast })e^{-(d_1+\frac{{\beta }_{1}M}{1+{\alpha }_{1}M})l\tau }]+\frac{1-D}{(d_2+\frac{{\beta }_{2}M}{1+{\alpha }_{2}M})}[{\lambda }_2-({\lambda }_2-(d_2+\frac{{\beta }_{2}M}{1+{\alpha }_{2}M})S_{52}^{\ast })e^{-(d_2+\frac{{\beta }_{2}M}{1+{\alpha }_{2}M})l\tau }]\text{,}\hfill \end{array}$$ (4.22)

where

$$A_{51}=(1-{\mu }_1)(1-D)e^{-(d_1+\frac{{\beta }_{1}M}{1+{\alpha }_{1}M})\tau }<1\text{,}$$

$$B_{51}=(1-{\mu }_1)D{e}^{-[(d_1+\frac{{\beta }_{1}M}{1+{\alpha }_{1}M})(1-l)+(d_2+\frac{{\beta }_{2}M}{1+{\alpha }_{2}M})l]\tau }<1\text{,}$$

$$A_{52}=(1-{\mu }_2)D{e}^{-[(d_1+\frac{{\beta }_{1}M}{1+{\alpha }_{1}M})l+(d_2+\frac{{\beta }_{2}M}{1+{\alpha }_{2}M})(1-l)]\tau }<1\text{,}$$

$$B_{52}=(1-{\mu }_2)(1-D)e^{-(d_2+\frac{{\beta }_{2}M}{1+{\alpha }_{2}M})\tau }<1\text{,}$$

$$A_5=(1-{\mu }_1)\times [\frac{{\lambda }_1(1-e^{-(d_1+\frac{{\beta }_{1}M}{1+{\alpha }_{1}M})l\tau })(1-(1-D)e^{-(d_1+\frac{{\beta }_{1}M}{1+{\alpha }_{1}M})(1-l)\tau })}{(d_1+\frac{{\beta }_{1}M}{1+{\alpha }_{1}M})}+\frac{D{\lambda }_2(1-e^{-(d_2+\frac{{\beta }_{2}M}{1+{\alpha }_{2}M})l\tau })e^{-(d_1+\frac{{\beta }_{1}M}{1+{\alpha }_{1}M})(1-l)\tau }}{(d_2+\frac{{\beta }_{2}M}{1+{\alpha }_{2}M})}]>0\text{,}$$

$$B_5=(1-{\mu }_2)\times [\frac{D{\lambda }_1(1-e^{-(d_1+\frac{{\beta }_{1}M}{1+{\alpha }_{1}M})l\tau })e^{-(d_2+\frac{{\beta }_{2}M}{1+{\alpha }_{2}M})(1-l)\tau }}{(d_1+\frac{{\beta }_{1}M}{1+{\alpha }_{1}M})}+\frac{{\lambda }_2(1-e^{-(d_2+\frac{{\beta }_{2}M}{1+{\alpha }_{2}M})l\tau })(1-(1-D)e^{-(d_2+\frac{{\beta }_{2}M}{1+{\alpha }_{2}M})(1-l)\tau })}{(d_2+\frac{{\beta }_{2}M}{1+{\alpha }_{2}M})}]>0\text{.}$$

For any ${\epsilon }_4$ small enough, we obtain

$$S_{51}\left(t\right)>\widetilde{S_{51}\left(t\right)}-{\epsilon }_4\text{,}$$

and

$$S_{52}\left(t\right)>\widetilde{S_{52}\left(t\right)}-{\epsilon }_4\text{.}$$

From the comparison theorem of impulsive differential equation, we have

$$S_1\left(t\right)>S_{51}\left(t\right)>\widetilde{S_{51}\left(t\right)}-{\epsilon }_4>(S_{51}{\left(t\right)}^{\ast }+S_{51}{\left(t\right)}^{\ast \ast })-{\epsilon }_4=m_{51}\text{,}$$

and

$$S_2\left(t\right)>S_{52}\left(t\right)>\widetilde{S_{52}\left(t\right)}-{\epsilon }_4>(S_{52}{\left(t\right)}^{\ast }+S_{52}{\left(t\right)}^{\ast \ast })-{\epsilon }_4=m_{52}\text{,}$$

i.e.  $S_1\left(t\right)>m_{51}$ and $S_2\left(t\right)>m_{52}$. This completes the proof.

Corollary 4.4

If

$$\underset{i=1,2}{min}\frac{{\beta }_i}{d_i}[{\lambda }_i\tau +\frac{{\lambda }_i-d_i\underset{i}{\overset{\ast }{S}}}{d_i}(e^{-d_{i}l\tau }-1)+\frac{{\lambda }_i-d_i\underset{i}{\overset{\ast \ast }{S}}}{d_i}(e^{-d_i\tau }-e^{-d_{i}l\tau })]-(r_i+d_i+b_i)\tau >0\text{,}$$ (4.23)

holds, system   (2.1)   is permanent. Where $S_i^{\ast }(i=1,2)$ and $S_i^{\ast \ast }(i=1,2)$ are defined as   (3.7), (3.12) .

5. Discussion

In this paper, we establish an SIR model with impulsive dispersal,vaccination and restricting infected individuals boarding transports. This SIR epidemic model for two regions, which are connected by transportation of non-infected individuals, portrays the evolvement of diseases. We prove that all solutions of the investigated system are uniformly ultimately bounded. From (4.1) and (4.2), if ${max}_{i=1,2}\frac{{\beta }_i}{d_i}[{\lambda }_i\tau +\frac{{\lambda }_i-d_i{S}_i^{\ast }}{d_i}(e^{-d_{i}l\tau }-1)+\frac{{\lambda }_i-d_i{S}_i^{\ast \ast }}{d_i}(e^{-d_i\tau }-e^{-d_{i}l\tau })]-(r_i+d_i+b_i)\tau <0$ holds, the infection-free boundary periodic solution $(\widetilde{S_1\left(t\right)},0,0,\widetilde{S_2\left(t\right)},0,0)$ of system (2.1) is globally asymptotically stable. From (4.12) or (4.23), if ${min}_{i=1,2}\frac{{\beta }_i}{d_i}[{\lambda }_i\tau +\frac{{\lambda }_i-d_i{S}_i^{\ast }}{d_i}(e^{-d_{i}l\tau }-1)+\frac{{\lambda }_i-d_i{S}_i^{\ast \ast }}{d_i}(e^{-d_i\tau }-e^{-d_{i}l\tau })]-(r_i+d_i+b_i)\tau >0$ holds, system (2.1) is permanent. It is concluded that the approach of impulsive vaccination and restricting infected individuals boarding transports provides reliable tactic basis for preventing disease spread. In the real world, after the etiology the way of propagating, the effective methods of medical control of a new disease are clarified in terms of medical science, the combining vaccination and social control policy are effective methods to reduce the number of infected people. So our works will pay an important role to study new epidemic model with vaccination, impulsive dispersal and restricting infected individuals boarding transports.

Competing interests

The authors declare that they have no competing interest. All authors have read and approved the final manuscript.