# Global dynamics in a class of discrete-time epidemic models with disease courses

- Lei Wang
^{1}, - Qianqian Cui
^{1}and - Zhidong Teng
^{1}Email author

**2013**:57

https://doi.org/10.1186/1687-1847-2013-57

© Wang et al.; licensee Springer 2013

**Received: **19 June 2012

**Accepted: **21 February 2013

**Published: **15 March 2013

## Abstract

In this paper, a class of discrete SIRS epidemic models with disease courses is studied. The basic reproduction number ${R}_{0}$ is computed. The main results on the permanence and extinction of the disease are established. That is, the disease-free equilibrium is globally attractive if ${R}_{0}<1$, and there exists a unique endemic equilibrium and the disease is also permanent if ${R}_{0}>1$.

**MSC:**39A30, 92D30.

## Keywords

## 1 Introduction

In recent years, more and more attention has been paid to the discrete-time epidemic models. There are several reasons for that. Firstly, since the statistic data about a disease is collected by day, week, month or year, it is more direct, more convenient and more accurate to describe the disease by using the discrete-time models than the continuous-time models; secondly, the discrete-time models have more wealthy dynamical behaviors; for example, the single-species discrete-time models have bifurcations, chaos and other more complex dynamical behaviors.

For a discrete-time epidemic model, we see that at the present time, the main research subjects are the computation of the basic reproduction number, the local and global stability of the disease-free equilibrium and endemic equilibrium, the extinction, persistence and permanence of the disease, and the bifurcations, chaos and more complex dynamical behaviors of the model, *etc.* Many important and interesting results can be found in articles [1–24] and the references cited therein.

The global stability of disease-free equilibrium and endemic equilibrium and the permanence of the disease were obtained.

However, we know that many diseases have different disease courses, for example, tuberculosis, syphilis, AIDS, *etc.* Therefore, taking into account the epidemic models with disease courses is very important since disease pathogen bacteria with different course may have different reproduction and survival capacities, which indirectly influences the population growth. Under a different disease course, the transmission rate, the mortality and other vital parameters will be different [25–27].

*m*disease courses. Let $x(n)$ be the number of susceptible individuals at the

*n*th generation, ${y}_{j}(n)$ ($j=1,2,\dots ,m$) denote the number of infectious individuals who are in the

*j*th course of a disease at

*n*th generation, and let $z(t)$ denote the number of recovered individuals at the

*n*th generation. We introduce the following assumptions.

- (1)
The susceptible

*x*has a constant input rate Λ and a natural death rate*d*. - (2)
The susceptible individuals of the $(n+1)$th generation are only infected by the infectious individuals of the

*n*th generation, and ${\beta}_{j}$ is the constant transmission coefficient of which the susceptible is infected by compartment ${y}_{j}$. - (3)
After a susceptible individual contacts infectives and is infected, he/she will firstly enter compartment ${y}_{1}$, and then turn into compartments ${y}_{2},{y}_{3},\dots $ , finally into compartment ${y}_{m}$.

- (4)
The infectious ${y}_{j}$ in the

*j*th disease course admits the constant natural death rate*d*, the constant death rate induced by disease ${\alpha}_{j}$, the constant recovery rate ${\gamma}_{j}$ and the constant transmit rate ${\epsilon}_{j}$ from compartment ${y}_{i}$ to ${y}_{i+1}$. - (5)
The recovered

*z*admits the constant natural death rate*d*, does not have permanent immunity, hence there is a constant transfer rate*δ*from the recovered class back to the susceptible class.

*m*disease courses can be established as follows:

where ${p}_{i}=d+{\epsilon}_{i}+{\alpha}_{i}+{\gamma}_{i}$, $i=1,2,\dots ,m-1$, ${p}_{m}=d+{\alpha}_{m}+{\gamma}_{m}$. For model (1), we always assume that the following basic hypotheses hold.

(H_{1}) For each $i=1,2,\dots ,m$, $d>0$, ${\epsilon}_{i}>0$, ${\alpha}_{i}\ge 0$, ${\beta}_{i}\ge 0$, ${\beta}_{m}>0$, ${\gamma}_{i}>0$, $\delta \ge 0$, $0<{p}_{i}<1$ and $0<\delta +d<1$.

_{2}) Any solution of model (1) satisfies the following initial conditions:

**Remark 1** For model (1), we can easily see that when ${\beta}_{1}>0$ then model (1) describes a discrete SIRS type epidemic model with disease courses, where ${y}_{i}(n)$ ($i=1,2,\dots ,m$) denotes the number of infectious individuals in the *i* th course of the disease; and when ${\beta}_{1}=0$ then model (1) describes a discrete SEIRS type epidemic model with disease courses, where ${y}_{1}(n)$ is exposed and ${y}_{i}(n)$ ($i=2,3,\dots ,m$) denotes the number of infectious individuals in the *i* th course of the disease.

**Remark 2** In model (1), based on the above assumption (1), we know that the disease incidence term is denoted by ${\sum}_{i=1}^{m}{\beta}_{i}{y}_{i}(n)x(n+1)$. This makes $x(n+1)$, *i.e.*, the susceptible number of the $(n+1)$th generation, appear on both sides of the first equation. The reason for the above arguments is based on two considerations. On the one hand, it is influenced by the works given in [11, 18]; on the other hand, for the sake of convenience for mathematical analysis, especially, the positivity of solutions in model (1).

In this paper, by developing the methods given in [10, 11, 18], we will give the explicit expression of the basic reproduction number ${R}_{0}$. The criteria on the permanence and extinction of the disease will be established. That is, the disease-free equilibrium is globally attractive if ${R}_{0}<1$, and there exists a unique endemic equilibrium and the disease is also permanent if ${R}_{0}>1$.

This paper is organized as follows. In Section 2, as preliminaries we will give several lemmas which will be used in the proofs of the main results. In Section 3, the basic reproduction number is calculated, the existence on the disease-free equilibrium and endemic equilibrium is given and the theorem on the globally asymptotic stability of the disease-free equilibrium is stated and proved. In Section 4, we will obtain the permanence of the disease. Conclusions are presented in the last section.

## 2 Preliminaries

*k*be any positive integer, we denote ${R}_{+}^{k}=\{({x}_{1},{x}_{2},\dots ,{x}_{k}):{x}_{i}\ge 0,i=1,2,\dots ,k\}$. For any sequence $\{f(n)\}$, we define

Firstly, on the positivity of solutions of model (1), we have the following result.

**Lemma 1** *For any solution* $(x(n),{y}_{1}(n),{y}_{2}(n),\dots ,{y}_{m}(n),z(n))$ *of model* (1), *it holds that* $x(n)>0$, ${y}_{j}(n)>0$, $z(n)>0$ ($j=1,2,\dots ,m$) *for all* $n>0$.

Therefore, by using the induction, we get $x(n)>0$, ${y}_{j}(n)>0$, $z(n)>0$ for all $n>0$ and $j=1,2,\dots ,m$. This completes the proof. □

**Lemma 2**

*For any solution*$(x(n),{y}_{1}(n),{y}_{2}(n),\dots ,{y}_{m}(n),z(n))$

*of model*(1),

*it follows that*

*Proof*

This completes the proof. □

On the weak permanence and permanence of the disease of model (1), we have the following definitions.

From Lemma 2, Theorem 1.1.3 and Theorem 1.10 given in [28], we can immediately obtain the following result.

**Lemma 3** *If the disease in model* (1) *is weak permanent*, *then it also is permanent*.

Similar to Lemma 2.3 in [11] and Lemma 5 in [18], we have the following result.

**Lemma 4**

*For any solution*$(x(n),{y}_{1}(n),{y}_{2}(n),\dots ,{y}_{m}(n),z(n))$

*of model*(1),

*the following inequalities hold*:

*Meanwhile*,

*we also have*

*where*

*Proof*From model (1) and Lemmas 1 and 2, we easily have that

Hence, we immediately obtain that ${\overline{y}}_{j}\ge {\epsilon}_{j-1}{\overline{y}}_{j-1}$ for $j=2,3,\dots ,m$.

Hence, inequality (5) holds.

Hence, inequality (6) holds. This completes the proof. □

## 3 Global attractivity of disease-free equilibrium

Firstly, on the existence of disease-free equilibrium and endemic equilibrium, we have the following result.

**Theorem 1**(1)

*Model*(1)

*always has a disease*-

*free equilibrium*${E}_{0}(\frac{\mathrm{\Lambda}}{d},0,\dots ,0)$.

- (2)

*Proof*The equilibrium of model (1) satisfies the following equations:

Having solved this equality, we obtain that ${y}_{1}=0$ and $x=\frac{1}{D}$.

When ${y}_{1}=0$, then from (12) we have ${y}_{i}=0$ for $i=2,3,\dots ,m$. Further, from the first and the last equations of (11), we have $z=0$ and $x=\frac{\mathrm{\Lambda}}{d}$. This shows that model (1) has a disease-free equilibrium ${E}_{0}(\frac{\mathrm{\Lambda}}{d},0,\dots ,0)$.

Thus, from (14) we obtain that ${y}_{1}>0$ if and only if ${R}_{0}>1$. Further, from (12) we obtain ${y}_{i}>0$ for $i=2,3,\dots ,m$. Finally, from (13), we also have $z>0$. Therefore, we prove that model (1) has a unique endemic equilibrium ${E}^{\ast}$. This completes the proof. □

**Remark 3**

The first term $\frac{{\beta}_{1}\mathrm{\Lambda}}{{p}_{1}d}$ denotes the ultimate number of the susceptible at the end of the first disease course which is infected by an infectious individual of the first disease course. The second term $\frac{{\epsilon}_{1}{\beta}_{2}\mathrm{\Lambda}}{{p}_{1}{p}_{2}d}$ denotes the ultimate number of the susceptible at the end of the second disease course which is infected by an infectious individual of the second disease course. And lastly, the final term $\frac{{\epsilon}_{1}\cdots {\epsilon}_{m-1}{\beta}_{m}\mathrm{\Lambda}}{{p}_{1}\cdots {p}_{m}d}$ denotes the ultimate number of the susceptible at the end of the *m* th disease course which is infected by an infectious individual of the *m* th disease course. We see that ${R}_{0}$ is the sum of these ultimate numbers. This shows that ${R}_{0}$ certainly is the basic reproduction number of model (1).

**Theorem 2**

*If*${R}_{0}<1$,

*then the disease*-

*free equilibrium*${E}_{0}$

*of model*(1)

*is globally attractive*.

*That is*,

*for any solution*$(x(n),{y}_{1}(n),{y}_{2}(n),\dots ,{y}_{m}(n),z(n))$

*of model*(1),

*we have*

*Proof*Since ${R}_{0}=D\frac{\mathrm{\Lambda}}{d}<1$, then by inequality (6) in Lemma 4, we can obtain ${\underline{y}}_{j}={\overline{y}}_{j}=0$ ($j=1,2,\dots ,m$). In fact, if for some $j\in \{1,2,\dots ,m\}$ such that ${\overline{y}}_{j}>0$, then by inequality (4) in Lemma 4, we can obtain ${\overline{y}}_{m}>0$. Hence, ${\sum}_{j=1}^{m}{\beta}_{j}{\overline{y}}_{j}>0$. From inequality (6) in Lemma 4 and ${R}_{0}<1$, we have

which leads to a contradiction. Hence, ${lim}_{n\to \mathrm{\infty}}{y}_{j}(n)=0$ ($j=1,2,\dots ,m$). Finally, from the expression of $x(n)$ and $z(n)$ of model (1), we can infer that (15) holds. This completes the proof. □

## 4 Permanence of disease

where ${\{g(n)\}}_{n=1}^{\mathrm{\infty}}$ is a given non-negative bounded real sequence, and parameters ${p}_{i}$, ${\beta}_{i}$, ${\gamma}_{i}$, ${\epsilon}_{j}$, *δ* and *d* ($i=1,2,\dots ,m$, $j=1,2,\dots ,m-1$) are defined as in model (1). We have the following result.

**Lemma 5**

*For any constants*$\eta >0$

*and*$M>0$,

*there exist a constant*$\xi =\xi (\eta )>0$

*and an integer*$T=T(M,\eta )>0$

*such that for any initial time*${n}_{0}\in {N}_{+}$

*and initial value*$({u}_{10},{u}_{20},\dots ,{u}_{m+1,0})\in {R}_{+}^{m+1}$

*with*$0<{u}_{i0}\le M$ ($i=1,2,\dots ,m+1$),

*if*${u}_{1}(n)\le \xi $

*for all*$n\ge {n}_{0}$,

*then we have*

*where* $({u}_{1}(n),{u}_{2}(n),\dots ,{u}_{m+1}(n))$ *is the solution of system* (16) *with the initial condition* $({u}_{1}({n}_{0}),{u}_{2}({n}_{0}),\dots ,{u}_{m+1}({n}_{0}))=({u}_{10},{u}_{20},\dots ,{u}_{m+1,0})$.

*Proof*Firstly, we consider the last equation of system (16)

*m*th equation of system (16)

This completes the proof. □

Therefore, we have the following result.

**Lemma 6**

*For any constants*$\epsilon >0$

*and*$M>0$,

*there exists an integer*${N}_{0}={N}_{0}(\epsilon ,M)>0$

*such that for any initial*${n}_{0}>0$

*and initial value*${v}_{0}$

*with*$0<{v}_{0}\le M$,

*we have*

*where* $v(n;{n}_{0},\eta )$ *is the solution of equation* (19) *with the initial condition* $v({n}_{0},\eta )={v}_{0}$.

*Proof*For any solution $v(n;{n}_{0},\eta )$ of equation (19) with the initial condition $v({n}_{0},\eta )={v}_{0}>0$, we define a function

*ε*, there exists an integer ${N}_{0}={N}_{0}(\epsilon ,M)>0$ such that for any initial time ${n}_{0}>0$ and initial value ${v}_{0}$ with $0<{v}_{0}\le M$, when $0<\eta \le {\eta}_{0}$, then for all $n\ge {n}_{0}+{N}_{0}$, we have

This shows that the conclusion of Lemma 6 holds. This completes the proof. □

For any $x,y\in {R}^{m}$ with $x=({x}_{1},{x}_{2},\dots ,{x}_{m})$ and $y=({y}_{1},{y}_{2},\dots ,{y}_{m})$, if ${x}_{i}\le {y}_{i}$ ($i=1,2,\dots ,m$), then we denote $x\le y$.

then the function $f(n,u)$ is said to be non-decreasing for $u\in D$.

**Lemma 7** (See [29])

*Let the domain*$D\subset {R}_{+}^{m}$

*and the function*$f(n,x)$

*defined on*$n\ge 0$

*and*$x\in D$

*be non*-

*decreasing for*$x\in D$.

*If the sequence*${\{x(n)\}}_{n=1}^{\mathrm{\infty}}\subset D$

*for all*$n\ge {n}_{0}$

*satisfies*$x(n+1)\ge f(n,y(n))$ ($x(n+1)\le f(n,y(n))$),

*then we have*

*where* $y(n)$ *is the solution of the difference equation* $y(n+1)=f(n,y(n))$ *with initial value* $x({n}_{0})\ge y({n}_{0})$ ($x({n}_{0})\le y({n}_{0})$).

where *A* is an $m\times m$ non-negative matrix and $x(n)\in {R}^{m}$. Then we have the following result.

**Lemma 8**

*Let*$r=r(A)$

*be the spectral radius of matrix*

*A*,

*then the following conclusions hold*.

- (1)
*There exists an**m*-*dimensional column vector*$e={({e}_{1},{e}_{2},\dots ,{e}_{m})}^{T}$*with*${e}_{i}>0$ ($i=1,2,\dots ,m$)*such that*$x(n)={\lambda}^{n}e$*is a solution of system*(22). - (2)
*For any*${x}_{0}=({x}_{10},{x}_{20},\dots ,{x}_{m0})$*with*${x}_{i0}>0$ ($i=1,2,\dots ,m$),*there exist constants*${a}_{i}>0$ ($i=1,2$)*such that*${\lambda}^{n-{n}_{0}}{a}_{1}e\le x(n)\le {\lambda}^{n-{n}_{0}}{a}_{2}e\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}\phantom{\rule{0.5em}{0ex}}n\ge {n}_{0},$

*where* $x(n)$ *is the solution of system* (22) *satisfying the initial condition* $x({n}_{0})={x}_{0}$.

*Proof*(1) Since

*λ*is an eigenvalue of matrix

*A*and matrix

*A*is a non-negative matrix, we can obtain that there is a vector $e>0$ corresponding to the eigenvalue

*λ*such that

- (2)Let $b={min}_{1\le i\le m}\{{x}_{i0}\}$, $B={max}_{1\le i\le m}\{{x}_{i0}\}$, $c={min}_{1\le i\le m}\{{e}_{i}\}$ and $C={max}_{1\le i\le m}\{{e}_{i}\}$. Further let ${a}_{1}=b/C$ and ${a}_{2}=B/c$. Then we have ${a}_{1}e\le {x}_{0}\le {a}_{2}e$. Let ${x}_{1}(n)={a}_{1}e{\lambda}^{n-{n}_{0}}$ and ${x}_{2}(n)={a}_{2}e{\lambda}^{n-{n}_{0}}$. Then ${x}_{1}(n)$ and ${x}_{2}(n)$ are the solutions of system (22) with initial value ${x}_{1}({n}_{0})={a}_{1}e$ and ${x}_{2}({n}_{0})={a}_{2}e$, respectively. Then from Lemma 7 it follows that${\lambda}^{n-{n}_{0}}{a}_{1}e\le {x}_{1}(n)\le x(n)\le {x}_{2}(n)\le {\lambda}^{n-{n}_{0}}{a}_{2}e$

for all $n\ge {n}_{0}$. □

**Lemma 9**

*If*${R}_{0}>1$,

*then there exists a constant*${h}_{1}>0$

*such that for any solution*$(x(n),{y}_{1}(n),{y}_{2}(n),\dots ,{y}_{m}(n),z(n))$

*of model*(1),

*we have*

*Proof*

*δ*and

*d*in system (29) are given as in model (1). Since $(x(n),{y}_{1}(n),\dots ,{y}_{m}(n),z(n))$ is the solution of model (1), then $({y}_{1}(n),\dots ,{y}_{m}(n),z(n))$ is the solution of system (29). From (28) and Lemma 5, for above ${\eta}_{0}>0$ and $M=\frac{\mathrm{\Lambda}}{d}+1$, there exist a constant ${\delta}^{\ast}>0$ and an integer ${N}_{2}>0$ with ${\delta}^{\ast}\le {\eta}_{0}$ such that for any initial time ${n}_{0}\ge {N}_{1}$ and initial value $({u}_{10},{u}_{20},\dots ,{u}_{m+1,0})\in {R}_{+0}^{m}$, if ${u}_{1}(n)\le {\delta}^{\ast}$ for all $n\ge {n}_{0}$, then we have ${u}_{j}(n)\le {\eta}_{0}$ for all $n\ge {n}_{0}+{N}_{2}$ and $j=1,2,\dots ,m+1$. Hence, if ${y}_{1}(n)\le {\delta}^{\ast}$ for all $n\ge {n}_{0}$, then we have

which leads to a contradiction. This completes the proof. □

Lastly, directly from Lemma 3, Lemma 4 and Lemma 9, we can obtain the following result on the permanence of model (1).

**Theorem 3** *If* ${R}_{0}>1$, *then the disease in model* (1) *is permanent*.

*Proof*In fact, from Lemma 9, we obtain that for any positive solution $(x(n),{y}_{1}(n),\dots ,{y}_{m}(n),z(n))$ of model (1),

This completes the proof. □

## 5 Conclusions

In this paper, we study a class of discrete epidemic models with disease courses, that is, model (1). The basic reproduction number ${R}_{0}$ is calculated. It is shown that the global dynamics of model (1) is determined by the basic reproduction number ${R}_{0}$. If ${R}_{0}<1$, then we obtain that the disease-free equilibrium of model (1) is globally asymptotically stable. This shows that when ${R}_{0}<1$ the disease in model (1) is extinct. If ${R}_{0}>1$, then we obtain that the endemic equilibrium of model (1) exists and the disease is permanent. Clearly, our condition given in this paper is the threshold condition between the extinction and the permanence of the disease. Hence, our results obtained in this paper extend the results given in [7, 11, 18] for the discrete epidemic models.

However, it is a pity that the case of the basic reproduction number ${R}_{0}=1$ is not discussed in this paper. From the results on the case ${R}_{0}=1$ obtained in [17, 18], we can guess that when ${R}_{0}=1$ then the disease-free equilibrium of model (1) is also globally asymptotically stable. This shows that when ${R}_{0}=1$ the disease in model (1) is also extinct. The other one which is not obtained in our this paper is the global stability of the endemic equilibrium of model (1). From the results on the global stability of the endemic equilibrium obtained in [7, 17, 18], we can guess that when ${R}_{0}>1$ the endemic equilibrium of model (1) is globally stable. We will discuss these problems in our future work.

## Declarations

### Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11271312, 11001235, 10901130), the China Postdoctoral Science Foundation (Grant Nos. 20110491750, 2012T50836) and the Natural Science Foundation of Xinjiang (Grant No. 2011211B08).

## Authors’ Affiliations

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