This section deals with the existence and stability of the endemic equilibrium of model (1). Let {E}^{\ast}({S}^{\ast},{I}^{\ast},{R}^{\ast}) be the endemic equilibrium of model (1). Then {S}^{\ast}, {I}^{\ast}, and {R}^{\ast} satisfy the following equations:

\mathrm{\Lambda}-\frac{\beta {S}^{\ast}{I}^{\ast}}{{N}^{\ast}}-\mu {S}^{\ast}=0,\phantom{\rule{2em}{0ex}}\frac{\beta {S}^{\ast}}{{N}^{\ast}}-(\mu +\delta +\gamma )=0,\phantom{\rule{2em}{0ex}}\gamma {I}^{\ast}-\mu {R}^{\ast}=0,

where {N}^{\ast}={S}^{\ast}+{I}^{\ast}+{R}^{\ast}. By straightforward and careful calculations, we know that model (1) has a unique endemic equilibrium when {R}_{0}>1 with

\begin{array}{c}{S}^{\ast}=\frac{\mathrm{\Lambda}(\mu +\gamma )}{\mu ((\mu +\delta +\gamma )({R}_{0}-1)+(\mu +\gamma ))},\hfill \\ {I}^{\ast}=\frac{\mathrm{\Lambda}({R}_{0}-1)}{(\mu +\delta +\gamma )({R}_{0}-1)+(\mu +\gamma )},\phantom{\rule{2em}{0ex}}{R}^{\ast}=\frac{\gamma {I}^{\ast}}{\mu}.\hfill \end{array}

The local stability of the endemic equilibrium {E}^{\ast} is given in the following theorem.

**Theorem 3** *If* {R}_{0}>1, *then the endemic equilibrium* {E}^{\ast}=({S}^{\ast},{I}^{\ast},{R}^{\ast}) *of model* (1) *is locally asymptotically stable*.

*Proof* In order to discuss the stability of the endemic equilibrium of model (1), we consider the following equivalent system:

\{\begin{array}{c}N(t+1)=N(t)+\mathrm{\Lambda}-\mu N(t)-\delta I(t),\hfill \\ I(t+1)=I(t)+(\mu +\delta +\gamma )({R}_{0}-1)I(t)-\frac{\beta I{(t)}^{2}}{N(t)}-\frac{\beta I(t)R(t)}{N(t)},\hfill \\ R(t+1)=R(t)+\gamma I(t)-\mu R(t).\hfill \end{array}

(15)

When {R}_{0}>1, the positive equilibrium of system (15) is ({N}^{\ast},{I}^{\ast},{R}^{\ast}), where

{N}^{\ast}=\frac{\mathrm{\Lambda}(\mu +\gamma ){R}_{0}}{\mu ((\mu +\delta +\gamma )({R}_{0}-1)+(\mu +\gamma ))}.

The linearization matrix of (15) at the positive equilibrium ({N}^{\ast},{I}^{\ast},{R}^{\ast}) is

{A}_{1}=\left(\begin{array}{ccc}1-\mu & -\delta & 0\\ \frac{\beta {I}^{\ast}({I}^{\ast}+{R}^{\ast})}{{{N}^{\ast}}^{2}}& 1-\frac{\beta {I}^{\ast}}{{N}^{\ast}}& -\frac{\beta {I}^{\ast}}{{N}^{\ast}}\\ 0& \gamma & 1-\mu \end{array}\right).

The characteristic equation of matrix {A}_{1} is

\varphi (\lambda )=(1-\mu -\lambda )((1-\mu -\lambda )(1-\frac{\beta {I}^{\ast}}{{N}^{\ast}}-\lambda )+\frac{\gamma \beta {I}^{\ast}}{{N}^{\ast}}+\frac{\delta \beta {I}^{\ast}({I}^{\ast}+{R}^{\ast})}{{{N}^{\ast}}^{2}}).

We see that the equation \varphi (\lambda )=0 has an eigenvalue 0<{\lambda}_{1}=1-\mu <1. Therefore, in order to determine the stability of the positive equilibrium of model (15), we discuss the roots of the following equation:

\begin{array}{rcl}{\varphi}_{1}(\lambda )& =& (1-\mu -\lambda )(1-\frac{\beta {I}^{\ast}}{{N}^{\ast}}-\lambda )+\frac{\gamma \beta {I}^{\ast}}{{N}^{\ast}}+\frac{\delta \beta {I}^{\ast}({I}^{\ast}+{R}^{\ast})}{{{N}^{\ast}}^{2}}\\ =& {\lambda}^{2}-(2-\mu -\frac{\beta {I}^{\ast}}{{N}^{\ast}})\lambda +(1-\mu )(1-\frac{\beta {I}^{\ast}}{{N}^{\ast}})+\frac{\gamma \beta {I}^{\ast}}{{N}^{\ast}}+\frac{\delta \beta {I}^{\ast}({I}^{\ast}+{R}^{\ast})}{{{N}^{\ast}}^{2}}.\end{array}

When {R}_{0}>1, the calculation yields

\begin{array}{c}\begin{array}{rl}{\varphi}_{1}(1)& =1-(2-\mu -\frac{\beta {I}^{\ast}}{{N}^{\ast}})+(1-\mu )(1-\frac{\beta {I}^{\ast}}{{N}^{\ast}})+\frac{\gamma \beta {I}^{\ast}}{{N}^{\ast}}+\frac{\delta \beta {I}^{\ast}({I}^{\ast}+{R}^{\ast})}{{{N}^{\ast}}^{2}}\\ =\frac{\mu \beta {I}^{\ast}}{{N}^{\ast}}+\frac{\gamma \beta {I}^{\ast}}{{N}^{\ast}}+\frac{\delta \beta {I}^{\ast}({I}^{\ast}+{R}^{\ast})}{{{N}^{\ast}}^{2}}>0,\end{array}\hfill \\ \begin{array}{rl}{\varphi}_{1}(-1)& =1+(2-\mu -\frac{\beta {I}^{\ast}}{{N}^{\ast}})+(1-\mu )(1-\frac{\beta {I}^{\ast}}{{N}^{\ast}})+\frac{\gamma \beta {I}^{\ast}}{{N}^{\ast}}+\frac{\delta \beta {I}^{\ast}({I}^{\ast}+{R}^{\ast})}{{{N}^{\ast}}^{2}}\\ =2(1-\mu -\frac{\beta {I}^{\ast}}{{N}^{\ast}})+2+\frac{\mu \beta {I}^{\ast}}{{N}^{\ast}}+\frac{\gamma \beta {I}^{\ast}}{{N}^{\ast}}+\frac{\delta \beta {I}^{\ast}({I}^{\ast}+{R}^{\ast})}{{{N}^{\ast}}^{2}}\\ >2(1-\mu -\beta )+2+\frac{\mu \beta {I}^{\ast}}{{N}^{\ast}}+\frac{\gamma \beta {I}^{\ast}}{{N}^{\ast}}+\frac{\delta \beta {I}^{\ast}({I}^{\ast}+{R}^{\ast})}{{{N}^{\ast}}^{2}}>0.\end{array}\hfill \end{array}

Furthermore, the constant term satisfies

\begin{array}{rcl}c& =& (1-\mu )(1-\frac{\beta {I}^{\ast}}{{N}^{\ast}})+\frac{\gamma \beta {I}^{\ast}}{{N}^{\ast}}+\frac{\delta \beta {I}^{\ast}({I}^{\ast}+{R}^{\ast})}{{{N}^{\ast}}^{2}}\\ =& 1-\mu -\frac{\beta {I}^{\ast}}{{N}^{\ast}}+\frac{\gamma \beta {I}^{\ast}}{{N}^{\ast}}+\frac{\delta \beta {I}^{\ast}}{{N}^{\ast}}\frac{{R}_{0}-1}{{R}_{0}}\\ <& 1-\mu -(1-\mu -\delta -\gamma )\frac{\beta {I}^{\ast}}{{N}^{\ast}}<1.\end{array}

The Jury criterion [16] implies that the two roots, {\lambda}_{2} and {\lambda}_{3}, of the equation {\varphi}_{1}(\lambda )=0 satisfy |{\lambda}_{2}|<1 and |{\lambda}_{3}|<1. The linearization theory implies that the positive equilibrium ({N}^{\ast},{I}^{\ast},{R}^{\ast}) of system (15) is locally asymptotically stable if {R}_{0}>1, *i.e.*, the endemic equilibrium {E}^{\ast} of system (1) is locally asymptotically stable. □

The global stability of the endemic equilibrium {E}^{\ast} of model (1) is quite difficult. Sufficient conditions are presented in two theorems for the special case \delta =0 and for the general case.

**Theorem 4** *Assume that* \delta =0. *The endemic equilibrium* {E}^{\ast}({S}^{\ast},{I}^{\ast},{R}^{\ast}) *of model* (1) *is globally asymptotically stable if* {R}_{0}>\frac{\mu +\gamma}{\mu} *and* \gamma <\mu.

*Proof* When \delta =0, the host population N(t)=S(t)+I(t)+R(t) satisfies

N(t+1)=\mathrm{\Lambda}+(1-\mu )N(t),\phantom{\rule{2em}{0ex}}N(0)={N}_{0}>0,

(16)

which has the solution

N(t)=\frac{\mathrm{\Lambda}(1-{(1-\mu )}^{t})}{\mu}+{(1-\mu )}^{t}{N}_{0}.

From the solution of the host population, we have that {lim}_{t\to \mathrm{\infty}}N(t)={N}^{\ast}=\frac{\mathrm{\Lambda}}{\mu}, and the limit of N(t) is independent of the initial value {N}_{0}.

The global stability of the endemic equilibrium, {E}^{\ast}, of model (1) is equivalent to the global stability of the endemic equilibrium {E}_{N}^{\ast}({N}^{\ast},{S}^{\ast},{I}^{\ast}) of the following model:

\{\begin{array}{c}N(t+1)=\mathrm{\Lambda}+(1-\mu )N(t),\hfill \\ S(t+1)=S(t)+\mathrm{\Lambda}-\frac{\beta S(t)I(t)}{N(t)}-\mu S(t),\hfill \\ I(t+1)=I(t)+\frac{\beta S(t)I(t)}{N(t)}-(\mu +\gamma )I(t).\hfill \end{array}

(17)

The first equation of model (17) is independent of other two state variables S(t) and I(t). The fact that {lim}_{t\to \mathrm{\infty}}N(t)={N}^{\ast} leads to the following limit model:

\{\begin{array}{c}S(t+1)=S(t)+\mathrm{\Lambda}-\frac{\beta \mu S(t)I(t)}{\mathrm{\Lambda}}-\mu S(t),\hfill \\ I(t+1)=I(t)+\frac{\beta \mu S(t)I(t)}{\mathrm{\Lambda}}-(\mu +\gamma )I(t).\hfill \end{array}

(18)

The limiting model (18) possesses the same dynamical property as that of model (17). Both equations of model (18) are nonlinear, but with a similar term \frac{\beta \mu S(t)I(t)}{\mathrm{\Lambda}}. We introduce the new variable L(t)=S(t)+I(t) to have one linear equation, then we have

\begin{array}{r}L(t+1)=L(t)+\mathrm{\Lambda}-\gamma I(t)-\mu L(t),\\ I(t+1)=I(t)+\frac{\beta \mu (L(t)-I(t))I(t)}{\mathrm{\Lambda}}-(\mu +\gamma )I(t).\end{array}

(19)

The global stability of the positive equilibrium of model (19) is equivalent to that of model (17). From the inequality 0\le I(t)\le L(t) and the first equation of (19), we have

\begin{array}{r}L(t+1)\le L(t)+\mathrm{\Lambda}-\mu L(t),\\ L(t+1)\ge L(t)+\mathrm{\Lambda}-(\mu +\gamma )L(t).\end{array}

(20)

From those two inequalities in (20) and the comparison theorem [17], we know that for any small \epsilon >0, there exists a positive integer {T}_{1l} such that {L}_{1}^{l}\le L(t)\le {L}_{1}^{m} for t>{T}_{1l}, where {L}_{1}^{l}=\frac{\mathrm{\Lambda}}{\mu +\gamma}-\epsilon and {L}_{1}^{m}=\frac{\mathrm{\Lambda}}{\mu}+\epsilon.

When t\ge {T}_{1l}, we substitute {L}_{1}^{l}\le L(t)\le {L}_{1}^{m} into the second equation of (19) to obtain

\begin{array}{r}I(t+1)\le I(t)+\frac{\beta \mu ({L}_{1}^{m}-I(t))I(t)}{\mathrm{\Lambda}}-(\mu +\gamma )I(t),\\ I(t+1)\ge I(t)+\frac{\beta \mu ({L}_{1}^{l}-I(t))I(t)}{\mathrm{\Lambda}}-(\mu +\gamma )I(t).\end{array}

(21)

We consider the following two difference equations corresponding to the inequalities in (21):

\{\begin{array}{c}{I}_{1}^{m}(t+1)={I}_{1}^{m}(t)+\frac{\beta \mu ({L}_{1}^{m}-{I}_{1}^{m}(t)){I}_{1}^{m}(t)}{\mathrm{\Lambda}}-(\mu +\gamma ){I}_{1}^{m}(t),\hfill \\ {I}_{1}^{l}(t+1)={I}_{1}^{l}(t)+\frac{\beta \mu ({L}_{1}^{l}-{I}_{1}^{l}(t)){I}_{1}^{l}(t)}{\mathrm{\Lambda}}-(\mu +\gamma ){I}_{1}^{l}(t).\hfill \end{array}

(22)

We substitute {I}_{1}^{m}(t)=\frac{\mathrm{\Lambda}+\beta \mu {L}_{1}^{m}-\mathrm{\Lambda}(\mu +\gamma )}{\beta \mu}{x}_{1}^{m}(t) and {I}_{1}^{l}(t)=\frac{\mathrm{\Lambda}+\beta \mu {L}_{1}^{l}-\mathrm{\Lambda}(\mu +\gamma )}{\beta \mu}{x}_{1}^{l}(t) into the first and the second equations of (22) to have

\{\begin{array}{cc}{x}_{1}^{m}(t+1)={r}_{1}^{m}{x}_{1}^{m}(t)(1-{x}_{1}^{m}(t)),\hfill & \text{with}{r}_{1}^{m}=1-(\mu +\gamma )+\frac{\beta \mu {L}_{1}^{m}}{\mathrm{\Lambda}},\hfill \\ {x}_{1}^{l}(t+1)={r}_{1}^{l}{x}_{1}^{l}(t)(1-{x}_{1}^{l}(t)),\hfill & \text{with}{r}_{1}^{l}=1-(\mu +\gamma )+\frac{\beta \mu {L}_{1}^{l}}{\mathrm{\Lambda}}.\hfill \end{array}

(23)

The variables {x}_{1}^{m}(t) and {x}_{1}^{l}(t) satisfy the quadratic difference equation. The well-known result of the famous population model x(t+1)=rx(t)(1-x(t)) says that {x}_{0}^{\ast}=0 is the unique and globally asymptotically stable equilibrium if 0<r<1, whereas {x}_{1}^{\ast}=1-\frac{1}{r} is the unique positive equilibrium if 1<r<3 and {x}_{1}^{\ast} is globally asymptotically stable. The result on the quadratic population model x(t+1)=rx(t)(1-x(t)) implies that the first equation in (22) has a positive equilibrium {I}_{1\ast}^{m}={L}_{1}^{m}-\frac{\mathrm{\Lambda}}{\mu {R}_{0}}, which is globally asymptotically stable. A similar argument implies that the second equation in (22) has a positive equilibrium {I}_{1\ast}^{l}={L}_{1}^{l}-\frac{\mathrm{\Lambda}}{\mu {R}_{0}}, which is globally asymptotically stable if {R}_{0}>\frac{\mu +\gamma}{\mu}.

The asymptotical stability and the comparison theory imply that there exists a {T}_{1i}\ge {T}_{1l} such that {I}_{1}^{l}\le I(t)\le {I}_{1}^{m} for all t>{T}_{1i}, where {I}_{1}^{l}={I}_{1\ast}^{l}-\epsilon and {I}_{1}^{m}={I}_{1\ast}^{m}+\epsilon. When t\ge {T}_{1i}, we substitute the inequality {I}_{1}^{l}\le I(t)\le {I}_{1}^{m} into the first equation of (19) and have

\begin{array}{r}L(t+1)\le L(t)+\mathrm{\Lambda}-\mu L(t)-\gamma {I}_{1}^{l},\\ L(t+1)\ge L(t)+\mathrm{\Lambda}-\mu L(t)-\gamma {I}_{1}^{m}.\end{array}

(24)

From (24) and a similar argument, we can obtain that there exists a positive integer {T}_{2l} such that {L}_{2}^{l}\le L(t)\le {L}_{2}^{m} for all t\ge {T}_{2l}, where {L}_{2}^{l}=\frac{\mathrm{\Lambda}-\gamma {I}_{1}^{m}}{\mu}-\epsilon, {L}_{2}^{m}=\frac{\mathrm{\Lambda}-\gamma {I}_{1}^{l}}{\mu}+\epsilon, and {L}_{1}^{l}<{L}_{2}^{l}<{L}_{2}^{m}<{L}_{1}^{m}.

When t\ge {T}_{2l}, the inequality {L}_{2}^{l}\le L(t)\le {L}_{2}^{m} and the second equation of (19) imply that

\begin{array}{r}I(t+1)\le I(t)+\frac{\beta \mu ({L}_{2}^{m}-I(t))I(t)}{\mathrm{\Lambda}}-(\mu +\gamma )I(t),\\ I(t+1)\ge I(t)+\frac{\beta \mu ({L}_{2}^{l}-I(t))I(t)}{\mathrm{\Lambda}}-(\mu +\gamma )I(t).\end{array}

(25)

A similar argument implies that there exists a {T}_{2i}\ge {T}_{2l} such that {I}_{2}^{l}\le I(t)\le {I}_{2}^{m} for all t>{T}_{2i}, where {I}_{2}^{l}={L}_{2}^{l}-\frac{\mathrm{\Lambda}}{\mu {R}_{0}}-\epsilon and {I}_{2}^{m}={L}_{2}^{m}-\frac{\mathrm{\Lambda}}{\mu {R}_{0}}+\epsilon. After substituting {L}_{2}^{m}=\frac{\mathrm{\Lambda}-\gamma {I}_{1}^{l}}{\mu}+\epsilon and {L}_{2}^{l}=\frac{\mathrm{\Lambda}-\gamma {I}_{1}^{m}}{\mu}-\epsilon into the equations {I}_{2}^{m}={L}_{2}^{m}-\frac{\mathrm{\Lambda}}{\mu {R}_{0}}+\epsilon and {I}_{2}^{l}={L}_{2}^{l}-\frac{\mathrm{\Lambda}}{\mu {R}_{0}}-\epsilon, we have

\{\begin{array}{c}{I}_{2}^{m}=-\frac{\gamma {I}_{1}^{l}}{\mu}+\frac{\mathrm{\Lambda}({R}_{0}-1)}{{R}_{0}\mu}+2\epsilon ,\hfill \\ {I}_{2}^{l}=-\frac{\gamma {I}_{1}^{m}}{\mu}+\frac{\mathrm{\Lambda}({R}_{0}-1)}{{R}_{0}\mu}-2\epsilon .\hfill \end{array}

(26)

Equations in (26) hold when t>{T}_{2i} and {I}_{2}^{l}\le I(t)\le {I}_{2}^{m}. From the mathematical induction, we know that there exist sequences {T}_{kl}, {T}_{ki}, {L}_{k}^{l}, {L}_{k}^{m}, {I}_{k}^{l}, and {I}_{k}^{m} such that {I}_{k}^{l}<I(t)<{I}_{k}^{m} for all t>{T}_{ki}. Furthermore, {I}_{k}^{l} and {I}_{k}^{m} satisfy the following recurrence equations:

\{\begin{array}{c}{I}_{k+1}^{m}=-\frac{\gamma {I}_{k}^{l}}{\mu}+\frac{\mathrm{\Lambda}({R}_{0}-1)}{{R}_{0}\mu}+2\epsilon ,\hfill \\ {I}_{k+1}^{l}=-\frac{\gamma {I}_{k}^{m}}{\mu}+\frac{\mathrm{\Lambda}({R}_{0}-1)}{{R}_{0}\mu}-2\epsilon .\hfill \end{array}

(27)

(27) is a linear system of difference equations. System (27) has a positive equilibrium {P}_{\ast}^{i}({I}_{\ast}^{l}(\epsilon ),{I}_{\ast}^{m}(\epsilon )), where

\begin{array}{r}{I}_{\ast}^{l}(\epsilon )=\frac{\mu \mathrm{\Lambda}-\mathrm{\Lambda}\gamma +\frac{\mathrm{\Lambda}({\gamma}^{2}-{\mu}^{2})}{\beta}-2\epsilon {\mu}^{2}-2\epsilon \mu \gamma}{{\mu}^{2}-{\gamma}^{2}},\\ {I}_{\ast}^{m}(\epsilon )=\frac{\mathrm{\Lambda}-\gamma {I}_{\ast}^{l}(\epsilon )}{\mu}-\frac{\mathrm{\Lambda}(\mu +\gamma )}{\beta \mu}+2\epsilon .\end{array}

(28)

Let {\lambda}_{1} and {\lambda}_{2} be two roots of the linearized matrix of system (27) at the equilibrium {P}_{\ast}^{i}. It is easy to see that |{\lambda}_{1}|=|{\lambda}_{2}|=\frac{\gamma}{\mu}<1 if \gamma <\mu. The stability theory of difference equations implies that the equilibrium {P}_{\ast}^{i} of (27) is globally asymptotically stable, *i.e.*, {lim}_{k\to \mathrm{\infty}}{I}_{k}^{l}={I}_{\ast}^{l}(\epsilon ) and {lim}_{k\to \mathrm{\infty}}{I}_{k}^{m}={I}_{\ast}^{m}(\epsilon ). From the expressions of {I}_{\ast}^{l}(\epsilon ) and {I}_{\ast}^{m}(\epsilon ), we have that {lim}_{\epsilon \to 0}{I}_{\ast}^{l}(\epsilon )={lim}_{\epsilon \to 0}{I}_{\ast}^{m}(\epsilon )=\frac{\mathrm{\Lambda}({R}_{0}-1)}{\beta}. From the inequality {I}_{k}^{l}<I(t)<{I}_{k}^{m} and those limits, we obtain that {lim}_{t\to \mathrm{\infty}}I(t)=\frac{\mathrm{\Lambda}({R}_{0}-1)}{\beta}.

Similarly, we can prove that the sequences \{{L}_{k}^{l}\} and \{{L}_{k}^{m}\} satisfy a linear system of difference equations. The sequences \{{L}_{k}^{l}\} and \{{L}_{k}^{m}\} tend to the same limit when *k* tends to infinity and *ε* tends to zero, *i.e.*,

\underset{k\to \mathrm{\infty}}{lim}{L}_{k}^{l}=\underset{k\to \mathrm{\infty}}{lim}{L}_{k}^{m}=\frac{\mathrm{\Lambda}(\beta \mu +\gamma (\mu +\gamma ))}{\mu (\mu +\gamma )\beta}\phantom{\rule{1em}{0ex}}\text{as}\epsilon \to 0.

The inequality {L}_{k}^{l}<L(t)<{L}_{k}^{m} and the limit imply {lim}_{t\to \mathrm{\infty}}L(t)=\frac{\mathrm{\Lambda}(\beta \mu +\gamma (\mu +\gamma ))}{\mu (\mu +\gamma )\beta}. Finally, we have

\begin{array}{c}\underset{t\to \mathrm{\infty}}{lim}S(t)=\underset{t\to \mathrm{\infty}}{lim}(L(t)-I(t))=\frac{\mathrm{\Lambda}(\mu +\gamma )}{\mu \beta}\phantom{\rule{1em}{0ex}}\text{and}\hfill \\ \underset{t\to \mathrm{\infty}}{lim}R(t)=\underset{t\to \mathrm{\infty}}{lim}(N(t)-L(t))=\frac{\mathrm{\Lambda}\gamma ({R}_{0}-1)}{\mu \beta}.\hfill \end{array}

Therefore, the endemic equilibrium of system (18) is globally asymptotic stable when {R}_{0}>\frac{\mu +\gamma}{\mu} and \gamma <\mu. Equivalently, the endemic equilibrium of system (1) is globally asymptotic stable. □

The comparison principle is the main idea to prove Theorem 4. The limitation of the method and the construction of the comparison equations may lead to the imposed conditions {R}_{0}>\frac{\mu +\gamma}{\mu} and \gamma \le \mu. We do hope that the global stability of the endemic equilibrium can be proved under the condition {R}_{0}>1. When \delta >0, the global stability condition of endemic equilibrium is more complicated. We use the same idea to get the sufficient stability conditions.

**Theorem 5** *If* \mu >\delta +\gamma, {\mu}^{2}>\frac{(\mu +\delta ){(\mu +\delta +\gamma )}^{2}}{2+\mu +\delta +\gamma} *and*

max\{\frac{\mu +\delta +\gamma}{\mu},\frac{\delta}{\mu -\delta -\gamma}\}<{R}_{0}<(1+\frac{2}{\mu +\delta +\gamma})\frac{\mu}{\mu +\delta},

*then the endemic equilibrium* {E}^{\ast}({S}^{\ast},{I}^{\ast},{R}^{\ast}) *of model* (1) *is globally asymptotically stable*.

*Proof* Let L(t)=S(t)+I(t) and N(t)=L(t)+R(t). We consider the equivalent model

The definitions of L(t) and N(t) show that 0\le I(t)\le L(t)\le N(t). The global stability of endemic equilibrium (29) is equivalent to that of model (1). From the first equation of system (29), we have

\begin{array}{r}N(t+1)\le N(t)+\mathrm{\Lambda}-\mu N(t),\\ N(t+1)\ge N(t)+\mathrm{\Lambda}-(\mu +\delta )N(t).\end{array}

(30)

From (30) and the comparison theorem, we know that for any small \epsilon >0, there exists a positive integer {T}_{1n} such that {N}_{1}^{l}\le N(t)\le {N}_{1}^{m} for all t>{T}_{1n}, where {N}_{1}^{l}=\frac{\mathrm{\Lambda}}{\mu +\delta}-\epsilon and {N}_{1}^{m}=\frac{\mathrm{\Lambda}}{\mu}+\epsilon. From the second equation of system (29), we have

\begin{array}{r}L(t+1)\le L(t)+\mathrm{\Lambda}-\mu L(t),\\ L(t+1)\ge L(t)+\mathrm{\Lambda}-(\mu +\delta +\gamma )L(t).\end{array}

(31)

From (31) and the comparison theorem [17], we know that there exists a positive integer {T}_{1l} such that {L}_{1}^{l}\le L(t)\le {L}_{1}^{m} for all t>{T}_{1l}, where {L}_{1}^{l}=\frac{\mathrm{\Lambda}}{\mu +\gamma +\delta}-\epsilon and {L}_{1}^{m}=\frac{\mathrm{\Lambda}}{\mu}+\epsilon.

When t>{T}_{1c}=max\{{T}_{1n},{T}_{1l}\}, the inequalities {N}_{1}^{l}\le N(t)\le {N}_{1}^{m}, {L}_{1}^{l}\le L(t)\le {L}_{1}^{m}, and the third equation of (29) can yield

\begin{array}{r}I(t+1)\le I(t)+\beta \frac{{L}_{1}^{m}-I(t)}{{N}_{1}^{l}}I(t)-(\mu +\delta +\gamma )I(t),\\ I(t+1)\ge I(t)+\beta \frac{{L}_{1}^{l}-I(t)}{{N}_{1}^{m}}I(t)-(\mu +\delta +\gamma )I(t).\end{array}

(32)

The comparison equations corresponding to those inequalities in (32),

\begin{array}{r}{I}_{1}^{m}(t+1)={I}_{1}^{m}(t)+\beta \frac{{L}_{1}^{m}-{I}_{1}^{m}(t)}{{N}_{1}^{l}}{I}_{1}^{m}(t)-(\mu +\delta +\gamma ){I}_{1}^{m}(t),\\ {I}_{1}^{l}(t+1)={I}_{1}^{l}(t)+\beta \frac{{L}_{1}^{l}-{I}_{1}^{l}(t)}{{N}_{1}^{m}}{I}_{1}^{l}(t)-(\mu +\delta +\gamma ){I}_{1}^{l}(t)\end{array}

(33)

are the quadratic difference equations of the form x(t+1)=rx(t)(1-x(t)). When {\mu}^{2}>\frac{(\mu +\delta ){(\mu +\delta +\gamma )}^{2}}{2+\mu +\delta +\gamma} and \frac{\mu +\delta +\gamma}{\mu}<{R}_{0}<(1+\frac{2}{\mu +\delta +\gamma})\frac{\mu}{\mu +\delta}, the well-know result on the population model x(t+1)=rx(t)(1-x(t)) and the comparison theorem imply that there exists an integer {T}_{1i}>{T}_{1c} such that {I}_{1}^{l}\le I(t)\le {I}_{1}^{m} for all t>{T}_{1i}, where {I}_{1}^{l}={L}_{1}^{l}-\frac{{N}_{1}^{m}}{{R}_{0}}-\epsilon and {I}_{1}^{m}={L}_{1}^{m}-\frac{{N}_{1}^{l}}{{R}_{0}}+\epsilon.

When t\ge {T}_{1i}, we use the inequality {I}_{1}^{l}\le I(t)\le {I}_{1}^{m} in the first two equations of system (29) to get

\begin{array}{r}N(t+1)\le N(t)+\mathrm{\Lambda}-\mu N(t)-\delta {I}_{1}^{l},\\ N(t+1)\ge N(t)+\mathrm{\Lambda}-\mu N(t)-\delta {I}_{1}^{m},\\ L(t+1)\le L(t)+\mathrm{\Lambda}-\mu L(t)-(\delta +\gamma ){I}_{1}^{l},\\ L(t+1)\ge L(t)+\mathrm{\Lambda}-\mu L(t)-(\delta +\gamma ){I}_{1}^{m}.\end{array}

(34)

A similar argument implies that there exists a positive integer {T}_{2c} such that {N}_{2}^{l}\le N(t)\le {N}_{2}^{m} and {L}_{2}^{l}\le L(t)\le {L}_{2}^{m} hold for t>{T}_{2c}, where {N}_{2}^{l}=\frac{\mathrm{\Lambda}-\delta {I}_{1}^{m}}{\mu}-\epsilon, {N}_{2}^{m}=\frac{\mathrm{\Lambda}-\delta {I}_{1}^{l}}{\mu}+\epsilon, {L}_{2}^{l}=\frac{\mathrm{\Lambda}-(\gamma +\delta ){I}_{1}^{m}}{\mu}-\epsilon, {L}_{2}^{m}=\frac{\mathrm{\Lambda}-(\gamma +\delta ){I}_{1}^{l}}{\mu}+\epsilon, {N}_{1}^{l}<{N}_{2}^{l}<{N}_{2}^{m}<{N}_{1}^{m}, and {L}_{1}^{l}<{L}_{2}^{l}<{L}_{2}^{m}<{L}_{1}^{m}. When t>{T}_{2c}, we use those estimates of N(t) and L(t) in the third equation of (29) and obtain the following inequalities:

\begin{array}{r}I(t+1)\le I(t)+\beta \frac{{L}_{2}^{m}-I(t)}{{N}_{2}^{l}}I(t)-(\mu +\delta +\gamma )I(t),\\ I(t+1)\ge I(t)+\beta \frac{{L}_{2}^{l}-I(t)}{{N}_{2}^{m}}I(t)-(\mu +\delta +\gamma )I(t).\end{array}

(35)

When {\mu}^{2}>\frac{(\mu +\delta ){(\mu +\delta +\gamma )}^{2}}{2+\mu +\delta +\gamma} and \frac{\mu +\delta +\gamma}{\mu}<{R}_{0}<(1+\frac{2}{\mu +\delta +\gamma})\frac{\mu}{\mu +\delta}, a similar procedure as aforementioned can imply that there exists an integer {T}_{2i}>{T}_{2c} such that {I}_{2}^{l}\le I(t)\le {I}_{2}^{m} for all t>{T}_{2i}, where {I}_{2}^{l}={L}_{2}^{l}-\frac{{N}_{2}^{m}}{{R}_{0}}-\epsilon and {I}_{2}^{m}={L}_{2}^{m}-\frac{{N}_{2}^{l}}{{R}_{0}}+\epsilon.

By using the mathematical induction, we obtain the sequences {T}_{ki}, {N}_{k}^{l}, {N}_{k}^{m}, {L}_{k}^{l}, {L}_{k}^{m}, {I}_{k}^{l}, and {I}_{k}^{m} such that

{N}_{k}^{l}\le N(t)\le {N}_{k}^{m},\phantom{\rule{2em}{0ex}}{L}_{k}^{l}\le L(t)\le {L}_{k}^{m},\phantom{\rule{2em}{0ex}}{I}_{k}^{l}\le I(t)\le {I}_{k}^{m},\phantom{\rule{1em}{0ex}}t>{T}_{ik}.

Furthermore, {I}_{k}^{l} and {I}_{k}^{m} satisfy the following recurrence equations:

\begin{array}{r}{I}_{k+1}^{m}=-\frac{\delta +\gamma}{\mu}{I}_{k}^{l}+\frac{\delta}{{R}_{0}\mu}{I}_{k}^{m}+\frac{\mathrm{\Lambda}({R}_{0}-1)}{{R}_{0}\mu}+\frac{\epsilon}{{R}_{0}}+2\epsilon ,\\ {I}_{k+1}^{l}=-\frac{\delta +\gamma}{\mu}{I}_{k}^{m}+\frac{\delta}{{R}_{0}\mu}{I}_{k}^{l}+\frac{\mathrm{\Lambda}({R}_{0}-1)}{{R}_{0}\mu}-\frac{\epsilon}{{R}_{0}}-2\epsilon .\end{array}

(36)

(36) is a linear system of difference equations. Let z(k)={({I}_{k}^{m},{I}_{k}^{l})}^{\tau}, b={(\frac{\mathrm{\Lambda}({R}_{0}-1)}{{R}_{0}\mu}+\frac{\epsilon}{{R}_{0}}+2\epsilon ,\frac{\mathrm{\Lambda}({R}_{0}-1)}{{R}_{0}\mu}+\frac{\epsilon}{{R}_{0}}+2\epsilon )}^{\tau}, and

B=\left(\begin{array}{cc}\frac{\delta}{{R}_{0}\mu}& -\frac{\delta +\gamma}{\mu}\\ -\frac{\delta +\gamma}{\mu}& \frac{\delta}{{R}_{0}\mu}\end{array}\right),

then system (36) becomes

z(k+1)=Bz(k)+b={B}^{k-1}b+\cdots +Bb+b+{B}^{k}z(1).

(37)

If the conditions of Theorem 5 hold, then we know that the two eigenvalues of matrix *B* satisfy |{\lambda}_{j}|<1 (j=1,2), and the matrix series {\sum}_{k=0}^{\mathrm{\infty}}{B}^{k} converges to {(I-B)}^{-1}. Under the conditions of Theorem 5, we can have {lim}_{k\to \mathrm{\infty}}z(k)={(I-B)}^{-1}b, *i.e.*, {z}^{\ast}=({I}_{\ast}^{m},{I}_{\ast}^{l})={({(I-B)}^{-1}b)}^{\tau} is the globally stable equilibrium of (37). Further calculation shows that

\underset{\epsilon \to 0}{lim}({I}_{\ast}^{m},{I}_{\ast}^{l})=(\frac{\mathrm{\Lambda}({R}_{0}-1)}{(\mu +\delta +\gamma )({R}_{0}-1)+(\mu +\gamma )},\frac{\mathrm{\Lambda}({R}_{0}-1)}{(\mu +\delta +\gamma )({R}_{0}-1)+(\mu +\gamma )}).

After taking \epsilon \to 0, we have

\begin{array}{c}\underset{k\to \mathrm{\infty}}{lim}{I}_{k}^{l}=\underset{k\to \mathrm{\infty}}{lim}{I}_{k}^{m}=\frac{\mathrm{\Lambda}({R}_{0}-1)}{(\mu +\delta +\gamma )({R}_{0}-1)+(\mu +\gamma )}\phantom{\rule{1em}{0ex}}\text{and}\hfill \\ \underset{t\to \mathrm{\infty}}{lim}I(t)=\frac{\mathrm{\Lambda}({R}_{0}-1)}{(\mu +\delta +\gamma )({R}_{0}-1)+(\mu +\gamma )}=\frac{\mathrm{\Lambda}({R}_{0}-1)}{\beta -\delta}.\hfill \end{array}

A similar argument implies that

\begin{array}{c}\underset{k\to \mathrm{\infty}}{lim}{N}_{k}^{l}=\underset{k\to \mathrm{\infty}}{lim}{N}_{k}^{m}=\frac{\mathrm{\Lambda}(\mu +\gamma ){R}_{0}}{\mu ((\mu +\delta +\gamma )({R}_{0}-1)+(\mu +\gamma ))},\hfill \\ \underset{t\to \mathrm{\infty}}{lim}N(t)=\frac{\mathrm{\Lambda}(\mu +\gamma ){R}_{0}}{\mu ((\mu +\delta +\gamma )({R}_{0}-1)+(\mu +\gamma ))},\hfill \\ \underset{k\to \mathrm{\infty}}{lim}{L}_{k}^{l}=\underset{k\to \mathrm{\infty}}{lim}{L}_{k}^{m}=\frac{\mathrm{\Lambda}(\mu +\gamma )+\mathrm{\Lambda}({R}_{0}-1)\mu}{\mu ((\mu +\delta +\gamma )({R}_{0}-1)+(\mu +\gamma ))},\hfill \\ \underset{t\to \mathrm{\infty}}{lim}L(t)=\frac{\mathrm{\Lambda}(\mu +\gamma )+\mathrm{\Lambda}({R}_{0}-1)\mu}{\mu ((\mu +\delta +\gamma )({R}_{0}-1)+(\mu +\gamma ))}.\hfill \end{array}

Those limits lead to

\begin{array}{c}\underset{t\to \mathrm{\infty}}{lim}S(t)=\underset{t\to \mathrm{\infty}}{lim}(L(t)-I(t))=\frac{\mathrm{\Lambda}(\mu +\gamma )}{\mu ((\mu +\delta +\gamma )({R}_{0}-1)+(\mu +\gamma ))}=\frac{\mathrm{\Lambda}(\gamma +\mu )({R}_{0}-1)}{\beta -\delta},\hfill \\ \underset{t\to \mathrm{\infty}}{lim}R(t)=\underset{t\to \mathrm{\infty}}{lim}(N(t)-L(t))=\frac{\mathrm{\Lambda}\gamma ({R}_{0}-1)}{\mu ((\mu +\delta +\gamma )({R}_{0}-1)+(\mu +\gamma ))}=\frac{\mathrm{\Lambda}\gamma ({R}_{0}-1)}{\mu (\beta -\delta )}.\hfill \end{array}

Therefore, the endemic equilibrium of system (1) is globally asymptotic stable when the conditions of Theorem 5 hold. □

There are some parameter values which can satisfy the conditions of Theorem 5. For example, if \delta =\gamma =0, those conditions become \mu >0, 1+\mu >0, 1<{R}_{0}<1+\frac{2}{\mu}. Then, for \mu >0, we know that the conditions of Theorem 5 will hold if *δ* and *γ* are small enough.