System (1) has three equilibrium points [32]: (i) when \(R_{0}=\frac{a_{21}b}{c(a_{11}+pb)}<1\), system (1) has an equilibrium point \(E_{1}(K,0,0)\); (ii) when \(R_{0}=\frac {a_{21}b}{c(a_{11}+pb)}>1\) and \(R_{1}=\frac{a_{21}b}{ (c+\frac{da_{22}}{\beta} ) (a_{11}+pb+\frac{qda_{11}}{\beta} )}<1\), system (1) has another disease free equilibrium point \(E_{2}(\overline{X},\overline {S},0)\); (iii) when \(R_{1}=\frac{a_{21}b}{ (c+\frac{da_{22}}{\beta } ) (a_{11}+pb+\frac{qda_{11}}{\beta} )}>1\), system (1) has a positive equilibrium point \(E_{3}(X^{*},S^{*},I^{*})\). For its stochastic system (2), however, these equilibrium points do not exist.
In this section, we study the asymptotic behaviors of model (2) around the three equilibrium points \(E_{1}(K,0,0)\), \(E_{2}(\overline {X},\overline{S},0)\), and \(E_{3}(X^{*},S^{*},I^{*})\) of its deterministic model (1), respectively.
3.1 Asymptotic behaviors around the equilibrium point \(E_{1}\) of system (1)
When \(R_{0}<1\), system (1) has an equilibrium point \(E_{1}(K,0,0)=(\frac{b}{a_{11}},0,0)\), but it is not the equilibrium point of system (2). In this subsection, we study the asymptotic behaviors of system (2) around \(E_{1}(K,0,0)\).
Theorem 3.1
Let
\((X(t),S(t),I(t))\)
be the solution of model (2) with initial value
\((X(0),S(0), I(0))\in R^{3}_{+}\). If
\(R_{0}<1\)
and
\(K=\frac {b}{a_{11}}\leq\frac{c}{a_{21}}\), then
$$ {\limsup_{t\rightarrow\infty}\frac{1}{t} \int_{0}^{t} \bigl[\bigl(X(\theta)-K \bigr)^{2}+S(\theta)^{2}+I^{2}(\theta) \bigr]\,d\theta \leq\frac{\sigma _{12}^{2}K}{2q^{2}W_{1}},} $$
where
\(W_{1}=\min \{a_{11},\frac{a_{12}a_{22}}{a_{21}},\frac {a_{12}a_{33}}{a_{21}} \}\).
Proof
Note that \((K,0,0)\) is the equilibrium point of system (1), where \(K=\frac{b}{a_{11}}\).
Define
$$ V(X,S,I)= \biggl(X-K-K\ln\frac{X}{K} \biggr)+\frac{a_{12}}{a_{21}} (S+I ). $$
Applying Itô’s formula to stochastic differential system (2) yields
$$ \begin{aligned} dV=&LV\,dt-\frac{\sigma_{12}(X-K)S}{1+pX+qS}\,dB_{1}(t)+ \frac{a_{12}\sigma _{21}SX}{a_{21}(1+pX+qS)}\,dB_{1}(t), \end{aligned} $$
(8)
where
$$ \begin{aligned} LV={}&(X-K) \biggl[b-a_{11}X- \frac{a_{12}S}{1+pX+qS} \biggr]+\frac{\sigma _{12}^{2}KS^{2}}{2(1+pX+qS)^{2}} \\ &{}+\frac{a_{12}}{a_{21}} \biggl[S \biggl(-c-a_{22}S+\frac {a_{21}X}{1+pX+qS}- \beta I \biggr)-I (d+a_{33}I-\beta S ) \biggr] \\ ={}&(X-K) \biggl[b-a_{11}(X-K)-a_{11}K-\frac{a_{12}S}{1+pX+qS} \biggr]+\frac {\sigma_{12}^{2}KS^{2}}{2(1+pX+qS)^{2}} \\ &{}+\frac{a_{12}}{a_{21}} \biggl[S \biggl(-c-a_{22}S+\frac {a_{21}X}{1+pX+qS}- \beta I \biggr)-I (d+a_{33}I-\beta S ) \biggr] \\ \leq{}&{-}a_{11}(X-K)^{2}+\frac{a_{12}KS}{1+pX+qS}+\frac{\sigma _{12}^{2}KS^{2}}{2(1+pX+qS)^{2}}- \frac{a_{12}}{a_{21}} \bigl(cS+a_{22}S^{2}+a_{33}I^{2} \bigr) \\ \leq{}&{-}a_{11}(X-K)^{2}+a_{12} \biggl(K- \frac{c}{a_{21}} \biggr)S+\frac{\sigma _{12}^{2}K}{2q^{2}}-\frac{a_{12}a_{22}}{a_{21}}S^{2}- \frac {a_{12}a_{33}}{a_{21}}I^{2} \\ \leq{}&{-}a_{11}(X-K)^{2}-\frac{a_{12}a_{22}}{a_{21}}S^{2}- \frac {a_{12}a_{33}}{a_{21}}I^{2}+\frac{\sigma_{12}^{2}K}{2q^{2}}. \end{aligned} $$
Integrating equation (8) from 0 to t, we obtain
$$ \begin{aligned}[b] V(t)-V(0)\leq{}&{-} \int_{0}^{t}a_{11}\bigl(X(\theta)-K \bigr)^{2}\,d\theta-\frac {a_{12}a_{22}}{a_{21}} \int_{0}^{t}S^{2}(\theta)\,d\theta\\ &{}- \frac{a_{12}a_{33}}{a_{21}} \int_{0}^{t}I^{2}(\theta)\,d\theta+\frac{\sigma_{12}^{2}K}{2q^{2}}t+M_{1}(t), \end{aligned} $$
(9)
where
$$ \begin{aligned} M_{1}(t)= \int_{0}^{t} \biggl[-\frac{\sigma_{12}(X(\theta)-K)S(\theta )}{1+pX(\theta)+qS(\theta)}+ \frac{a_{12}\sigma_{21}S(\theta)X(\theta )}{a_{21}(1+pX(\theta)+qS(\theta))} \biggr]\,dB_{1}(\theta) \end{aligned} $$
is a real-valued continuous local martingale.
Thus
$$ \begin{aligned} \limsup_{t\rightarrow+\infty}\frac{\langle M_{1},M_{1}\rangle_{t}}{t}&= \limsup_{t\rightarrow+\infty}\frac{1}{t} \int_{0}^{t} \biggl[-\frac{\sigma _{12}(X(\theta)-K)S(\theta)}{1+pX(\theta)+qS(\theta)}+ \frac{a_{12}\sigma _{21}S(\theta)X(\theta)}{a_{21}(1+pX(\theta)+qS(\theta))} \biggr]^{2}\,d\theta \\ &\leq2C^{2} \biggl[\frac{\sigma_{12}^{2}}{q^{2}}+\frac{a_{12}^{2}\sigma _{21}^{2}}{a_{21}^{2}q^{2}} \biggr]< + \infty. \end{aligned} $$
Applying the strong law of large numbers, we obtain \({\lim_{t\rightarrow+\infty}}\frac{M_{1}(t)}{t}=0\).
Dividing equation (9) by t and taking the limit superior, we have
$$ {\limsup_{t\rightarrow\infty}\frac{1}{t} \int_{0}^{t} \biggl[a_{11}\bigl(X( \theta)-K\bigr)^{2}+\frac{a_{12}a_{22}}{a_{21}}S^{2}(\theta)+ \frac {a_{12}a_{33}}{a_{21}}I^{2}(\theta) \biggr]\,d\theta\leq\frac{\sigma_{12}^{2}K}{2q^{2}},} $$
thus
$$ {\limsup_{t\rightarrow\infty}\frac{1}{t} \int_{0}^{t} \bigl[a_{11}\bigl(X(\theta)-K \bigr)^{2}+S^{2}(\theta)+I^{2}(\theta) \bigr]\,d\theta \leq\frac {\sigma_{12}^{2}K}{2q^{2}W_{1}}.} $$
□
Corollary 3.1
From Theorem
3.1, when
\(\sigma_{12}=0\), we have
$$LV\leq-a_{11}(X-K)^{2}-\frac{a_{12}a_{22}}{a_{21}}S^{2}- \frac {a_{12}a_{33}}{a_{21}}I^{2}\leq0, $$
thus when
\(R_{0}<1\)
and
\(K=\frac{b}{a_{11}}\leq\frac{c}{a_{21}}\)
hold, the equilibrium point
\(E_{1}(K,0,0)\)
of system (1) is globally asymptotically stable.
Remark 3.1
From Theorem 3.1, if the interference intensity is sufficiently small, the solution of model (2) will fluctuate around the equilibrium point \(E_{1}(K,0,0)\). Moreover, the fluctuation intensity is related with the disturbance intensity: the fluctuation intensity is positively correlated with the value of \(\sigma_{12}\).
3.2 Asymptotic behaviors around the equilibrium point \(E_{2}\) of system (1)
When \(R_{0}>1\) and \(R_{1}<1\), system (1) has an equilibrium point \(E_{2}(\overline{X},\overline{S},0)\), but it is not the equilibrium point of system (2). In this subsection, we study the asymptotic behaviors of system (2) around \(E_{2}(\overline{X},\overline{S},0)\).
Theorem 3.2
Let
\((X(t),S(t),I(t))\)
be the solution of model (2) with initial value
\((X(0),S(0), I(0))\in R^{3}_{+}\). If
\(R_{0}>1, R_{1}<1\)
and
\(a_{11}q>a_{22}p\), then we have
$$ \begin{aligned} \limsup_{t\rightarrow+\infty}\frac{1}{t} \int_{0}^{t} \bigl[\bigl(X(\theta )-\overline{X} \bigr)^{2}+\bigl(S(\theta)-\overline{S}\bigr)^{2}+I^{2}( \theta) \bigr]\,d\theta \leq\frac{U_{2}}{W_{2}}, \end{aligned} $$
where
$$U_{2}=\frac{\sigma_{12}^{2}\overline{X}}{2q^{2}}+\frac{a_{12}(1+p\overline {X})}{a_{21}(1+q\overline{S})} \biggl(\frac{\sigma_{21}^{2}\overline {S}}{2p^{2}}+ \frac{\sigma^{2}\overline{S}}{2}C_{0}^{2} \biggr) $$
and
$$W_{2}=\min \biggl\{ a_{11}-\frac{a_{12}p}{q}, \frac{a_{12}a_{22}(1+p\overline {X})}{a_{21}(1+q\overline{S})},\frac{a_{12}a_{33}(1+p\overline {X})}{a_{21}(1+q\overline{S})} \biggr\} . $$
Proof
Noting that \((\overline{X},\overline{S},0)\) is the equilibrium point of system (1), thus
$$b-a_{11}\overline{X}-\frac{a_{12}\overline{S}}{1+p\overline {X}+q\overline{S}}=0,\qquad c+a_{22} \overline{S}-\frac{a_{21}\overline {X}}{1+p\overline{X}+q\overline{S}}=0. $$
Define
$$ \begin{aligned}[b] V(X,S,I)&= \biggl(X-\overline{X}-\overline{X}\ln \frac{X}{\overline {X}} \biggr)+\frac{a_{12}(1+p\overline{X})}{a_{21}(1+q\overline{S})} \biggl(S-\overline{S}-\overline{S} \ln\frac{S}{\overline{S}} \biggr)+\frac {a_{12}(1+p\overline{X})}{a_{21}(1+q\overline{S})}I\\ &:=V_{1}+\frac{a_{12}(1+p\overline{X})}{a_{21}(1+q\overline{S})}V_{2}+\frac {a_{12}(1+p\overline{X})}{a_{21}(1+q\overline{S})}V_{3}. \end{aligned} $$
Applying Itô’s formula to stochastic differential system (2) yields
$$ dV_{1}=LV_{1}\,dt- \frac{\sigma_{11}(X-\overline{X})S}{1+pX+qS}\,dB_{1}(t), $$
where
$$ \begin{aligned} LV_{1}={}&(X-\overline{X}) \biggl[b-a_{11}X-\frac{a_{12}S}{1+pX+qS} \biggr] +\frac{\sigma_{12}^{2}\overline{X}S^{2}}{2(1+pX+qS)^{2}} \\ ={}&(X-\overline{X}) \biggl[b-a_{11}(X-\overline{X})-a_{11} \overline {X}-\frac{a_{12}S}{1+pX+qS} \biggr]+\frac{\sigma_{12}^{2}\overline {X}S^{2}}{2(1+pX+qS)^{2}} \\ ={}&(X-\overline{X}) \biggl[-a_{11}(X-\overline{X})+a_{12} \frac{p\overline {S}(X-\overline{X})-(S-\overline{S})(1+p\overline{X})}{(1+p\overline {X}+q\overline{S})(1+pX+qS)} \biggr]+\frac{\sigma_{12}^{2}\overline {X}S^{2}}{2(1+pX+qS)^{2}} \\ ={}&-a_{11}(X-\overline{X})^{2}+\frac{a_{12}p\overline{S}(X-\overline {X})^{2}}{(1+p\overline{X}+q\overline{S})(1+pX+qS)}- \frac {a_{12}(1+p\overline{X})(S-\overline{S})(X-\overline{X})}{(1+p\overline {X}+q\overline{S})(1+pX+qS)}\\ &{}+\frac{\sigma_{12}^{2}\overline{X}S^{2}}{2(1+pX+qS)^{2}}. \end{aligned} $$
Similarly,
$$ dV_{2}=LV_{2}\,dt+\frac{\sigma_{21}(S-\overline{S})X}{1+pX+qS}\,dB_{1}(t)- \sigma (S-\overline{S})I\,dB_{2}(t), $$
where
$$ \begin{aligned} LV_{2}={}&(S-\overline{S}) \biggl[-c-a_{22}S+\frac{a_{21}X}{1+pX+qS}-\beta I \biggr]+ \frac{\sigma_{21}^{2}\overline{S}X^{2}}{2(1+pX+qS)^{2}}+\frac{\sigma ^{2}\overline{S}}{2}I^{2} \\ ={}&(S-\overline{S}) \biggl[-c-a_{22}(S-\overline{S})-a_{22} \overline {S}+\frac{a_{21}X}{1+pX+qS}-\beta I \biggr]+\frac{\sigma_{21}^{2}\overline {S}X^{2}}{2(1+pX+qS)^{2}}+ \frac{\sigma^{2}\overline{S}}{2}I^{2} \\ ={}&(S-\overline{S}) \biggl[-a_{22}(S-\overline{S})+a_{21} \frac{(X-\overline {X})(1+q\overline{S})-q\overline{X}(S-\overline{S})}{(1+p\overline {X}+q\overline{S})(1+pX+qS)}-\beta I \biggr] \\ &{}+\frac{\sigma_{21}^{2}\overline{S}X^{2}}{2(1+pX+qS)^{2}}+\frac{\sigma ^{2}\overline{S}}{2}I^{2} \\ ={}&-a_{22}(S-\overline{S})^{2}+a_{21} \frac{(1+q\overline{S})(X-\overline {X})(S-\overline{S})}{(1+p\overline{X}+q\overline {S})(1+pX+qS)}-a_{21}\frac{q\overline{X}(S-\overline {S})^{2}}{(1+p\overline{X}+q\overline{S})(1+pX+qS)} \\ &{}-\beta I(S-\overline{S})+\frac{\sigma_{21}^{2}\overline {S}X^{2}}{2(1+pX+qS)^{2}}+\frac{\sigma^{2}\overline{S}}{2}I^{2}. \end{aligned} $$
Also, we have
$$ dV_{3}=I(t) [-d-a_{33}I+\beta S ]\,dt+\sigma SI\,dB_{2}(t). $$
Hence
$$ \begin{aligned}[b] dV={}&LV\,dt-\frac{\sigma_{12}(X-\overline{X})S}{1+pX+qS}\,dB_{1}(t)+ \frac {a_{12}(1+p\overline{X})}{a_{21}(1+q\overline{S})} \biggl[\frac{\sigma _{21}(S-\overline{S})X}{1+pX+qS}\,dB_{1}(t)\\ &{}-\sigma(S-\overline{S})I\,dB_{2}(t)+\sigma SI\,dB_{2}(t)\biggr], \end{aligned} $$
(10)
where
$$ \begin{aligned} LV={}&LV_{1}+\frac{a_{12}(1+p\overline{X})}{a_{21}(1+q\overline {S})}LV_{2}+ \frac{a_{12}(1+p\overline{X})}{a_{21}(1+q\overline{S})}LV_{3} \\ ={}&-a_{11}(X-\overline{X})^{2}+\frac{a_{12}p\overline{S}(X-\overline {X})^{2}}{(1+p\overline{X}+q\overline{S})(1+pX+qS)}- \frac {a_{12}(1+p\overline{X})(S-\overline{S})(X-\overline{X})}{(1+p\overline {X}+q\overline{S})(1+pX+qS)}\\ &{}+\frac{\sigma_{12}^{2}\overline{X}S^{2}}{2(1+pX+qS)^{2}}+\frac {a_{12}(1+p\overline{X})}{a_{21}(1+q\overline{S})} \biggl[-a_{22}(S- \overline{S})^{2}+a_{21}\frac{(1+q\overline{S})(X-\overline {X})(S-\overline{S})}{(1+p\overline{X}+q\overline{S})(1+pX+qS)}\\ &{}-a_{21}\frac{q\overline{X}(S-\overline{S})^{2}}{(1+p\overline {X}+q\overline{S})(1+pX+qS)}-\beta I(S-\overline{S})+ \frac{\sigma _{21}^{2}\overline{S}X^{2}}{2(1+pX+qS)^{2}}+\frac{\sigma^{2}\overline {S}}{2}I^{2}\\ &{}+I (-d-a_{33}I+\beta S ) \biggr]\\ \leq{}&- \biggl(a_{11}-\frac{a_{12}p}{q} \biggr) (X- \overline{X})^{2}-\frac {a_{12}a_{22}(1+p\overline{X})}{a_{21}(1+q\overline{S})}(S-\overline {S})^{2}- \frac{a_{12}a_{33}(1+p\overline{X})}{a_{21}(1+q\overline {S})}I^{2}\\ &{}+\frac{a_{12}(1+p\overline{X})}{a_{21}(1+q\overline{S})} (\beta \overline{S}-d)I+\frac{\sigma_{12}^{2}\overline{X}S^{2}}{2(1+pX+qS)^{2}}\\ &{}+ \frac {a_{12}(1+p\overline{X})}{a_{21}(1+q\overline{S})} \biggl(\frac{\sigma _{21}^{2}\overline{S}X^{2}}{2(1+pX+qS)^{2}}+\frac{\sigma^{2}\overline {S}}{2}I^{2} \biggr). \end{aligned} $$
Since \(\beta\overline{S}< d\), thus
$$ \begin{aligned} LV\leq{}&{-}\biggl(a_{11}-\frac{a_{12}p}{q} \biggr) (X-\overline{X})^{2}-\frac {a_{12}a_{22}(1+p\overline{X})}{a_{21}(1+q\overline{S})}(S-\overline {S})^{2}-\frac{a_{12}a_{33}(1+p\overline{X})}{a_{21}(1+q\overline {S})}I^{2}\\ &{}+\frac{\sigma_{11}^{2}\overline{X}}{2q^{2}}+\frac{a_{12}(1+p\overline {X})}{a_{21}(1+q\overline{S})} \biggl(\frac{\sigma_{12}^{2}\overline {S}}{2p^{2}}+ \frac{\sigma_{2}^{2}\overline{S}}{2}C_{0}^{2} \biggr). \end{aligned} $$
Integrating both sides of equation (10) from 0 to t yields
$$ \begin{aligned}[b] V(t)-V(0)\leq{}& \int_{0}^{t} \biggl[- \biggl(a_{11}- \frac{a_{12}p}{q} \biggr) \bigl(X(\theta)-\overline {X}\bigr)^{2}\\ &{}- \frac{a_{12}a_{22}(1+p\overline{X})}{a_{21}(1+q\overline {S})}\bigl(S(\theta)-\overline{S}\bigr)^{2} -\frac{a_{12}a_{33}(1+p\overline{X})}{a_{21}(1+q\overline {S})}I^{2}(\theta) \biggr]\,d\theta\\ &{}+ \biggl[ \frac{\sigma_{12}^{2}\overline {X}}{2q^{2}}+\frac{a_{12}(1+p\overline{X})}{a_{21}(1+q\overline{S})} \biggl(\frac{\sigma_{21}^{2}\overline{S}}{2p^{2}}+ \frac{\sigma^{2}\overline {S}}{2}C_{0}^{2} \biggr) \biggr]t +M_{2}(t)+M_{3}(t), \end{aligned} $$
(11)
where
$$ M_{2}(t)= \int_{0}^{t} \biggl[-\frac{\sigma_{12}(X(\theta)-\overline{X})S(\theta )}{1+pX(\theta)+qS(\theta)}+ \frac{a_{12}(1+p\overline {X})}{a_{21}(1+q\overline{S})}\frac{\sigma_{21}(S(\theta)-\overline {S})X(\theta)}{1+pX(\theta)+qS(\theta)} \biggr]\,dB_{1}(\theta) $$
and
$$ M_{3}(t)= \int_{0}^{t}\frac{a_{12}\sigma(1+p\overline{X})\overline{S}I(\theta )}{a_{21}(1+q\overline{S})}\,dB_{2}( \theta) $$
are real-valued continuous local martingales.
Thus
$$\begin{aligned} &\limsup_{t\rightarrow+\infty}\frac{\langle M_{2},M_{2}\rangle_{t}}{t}\\ &\quad= \limsup_{t\rightarrow+\infty}\frac{1}{t} \int_{0}^{t} \biggl[-\frac{\sigma _{12}(X(\theta)-\overline{X})S(\theta)}{1+pX(\theta)+qS(\theta)}+ \frac {a_{12}(1+p\overline{X})}{a_{21}(1+q\overline{S})}\frac{\sigma _{21}(S(\theta)-\overline{S})X(\theta)}{1+pX(\theta)+qS(\theta)} \biggr]^{2}\,d\theta \\ &\quad\leq2C_{0}^{2} \biggl[\frac{\sigma_{12}^{2}}{q^{2}}+ \frac{a_{12}^{4}\sigma _{21}^{2}(1+p\overline{X})^{2}}{a_{21}^{2}p^{2}(1+q\overline{S})^{2}} \biggr]< +\infty \end{aligned}$$
and
$$ \begin{aligned} \limsup_{t\rightarrow+\infty}\frac{\langle M_{3},M_{3}\rangle_{t}}{t}&= \limsup_{t\rightarrow+\infty}\frac{1}{t} \int_{0}^{t} \biggl[\frac{a_{12}\sigma (1+p\overline{X})\overline{S}I(\theta)}{a_{21}(1+q\overline{S})} \biggr]^{2}\,d\theta\\ &\leq C_{0}^{2} \biggl[\frac{a_{12}^{2}\sigma^{2}\overline{S}^{2}(1+p\overline {X})^{2}}{(1+q\overline{S})^{2}} \biggr]< +\infty. \end{aligned} $$
Applying the strong law of large numbers, we have \(\lim_{t\rightarrow+\infty}\frac{M_{i}(t)}{t}=0 \) (\(i=2,3\)).
Dividing equation (11) by t and taking the limit superior, we have
$$ \begin{aligned} &\limsup_{t\rightarrow+\infty}\frac{1}{t} \int_{0}^{t} \biggl[ \biggl(a_{11}- \frac{a_{12}p}{q} \biggr) \bigl(X(\theta)-\overline{X}\bigr)^{2}+ \frac {a_{12}a_{22}(1+p\overline{X})}{a_{21}(1+q\overline{S})}\bigl(S(\theta ) -\overline{S}\bigr)^{2}\\ &\qquad{}+\frac{a_{12}a_{33}(1+p\overline{X})}{a_{21}(1+q\overline {S})}I^{2}(\theta) \biggr]\,d\theta\leq \frac{\sigma_{12}^{2}\overline {X}}{2q^{2}}+\frac{a_{12}(1+p\overline{X})}{a_{21}(1+q\overline{S})} \biggl(\frac{\sigma_{21}^{2}\overline{S}}{2p^{2}}+ \frac{\sigma^{2}\overline {S}}{2}C_{0}^{2} \biggr). \end{aligned} $$
Thus
$$ \limsup_{t\rightarrow+\infty}\frac{1}{t} \int_{0}^{t} \bigl[\bigl(X(\theta )-\overline{X} \bigr)^{2}+\bigl(S(\theta)-\overline{S}\bigr)^{2}+I^{2}( \theta) \bigr]\,d\theta \leq\frac{U_{2}}{W_{2}}. $$
□
Corollary 3.2
From Theorem
3.2, when
\(\sigma_{12}=\sigma_{21}=\sigma=0\), we have
$$LV\leq- \biggl(a_{11}-\frac{a_{12}p}{q} \biggr) (X- \overline{X})^{2}-\frac {a_{12}a_{22}(1+p\overline{X})}{a_{21}(1+q\overline{S})}(S-\overline {S})^{2}- \frac{a_{12}a_{33}(1+p\overline{X})}{a_{21}(1+q\overline {S})}I^{2}\leq0, $$
thus when
\(a_{11}q>a_{22}p\), \(R_{0}>1\)
and
\(R_{1}<1\)
hold, the equilibrium point
\(E_{2}(\overline{X},\overline{S},0)\)
of system (1) is globally asymptotically stable.
Remark 3.2
From Theorem 3.2, if the interference intensity is sufficiently small, the solution of model (2) will fluctuates around the equilibrium point \(E_{2}(\overline{X},\overline{S},0)\). Moreover, the fluctuation intensity is related with the disturbance intensity: the fluctuation intensity is positively correlated with the value of \(\sigma_{12},\sigma_{21}\) and σ.
3.3 Asymptotic behaviors around the positive equilibrium point \(E_{3}\) of system (1)
When \(R_{1}>1\), system (1) has a positive equilibrium point \(E_{3}(X^{*},S^{*},I^{*})\), but it is not the equilibrium point of model (2). Now, we explore the asymptotic behaviors of system (2) around \(E_{3}(X^{*},S^{*},I^{*})\).
\(X(t)\) is a temporally homogeneous Markov process in \(E_{l}\), which is given by the stochastic differential equation
$$dX(t)=b(X)\,dt+ {\sum^{k}_{m=1}} \sigma_{m}(x)\,dB_{m}(t), $$
where \(E_{l}\subset R^{l}\) represents a l-dimensional Euclidean space.
The diffusion matrix of \(X(t)\) is given by
$$\Lambda(x)=\bigl(a_{i,j}(x)\bigr),a_{i,j}(x)= {\sum ^{k}_{m=1}}\sigma _{m}^{i}(x) \sigma_{m}^{j}(x). $$
Assumption 3.1
[33]
Assume that there is a bounded domain \(U\subset E_{l}\) with regular boundary, satisfying the following conditions:
-
(1)
In the domain U and some of its neighbors, the minimum eigenvalue of the diffusion matrix \(A(x)\) is nonzero.
-
(2)
When \(x\in E_{l}\backslash U\), the mean time τ at which a path starting from x to the set U is limited, and \(\sup_{x\in H} E_{x}\tau<\infty\) for every compact subset \(H \subset E_{l}\).
Lemma 3.1
[33]
When Assumption
3.1
holds, the Markov process
\(X(t)\)
has a stationary distribution
\(\mu(\cdot)\)
with density in
\(E_{l}\). Let
\(f(x)\)
be a function integrable with respect to the measure
μ, where
\(x\in E_{l}\), then, for any Borel set
\(B\subset E_{l}\), we have
$${\lim_{t\rightarrow\infty}}P(t,x,B)=\mu(B) $$
and
$$P_{x} \biggl\{ {\lim_{T\rightarrow\infty}}\frac{1}{T} \int _{0}^{T}f\bigl(x(t)\bigr)\,dt= \int_{E_{l}}f(x)\mu(dx) \biggr\} =1. $$
Theorem 3.3
Let
\((X(t),S(t),I(t))\)
be the solution of model (2) with initial value
\((X(0),S(0), I(0))\in R^{3}_{+}\). If
\(a_{11}q>a_{12}p\)
and
\(R_{1}>1\)
hold, then
$$ \limsup_{t\rightarrow+\infty}\frac{1}{t} \int_{0}^{t} \bigl[\bigl(X(\theta )-X^{*} \bigr)^{2}+\bigl(S(\theta)-S^{*}\bigr)^{2}+\bigl(I(\theta)-I^{*} \bigr)^{2} \bigr]\,d\theta\leq\frac {U_{3}}{W_{3}}, $$
where
$$U_{3}=\frac{\sigma_{12}^{2}X^{*}}{2q^{2}}+\frac {a_{12}(1+pX^{*})}{a_{21}(1+qS^{*})} \biggl( \frac{\sigma _{21}^{2}S^{*}}{2p^{2}}+\frac{C_{0}^{2}\sigma^{2}}{2}\bigl(S^{*}+I^{*}\bigr) \biggr) $$
and
$$W_{3}=\min \biggl\{ a_{11}-\frac{a_{12}p}{q}, \frac {a_{12}a_{22}(1+pX^{*})}{a_{21}(1+qS^{*})},\frac {a_{12}a_{33}(1+pX^{*})}{a_{21}(1+qS^{*})} \biggr\} . $$
Proof
Noting that \((X^{*},S^{*},I^{*})\) is the equilibrium point of system (1), thus
$$ \textstyle\begin{cases} b-a_{11}X^{*}-\frac{a_{12}S^{*}}{1+pX^{*}+qS^{*}}=0,\\ -c-a_{22}S^{*}+\frac{a_{21}X^{*}}{1+pX^{*}+qS^{*}}-\beta I^{*}=0,\\ \beta S^{*}-d-a_{33}I^{*}=0. \end{cases} $$
Define
$$ \begin{aligned} V(X,S,I)={}& \biggl(X-X^{*}-X^{*}\ln\frac{X}{X^{*}} \biggr)+\frac {a_{12}(1+pX^{*})}{a_{21}(1+qS^{*})} \biggl(S-S^{*}-S^{*}\ln\frac{S}{S^{*}} \biggr)\\ &{}+\frac{a_{12}(1+pX^{*})}{a_{21}(1+qS^{*})} \biggl(I-I^{*}-I^{*}\ln\frac {I}{I^{*}} \biggr) \\ :={}&V_{1}+\frac{a_{12}(1+pX^{*})}{a_{21}(1+qS^{*})}V_{2}+\frac {a_{12}(1+pX^{*})}{a_{21}(1+qS^{*})}V_{3}. \end{aligned} $$
Applying Itô’s formula to the stochastic differential system (2) yields
$$ dV_{1}=LV_{1}\,dt- \frac{\sigma_{12}(X-X^{*})S}{1+pX+qS}\,dB_{1}(t), $$
where
$$ \begin{aligned} LV_{1}={}&\bigl(X-X^{*}\bigr) \biggl[b-a_{11}X-\frac{a_{12}S}{1+pX+qS} \biggr]+\frac{\sigma _{12}^{2}X^{*}S^{2}}{2(1+pX+qS)^{2}} \\ ={}&\bigl(X-X^{*}\bigr) \biggl[b-a_{11}\bigl(X-X^{*}\bigr)-a_{11}X^{*}- \frac{a_{12}S}{1+pX+qS} \biggr]+\frac{\sigma_{12}^{2}X^{*}S^{2}}{2(1+pX+qS)^{2}} \\ ={}&\bigl(X-X^{*}\bigr) \biggl[-a_{11}\bigl(X-X^{*}\bigr)+a_{12} \frac {pS^{*}(X-X^{*})-(S-S^{*})(1+pX^{*})}{(1+pX^{*}+qS^{*})(1+pX+qS)} \biggr] \\ &{}+\frac{\sigma_{12}^{2}X^{*}S^{2}}{2(1+pX+qS)^{2}} \\ ={}&{-}a_{11}\bigl(X-X^{*}\bigr)^{2}+\frac {a_{12}pS^{*}(X-X^{*})^{2}}{(1+pX^{*}+qS^{*})(1+pX+qS)}+ \frac{\sigma _{12}^{2}X^{*}S^{2}}{2(1+pX+qS)^{2}} \\ &{}-\frac{a_{12}(1+pX^{*})(S-S^{*})(X-X^{*})}{(1+pX^{*}+qS^{*})(1+pX+qS)}. \end{aligned} $$
Similarly,
$$ dV_{2}=LV_{2}\,dt+ \frac{\sigma_{21}(S-S^{*})X}{1+pX+qS}\,dB_{1}(t)-\sigma \bigl(S-S^{*}\bigr)I\,dB_{2}(t), $$
where
$$\begin{aligned} LV_{2}={}&\bigl(S-S^{*}\bigr) \biggl[-c-a_{22}S+\frac{a_{21}X}{1+pX+qS}-\beta I \biggr]+ \frac{\sigma_{21}^{2}S^{*}X^{2}}{2(1+pX+qS)^{2}}+\frac{1}{2}\sigma^{2}S^{*}I^{2} \\ ={}&\bigl(S-S^{*}\bigr) \biggl[-c-a_{22}\bigl(S-S^{*}\bigr)-a_{22}S^{*}+ \frac{a_{21}X}{1+pX+qS}-\beta \bigl(I-I^{*}\bigr)-\beta I^{*} \biggr] \\ &{}+\frac{\sigma_{21}^{2}S^{*}X^{2}}{2(1+pX+qS)^{2}}+\frac{1}{2}\sigma^{2}S^{*}I^{2} \\ ={}&\bigl(S-S^{*}\bigr) \biggl[-a_{22}\bigl(S-S^{*}\bigr)+a_{21} \frac {(X-X^{*})(1+qS^{*})-qX^{*}(S-S^{*})}{(1+pX^{*}+qS^{*})(1+pX+qS)}-\beta \bigl(I-I^{*}\bigr) \biggr] \\ &{}+\frac{\sigma_{21}^{2}S^{*}X^{2}}{2(1+pX+qS)^{2}}+\frac{1}{2}\sigma^{2}S^{*}I^{2} \\ ={}&-a_{22}\bigl(S-S^{*}\bigr)^{2}+a_{21} \frac {(1+qS^{*})(X-X^{*})(S-S^{*})}{(1+pX^{*}+qS^{*})(1+pX+qS)}-\beta\bigl(I-I^{*}\bigr) \bigl(S-S^{*}\bigr) \\ &{}-a_{21}\frac{qX^{*}(S-S^{*})^{2}}{(1+pX^{*}+qS^{*})(1+pX+qS)}+\frac{\sigma _{21}^{2}S^{*}X^{2}}{2(1+pX+qS)^{2}}+\frac{1}{2} \sigma^{2}S^{*}I^{2}. \end{aligned}$$
Also, we have
$$ dV_{3}=LV_{3}\,dt+\sigma\bigl(I-I^{*}\bigr)S\,dB_{2}(t), $$
where
$$ \begin{aligned} LV_{3}&=\bigl(I-I^{*}\bigr) [-d-a_{33}I+\beta S ]+\frac{1}{2}\sigma^{2}I^{*}S^{2}\\ &=\bigl(I-I^{*}\bigr) \bigl[-d-a_{33}\bigl(I-I^{*}\bigr)-a_{33}I^{*}+ \beta\bigl(S-S^{*}\bigr)+\beta S^{*} \bigr]+\frac{1}{2}\sigma^{2}I^{*}S^{2}\\ &=-a_{33}\bigl(I-I^{*}\bigr)^{2}+\beta\bigl(S-S^{*}\bigr) \bigl(I-I^{*}\bigr)+\frac{1}{2}\sigma^{2}I^{*}S^{2}. \end{aligned} $$
Then we have
$$ \begin{aligned} dV={}&LV\,dt-\frac{\sigma_{12}(X-X^{*})S}{1+pX+qS}\,dB_{1}(t)+ \frac {a_{12}(1+pX^{*})}{a_{21}(1+qS^{*})} \biggl[\frac{\sigma _{21}(S-S^{*})X}{1+pX+qS}\,dB_{1}(t)\\ &{}-\sigma\bigl(S-S^{*}\bigr)I\,dB_{2}(t)+\sigma\bigl(I-I^{*} \bigr)S\,dB_{2}(t) \biggr], \end{aligned} $$
(12)
where
$$\begin{aligned} LV={}&LV_{1}+\frac{a_{12}(1+pX^{*})}{a_{21}(1+qS^{*})}LV_{2}+ \frac {a_{12}(1+pX^{*})}{a_{21}(1+qS^{*})}LV_{3} \\ ={}&-a_{11}\bigl(X-X^{*}\bigr)^{2}+\frac {a_{12}pS^{*}(X-X^{*})^{2}}{(1+pX^{*}+qS^{*})(1+pX+qS)}+ \frac{\sigma _{12}^{2}X^{*}S^{2}}{2(1+pX+qS)^{2}} \\ &{}-\frac{a_{12}(1+pX^{*})(S-S^{*})(X-X^{*})}{(1+pX^{*}+qS^{*})(1+pX+qS)}+\frac {a_{12}(1+pX^{*})}{a_{21}(1+qS^{*})} \biggl[-a_{22}\bigl(S-S^{*} \bigr)^{2}-a_{33}\bigl(I-I^{*}\bigr)^{2} \\ &{}+a_{21}\frac {(1+qS^{*})(X-X^{*})(S-S^{*})}{(1+pX^{*}+qS^{*})(1+pX+qS)}-a_{21}\frac {qX^{*}(S-S^{*})^{2}}{(1+pX^{*}+qS^{*})(1+pX+qS)} \\ &{}+\frac{\sigma_{21}^{2}S^{*}X^{2}}{2(1+pX+qS)^{2}}+\frac{\sigma ^{2}S^{*}}{2}I^{2}+\frac{\sigma^{2}I^{*}}{2}S^{2} \biggr] \\ \leq{}&{-} \biggl(a_{11}-\frac{a_{12}p}{q} \biggr) \bigl(X-X^{*} \bigr)^{2}-\frac {a_{12}a_{22}(1+pX^{*})}{a_{21}(1+qS^{*})}\bigl(S-S^{*}\bigr)^{2}- \frac {a_{12}a_{33}(1+pX^{*})}{a_{21}(1+qS^{*})}\bigl(I-I^{*}\bigr)^{2} \\ &{}+\frac{\sigma_{12}^{2}X^{*}}{2q^{2}}+\frac {a_{12}(1+pX^{*})}{a_{21}(1+qS^{*})} \biggl[\frac{\sigma _{21}^{2}S^{*}}{2p^{2}}+ \frac{C^{2}\sigma^{2}}{2}\bigl(S^{*}+I^{*}\bigr) \biggr]. \end{aligned}$$
It is easy to see that, for any
$$\phi< \min \biggl\{ \biggl(a_{11}-\frac{a_{12}p}{q} \biggr)X^{*}, \frac {a_{12}a_{22}(1+pX^{*})}{a_{21}(1+qS^{*})}S^{*},\frac {a_{12}a_{33}(1+pX^{*})}{a_{21}(1+qS^{*})}I^{*} \biggr\} , $$
the ellipsoid
$$ \begin{aligned} &{-} \biggl(a_{11}-\frac{a_{12}p}{q} \biggr) \bigl(X-X^{*}\bigr)^{2}-\frac {a_{12}a_{22}(1+pX^{*})}{a_{21}(1+qS^{*})}\bigl(S-S^{*} \bigr)^{2}\\ &\quad{}-\frac {a_{12}a_{33}(1+pX^{*})}{a_{21}(1+qS^{*})}\bigl(I-I^{*}\bigr)^{2}+\phi=0 \end{aligned} $$
lies entirely in \(R_{+}^{3}\). Let U to be any neighborhood of the ellipsoid with \(\bar{U}\subseteq E_{3}=R_{+}^{3}\), thus for any \(x\in U\backslash E_{l}\), we have \(LV\leq-\overline{M}\) (M̅ is a positive constant). Therefore, condition (2) in Assumption 3.1 is satisfied. Moreover, there exists a \(G=\min\{\sigma _{1}^{2}x_{1}^{2},\sigma_{2}^{2}x_{2}^{2},\sigma_{3}^{2}x_{3}^{2},(x_{1},x_{2},x_{3})\in\overline {U}\}>0\) such that
$${\sum^{3}_{i,j=1}} \Biggl( {\sum ^{3}_{k=1}a_{ik}(x)a_{jk}(x) \Biggr)}\xi_{i}\xi_{j}=\sigma_{1}^{2}x_{1}^{2} \xi _{1}^{2}+\sigma_{2}^{2}x_{2}^{2} \xi_{2}^{2}+\sigma_{3}^{2}x_{3}^{2} \xi_{3}^{2}\geq G\|\xi\|^{2} $$
for all \(x\in\bar{U}, \xi\in R^{3}\), which means condition (1) in Assumption 3.1 is satisfied. Therefore, the stochastic model (2) has a unique stationary distribution \(\mu(\cdot)\), it also has the ergodic property.
Integrating equation (12) from 0 to t on both sides yields
$$ \begin{aligned}[b] V(t)-V(0)\leq{}& \int_{0}^{t} \biggl[- \biggl(a_{11}- \frac{a_{12}p}{q} \biggr) \bigl(X(\theta)-X^{*}\bigr)^{2}- \frac {a_{12}a_{22}(1+pX^{*})}{a_{21}(1+qS^{*})}\bigl(S(\theta)-S^{*}\bigr)^{2} \\ &{}-\frac{a_{12}a_{33}(1+pX^{*})}{a_{21}(1+qS^{*})}\bigl(I(\theta)-I^{*}\bigr)^{2} \biggr]\,d\theta \\ &{}+ \biggl[\frac{\sigma_{12}^{2}X^{*}}{2q^{2}}+\frac {a_{12}(1+pX^{*})}{a_{21}(1+qS^{*})} \biggl(\frac{\sigma _{21}^{2}S^{*}}{2p^{2}}+ \frac{C_{0}^{2}\sigma^{2}}{2}\bigl(S^{*}+I^{*}\bigr) \biggr) \biggr]\\ &{}+M_{4}(t)+M_{5}(t), \end{aligned} $$
(13)
where
$$ M_{4}(t)= \int_{0}^{t} \biggl[-\frac{\sigma_{12}(X(\theta)-X^{*})S(\theta )}{1+pX(\theta)+qS(\theta)}+ \frac{a_{12}(1+pX^{*})}{a_{21}(1+qS^{*})}\frac {\sigma_{21}(S(\theta)-S^{*})X(\theta)}{1+pX(\theta)+qS(\theta)} \biggr]\,dB_{1}(\theta) $$
and
$$ M_{5}(t)= \int_{0}^{t}\frac{a_{12}(1+pX^{*})}{a_{21}(1+qS^{*})} \bigl[-\sigma \bigl(S(\theta)-S^{*}\bigr)I(\theta)+\sigma\bigl(I(\theta)-I^{*}\bigr)S(\theta) \bigr]\,dB_{2}(\theta) $$
are real-valued continuous local martingales.
Thus
$$\begin{aligned} &\limsup_{t\rightarrow+\infty} \frac{\langle M_{4},M_{4}\rangle_{t}}{t}\\ &\quad= \limsup_{t\rightarrow+\infty}\frac{1}{t} \int_{0}^{t} \biggl[\frac {a_{12}(1+pX^{*})}{a_{21}(1+qS^{*})} \frac{\sigma_{21}(S(\theta)-S^{*})X(\theta )}{1+pX(\theta)+qS(\theta)}-\frac{\sigma_{12}(X(\theta)-X^{*})S(\theta )}{1+pX(\theta)+qS(\theta)} \biggr]^{2}\,d\theta\\ &\quad\leq2C_{0}^{2} \biggl[\frac{\sigma_{12}^{2}}{q^{2}}+ \frac{a_{12}^{4}\sigma _{21}^{2}(1+pX^{*})^{2}}{a_{21}^{2}p^{2}(1+qS^{*})^{2}} \biggr]< +\infty \end{aligned}$$
and
$$ \limsup_{t\rightarrow+\infty}\frac{\langle M_{5},M_{5}\rangle_{t}}{t}= \limsup_{t\rightarrow+\infty}\frac{1}{t} \int_{0}^{t} \bigl(\sigma S^{*}I(\theta )-\sigma I^{*}S( \theta) \bigr)^{2}\,d\theta\leq2C_{0}^{2} \sigma^{2} \bigl(S^{*}+I^{*} \bigr)< +\infty. $$
Applying the strong law of large numbers, we have \(\lim_{t\rightarrow+\infty}\frac{M_{i}(t)}{t}=0\) (\(i=4,5\)).
Dividing equation (13) by t and taking the limit superior, we have
$$ \begin{aligned} &\limsup_{t\rightarrow+\infty}\frac{1}{t} \int_{0}^{t} \biggl[ \biggl(a_{11}- \frac{a_{12}p}{q} \biggr) \bigl(X(\theta)-X^{*}\bigr)^{2}+ \frac {a_{12}a_{22}(1+pX^{*})}{a_{21}(1+qS^{*})}\bigl(S(\theta)-S^{*}\bigr)^{2} \\ &\qquad{}+\frac{a_{12}a_{33}(1+pX^{*})}{a_{21}(1+qS^{*})}\bigl(I(\theta)-I^{*}\bigr)^{2} \biggr]\,d\theta \\ &\quad\leq\frac{\sigma_{12}^{2}X^{*}}{2q^{2}}+\frac {a_{12}(1+pX^{*})}{a_{21}(1+qS^{*})} \biggl[\frac{\sigma _{21}^{2}S^{*}}{2p^{2}}+ \frac{C_{0}^{2}\sigma^{2}}{2}\bigl(S^{*}+I^{*}\bigr) \biggr], \end{aligned} $$
thus
$$ \limsup_{t\rightarrow+\infty}\frac{1}{t} \int_{0}^{t} \bigl[\bigl(X(\theta )-X^{*} \bigr)^{2}+\bigl(S(\theta)-S^{*}\bigr)^{2}+\bigl(I(\theta)-I^{*} \bigr)^{2} \bigr]\,d\theta\leq\frac {U_{3}}{W_{3}}. $$
(14)
□
Corollary 3.3
From Theorem
3.3, when
\(\sigma_{12}=\sigma_{21}=\sigma=0\), we have
$$\begin{aligned} LV&\leq- \biggl(a_{11}-\frac{a_{12}p}{q} \biggr) \bigl(X-X^{*} \bigr)^{2}-\frac {a_{12}a_{22}(1+pX^{*})}{a_{21}(1+qS^{*})}\bigl(S-S^{*}\bigr)^{2}- \frac {a_{12}a_{33}(1+pX^{*})}{a_{21}(1+qS^{*})}\bigl(I-I^{*}\bigr)^{2}\\ &\leq0. \end{aligned}$$
Thus when
\(a_{11}q>a_{22}p\)
and
\(R_{1}>1\)
hold, the positive equilibrium point
\(E_{3}(X^{*},S^{*},I^{*})\)
of system (1) is globally asymptotically stable.
Remark 3.3
From Theorem 3.3, if the interference intensity is sufficiently small, the solution of model (2) will fluctuates around the equilibrium point \(E_{3}(X^{*},S^{*},I^{*})\). Moreover, the fluctuation intensity is related with the disturbance intensity: the fluctuation intensity is positively correlated with the value of \(\sigma _{12},\sigma_{21}\) and σ.
Remark 3.4
If the conditions in Theorem 3.3 are hold, then the solution of model (2) has a unique stationary distribution, it also has the ergodic property.