Our first concern is whether the solution has a global existence. Moreover, we also consider whether, as a population dynamic model, the value is nonnegative. Therefore, we guarantee the existence of a global positive solution under some assumptions.
For the jump diffusion coefficient, we assume that, for each \(m>0\), there exists \(L_{m}>0\) such that
-
(H1)
\(\int_{U}\vert H_{i}(x,u,k)-H_{j}(y,u,k)\vert ^{2}\nu (du)\leq L_{m}\vert x-y\vert ^{2}\) (\(i=1,2,3\), \(k\in \mathbb{M}\)),
where \(H_{1}(x,u,k)=D_{1}(k,u)S(t)\), \(H_{2}(x,u,k)=D_{2}(k,u)I(t)\), \(H _{3}(x,u,k)=D_{3}(k,u)V(t)\) with \(\vert x\vert \vee \vert y\vert \leq m\);
-
(H2)
\(\vert \log (1+D_{i}(k,u))\vert <\infty \) for \(D_{i}(k,u)>-1\) (\(i=1,2,3\), \(k\in \mathbb{M}\)).
Theorem 3.1
Let assumptions (H1) and (H2) hold. Then, for any given initial value
\((S(0),I(0),V(0)) \in R_{+}^{3}\), there is a unique solution
\((S(t),I(t),V(t))\)
of system (1.3) on
\(t\geq 0\)
almost surely, and the solution remains in
\(R_{+}^{3}\)
with probability 1.
Proof
Since the drift and the diffusion of system (1.3) are both locally Lipschitz, for any given initial value \((S(0),I(0),V(0))\in R_{+}^{3}\), there is a unique local solution \((S(t),I(t),V(t))\in R_{+}^{3}\) for any \(t\in [0,\tau_{e})\), where \(\tau_{e}\) is the explosion time [25]. Let \(\eta_{0}>0\) be sufficiently large such that
$$\tau_{\eta }=\inf \biggl\{ t\in [0,\tau_{e}):S(t)\notin \biggl( \frac{1}{\eta }, \eta \biggr), I(t)\notin \biggl(\frac{1}{\eta },\eta \biggr), \mbox{ or } V(t)\notin \biggl(\frac{1}{\eta },\eta \biggr)\biggr\} . $$
Obviously, \(\tau_{\eta }\) is increasing as \(\eta \rightarrow \infty \), and \(\tau_{\infty }=\lim_{\eta \rightarrow \infty }\tau_{ \eta }\leq \tau_{e}\) a.s. To show that the solution is global, it suffices to show that \(\tau_{\infty }=\infty \) a.s.
Consider the following Lyapunov function:
$$ W(S,I,V,k)=c_{1}(k) (S-1-\log S)+c_{2}(k) (I-1-\log I)+c_{3}(k) (V-1- \log V), $$
(3.1)
where \(c_{i}(k)\) (\(i=1,2,3\)) are positive constants for all \(k\in \mathbb{M}\).
Set \(T>0\) be arbitrary. Then, for any \(0< t<\tau_{\eta }\wedge T\), we have
$$\begin{aligned}& dW(S,I,V,k) \\& \quad =LW(S,I,V,k)\,dt+c_{1}(k) \sigma_{1}(k) (S-1)\,dB_{1}(t)+c_{2}(k) \sigma_{2}(k) (I-1)\,dB_{2}(t) \\& \qquad {}+c_{3}(k)\sigma_{3}(k) (V-1)\,dB_{3}(t)+ \int_{U}\bigl[c_{1}(k) \bigl(D_{1}(k,u)S- \log \bigl(1+D_{1}(k,u)\bigr)\bigr) \\& \qquad {}+c_{2}(k) \bigl(D _{2}(k,u)I-\log \bigl(1+D_{2}(k,u) \bigr)\bigr) +c_{3}(k) \bigl(D_{3}(k,u)V \\& \qquad {}-\log \bigl(1+D_{3}(k,u)\bigr)\bigr)\bigr] \tilde{N}(dt,du), \end{aligned}$$
(3.2)
where
$$\begin{aligned}& LW(S,I,V,k) \\& \quad \leq c_{1}(k) \bigl(1-q(k) \bigr)A(k)+c_{1}(k) \bigl(\mu (k)+p(k)\bigr)+c_{2}(k) \bigl( \mu (k)+\gamma (k)+\alpha (k)\bigr) \\& \qquad {}+c_{3}(k) \bigl(q(k)A(k)+\mu (k)+\varepsilon (k)\bigr)- \bigl(c_{1}(k)-c_{2}(k)\bigr)\beta (k) SI - \bigl(c_{1}(k) \bigl(\mu (k) \\& \qquad {}+p(k)\bigr)+c _{2}(k)\beta (k)-c_{3}(k)p(k)\bigr)S- \bigl(c_{2}(k) \bigl(\mu (k)+\gamma (k)+\alpha (k)\bigr)-c _{1}(k) \\& \qquad {}\times \bigl(\gamma (k)+\beta (k)\bigr)\bigr)I -\bigl(c_{3}(k) \bigl(\mu (k)+\varepsilon (k)\bigr)-c _{1}(k)\varepsilon (k) \bigr)V-c_{1}(k) \bigl(1-q(k)\bigr)A(k)\frac{1}{S} \\& \qquad {}-c_{1}(k) \gamma (k)\frac{I}{S}-c_{1}(k) \varepsilon (k)\frac{V}{S}-c_{3}(k)q(k)A(k) \frac{1}{V}-c_{3}(k)p(k) \frac{S}{V}+\frac{1}{2}c_{1}(k)\sigma_{1}^{2}(k) \\& \qquad {}+\frac{1}{2}c_{2}(k)\sigma_{2}^{2}(k)+ \frac{1}{2}c_{3}(k)\sigma _{3}^{2}(k)+ \int_{U}\bigl[c_{1}(k) \bigl(D_{1}(k,u)- \log \bigl(1+D_{1}(k,u)\bigr)\bigr) \\& \qquad {}+c _{2}(k) \bigl(D_{2}(k,u)-\log \bigl(1+D_{2}(k,u) \bigr)\bigr)+c_{3}(k) \bigl(D_{3}(k,u)-\log \bigl(1+D _{3}(k,u)\bigr)\bigr)\bigr]\nu (du) \\& \qquad {}+\sum_{l=1}^{N}\gamma_{kl}W(S,I,V,l). \end{aligned}$$
Choose
$$\begin{aligned}& c_{1}(k)>c_{2}(k),\qquad c_{1}(k)\bigl(\mu (k)+p(k)\bigr)+c_{2}(k)\beta (k)>c_{3}(k)p(k), \\& c_{3}(k)\bigl(\mu (k)+\varepsilon (k)\bigr)>c_{1}(k)\varepsilon (k), \quad \mbox{and} \\& c_{2}(k)\bigl(\mu (k)+\gamma (k)+\alpha (k)\bigr)>c_{1}(k)\bigl( \gamma (k)+\beta (k)\bigr) \quad \mbox{for }k\in \mathbb{M}. \end{aligned}$$
By Assumption (H2) and the inequality \(D_{i}(k,u)-\log (1+D_{i}(k,u)) \geq 0\) for \(D_{i}(k,u)>-1\), we have
$$\begin{aligned} LW \leq &c_{1}(k) \bigl(1-q(k)\bigr)A(k)+c_{1}(k) \bigl(\mu (k)+p(k)\bigr)+c_{2}(k) \bigl(\mu (k)+ \gamma (k)+\alpha (k) \bigr) \\ &{}+c_{3}(k) \bigl(q(k)A(k) +\mu (k)+\varepsilon (k)\bigr)+ \frac{1}{2}c_{1}(k) \sigma_{1}^{2}(k)+\frac{1}{2}c_{2}(k) \sigma_{2} ^{2}(k)\\ &{}+\frac{1}{2}c_{3}(k) \sigma_{3}^{2}(k)+3K+\sum_{l=1}^{N} \gamma _{kl}V(S,I,V,l), \end{aligned}$$
where \(K=\max_{1\leq i\leq 3}\{\int_{U}c_{i}(k)(D_{i}(k,u)- \log (1+D_{i}(k,u)))\nu (du)\}\).
Let \(\check{c}=\max \{\frac{c_{i}(l)}{c_{i}(k)}:1\leq i\leq 3,1\leq l,k \leq N\}\). Then, for any \(l,k\in \mathbb{M}\), we get
$$\begin{aligned} W(S,I,V,l) \leq& \check{c}\bigl[c_{1}(k) (S-1-\log S)+c_{2}(k) (I-1-\log I)+c _{3}(k) (V-1-\log V)\bigr] \\ =&\check{c}W(S,I,V,k). \end{aligned}$$
Therefore
$$\sum_{l=1}^{N}\gamma_{kl}W(S,I,V,l) \leq \check{c}\Biggl(\sum_{l=1}^{N}\vert \gamma_{kl}\vert \Biggr)W(S,I,V,k), $$
and thus
$$\begin{aligned} LW(S,I,V,k) :=&\bar{K}+\sum_{l=1}^{N} \gamma_{kl}W(S,I,V,k) \\ \leq& \tilde{K}\bigl[1+W(S,I,V,k)\bigr], \end{aligned}$$
(3.3)
where
$$\begin{aligned}& \tilde{K} =\max \Biggl\{ \bar{K}, \check{c}\Biggl(\sum _{l=1}^{N}\vert \gamma_{kl}\vert \Biggr) \Biggr\} , \\& \bar{K}=c_{1}(k) \bigl(1-q(k)\bigr)A(k)+c _{1}(k) \bigl( \mu (k)+p(k)\bigr)+c_{2}(k) \bigl(\mu (k)+\gamma (k)+\alpha (k)\bigr) \\& \hphantom{\bar{K}=}{}+c _{3}(k) \bigl(q(k)A(k)+\mu (k)+\varepsilon (k)\bigr)+ \frac{1}{2}c_{1}(k)\sigma _{1}^{2}(k)+ \frac{1}{2}c_{2}(k)\sigma_{2}^{2}(k)+ \frac{1}{2}c_{3}(k) \sigma_{3}^{2}(k)+3K. \end{aligned}$$
Integrating both sides of (3.2) from 0 to \(\tau_{\eta }\wedge T\) and taking expectation yield
$$\begin{aligned}& EW\bigl(S(\tau_{\eta }\wedge T),I(\tau_{\eta } \wedge T),V(\tau_{\eta } \wedge T),r(\tau_{\eta }\wedge T)\bigr) \\& \quad \leq W\bigl(S(0),I(0),V(0),r(0)\bigr)+E \int_{0}^{\tau_{\eta }\wedge T}LW\bigl(S(\tau),I(\tau),V(\tau),r(\tau)\bigr)\,d\tau \\& \quad \leq \bigl[W\bigl(S(0),I(0),V(0),r(0)\bigr)+\tilde{K}T\bigr]e^{\tilde{K}T}. \end{aligned}$$
In fact, we find that
$$\begin{aligned}& \bigl[W\bigl(S(0),I(0),V(0),r(0)\bigr)+\tilde{K}T \bigr]e^{\tilde{K}T} \\& \quad \geq E\bigl[1_{\{ \tau_{\eta }\wedge T\}}W\bigl(S(\tau_{m},\omega),I( \tau_{m},\omega),V( \tau_{m},\omega),r( \tau_{m},\omega)\bigr)\bigr] \\& \quad \geq \min_{1\leq i\leq 3}\biggl\{ c_{i}(k) (\eta -1-\log \eta), c_{i}(k) \biggl(\frac{1}{ \eta }-1+\log \eta \biggr)\biggr\} P(\tau_{\eta }\leq T), \end{aligned}$$
where \(1_{\{\tau_{\eta }\wedge T\}}\) is the indicator function of \(\{\tau_{\eta }\wedge T\}\). Letting \(\eta \rightarrow \infty \) implies
$$P(\tau_{\infty }\leq T)=0. $$
By the arbitrariness of T we can see that
$$P(\tau_{\infty }=\infty)=1. $$
Thus, the proof is complete. □