## 2 Existence and Uniqueness of the Global Positive Solution  
In this section, we first define \(\Delta\) as follows:  
\[\Delta:=\left\{\left(S;I;C;R\right)\in\mathbb{R}_{+}^{4}:\frac{\lambda}{\mu+ \mathfrak{d}}\leq S+I+C+R\leqq\frac{\lambda}{\mu}\right\}\]  
and we consider the following assumption:  
\[(\text{A})\ :\ 0<Q(u)\leqq\frac{\mu}{\lambda}\quad(u\in U).\]  
**Theorem 1**. : _With the initial condition \(\left(S(0);I(0);C(0);R(0)\right)\in\Delta\), the system (1) has a unique global solution \(\left(S(t);I(t);C(t);R(t)\right)\in\Delta\) for any \(t\geqq 0\) a.s. Moreover, each of the following inequalities holds true:_  
\[\limsup_{t\to\infty}S(t)\leqq\frac{\lambda}{\mu}\ \ \text{a.s.}\qquad\text{and} \qquad\liminf_{t\to\infty}S(t)\geqq\frac{\lambda}{\mu+\mathfrak{d}}\ \ \text{a.s.},\] \[\limsup_{t\to\infty}I(t)\leqq\frac{\lambda}{\mu}\ \ \text{a.s.}\qquad\text{and} \qquad\liminf_{t\to\infty}I(t)\geqq\frac{\lambda}{\mu+\mathfrak{d}} \text{a.s.},\] \[\limsup_{t\to\infty}C(t)\leqq\frac{\lambda}{\mu}\ \ \text{a.s.}\qquad\text{and} \qquad\liminf_{t\to\infty}C(t)\geqq\frac{\lambda}{\mu+\mathfrak{d}}\ \ \text{a.s.},\]  
_and_  
\[\limsup_{t\to\infty}R(t)\leqq\frac{\lambda}{\mu}\ \ \text{a.s.}\qquad\text{and} \qquad\liminf_{t\to\infty}R(t)\geqq\frac{\lambda}{\mu+\mathfrak{d}}\ \ \text{a.s.}\]  
Proof. : Since the drift and the diffusion are local Lipschitzian, and in view of the initial condition \(\left(S(0);I(0);C(0);R(0)\right)\in\Delta\), we prove the existence and uniqueness of the local \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Meaning** \\ \hline \(\lambda\) & The recruitment rate \\ \hline \(\mu\) & Natural death rate \\ \hline \(\beta\) & The transmission rate \\ \hline \(\gamma_{2}\) & The average to the infected \(I\) individuals becoming chronically infected \\ \hline \(\gamma_{1}\) & Default treatment rate for \(I\) individuals \\ \hline \(\epsilon\) & The average of the recovered individuals returning infected \\ \hline \(\delta\) & The average the chronically infected individuals returning infected \\ \hline \(\mathfrak{d}\) & Death rate due to the infection in chronic stage \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters and their meanings in the suggested SICR model. Figure 1: The diagram of the SICR model. solution \(\big{(}S(t);I(t);C(t);R(t)\big{)}\in\Delta\) for \(t\in[0,t_{e})\), where \(t_{e}\) is the explosion time. In order to prove that this solution is global, we define the stopping time \(\tau\) given by  
\[\tau=\{t\in[0,t_{e}):S(t)\leqq 0,\ I(t)\leqq 0,\ C(t)\leqq 0\ \ \text{and}\ \ R(t)\leqq 0\}.\]  
If we suppose that \(t_{e}<\infty\), then \(\tau\leqq t_{e}\), so there exist \(T>0\) and \(\epsilon>0\) such that \(P(\tau\leqq T)>\epsilon\). Consider the function \(V\) on \(\mathbb{R}^{4}_{+}\) defined by  
\[V(S,I,C,R)=\log(SICR).\]  
Using the [PERSON]'s formula, we get  
\[dV=LVdt+\bigg{(}\frac{\lambda}{S}-\beta I+\beta S-\gamma_{1}- \gamma_{2}-2\mu+\epsilon\frac{R}{I}+\delta\frac{C}{I}\bigg{)}dW_{t}\] \[\qquad\qquad+\int_{U}\log(1-QI)(1+QS)\tilde{N}(dt,du),\]  
where  
\[LV= (\lambda-\beta IS-\mu S)\frac{1}{S}+\left(\beta IS-(\gamma_{1}+ \gamma_{2}+\mu)\,I+\epsilon R+\delta C\right)\,\frac{1}{I}-\frac{\sigma^{2}I^ {2}}{2}\] \[-\frac{\sigma^{2}S^{2}}{2}+\int_{U}[\log(1-QI)(1+QS)-QI+QS]\
u(du).\]  
Noting that \(1-QI>0\) from Assumption (A), this fact implies that  
\[LV\geqq-\frac{\beta\lambda}{\mu}-2\mu-\gamma_{1}-\gamma_{2}- \frac{\sigma^{2}\lambda^{2}}{\mu^{2}}+\int_{U}[\log(1-QI)+QI]\
u(du)\] \[\qquad\qquad+\int_{U}[\log(1+QS)-QS]\
u(du)\] Figure 1:0  
Since \(x\longmapsto\log(1-x)+x\) and \(x\longmapsto\log(1+x)-x\) are nonpositive functions, we find that  
Figure 1:1 Figure 1:2  
which, upon integrating from \(0\) to \(t\), yields  
Figure 1:3 Figure 1:4  
Now, by using the continuity property, some components of \(\big{(}S(\tau),I(\tau),C(\tau),R(\tau)\big{)}\) will be equal to \(0\), so we obtain  
Figure 1:5For \(t\to\tau\), we get  
Figure 1:6 Figure 1:7 Figure 1:8  
which contradicts our hypothesis.