## 5 Numerical Results  
In order to illustrate the numerical simulations of the model (1), we consider the following problem:  
\[dX(t)=f\big{(}t,X(t)\big{)}dt+\sigma\big{(}t,X(t)\big{)}dW_{t}+\int_{U}Q(t,u)g \big{(}t-,X(t-)\big{)}\tilde{N}(dt,du). \tag{2}\]  
The solution of the system (2) will now be given by  
\[X(t)=X(0) +\underbrace{\int_{0}^{t}f\big{(}s,X(s)\big{)}ds+\int_{0}^{t} \sigma\big{(}s,X(s)\big{)}dW_{s}}_{\text{Part 1}}\] \[+\underbrace{g\big{(}t-,X(t-)\big{)}\int_{0}^{t}\int_{U}Q(s,u) \tilde{N}(ds,du)}_{\text{Part 2}}, \tag{3}\]  
The Milsteins Higher Order Method consists of approximating Part 1 in the system (2) in the following form:  
\[X_{k+1}=X_{k} +f(t_{k},X_{k})\Delta t+\sigma(t_{k},X_{k})(W_{k+1}-W_{k})\] \[\ \ \ \ \ \ \ +\frac{1}{2}\sigma^{2}(t_{k},X_{k})\Big{[}(W_{k+1}-W_{k })^{2}-(t_{k+1}-t_{k})\Big{]}.\]  
About the approximation of Part 2, we have two cases. Consider any infinitesimal interval \([T_{i},T_{i+1})\subset(\tau_{i},\tau_{i+1})\).  
There is no jump point in this interval:  
\[g\big{(}t-,X(t-)\big{)}\int_{T_{i}}^{T_{i+1}}\int_{U}Q(s,u)\tilde{N}(ds,du)=0.\]  
If there is only one jump point \(\tau_{i}\in[T_{i},T_{i+1})\), then  
\[g\big{(}t-,X(t-)\big{)}\int_{T_{i}}^{T_{i+1}}\int_{U}Q(u)\tilde{N}(ds,du)=g \big{(}\tau_{i}-,X(\tau_{i}-)\big{)}Q\big{(}\tau_{i},\zeta(\tau_{i})\big{)},\]  
so that  
\[g\big{(}t-,X(t-)\big{)}\int_{0}^{t}\int_{U}Q(s,u)\tilde{N}(ds,du)=\sum_{i=0}^{ i=n}g\big{(}\tau_{i}-,X(\tau_{i}-)\big{)}Q\big{(}\tau_{i},\zeta(\tau_{i}) \big{)}.\]  
Therefore, the Milsteins Higher Order Method of the system (2) will be given by  
\[X_{k+1}=X_{k} +f\big{(}t_{k},X_{k}\big{)}\big{)}\Delta t+\sigma(t_{k},X_{k})(W_ {k+1}-W_{k})\] \[+\frac{1}{2}\sigma^{2}(t_{k},X_{k})\Big{(}(W_{k+1}-W_{k})^{2}- \Delta t\Big{)}+g\big{(}t_{k}-,X(t_{k})\big{)}Q\big{(}t_{k},\zeta(t_{k})\big{)}\]  
We now use the previous method to solve the system (1).  
Figures 2 and 3 show the dynamics of the susceptible population for the case of the extinction of the infection. From these figures, we observe clearly that the infected compartment converges to 0. It should be noted that, in this case, the susceptible class increases to attain their maximum, which means the the disease dies out. This is consistent with our theoretical results for the extinction of the infection.  
The behavior of the infection for both the infected and susceptible population with the Levy jump process is graphically illustrated in Figures 4 and 5 in the case of the persistence of the infection. In this epidemic scenario, we note from all of the four SICR compartments, that is, the infected, the susceptible, the chronic sub-population and the recovered, that the infection persists. This shows that our theoretical and numerical results are compatible.  
Figure 2: The Susceptible population as a function of time when \(\lambda=10\), \(\mu=0.0125\), \(\beta=0.0001\), \(\gamma_{2}=1\), \(\gamma_{1}=0.1\), \(\epsilon=0.33\), \(\delta=0.09\) and \(\vartheta=0.025\).  
Figure 4: The susceptible and infected populations as functions of time when \(\lambda=100\), \(\mu=0.0013\), \(\beta=0.1\), \(\gamma_{2}=1\), \(\gamma_{1}=0.1\), \(\epsilon=0.33\), \(\delta=0.09\) and \(\mathfrak{d}=0.025\).