Fig. 3.

The smooth curve in each figure is the solution of the host-vector ODE model. The other curves are one sample path of the CTMC model (top two figures) and one sample path of the SDE model (bottom two figures). The parameters are $H=100,M=1000$, $\alpha =0.2$, $\gamma =0.1$, $\beta =0.2$, and $\mu =0.5$. Initial conditions are $I\left(0\right)=0$, $V\left(0\right)=10$, $S\left(0\right)=100$ and $U\left(0\right)=990.$ The value of ${\mathcal{R}}_0=8$ with $q_1=1/6$ and $q_2=3/4$. The probability of extinction in the CTMC is approximately $\mathrm{\mathbb{P}}\approx 0.06$. The stable endemic equilibrium in the ODE model is $\left(\overline{S},\overline{I},\overline{U},\overline{V}\right)$ = (16.7, 83.3, 750, 250).