Large-population approximation for the variable-vector epidemic model

Let $n$ be the host population size, let $I_h^n(t)$ denote the number of infected hosts at time $t$, and let $\mathcal{R}_0$ be the basic reproduction number. When $\mathcal{R}_0>1$, let $E^e$ be the endemic equilibrium and let $\Sigma^*$ be the limiting covariance defined by $\Sigma^*=-\frac{\sigma_e^2}{2C_e}$, with $C_e$ and $\sigma_e$ as above. Large-population approximation. For large $n$, (i) if $\mathcal{R}_0<1$, $I_h^n(t)$ approaches $0$ exponentially fast as $t\to\infty$; and (ii) if $\mathcal{R}_0>1$, the quasi-stationary distribution of $I_h^n$ can be approximated by a normal distribution with mean $nE^e$ and covariance matrix $n\Sigma^*$, where

$$E^e=\frac{C_0\beta_h\beta_v-\gamma_h\gamma_v}{C_0\beta_h\beta_v+\beta_v\gamma_h}.$$

This gives a large-population description of extinction and endemic fluctuations for the model. The statement is presented as an approximation based on the fluid and diffusion limits; its resolution is not specified in the source.