Convergence to endemic equilibrium in the stochastic epidemic model

Under Assumptions~--, let $R_0 > \mathbb{E}\left[\frac{1}{\gamma_*}\right]$ and suppose that $\overline{\mathfrak F}(0)>0$. The quantities $\overline{\mathfrak S}(t)$ and $\overline{\mathfrak F}(t)$ denote the mean susceptibility and force of infection, respectively. The endemic equilibrium is the pair $(\overline{\mathfrak F}_*,\overline{\mathfrak S}_*)$, where

$$\overline{\mathfrak S}_*=1/R_0$$

and $\overline{\mathfrak F}_*$ is the unique positive solution of the equilibrium equation.

Convergence conjecture. Under these assumptions,

$$(\overline{\mathfrak F}(t),\overline{\mathfrak S}(t))\to(\overline{\mathfrak F}_*,\overline{\mathfrak S}_*)\quad\text{as }t\to\infty.$$

The conjecture asserts convergence to the unique endemic equilibrium when the basic reproduction parameter exceeds the threshold $\mathbb{E}[1/\gamma_*]$. The preceding lemma and corollary establish persistence and identify the equilibrium, while convergence itself remains conjectural.