Quasi-stationary distributions in fixed and random vector-population epidemic models

Let $n$ be the population-size scaling parameter, let $I_h^n(t)$ and $I_v^n(t)$ denote infected hosts and infected vectors, and let $\mathcal{R}_0$ be the basic reproduction number. Consider both the model with fixed host and random vector population sizes and the model with fixed host and vector population sizes. Quasi-stationary distribution conjecture. For large $n$: (i) if $\mathcal{R}_0<1$, $(I_h^n(t),I_v^n(t))$ approaches $(0,0)$ exponentially fast as $t\to\infty$; (ii) if $\mathcal{R}_0>1$, the model with fixed host and random vector population size has no quasi-stationary distribution; and (iii) if $\mathcal{R}_0>1$, the model with fixed host and vector population sizes admits a quasi-stationary distribution that can be approximated by a normal distribution with mean $nE^e$ and variance $n\Sigma^*$ as specified in the source. The claim contrasts the effects of random and fixed vector population sizes on long-term epidemic behavior. The preceding discussion explains the absence of a quasi-stationary distribution in the random-vector case and the approximate normal law in the fixed-vector case; no separate resolution status is given.