Criterion for moderate infection under increasing division rates

Let $X_t$ be the total quantity of parasites at time $t$, let $V_t$ be the set of cells at time $t$, let $N_t=\#V_t$, and let $X_t^i$ denote the parasite quantity in cell $i\in V_t$. Assume that the division rate $r$ is increasing, and write

$$r^*=\sup\{r(x):x\geq 0\}.$$

Let $g>0$ be the parasite-growth drift and let $\Theta$ be the random fraction governing parasite sharing. Criterion for moderate infection. (i) If

$$g<2r^*\mathbb{E}(\log(1/\Theta)),$$

then, almost surely,

$$\lim_{A\rightarrow\infty}\lim_{t\rightarrow+\infty}\frac{\#\{i\in V_t:X_t^i\geq A\}}{N_t}=0.$$

(ii) If

$$g>2r^*\mathbb{E}(\log(1/\Theta)),$$

then for every $A>0$,

$$\left\{\limsup_{t\rightarrow+\infty}\frac{\#\{i\in V_t:X_t^i\geq A\}}{N_t}>0\right\}=\left\{\forall t>0,\ X_t>0\right\}\quad\text{almost surely}.$$

This criterion separates moderate infection, where the proportion of cells carrying at least $A$ parasites vanishes as $A$ tends to infinity, from persistent infection in a positive proportion of cells. It is proposed using an approximation of the evolution of the cell proportions and extends the recovery criterion from the constant-division-rate case; the source presents it as a conjectural criterion rather than a proved result.