Explosiveness of age-dependent branching processes with contagious and incubation periods

We study explosiveness of age-dependent branching processes describing the early stages of an epidemic-spread: both forward- and backward process are analysed. For the classical age-dependent branching process $(h,G)$, where the offspring has probability generating function $h$ and all individuals have life-lengths independently picked from a distribution $G$, we focus on the setting $h = h_{\alpha}^L$, with $L$ a function varying slowly at infinity and $\alpha \in (0,1)$. Here, $h^L_{\alpha}(s) = 1 - (1-s)^{\alpha} L(\frac{1}{1-s}),$ as $s \to 1$. For a fixed $G$, the process $(h^L_{\alpha},G)$ explodes either for all $\alpha \in (0,1)$ or for no $\alpha \in (0,1)$, regardless of $L$. Next, we add contagious periods to all individuals and let their offspring survive only if their life-length is smaller than the contagious period of their mother: a forward process. An explosive process $(h^L_{\alpha},G)$, as above, stays explosive when adding a non-zero contagious period. We extend this setting to backward processes with contagious periods. Further, we consider processes with incubation periods during which an individual has already contracted the disease but is not able yet to infect her acquaintances. We let these incubation periods follow a distribution $I$. In the forward process $(h^L_{\alpha},G,I)_{f}$, every individual possesses an incubation period and only her offspring with life-time larger than this period survives. In the backward process $(h^L_{\alpha},G,I)_{b}$, individuals survive only if their life-time exceeds their own incubation period. These two processes are the content of the third main result that we establish: under a mild condition on $G$ and $I$, explosiveness of both $(h,G)$ and $(h,I)$ is necessary and sufficient for processes $(h^L_{\alpha},G,I)_{f}$ and $(h^L_{\alpha},G,I)_{b}$ to explode.