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We study explosiveness of age-dependent branching processes describing theearly stages of an epidemic-spread: both forward- and backward process areanalysed. For the classical age-dependent branching process $(h,G)$, where theoffspring has probability generating function $h$ and all individuals havelife-lengths independently picked from a distribution $G$, we focus on thesetting $h = h_{\alpha}^L$, 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 allindividuals and let their offspring survive only if their life-length issmaller than the contagious period of their mother: a forward process. Anexplosive process $(h^L_{\alpha},G)$, as above, stays explosive when adding anon-zero contagious period. We extend this setting to backward processes withcontagious periods. Further, we consider processes with incubation periodsduring which an individual has already contracted the disease but is not ableyet to infect her acquaintances. We let these incubation periods follow adistribution $I$. In the forward process $(h^L_{\alpha},G,I)_{f}$, everyindividual possesses an incubation period and only her offspring with life-timelarger than this period survives. In the backward process$(h^L_{\alpha},G,I)_{b}$, individuals survive only if their life-time exceedstheir own incubation period. These two processes are the content of the thirdmain result that we establish: under a mild condition on $G$ and $I$,explosiveness of both $(h,G)$ and $(h,I)$ is necessary and sufficient forprocesses $(h^L_{\alpha},G,I)_{f}$ and $(h^L_{\alpha},G,I)_{b}$ to explode.

Explosiveness of Age-Dependent Branching Processes with Contagious and Incubation Periods