  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.
