  In this paper we initiate the theory of Crump-Mode-Jagers branching processes
(BP) in the setting where no Malthusian parameter exist, i.e., the process
grows faster than exponential. A Crump-Mode-Jagers BP is a branching process
(in continuous time) where arbitrary dependencies are allowed between the
birth-times of the children of a single individual in the population. It is
however assumed that these reproduction processes are i.i.d. point processes
for different individuals. This paper focuses on determining whether this
branching process explodes, that is, the process reaches infinitely many
individuals in finite time. We develop comparison techniques between
reproduction processes. We study special cases in terms of explosivity such as
age-dependent BPs, and epidemic models with contagious intervals. For this, we
superimpose a random contagious interval $[I, C]$ on every individual in the BP
and keep only the children with birth-times that fall in this interval of the
parent. We show that the distribution of the end $C$ of the contagious interval
does not matter in terms of explosion, while the distribution of $I$ does: the
epidemic explodes if and only if the two age-dependent BPs with the original
birth-times and birth-times $I$ explode. We finish studying some pathological
examples such as birth-time distributions that are singular to the
Lebesque-measure yet they produce an explosive BP with arbitrary power-law
offspring distributions.
