Explosive Crump-Mode-Jagers branching processes

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.