Semi-semi-Markov processes : a new class of processes for formalizing and generalizing state-dependent individual-based models

Individual-based models are a “bottom-up” approach for calculating empirical distributions at the level of the population from simulated individual trajecto- ries. We build a new class of stochastic processes for mathematically formalizing and generalizing these simulation models according to a “top-down” approach, when the individual state changes occur at countable random times. We allow individual population-dependent semi-Markovian transitions in a non closed population such as a branching population. These new processes are called Semi-Semi-Markov Processes (SSMP) and are generalizations of Semi-Markov processes. We calculate their kernel and their probability law, and we build a simulation algorithm from the kernel. The starting point of this work was the modelling of the propagation of a disease (stochastic process) in a branching population with interactions (nonbounded random graph)