Modeling reinfections in epidemics appeared of importance during the recent stages of the Covid-19 pandemic (2021-22); they were frequent, especially due to the Delta and Omicron strains. The classical Poisson distribution describes reinfections without the impact of immunity. However, immunity is important here, and a hardcore lattice version of the Poisson distribution is needed, a discrete variant of the so-called Matern II process in 1D. Combinatorially, segments of one or several different lengths (protective immunity intervals) are placed in a bigger segment (the epidemic cycle) with spaces between them. The role of the edges is significant: the duration of immunity is comparable with that of the epidemic cycles. We provide exact formulas for the corresponding distributions, the lattice one and its continuous limit.
Discrete Poisson hardcore 1D model and reinfections