A Stochastic Jump Model for Epidemics With Demography, and Confinement and Vaccination Controls: Safety Zones and Algorithms

This short paper aims at describing, from the mathematical point of view, two useful tools for the analysis of mixed pharmaceutical (vaccination) and non-pharmaceutical (confinement) measures. The compartmental model considered hereafter is derived with the standing assumption that deaths occur at a different scale than the transitions between the epidemiological compartments. Such mortality events will be designed as pure-jump ones, while the remaining transitions are designed using the law of mass action from the reaction (or interaction) network. The first objective of our paper is to describe feasible (yellow) zones in which non-pharmaceutical solutions exist maintaining the infections within intensive care units capacities. This is done independently of (or uniformly in) the vaccination/pharmaceutical measures. The analysis is complemented with a description of safe (green) zones in which the herd immunity is achieved without confinement and, again, uniformly with respect to vaccination policies. The second objective is to present a linear programming dual algorithm to optimize possible discontinuous costs with respect to the vaccination schemes and with state constraints induced by the feasible region. Attention will be paid to the particularities of our jump-type generators.