Positive and conservative geco Methods for chemical and Epidemiological models

Many important phenomena in biology, chemistry, epidemiology and ecology are modelled by differential equations with two main qualitative characteristics: positive solutions and conservation of linear invariants, such us the total density. A suitable numerical integrator for this models should therefore be featured by the same two characteristics. GeCo (Geometric-Conservative) schemes [4] are explicit numerical integrators, up to second order, which preserve both linear invariants and positivity, independently on the choice of the step size. They falls in the general class of nonstandard schemes, where the advancement in time is lead by a non linear function of the temporal step size. We show the performance of GeCo methods in solving the following models: • a vibrational kinetics system, where the relaxation of the vibrational levels of diatomic molecules is considered [1]. This problem is an interesting test case from the numerical point of view, because of the large stiffness and the variation of the concentration over many orders of magnitude; • a SEIR model for Covid-19 pandemic analysed in [3], where the potential of awareness to produce social distancing and self-isolation among susceptibles are exploited, and the Z-control approach is applied to detect what trend must awareness display over time in order to eradicate the disease; • the SIDARTHE model introduced in [2] for predicting the course of the Covid-19 epidemic in Italy to help the planning of an effective control strategy.