On matrix-SIR Arino models with linear birth rate, loss of immunity, disease and vaccination fatalities, and their approximations

Abstract

In this work we study the stability properties of the equilibrium points of deterministic epidemic models with nonconstant population size. Models with nonconstant population have been studied in the past only in particular cases, two of which we review and combine. Our main result shows that for simple “matrix epidemic models” introduced in [1], an explicit general formula for the reproduction number R and the corresponding “weak stability alternative” [2, Thm 1] still holds, under small modifications, for models with nonconstant population size, and even when the model allows for vaccination and loss of immunity. The importance of this result is clear once we note that the models of [1] include a large number of viral and bacterial models of epidemic propagation, including for example the totality of homogeneous COVID-19 models. To better understand the nature of the result, we emphasize that the models proposed in [1] and considered here are extensions of the SIR-PH model [3], which is essentially characterized by a phase-type distribution (~α, A) that models transitions between the “disease/infectious compartments”. In these cases, the reproduction number R and a certain Lyapunov function for the disease free equilibrium are explicitly expressible in terms of (~α, A). Not surprisingly, accounting for varying demography, loss of immunity, and vaccinations lead to several challenges. One of the most important is that a varying population size leads to multiple endemic equilibrium points: this is in contrast with “classic models,” which in general admit unique disease-free and endemic equilibria. As a special case of our analysis, we consider a “first approximation” (FA) of our model, which coincides with the constant-demography model often studied in the literature, and for which more explicit results are available. Furthermore, we propose a second heuristic approximation named “intermediate approximation” (IA). We hope that more light on varying population models with loss of immunity and vaccination, which have been largely avoided until now – see though [4–11] – will emerge in the future.