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In this work we study the stability properties of the equilibrium points ofdeterministic epidemic models with nonconstant population size. Models withnonconstant 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 thereproduction number and the corresponding "weak stability alternative" stillholds, under small modifications, for models with nonconstant population size,and even when the model allows for vaccination and loss of immunity. Theimportance of this result is clear once we note that the models of [1] includea large number of viral and bacterial models of epidemic propagation, includingfor example the totality of homogeneous COVID-19 models. To better understandthe nature of the result, we emphasize that the models proposed in [1] andconsidered here are extensions of the SIR-PH model, which is essentiallycharacterized by a phase-type distribution that models transitions between the"disease/infectious compartments". In these cases, the reproduction number anda certain Lyapunov function for the disease free equilibrium are explicitlyexpressible. Not surprisingly, accounting for varying demography, loss ofimmunity, and vaccinations lead to several challenges. One of the mostimportant is that a varying population size leads to multiple endemicequilibrium points: this is in contrast with "classic models" which in generaladmit unique disease-free and endemic equilibria. As a special case of ouranalysis, we consider a "first approximation" (FA) of our model, whichcoincides with the constant-demography model often studied in the literature,and for which more explicit results are available. Furthermore, we propose asecond heuristic approximation named "intermediate approximation" (IA).

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