New Results and Open Questions for SIR-PH Epidemic Models with Linear Birth Rate, Loss of Immunity, Vaccination, and Disease and Vaccination Fatalities

Our paper presents three new classes of models: SIR-PH, SIR-PH-FA, and SIR-PH-IA, states two problems we would like to solve about them. Recall that deterministic mathematical epidemiology has one basic general law, the “R0 alternative” Van den Driessche Watmough, which local stability condition disease-free equilibrium may be expressed as R0<1, where R0 is famous reproduction number, also plays a major role in theory branching processes. The literature suggests it impossible find laws concerning endemic points. However, quite common 1. When R0>1, there exists unique fixed point, 2. point locally stable when R0>1. One establish these properties for large class realistic epidemic models (and do not include here epidemics without casualties). We have introduced recently “simple” but broad “SIR-PH models” with varying populations, express purpose establishing processes above. Since seemed still hard, further “SIR-PH-FA” models, interpreted approximations SIR-PH simpler typically studied (with constant population, loss immunity, etc.). For this class, first “endemic law” above “almost established”, explicit formulas are available, independently number infectious compartments, only remains check its belonging invariant domain. This yet turn out always verified, been able that. second property, sufficiency R0>1 an open even SIR-PH-FA despite numerous particular cases was checked hold (via Routh–Hurwitz time-onerous computations, or Lyapunov functions). goal our draw attention above, second, more refined “intermediate approximation” SIR-PH-IA. illustrate current status-quo by presenting results on generalization SAIRS model.