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Our paper presents three new classes of models: SIR-PH, SIR-PH-FA, andSIR-PH-IA, and states two problems we would like to solve about them. Recallthat deterministic mathematical epidemiology has one basic general law, the R0alternative" of [52, 51], which states that the local stability condition ofthe disease free equilibrium may be expressed as R0 < 1, where R0 is the famousbasic reproduction number, which plays also a major role in the theory ofbranching processes. The literature suggests that it is impossible to findgeneral laws concerning the endemic points. However, it is quite common that  1. When R0 > 1, there exists a unique fixed endemic point, and  2. the endemic point is locally stable when R0 > 1.  One would like to establish these properties for a large class of realisticepidemic models (and we do not include here epidemics without casualties). Wehave introduced in [7, 5] a "simple", but broad class of "SIR-PH models" withvarying population, with the express purpose of establishing for theseprocesses the two properties above. Since that seemed still hard, we haveintroduced a further class of "SIR-PH-FA" models, which may be interpreted asapproximations for the SIR-PH models, and which includes simpler modelstypically studied in the literature (with constant population, without loss ofimmunity, etc). The goal of our paper is to draw attention to the two openproblems above, for the SIR-PH, SIR-PH-FA, and also for a second, more refined"intermediate approximation" SIR-PH-IA. We illustrate the current status-quo bypresenting new results on a generalization of the SAIRS epidemic model of [44,40].

New results and open questions for SIR-PH epidemic models with linear  birth rate, loss of immunity, vaccination, and disease and vaccination  fatalities