  Our paper presents three new classes of models: SIR-PH, SIR-PH-FA, and
SIR-PH-IA, and states two problems we would like to solve about them. Recall
that deterministic mathematical epidemiology has one basic general law, the R0
alternative" of [52, 51], which states that the local stability condition of
the disease free equilibrium may be expressed as R0 < 1, where R0 is the famous
basic reproduction number, which plays also a major role in the theory of
branching processes. The literature suggests that it is impossible to find
general 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 realistic
epidemic models (and we do not include here epidemics without casualties). We
have introduced in [7, 5] a "simple", but broad class of "SIR-PH models" with
varying population, with the express purpose of establishing for these
processes the two properties above. Since that seemed still hard, we have
introduced a further class of "SIR-PH-FA" models, which may be interpreted as
approximations for the SIR-PH models, and which includes simpler models
typically studied in the literature (with constant population, without loss of
immunity, etc). The goal of our paper is to draw attention to the two open
problems 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 by
presenting new results on a generalization of the SAIRS epidemic model of [44,
40].
