We revisit here a landmark five-parameter SIR-type model, which is maybe the simplest example where a complete picture of all cases, including non-trivial bistability behavior, may be obtained using simple tools. We also generalize it by adding essential vaccination and vaccination-induced death parameters, with the aim of revealing the role of vaccination and its possible failure. The main result is Theorem 1, which describes the stability behavior of our model in all possible cases.

Many of the models used nowadays in mathematical epidemiology, in particular in COVID-19 research, belong to a certain subclass of compartmental models whose classes may be divided into three “(x,y,z)” groups, which we will call respectively “susceptible/entrance, diseased, and output” (in the classic SIR case, there is only one class of each type). Roughly, the ODE dynamics of these models contains only linear terms, with the exception of products between x and y terms. It has long been noticed that the reproduction number R has a very simple Formula in terms of the matrices which define the model, and an explicit first integral Formula is also available. These results can be traced back at least to Arino, Brauer, van den Driessche, Watmough, and Wu (2007) and to Feng (2007), respectively, and may be viewed as the “basic laws of SIR-type epidemics”. However, many papers continue to reprove them in particular instances. This motivated us to redraw attention to these basic laws and provide a self-contained reference of related formulas for (x,y,z) models. For the case of one susceptible class, we propose to use the name SIR-PH, due to a simple probabilistic interpretation as SIR models where the exponential infection time has been replaced by a PH-type distribution. Note that to each SIR-PH model, one may associate a scalar quantity Y(t) which satisfies “classic SIR relations”, which may be useful to obtain approximate control policies.

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][5][6][7][8][9][10][11] -will emerge in the future.

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In this work, we first introduce a class of deterministic epidemic models with varying populations inspired by Arino et al. (2007), the parameterization of two matrices, demography, the waning of immunity, and vaccination parameters. Similar models have been focused on by Julien Arino, Fred Brauer, Odo Diekmann, and their coauthors, but mostly in the case of “closed populations” (models with varying populations have been studied in the past only in particular cases, due to the difficulty of this endeavor). Our Arino–Brauer models contain SIR–PH models of Riano (2020), which are characterized by the phase-type distribution (α→,A), modeling transitions in “disease/infectious compartments”. The A matrix is simply the Metzler/sub-generator matrix intervening in the linear system obtained by making all new infectious terms 0. The simplest way to define the probability row vector α→ is to restrict it to the case where there is only one susceptible class s, and when matrix B (given by the part of the new infection matrix, with respect to s) is of rank one, with B=bα→. For this case, the first result we obtained was an explicit formula (12) for the replacement number (not surprisingly, accounting for varying demography, waning immunity and vaccinations led to several nontrivial modifications of the Arino et al. (2007) formula). The analysis of (A,B) Arino–Brauer models is very challenging. As obtaining further general results seems very hard, we propose studying them at three levels: (A) the exact model, where only a few results are available—see Proposition 2; and (B) a “first approximation” (FA) of our model, which is related to the usually closed population model often studied in the literature. Notably, for this approximation, an associated renewal function is obtained in (7); this is related to the previous works of Breda, Diekmann, Graaf, Pugliese, Vermiglio, Champredon, Dushoff, and Earn. (C) Finally, we propose studying a second heuristic “intermediate approximation” (IA). Perhaps our main contribution is to draw attention to the importance of (A,B) Arino–Brauer models and that the FA approximation is not the only way to tackle them. As for the practical importance of our results, this is evident, once we observe that the (A,B) Arino–Brauer models include a large number of epidemic models (COVID, ILI, influenza, illnesses, etc.).

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 Van den Driessche and Watmough, 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 also plays 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 recently a “simple” but broad class of “SIR-PH models” with varying populations, 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 include simpler models typically studied in the literature (with constant population, without loss of immunity, etc.). For this class, the first “endemic law” above is “almost established”, as explicit formulas for a unique endemic point are available, independently of the number of infectious compartments, and it only remains to check its belonging to the invariant domain. This may yet turn out to be always verified, but we have not been able to establish that. However, the second property, the sufficiency of R0>1 for the local stability of an endemic point, remains open even for SIR-PH-FA models, despite the numerous particular cases in which it was checked to hold (via Routh–Hurwitz time-onerous computations, or Lyapunov functions). The goal of our paper is to draw attention to the two open problems above, for the SIR-PH and 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.

Many of the models used nowadays in mathematical epidemiology, in particular in COVID-19 research, belong to a certain sub-class of compartmental models whose classes may be divided into three "(x, y, z)" groups, which we will call respectively"susceptible/entrance, diseased, and output" (in the classic SIR case, there is only one class of each type). Roughly, the ODE dynamics of these models contains only linear terms, with the exception of products between x and y terms. It has long been noticed that the basic reproduction number R has a very simple formula (3.3) in terms of the matrices which define the model, and an explicit first integral formula (3.8) is also available. These results can be traced back at least to [ABvdD + 07] and [Fen07], respectively, and may be viewed as the "basic laws of SIR-type epidemics"; however many papers continue to reprove them in particular instances (by the nextgeneration matrix method or by direct computations, which are unnecessary). This motivated us to redraw the attention to these basic laws and provide a self-contained reference of related formulas for (x, y, z) models. We propose to rebaptize the class to which they apply as matrix SIR epidemic models, abbreviated as SYR, to emphasize the similarity to the classic SIR case. For the case of one susceptible class we propose to use the name SIR-PH, due to a simple probabilistic interpretation as SIR models where the exponential infection time has been replaced by a PH-type distribution. We note that to each SIR-PH model, one may associate a scalar quantity Y (t) which satisfies"classic SIR relations" -see (3.8). In the case of several susceptible classes this generalizes to (5.10); in a future paper, we will show that (3.8), (5.10) may be used to obtain approximate control policies which compare well with the optimal control of the original model.

We revisit a five parameter SIR-type model of [DvdD93, Sec. 4], and generalize it by adding essential vaccination and vaccination induced death parameters, with the aim of revealing the role of vaccination and its possible failure.