On the Carrying and Evolution Matrices in Epidemic Models

Sen, M. De La; Ibeas, Asier; Alonso‐Quesada, S.; Nistal, Raúl

doi:10.1088/1742-6596/1746/1/012015

Cited by 1 publication

(2 citation statements)

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“…At this point, let us mention that we believe that epidemic models should be ideally parameterized by the two matrices F, V, which intervene in the next generation matrix approach, which has been called disease-carrying and state evolution matrices [6,7]. A foundational paper in this direction is [8], Arino shows that further simplifications arise for models having only one susceptible class and disease-carrying matrix of rank one.…”

Section: Introductionmentioning

confidence: 99%

“…Note that the matrices F, V are featured in the famous the (Next Generation Matrix) approach [13], that some authors refer to them as "new infections" and "transmission matrices", that [7] call them the disease-carrying and state evolution matrices, and that ( [27], Ch 5) gives a way to define an associated stochastic birth and death model associated to these matrices. The matrix B is further useful in defining and studying more general SIR-PH models-see [9,10] and below, and see also [28], ([29], (2.1)), [6] for related works.…”

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confidence: 99%

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

2022

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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.

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Section: Introductionmentioning

confidence: 99%