An Age of Infection Kernel, an <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">R</mi></mrow></semantics></math></inline-formula> Formula, and Further Results for Arino–Brauer <i>A</i>, <i>B</i> Matrix Epidemic Models with Varying Populations, Waning Immunity, and Disease and Vaccination Fatalities

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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mover accent="true"><mi>α</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>A</mi><mo>)</mo></mrow></semantics></math></inline-formula>, modeling transitions in “disease/infectious compartments”. The <i>A</i> 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>α</mi><mo stretchy="false">→</mo></mover></semantics></math></inline-formula> is to restrict it to the case where there is only one susceptible class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">s</mi></semantics></math></inline-formula>, and when matrix <i>B</i> (given by the part of the new infection matrix, with respect to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">s</mi></semantics></math></inline-formula>) is of rank one, with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mo>=</mo><mi>b</mi><mover accent="true"><mi>α</mi><mo stretchy="false">→</mo></mover></mrow></semantics></math></inline-formula>. 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo></mrow></semantics></math></inline-formula> 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo></mrow></semantics></math></inline-formula> 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo></mrow></semantics></math></inline-formula> Arino–Brauer models include a large number of epidemic models (COVID, ILI, influenza, illnesses, etc.).