Some Probabilistic Interpretations Related to the Next-Generation Matrix Theory: A Review with Examples

Florin Avram, Rim Adenane () and Lasko Basnarkov
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Florin Avram: Laboratoire de Mathématiques Appliquées, Université de Pau, 64000 Pau, France
Rim Adenane: Département des Mathématiques, Université Ibn-Tofail, Kenitra 14000, Morocco
Lasko Basnarkov: Faculty of Computer Science and Engineering, Ss. Cyril and Methodius University, 1000 Skopje, North Macedonia

Mathematics, 2024, vol. 12, issue 15, 1-16

Abstract: The fact that the famous basic reproduction number R 0 , i.e., the largest eigenvalue of the next generation matrix F V − 1 , sometimes has a probabilistic interpretation is not as well known as it deserves to be. It is well understood that half of this formula, − V , is a Markovian generating matrix of a continuous-time Markov chain (CTMC) modeling the evolution of one individual on the compartments. It has also been noted that the not well-enough-known rank-one formula for R 0 of Arino et al. (2007) may be interpreted as an expected final reward of a CTMC, whose initial distribution is specified by the rank-one factorization of F . Here, we show that for a large class of ODE epidemic models introduced in Avram et al. (2023), besides the rank-one formula, we may also provide an integral renewal representation of R 0 with respect to explicit “age kernels” a ( t ) , which have a matrix exponential form.This latter formula may be also interpreted as an expected reward of a probabilistic continuous Markov chain (CTMC) model. Besides the rather extensively studied rank one case, we also provide an extension to a case with several susceptible classes.

Keywords: stability; basic replacement number; basic reproduction number; age of infection kernel; several susceptible compartments; Diekmann matrix kernel (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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