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

Abstract

The fact that the famous basic reproduction number (Formula presented.), i.e., the largest eigenvalue of the next generation matrix (Formula presented.), sometimes has a probabilistic interpretation is not as well known as it deserves to be. It is well understood that half of this formula, (Formula presented.), 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 (Formula presented.) 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 (Formula presented.) with respect to explicit “age kernels” (Formula presented.), 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.