Explicit mathematical epidemiology results on age renewal kernels and R0 formulas are often consequences of the rank one property of the next generation matrix

TL;DR

This paper demonstrates that many ODE epidemic models with rank-one infection matrices can be reformulated as integral renewal processes, revealing probabilistic properties and generalizing R0 formulas.

Contribution

The paper provides a simple proof that epidemic models with rank-one infection matrices admit an integral renewal formulation and extends these results to multiple susceptible classes.

Findings

The integral renewal kernel a(t) has a matrix exponential form.

The Laplace transform of a(t) generalizes the basic reproduction number R0.

Models with multiple susceptible classes can also have rank-one infection matrices.

Abstract

A very large class of ODE epidemic models (2.2) discussed in this paper enjoys the property of admitting also an integral renewal formulation, with respect to an "age of infection kernel" a(t) which has a matrix exponential form (3.2). We observe first that a very short proof of this fact is available when there is only one susceptible compartment, and when its associated "new infections" matrix has rank one. In this case, a(t) normalized to have integral 1, is precisely the probabilistic law which governs the time spent in all the "infectious states associated to the susceptible compartment", and the normalization is precisely the basic replacement number. The Laplace transform (LT) of a(t) is a generalization of the basic replacement number, and its structure reflects the laws of the times spent in each infectious state. Subsequently, we show that these facts admit extensions to processes with several susceptible classes, provided that all of them have a new infections matrix of rank one. These results reveal that the ODE epidemic models highlighted below have also interesting probabilistic properties.