A Review of Matrix SIR Arino Epidemic Models

Florin Avram,David I Ketcheson,Rim Adenane

doi:10.3390/math9131513

Abstract

Many of the models used nowadays in mathematical epidemiology, in particular in COVID-19 research, belong to a certain subclass of compartmental models whose classes may be divided into three “(x,y,z)” groups, which we will call respectively “susceptible/entrance, diseased, and output” (in the classic SIR case, there is only one class of each type). Roughly, the ODE dynamics of these models contains only linear terms, with the exception of products between x and y terms. It has long been noticed that the reproduction number R has a very simple Formula in terms of the matrices which define the model, and an explicit first integral Formula is also available. These results can be traced back at least to Arino, Brauer, van den Driessche, Watmough, and Wu (2007) and to Feng (2007), respectively, and may be viewed as the “basic laws of SIR-type epidemics”. However, many papers continue to reprove them in particular instances. This motivated us to redraw attention to these basic laws and provide a self-contained reference of related formulas for (x,y,z) models. For the case of one susceptible class, we propose to use the name SIR-PH, due to a simple probabilistic interpretation as SIR models where the exponential infection time has been replaced by a PH-type distribution. Note that to each SIR-PH model, one may associate a scalar quantity Y(t) which satisfies “classic SIR relations”,which may be useful to obtain approximate control policies.

Highlights

For the case of one susceptible class, we propose to use the name SIR-PH, due to a simple probabilistic interpretation as SIR models where the exponential infection time has been replaced by a PH-type distribution
SIR-PH Epidemics with One Susceptible Class. It has been known for a long while that R and the final size for many compartmental model epidemics may be explicitly expressed in terms of the matrices which define the model, and the works in [19,20,46,47] offer a quite general framework of “xyz” models which ensures this
Using the well-known convolution formula—see, for example, [53] (Theorem 3.1.26) we find that formulas like (12) still apply, with (~α, A, β) given by

Summary

Introduction

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. This is just a sample chosen from some of our favorite COVID papers. This topic requires further development; we include it due to the recognized importance of heterogeneity factors

The Classic Kermack–McKendrick SIR Epidemic Model

Examples of SIR-PH Models Used in COVID-19 Modelling

A Generalization of Heterogeneous SEIR

Conclusions