mathematics Article A Review of Matrix SIR Arino Epidemic Models Florin Avram 1, * , Rim Adenane 2 and David I. Ketcheson 3   Citation: Avram, F.; Adenane, R.; Ketcheson, D.I. A Review of Matrix SIR Arino Epidemic Models. Mathematics 2021, 9, 1513. https://doi.org/10.3390/math9131513 Academic Editor: Jan Awrejcewicz Received: 26 May 2021 Accepted: 21 June 2021 Published: 28 June 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Laboratoire de Mathématiques Appliquées, Université de Pau, 64000 Pau, France 2 Département des Mathématiques, Université Ibn-Tofail, Kenitra 14000, Morocco; rim.adenane@uit.ac.ma 3 Department of Applied Mathematics and Computational Science, King Abdullah University of Science and Technology, Thuwal 23955, Saudi Arabia; david.ketcheson@kaust.edu.ma * Correspondence: Florin.Avram@univ-Pau.fr 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the name SIR-PH, due to a simple probabilistic interpretation as SIR models where the exponential infection time has been replaced by a PH-type distribution.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. Keywords: epidemiological modeling; COVID-19; SIR-PH model; matrix SIR model; reproduction number; first integral 1. Introduction Motivation. Mathematical epidemiology may be said to have started with the SIR ODE model, which saw its birth in the works of Kermack–McKendrick [1]. This was initially applied to model the Bombay plague of 1905–06, and later to measles [2], small- pox, chickenpox, mumps, typhoid fever and diphtheria, and recently to the COVID-19 pandemic—see, for example in [3–18], to cite just a few representatives of a huge literature. Note that during the COVID-19 pandemic, researchers have relied mostly on models with quadratic interactions (linear force of infection), which belong furthermore to a par- ticular class [19–22] of “( x, y, z)” models. Here, x denotes “entrance/susceptible” classes; y denotes diseased classes, which must converge asymptotically to 0; and z denotes output classes. These models are very useful; to make references to them easier, we propose to call them matrix-SIR (SYR) models, and also SIR-PH [21], when x ∈ R 1 . Contents. We begin by recalling in Section 2 several basic explicit formulas for the SIR model. Section 3 presents the corresponding SIR-PH generalizations, and Section 4 offers some applications: the SEIHRD model [23–27], which adds to the classic SEIR (susceptible + exposed + infectious + recovered) a hospitalized (H) class and a dead class (D); the SEICHRD model [28] which adds a critically ill class (C); the SEIARD [29] and SEIAHR/SEIRAH(D) models [30–36], which add an asymptomatic class (A); and the S I 3 QR model [37]. This is just a sample chosen from some of our favorite COVID papers. We note in passing that they seem though all unaware of the existence of the Arino and Feng Formulas (12) and (17). Mathematics 2021, 9, 1513. https://doi.org/10.3390/math9131513 https://www.mdpi.com/journal/mathematics