FINAL SIZE OF AN EPIDEMIC FOR A TWO-GROUP SIR MODEL ∗ 1 PIERRE MAGAL † , OUSMANE SEYDI ‡ , AND GLENN WEBB § 2 Abstract. In this paper we consider a two-group SIR epidemic model. We study the finale 3 size of the epidemic for each sub-population. The qualitative behavior of the infected classes at the 4 earlier stage of the epidemic is described with respect to the basic reproduction number. Numerical 5 simulations are also preformed to illustrate our results. 6 Key words. Epidemic models, final size, two-group, criss-cross transmission. 7 AMS subject classifications. 92D25, 92D30. 8 1. Introduction. In this article we study a two-group epidemic model. In order 9 to focus on the dynamical properties of an infectious disease itself, here we neglect the 10 demography, namely the birth and death processes, and the immigration/emigration 11 process. The classical SIR model takes the following form (Anderson and May [1]) 12 (1)            dS(t) dt = −βS(t)I (t) dI (t) dt = βS(t)I (t) − ηI (t) dR(t) dt = ηI (t) 13 with the initial distributions 14 S(0) = S 0 ∈ R + ,I (0) = I 0 ∈ R + and R(0) = R 0 ∈ R + 15 where S(t) is the number of susceptible individuals, I (t) is the number of infectious 16 individuals (i.e. individuals who are infected and capable to transmit the disease), 17 R(t) is the number of recovered individuals at time t, respectively. The parameter 18 β> 0 is called the infection rate (i.e. the contact rate times the probability of 19 infection, see Thieme [40]), and η> 0 is the recovery rate (i.e. the rate at which 20 infectious individuals recover). 21 Epidemic model have a long history and starts with the pioneering work of 22 Bernoulli [7] in 1760 in which he aimed at evaluating the effectiveness of inocula- 23 tion against smallpox. The susceptible-infectious-recovered (SIR) model as we know 24 today takes its origin in the fundamental works on “a priori pathometry” by Ross [38] 25 and Ross and Hudson [37, 36] in 1916-1917 in which a system of ordinary differential 26 equations was used to describe the transmission of infectious diseases between suscep- 27 tible and infected individuals. In 1927-1933, Kermack and McKendrick [22, 23, 24] 28 extended Ross’s ideas and model, proposed the cross quadratic term βIS linking the 29 sizes of the susceptible (S) and infectious (I) populations from a probabilistic analysis 30 of the microscopic interactions between infectious agents and/or vectors and hosts in 31 the dynamics of contacts, and established the threshold theorem. Since then epidemic 32 * Submitted to the editors DATE. Funding: None † Univ. Bordeaux, IMB, UMR 5251, F-33076 Bordeaux, France and CNRS, IMB, UMR 5251, F-33400 Talence, France (pierre.magal@u-bordeaux.fr, https://www.math.u-bordeaux.fr/ ∼ pmagal100p/). ‡ D´ epartement Tronc Commun, ´ Ecole Polytechnique de Thi` es, S´ en´ egal (oseydi@ept.sn) § Vanderbilt University, Nashville, TN 37240, USA (glenn.f.webb@Vanderbilt.Edu). 1 This manuscript is for review purposes only.