In the first part of this presentation we will consider SIR epidemic models without entering flux of new susceptible. For the single group case, we will discuss the comparison of the with real data [1]. Therefore we will identify the parameter of the system as well as the initial value of the system. We will note that our analysis is strongly based on the computation of the final size for the SIR model with classical mass action law. Therefore the similar question are still fully open for a general non-linear functional response.

Next, we will discuss SIR epidemic with multiple groups [2,3].  The main question addressed in our the computation of the final size of the epidemic. We will also discuss the asymptotic behavior, and we will apply the results to the SARS epidemic in Singapore in 2003, where it is shown that the two-peak evolution of the infected population can be attributed to a two-group formulation of transmission.

Next we will a model to describe the spreading of influenza in Puerto-Rico including the real map to compare the model with real data  [4]. The spreading of pathogen locally in space will be discussed.

The second part of the talk will be devoted to more the mathematical analysis of SIR with input flux of new susceptible and with inflection [5,6]. This work combine integrated semigroup analysis to derive the major properties of the semiflow generated by such a system. The main result presented in this second part is combining global attractors, uniform persistence and Liapunov functional. The major mathematical difficulty is that the Liapunov function are only defined on a dense subset of .

[1] P. Magal and G. Webb, The Parameter Identification Problem for SIR Epidemic Models: Identifying Unreported Cases, to appear in J. Math. Biol.

[2] P. Magal, O. Seydi and G. Webb, Final size of a multi-group SIR epidemic model: Irreducible and non-irreducible modes of transmission (Submitted).

[3] P. Magal, O. Seydi and G. Webb (2016), Final size of an epidemic for a two group SIR model, SIAM Journal on Applied Mathematics, 76, 2042-2059.

[4] P. Magal, G. Webb and Y. Wu, Spatial Spread of Epidemic Diseases in Geographical Settings: Seasonal Inuenza Epidemics in Puerto Rico, (In preparation).

[5] P. Magal, C. C. McCluskey, and G. F. Webb (2010), Liapunov functional and global asymptotic stability for an infection-age model, Applicable Analysis 89, 1109 -1140.

[6] P. Magal and C.C. McCluskey (2013), Two group infection age model: an application to nosocomial infection, SIAM J. Appl. Math., 73(2), 1058-1095.