Section: New Results

Multi-stages and multi-strains epidemic models

The model SI (Susceptible-Infected) is one of the most important and used epidemiological models. We gave a complete analysis of the stability of the model with a non-linear incidence and two classes of infected individuals [12] .

We have also studied SIS, SIR and MSIR models with bilinear incidence and varying population, with $n$ different pathogen strains of an infectious disease, with or without vertical transmission. For these classes of models, we have proved that under generic conditions a competitive exclusion principle holds. To each strain a basic reproduction ratio can be associated. It corresponds to the case where only this strain exists. The basic reproduction ratio of the complete system is the maximum of the individual basic reproduction ratios. Actually we have also defined an equivalent threshold for each strain. The winner of the competition is the strain with the maximum threshold. It turns out that this strain is the most virulent, i.e., this is the strain for which the endemic equilibrium gives the minimum population for the susceptible host population. This can be interpreted as a pessimization principle [10] .

A mathematical multi-patches model for highland malaria in Kenya has been developed and analysed in [13] and [14] .