Global stability of a SI epidemic model with two infected stages and mass-action incidence

Abstract : The work done in this paper consists in the establishment of the global stability of the model SI containing two classes of infected stages. The incidence used is non-linear and given by $(\beta_1 I_1+\beta_2 I_2)\dfrac{S}{N}$.\\ Existence and uniqueness of the endemic equilibrium is established. A Lyapunov function is used to prove the stability of the disease free equilibrium, and the Poincarré-Bendixson theorem allows to prove the global asymptotic stability of the endemic equilibrium when it exists.