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

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.