A Susceptible–Infectious (SI) model with two infective stages and an endemic equilibrium

The focus of this article is on the dynamics of a susceptible–infected model which consists of a susceptible group (S) and two different infectious groups (I₁ and I₂). Once infected, an individual becomes a member of one of these infectious groups which have different clinical forms of infection. In addition, during the progress of the illness, an infected individual in group I₁ may pass to the infectious group I₂ which has a higher mortality rate. The infection is deadly and it has no cure. In this study, positiveness of the solutions for the model is proved. Stability analysis of species extinction, I₁-free equilibrium and endemic equilibrium as well as disease-free equilibrium is studied, and it is shown that the disease-free equilibrium is stable whereas all other equilibrium points are asymptotically stable for parameter ranges determined by certain inequalities. In addition, relations between the basic reproduction number of the disease and the basic reproduction number of each infectious stage are examined. Furthermore, the case where all newborns from infected mothers are also infected is analysed. For this type of vertical transmission, endemic equilibrium is asymptotically stable for certain parameter ranges. Finally, a special case which refers to the disease without vital dynamics is investigated and its exact solution is obtained.