An SEnIS epidemiological model with vital dynamics in an exponentially growing population is dis- cussed. Without treatment three threshold parameters R0,R1 and R2 determine the dynamic of compartments sizes and that of the fractions. With the treatment the dynamics of the population and that of the epidemic depend on three other threshold parameters RT , R1T and R2T . We made a link between the models with one latent stage and the models with multiple latent stages by defining and deriving the ”effective” activation rate and the ”effective” treatment rate for the latent individuals. We defined the treatment force, the relative treatment force and deduced the critical treatment force needed to eradicate the disease. The theoretical results are validated by numerical simulations.

mathematical model; epidemiological model; Lyapunov function; numerical simulations