Biomathematical analysis and extension of the new class of epidemic models proposed by Satsuma et al. (2004)

The aim of this paper is to discuss the new class of epidemic models proposed by Satsuma et al., which are characterized by incidence rates which are nonlinearly dependent on the number of susceptibles as follows: infection rate (S, I) = g(S)I. By adding the biologically plausible constraint g′(S) > 0, we study the SIR and the SEIR models with vital dynamics with such infection rate, and results are done on the global asymptotic stability of the disease free and of the endemic equilibria, similarly to the ones of the classical models, also in presence of traditional and pulse vaccination strategies. Relaxing the constraint g′(S) > 0, we show that the epidemic system may exhibit multiple endemic equilibria.