Analysis of a delayed SIR model subject to multiple infectious stages and nonlinear incidence rate

We investigate the threshold dynamics problem of a delayed Susceptible-Infected-Recovered (SIR) model with general nonlinear incidence and multiple parallel infectious stages. Biologically, the model contains the following aspects: (i) once infection occurs, a fraction of the infected individuals is detected and treated, while the rest of the infected remains undetected and untreated; (ii) distributed delays governed by a general nonlinear incidence function are included into the model due to the complexity of disease transmissions. Mathematically, under some suitable assumptions on nonlinear incidence rate, we prove that the reproduction number R0 can be used to govern the the global dynamics of the model. The proofs of global attractivity of disease-free equilibrium (which means the extinction of disease) and endemic equilibrium (which means the persistence of the disease) are achieved by constructing suitable Lyapunov functionals.