Final epidemic size and critical times for susceptible-infectious-recovered models with a generalized contact rate

During the spread of an infectious disease, the contact rate or the incidence rate may affect disease characteristics. For simplicity, most disease models assume standard incidence or mass action rates to calculate the basic reproduction number, final epidemic size, and peak time of an epidemic. For standard incidence, the contact rate remains constant resulting in the incidence rate is inversely proportional to the population size, while for the mass action rate, this contact rate is proportional to the total population size resulting in the incidence rate is independent of the population size. In this paper, we consider susceptible-infectious-recovered epidemic models with a generalized contact rate C(N) and a nonlinear incidence rate in view of the behavioral change from susceptible or infectious individuals when an infectious disease appears. The basic reproduction number and the final size equation are derived. The impact of different types of contact rates on them is studied. Moreover, two critical times (peak time and epidemic duration) of an epidemic are considered. Explicit formulas for the peak time and epidemic duration are obtained. These formulas are helpful not only for taking early effective epidemic precautions but also for understanding how the epidemic duration can be changed by acting on the model parameters, especially when the epidemic model is used to make public health policy.