Differential susceptibility and infectivity epidemic models

We formulate differential susceptibility and differential infectivity models for disease transmission in this paper. The susceptibles are divided into n groups based on their susceptibilities, and the infectives are divided into m groups according to their infectivities. Both the standard incidence and the bilinear incidence are considered for different diseases. We obtain explicit formulas for the reproductive number. We define the reproductive number for each subgroup. Then the reproductive number for the entire population is a weighted average of those reproductive numbers for the subgroups. The formulas for the reproductive number are derived from the local stability of the infection-free equilibrium. We show that the infection-free equilibrium is globally stable as the reproductive number is less than one for the models with the bilinear incidence or with the standard incidence but no disease-induced death. We then show that if the reproductive number is greater than one, there exists a unique endemic equilibrium for these models. For the general cases of the models with the standard incidence and death, conditions are derived to ensure the uniqueness of the endemic equilibrium. We also provide numerical examples to demonstrate that the unique endemic equilibrium is asymptotically stable if it exists.