Differential susceptibility and infectivity epidemic models - James M. Hyman; Jia Li (Mathematical Biosciences and Engineering)
We formulate differential susceptibility and differential infectivitymodels for disease transmission in this paper. The susceptibles are divided inton groups based on their susceptibilities, and the infectives are divided into mgroups according to their infectivities. Both the standard incidence and thebilinear incidence are considered for different diseases. We obtain explicitformulas for the reproductive number. We define the reproductive numberfor each subgroup. Then the reproductive number for the entire populationis a weighted average of those reproductive numbers for the subgroups. Theformulas for the reproductive number are derived from the local stability ofthe infection-free equilibrium. We show that the infection-free equilibrium isglobally stable as the reproductive number is less than one for the models withthe bilinear incidence or with the standard incidence but no disease-induceddeath. We then show that if the reproductive number is greater than one,there exists a unique endemic equilibrium for these models. For the generalcases of the models with the standard incidence and death, conditions arederived to ensure the uniqueness of the endemic equilibrium. We also providenumerical examples to demonstrate that the unique endemic equilibrium isasymptotically stable if it exists.