Mathematical analysis of a social hierarchy-structured model for malaria transmission dynamics

Abstract

A vector-borne disease such as malaria has the potential to infect anyone regardless of the social classes to which the individuals belong, but the degree of disease transmission may be higher in one social class than the other. This paper presents and analyzes a time-dependent social hierarchy-structured deterministic model with a view to preventing and controlling the effects of social class disparity on the transmission dynamics of malaria disease in the interacting populations of humans and mosquitoes. The total human population is broadly stratified into low and high social classes with each consisting of three mutually exclusive compartments, namely susceptible, infectious and recovered humans with temporary immunity. The total vector population is sub-divided into susceptible and infectious mosquitoes. The derived eight-dimensional system of differential equations is rigorously analyzed under optimal control framework with four control variables. Using Pontryagin’s maximum principle, the existence of the control quadruple is proved. Efficiency analysis carried out shows that combination of all the controls is the most efficient intervention, while the cost-effectiveness analysis reveals the most cost-effective single, double and triple interventions to curtail malaria spread in a social hierarchy-structured population.