Mathematical models and analysis for the transmission dynamics of malaria

Abstract

Malaria is one of the most widespread and complex parasitic diseases in the world. According to the World Health Organization's records for the year 2013, there were 207 million malaria cases with 627,000 deaths in 2012 globally. Although its control and prevention has been pursued for a long time, however, because the parasite developed resistance to many of the standard treatments, it is becoming more di cult for researchers to stay ahead of the disease. In this dissertation, two deterministic models for the transmission dynamics of malaria are presented. First we comprehensively studied the dynamical interaction of sporozoites with humans, production of merozoites, and the invasion of red blood cells during erythrocytic stage of malaria infection. Then we construct a model, which takes the form of an autonomous deterministic system of non-linear di erential equations with standard incidence, consisting of seven mutually-exclusive compartments representing the human and vector dynamics. The model is then extended to incorporate additional compartment of vaccinated individuals. Rigorous analysis of the two models (with and without vaccine) shows that, both the non-vaccinated and vaccinated models have a locally asymptotically stable disease-free equilibrium (DFE) whenever their respective threshold parameters, known as the basic reproduction number and the vaccinated reproduction number are respectively less than unity, and the DFE is unstable when they are greater than unity. In addition, the models exhibit the phenomenon of backward bifurcation, where the stable disease-free equilibrium coexists with a stable endemic equilibrium when the associated reproduction numbers are less than unity. Furthermore, it was shown that, the backward bifurcation phenomenon can be removed by substituting the associated standard incidence function with the mass action incidence, this is achieved using Lyapunov functions in conjunction with LaSalle invariance principle. We further presented numerical simulations using parameter values for both low and high malaria incidence regions.