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.
