Abstract:
In this thesis, a deterministic mathematical model for the transmission and control
of malaria, incorporating prevention and treatment as control parameters
has being developed. A novel addition in our model is that, a proportion ca; (0
c 1); of the prevention effort (a), reduces the vector population. The model
has two unique equilibrium points namely, a disease-free equilibrium point,
which is locally and globally asymptotically stable when R0 < 1; and an endemic
equilibrium point which is locally and globally asymptotically stable
when R0 > 1: The parameters of the model were estimated using yearly malaria
transmission data for Ghana, (from 2004 to 2017), obtained from the World
Health Organization. Simulations of our model using various combinations of
treatment and prevention, with increasing values of the constant c; show that, infected
vector and human populations can be drastically reduced, thus effectively
controlling the transmission of Malaria. To determine an optimal combination
of prevention and treatment, we formulated an optimal control problem, with
an appropriate cost functional, using 0 u1 1 (prevention), and 0 u2 1
(treatment) as controls. Pontryagin’s Maximum Principle was used to determine
the optimality system. Solutions of the optimality system, with u1max = 0:5; and
u2max = 0:2; (representing maximum prevention effort and treatment rate respectively),
show a dramatic reduction in both infected human and vector populations.
Further simulations show that, malaria can be eradicated by increasing
prevention efforts (u1max > 0:5), combined with treatment made accessible to
everyone diagnosed with malaria.
