Optimal Strategy for Effective Control and Possible Eradication of Malaria

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.