Transmission Dynamics and Optimal Control of Malaria in Kenya

Abstract

This paper proposes and analyses a mathematical model for the transmission dynamics of malaria with four-time dependent control measures in Kenya: insecticide treated bed nets (ITNs), treatment, indoor residual spray (IRS), and intermittent preventive treatment of malaria in pregnancy (IPTp). We first considered constant control parameters and calculate the basic reproduction number and investigate existence and stability of equilibria as well as stability analysis. We proved that if Ro ≤ 1 , the disease-free equilibrium is globally asymptotically stable in D . If Ro > 1, the unique endemic equilibrium exists and is globally asymptotically stable. The model also exhibits backward bifurcation at Ro = 1. If R0 > 1, the model admits a unique endemic equilibrium which is globally asymptotically stable in the interior of feasible region D . The sensitivity results showed that the most sensitive parameters are mosquito death rate and mosquito biting rates. We then consider the time-dependent control case and use Pontryagin’s Maximum Principle to derive the necessary conditions for the optimal control of the disease using the proposed model. The existence of optimal control problem is proved. Numerical simulations of the optimal control problem using a set of reasonable parameter values suggest that the optimal control strategy for malaria control in endemic areas is the combined use of treatment and IRS; for epidemic prone areas is the use of treatment and IRS; for seasonal areas is the use of treatment; and for low risk areas is the use of ITNs and treatment. Control programs that follow these strategies can effectively reduce the spread of malaria disease in different malaria transmission settings in Kenya.