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Discrete Dynamics in Nature and Society
Volume 2016, Article ID 8013574, 27 pages
http://dx.doi.org/10.1155/2016/8013574
Research Article

Transmission Dynamics and Optimal Control of Malaria in Kenya

Department of Statistics and Computer Science, Moi University, P.O. Box 3900, Eldoret 30100, Kenya

Received 16 November 2015; Revised 5 January 2016; Accepted 6 April 2016

Academic Editor: Xiaohua Ding

Copyright © 2016 Gabriel Otieno et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 , the disease-free equilibrium is globally asymptotically stable in . If , the unique endemic equilibrium exists and is globally asymptotically stable. The model also exhibits backward bifurcation at . If , the model admits a unique endemic equilibrium which is globally asymptotically stable in the interior of feasible region . 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.