Modelling the Optimal Control of Transmission Dynamics of Mycobacterium ulceran Infection

Magreth Anga Kimaro¹, Estomih S. Massawe^1*, Daniel Oluwole Makinde²

¹Mathematics Department, University of Dar es Salaam, Dar es Salaam, Tanzania.
²Faculty of Military Science, Stellenbosch University, Stellenbosch, South Africa.

DOI: 10.4236/ojepi.2015.54027                

Abstract

This paper examines optimal control of transmission dynamics of Mycobacterium ulceran (MU) infection. A nonlinear mathematical model for the problem is proposed and analysed qualitatively using the stability theory of the differential equations, optimal control and computer simulation. The basic reproduction number of the reduced model system is obtained by using the next generation operator method. It is found that by using Ruth Hurwitz criteria, the disease free equilibrium point is locally asymptotically stable and using centre manifold theory, the model shows the transcritical (forward) bifurcation. Optimal control is applied to the model seeking to minimize the transmission dynamics of MU infection on human and water-bugs. Pontryagin’s maximum principle is used to characterize the optimal levels of the controls. The results of optimality are solved numerically using MATLAB software and the results show that optimal combination of two controls (environmental and health education for prevention) and (water and environmental purification) minimizes the MU infection in the population.

Keywords

MU, Optimal Control, Stability, Pontryagin’s Maximum Principle, Reproduction Number

Conflicts of Interest

The authors declare no conflicts of interest.

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