MATHEMATICAL MODELLING OF ANTHRAX WITH OPTIMAL CONTROL

  • S. Osman Department of Mathematics, Pan African University, Institute for Basic Sciences, Technology and Innovations, Nairobi, Kenya
  • O.D. Makinde Faculty of Military Science, Stellenbosch University, Saldanha, South Africa
  • D.M. Theuri Department of Mathematics, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya
Keywords: Basic reproduction number, stability analysis, equilibrium points, sensitivity analysis, optimal control.

Abstract

Anthrax is an infectious notifiable disease that is caused by the bacteria Bacillus anthraces. The disease affects wild, domestic animals and humans. Susceptible individuals contract the anthrax disease if they interact with infected animals or consumed contaminated dairy and animal products. This research developed a mathematical model that explains the transmission dynamics of anthrax as a zoonotic disease. Ordinary differential equations are used in the formulation systems of equations from the model’s flow diagram. The model was analysed qualitatively and quantitatively. A vaccination compartment with waning immunity was incorporated into the model. The local and global stability analysis of the equilibrium points were found to be locally asymptotically stable if the basic reproductive number is less than one and unstable if the basic reproductive number is greater than one. The sensitivity analysis of the model’s parameters was performed to determine the contribution of each parameter to the basic reproduction number. The analysis revealed that, by decreasing human and animal contact rate, it would cause a decrease in the basic reproduction number. The model was extended to optimal control to determine the best control measure in combating the disease. Qualitative analysis of optimal control of the model was performed and derived the necessary conditions for the optimal control of the anthrax disease. The most effective strategy is the vaccination of susceptible animals and the treatment of infectious animals. The combination of these two controls would help combat the spread of anthrax.

Published
2019-07-08