MATHEMATICALMODELLINGOFANTHRAXANDLISTERIOSISCO-DYNAMICS WITH OPTIMAL CONTROL

Abstract

In this research, three mathematical models were developed to explain the transmission dynamics of zoonotic diseases that are bacteria related, specifically Listeriotic, Anthrax and their co-infection. Ordinary differential equations were used in the formulation of the Anthrax, Listeriosis and Anthrax-Listeriosisco-infection deterministic models. These deterministic models were formulated and analyzed qualitatively and quantitatively. A vaccination compartment with waning immunity was incorporated into the Anthrax model. The local and global stability analysis of equilibrium points were analyzed and found to be locally asymptotically stable whenever the basic reproductive number is less than one and unstable if the basic reproductive number is greater than one. The basic reproductive numbers were computed and used as the threshold parameter that governs the spread of Anthrax, Listeriosis and Anthrax-Listeriosis coinfection. Moreover, the extension of the Anthrax, Listeriosis and the co-infection models to optimal control theory seek to minimize the objective functional subject to some controls variables were determined. The resulting control problem was solved numerically in order to determine the most effective control measure in combating the Anthrax infections. It established that Anthrax and Listeriosis models exhibited multiple endemic equilibrium. Sensitivity analysis of the basic reproduction numbers of all the three models were determined. It was established that, increasing animal (livestock) recovery rate, would lead a decrease in the basic reproduction number. The qualitative analysis of optimal control of the Anthrax, Listeriosis and Anthrax-Listeriosis co-infection were performed and the necessary conditions for the optimality of Anthrax, Listeriosis and Anthrax-Listeriosisinfectionswereanalysed. The most effective control strategies of the Listeriosis model were the combination of treatment of infectious vectors and treatment of infectious humans, combination of prevention of susceptible humans and the treatment of infectious vectors. Anthrax-Listeriosis co-infection model analysis reveals that the disease free equilibrium was locally asymptotically stable whenever the reproduction number is less one. The co-infection model exhibited the phenomenon of backward bifurcation. Mathematically, it implies that the idea of the model been locally asymptotically stable whenever the reproduction number is less than unity and unstable otherwise is not a sufficient condition for disease eradication. The impact of Listeriosis on Anthrax infections reveals that Anthrax infections can be attributed to increased risk of Listeriosis butthereverseisnotthesame. However, optimal prevention and treatment of Anthrax and not keeping Listeriosis under control is not the best strategy for eradicating either of the disease.