Mathematical Model and Optimal Control of the Transmission Dynamics of Avian Spirochaetosis (TICK Fever)

Abstract

Avian Spirochaetosis is an acute endemic tick-borne disease of birds, caused by Borrelia anserins, a species of Borrelia bacteria. In this paper, we present a compartmental Mathematical model of the disease for the bird population and Tick population. The model so constructed was analyzed using methods from dynamical systems theory. \tcr{The disease steady (equilibrium) state was determined and the conditions for the disease-free steady state to be stable were determined}. The analysis showed that the disease-free steady state is locally stable if $d\geq \tau_B$ and $\delta \geq \tau_T$, that is, the natural death rate of birds (d) will be greater than the per capita birth rate of birds $\tau_B$ and the death rate of tick $\delta)$ is greater than the per capita birth rate of tick $\tau_T$. This means that for the disease to be under control and eradicated within a while from its outbreak, the natural death rate of birds $d$ will be greater than the per capita birth rate of bird $\tau_B$ and the death rate of tick $\delta$ is greater than the per capita birth rate of tick $\tau_T$. It was also proved the disease-free equilibrium (DFE) and the endemic equilibrium (EE) are globally stable using Lyaponov method. Three control measures were introduced into the model. The optimality system of the three controls is characterized using optimal control theory and the existence and uniqueness of the optimal control are established. Then, the effect of the incorporation of the three controls is investigated by performing numerical simulation.