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Avian Spirochaetosis is an acute endemic tick-borne disease of birds, causedby Borrelia anserins, a species of Borrelia bacteria. In this paper, we presenta compartmental Mathematical model of the disease for the bird population andTick population. The model so constructed was analyzed using methods fromdynamical systems theory. \tcr{The disease steady (equilibrium) state wasdetermined and the conditions for the disease-free steady state to be stablewere determined}. The analysis showed that the disease-free steady state islocally stable if $d\geq \tau_B$ and $\delta \geq \tau_T$, that is, the naturaldeath 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 capitabirth rate of tick $\tau_T$. This means that for the disease to be undercontrol and eradicated within a while from its outbreak, the natural death rateof 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 rateof tick $\tau_T$. It was also proved the disease-free equilibrium (DFE) and theendemic equilibrium (EE) are globally stable using Lyaponov method. Threecontrol measures were introduced into the model. The optimality system of thethree controls is characterized using optimal control theory and the existenceand uniqueness of the optimal control are established. Then, the effect of theincorporation of the three controls is investigated by performing numericalsimulation.

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