The dynamics of vector-borne relapsing diseases

In this paper, we describe the dynamics of a vector-borne relapsing disease, such as tick-borne relapsing fever, using the methods of compartmental models. After some motivation and model description we provide a proof of a conjectured general form of the reproductive ratio R0, which is the average number of new infections produced by a single infected individual. A disease free equilibrium undergoes a bifurcation at R0=1 and we show that for an arbitrary number of relapses it is a transcritical bifurcation with a single branch of endemic equilibria that is locally asymptotically stable for R0 sufficiently close to 1. Furthermore, we show there is no backwards bifurcation. We then show that these results can be extended to variants of the model with an example that allows for variation in the number of relapses before recovery. Finally, we discuss implications of our results and directions for future research.