The Dynamics of Vector-Borne Relapsing Diseases

TL;DR

This paper models the spread of relapsing vector-borne diseases like tick-borne fever using compartmental models, deriving the basic reproductive ratio and analyzing stability and bifurcations of disease-free and endemic states.

Contribution

It provides a general formulation of the reproductive ratio and analyzes bifurcation behavior in relapsing diseases within a compartmental modeling framework.

Findings

Derivation of a general reproductive ratio $R_0$ for relapsing diseases

Identification of a transcritical bifurcation at $R_0=1$

Endemic equilibrium stability analysis near $R_0=1$

Abstract

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, model description, and a brief overview of the theory of compartmental models, we compute a general form of the reproductive ratio $R_0$, which is the average number of new infections produced by a single infected individual. A disease free equilibrium undergoes a bifurcation at $R_0 =1$ 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 $R_0$ sufficiently close to 1. We close with some discussion and directions for future research.