Department of Mathematical Sciences, University of Montana, USA.
Division of Biological Sciences, University of Montana, USA.
Math Biosci. 2018 Mar;297:32-42. doi: 10.1016/j.mbs.2018.01.001. Epub 2018 Jan 12.
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 R, which is the average number of new infections produced by a single infected individual. A disease free equilibrium undergoes a bifurcation at R=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 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.
在本文中，我们使用房室模型的方法描述了一种虫媒传染病（如蜱传回归热）的动态。在进行了一些动机和模型描述之后，我们提供了一个关于繁殖率 R 的猜想的一般形式的证明，R 是指单个感染个体产生的新感染数量的平均值。在 R=1 时，无病平衡点会发生分岔，我们证明对于任意数量的复发，它是一个具有单个地方渐近稳定的地方渐近稳定的局部分叉。此外，我们证明不存在向后分岔。然后，我们证明这些结果可以扩展到具有恢复前复发次数变化的模型变体，并用一个例子来说明。最后，我们讨论了我们的结果的意义和未来研究的方向。
