Department of Mathematics and Statistics, University of Strathclyde, Livingstone Tower, Glasgow G1 1XH, UK.
Math Biosci. 2009 Oct;221(2):136-49. doi: 10.1016/j.mbs.2009.08.003. Epub 2009 Aug 13.
In order to obtain a reasonably accurate model for the spread of a particular infectious disease through a population, it may be necessary for this model to possess some degree of structural complexity. Many such models have, in recent years, been found to exhibit a phenomenon known as backward bifurcation, which generally implies the existence of two subcritical endemic equilibria. It is often possible to refine these models yet further, and we investigate here the influence such a refinement may have on the dynamic behaviour of a system in the region of the parameter space near R(0)=1. We consider a natural extension to a so-called Core Group model for the spread of a sexually transmitted disease, arguing that this may in fact give rise to a more realistic model. From the deterministic viewpoint we study the possible shapes of the resulting bifurcation diagrams and the associated stability patterns. Stochastic versions of both the original and the extended models are also developed so that the probability of extinction and time to extinction may be examined, allowing us to gain further insights into the complex system dynamics near R(0)=1. A number of interesting phenomena are observed, for which heuristic explanations are provided.
为了获得一个通过人群传播特定传染病的合理准确模型，这个模型可能需要具有一定程度的结构复杂性。近年来，许多此类模型都发现了一种称为反向分歧的现象，这通常意味着存在两个亚临界地方性平衡点。通常可以进一步完善这些模型，我们在这里研究这种细化对接近 R(0)=1 的参数空间区域系统动态行为的影响。我们考虑对传播性传播疾病的所谓核心群体模型进行自然扩展，认为这实际上可能会产生更现实的模型。从确定性的角度来看，我们研究了产生的分歧图的可能形状和相关的稳定性模式。还开发了原始和扩展模型的随机版本，以便可以检查灭绝的概率和灭绝时间，从而使我们能够更深入地了解接近 R(0)=1 的复杂系统动态。观察到了一些有趣的现象，并提供了启发式解释。
