Andreasen V, Christiansen F B
Department of Mathematics and Physics, Roskilde University, Denmark.
Philos Trans R Soc Lond B Biol Sci. 1995 May 30;348(1325):341-54. doi: 10.1098/rstb.1995.0072.
We study a population exposed to a lethal infectious disease. Host response is carried at one locus with two alleles while the pathogen occurs in two variants. Based on an SI-type epidemic model we derive explicit equations for the dynamics of each genotype. By assuming small variations in both host and disease, we obtain a separation in time scales between epidemic and evolutionary processes. This allows us to describe explicitly the changes in host and disease gene frequencies. The resulting model has a rich behaviour including multiple stable states and oscillations. However, in the oscillatory situation the model is degenerate excluding the possibility of limit cycles. We show that the degeneracy can only be removed by frequency dependent selection in the pathogen, for example by including direct interaction of virus in a free-living stage. The qualitative conclusions extend to an SIR-type epidemic model, where recovery with immunity from the disease is possible.
我们研究了一个暴露于致死性传染病的种群。宿主反应由一个具有两个等位基因的位点控制，而病原体有两种变体。基于一个SI型流行病模型，我们推导出了每种基因型动态变化的显式方程。通过假设宿主和疾病的微小变化，我们在时间尺度上区分了流行过程和进化过程。这使我们能够明确描述宿主和疾病基因频率的变化。由此产生的模型具有丰富的行为，包括多个稳定状态和振荡。然而，在振荡情况下，模型是退化的，排除了极限环的可能性。我们表明，只有通过病原体中频率依赖的选择，例如通过在自由生活阶段纳入病毒的直接相互作用，才能消除退化。定性结论扩展到SIR型流行病模型，在该模型中，从疾病中康复并获得免疫力是可能的。
