White L J, Cox M J, Medley G F
Department of Biological Sciences, University of Warwick, Coventry, UK.
IMA J Math Appl Med Biol. 1998 Sep;15(3):211-33.
We explore the equilibrium properties of a series of compartmental, ODE models describing the interaction between different strains of pathogen. The interaction is conceptualized as acting through shared antigens: infection and recovery from one strain leaves the host with a primed immune response against subsequent strains. The models consider the effect of this priming on susceptibility (the ability to be infected) and transmission (the ability to infect) in an SIR model. In these models, the specific past history of infection is encapsulated in different susceptible compartments within the model. In a third, SIS, model, specific past history is not included, but strains have differential abilities to infect previously infected hosts. Equilibrium results include criteria for the coexistence of strains. For the SIR models, the region of coexistence defined by parameters shrinks as the effect of strains on each other (increased antigenic similarity) increases. For the SIS model, coexistence depends critically on the rate at which complete susceptibility is recovered following infection, and coexisting strains must have differential abilities to infect completely and partially susceptible hosts. Interestingly, this model provides analogies to commensalism (the first species gains from the presence of the second; the second neither gains nor loses from the interaction) and symbiosis (the presence of both species benefits the other). Additionally, we show that the maximum number of coexisting strains is two in this model. The effect of vaccination depends on the initial strain structure, the ability of vaccination to mount protection to both strains and the coverage. Vaccination may allow a previously excluded strain to coexist or exist alone, and may allow a previously rarer strain to become more common with the possibility of increasing incidence of disease. We discuss the dynamics of these models, compare model results to observed patterns and consider additional model structures. The importance of these results to specific multi-strain pathogens, in particular rotavirus, is considered.
我们探究了一系列描述不同病原体菌株间相互作用的房室常微分方程模型的平衡特性。这种相互作用被概念化为通过共享抗原起作用：感染一种菌株并从中恢复后，宿主会对后续菌株产生免疫反应。这些模型在SIR模型中考虑了这种免疫反应对易感性（被感染的能力）和传播性（感染的能力）的影响。在这些模型中，特定的既往感染史被封装在模型内不同的易感房室中。在第三个SIS模型中，不包括特定的既往感染史，但菌株对先前感染宿主的感染能力有所不同。平衡结果包括菌株共存的标准。对于SIR模型，随着菌株间相互作用的影响（抗原相似性增加）增强，由参数定义的共存区域会缩小。对于SIS模型，共存关键取决于感染后完全易感性恢复的速率，并且共存菌株必须对完全易感和部分易感宿主具有不同的感染能力。有趣的是，该模型提供了与共生关系（第一种物种从第二种物种的存在中获益；第二种物种在相互作用中既不获益也不损失）和共生现象（两种物种的存在对彼此都有益）的类比。此外，我们表明在该模型中共存菌株的最大数量为两个。疫苗接种的效果取决于初始菌株结构、疫苗接种对两种菌株产生保护的能力以及覆盖率。疫苗接种可能会使先前被排除的菌株共存或单独存在，并可能使先前较罕见的菌株变得更常见，从而增加疾病发病率。我们讨论了这些模型的动态变化，将模型结果与观察到的模式进行比较，并考虑了其他模型结构。还考虑了这些结果对特定多菌株病原体，特别是轮状病毒的重要性。
