Mathematical models in epidemiology

?对于流行病模型，有一个关系，称为最终规模关系，它与繁殖数（易感人群中单个感染个体引起的二次感染的平均数）有关， 在流行病期间被感染的人口比例。我们已经开发了一个最终的大小关系的一般类的年龄感染流行病模型，并将使用它来检查控制参数的敏感性，并比较不同的管理策略的效果。 流行病中的治疗可能导致耐药性的发展，已经开发了流行病模型来描述这种情况。数值模拟表明，增加治疗率可能会增加流行规模，但没有这种性质的分析结果。我们将研究这样的模型，并寻找额外的关系，以导致最终的大小关系，给出每种菌株的情况下的数量。这将导致最佳治疗率的标准，将最大限度地减少总的流行病规模。 在2009年的流感大流行中，与以往的大流行一样，出现了多个流行波。一种可能的解释是接触率的季节性变化，模拟结果证实了单波或两波的可能性。哪一种可能性发生，关键取决于第一波的开始时间。我们建议调查模型与周期性的接触率，并试图发展一个一般的理论类似的最终大小关系结构的恒定接触率。 流行病期间行为变化的一个简单模型假设，感染者可能会因为疾病的影响而减少接触，而未感染者可能会因为其他原因减少接触，例如试图避免感染。这导致了具有接触异质性的模型。将这些结果扩展到感染模型的年龄，将允许推广到具有更一般的房室结构的疾病。我们希望从社会学的角度了解哪些信息实际上会影响行为，并分析包括这种影响在内的模型。

?There is a relation, called the final size relation, for epidemic models relating the reproduction number (the mean number of secondary infections caused by a single infectious individual in a susceptible population), to the fraction of the population infected during an epidemic. We have developed a final size relation for a general class of age of infection epidemic models and will use it to examine sensitivity to control parameters and to compare the effects of different management strategies. Treatment in an epidemic may lead to the development of drug resistance, and epidemic models have been developed to describe this situation. Numerical simulations indicate that increasing the treatment rate may increase the epidemic size, but there are no analytic results of this nature. We will study such models and search for additional relations to lead to final size relations giving the number of cases of each strain. This should lead to criteria for optimal treatment rates that will minimize the overall epidemic size. In the 2009 influenza pandemic, as in past pandemics, there were multiple epidemic waves. One possible explanation is seasonal variation in contact rates, confirmed by simulations showing the possibility of a either a single wave or two waves. Which possibility occurs depends critically on the starting time of the first wave. We propose to investigate models with periodic contact rates and attempt to develop a general theory analogous to the final size relation structure for constant contact rates. A simple model of behavioural change during an epidemic assumes that infectious individuals might reduce contacts because of the effects of illness, while uninfected individuals might reduce contacts for other reasons such as an attempt to avoid infection. This leads to a model with heterogeneity of contacts. Extension of these results to an age of infection model will allow generalization to diseases with a more general compartmental structure. We hope to learn from a sociological point of view what information actually influences behaviour and to analyze models including this influence.