Mathematical models in epidemiology

Kermack和McKendrick（1927）的感染年龄流行病模型多年来一直被忽视，但它已成为研究流行病的有用工具。它提供了包括检疫、隔离和治疗在内的一般隔间结构，并提供了一种比较不同控制策略有效性的方法。然而，它不包括混合的异质性、通过接触受感染个体传播的病原体而间接传播疾病、以及在人群中产生耐药菌株（包括对受感染个体进行抗病毒治疗）等方面。我们计划研究感染年龄模型的扩展，以包括这些方面。******如果可以用实验方法估计流行病的初始指数增长率，并且知道平均传染性作为感染年龄的函数，就有可能估计基本繁殖数。无论模型中的混合是均匀的还是非均匀的，这个估计都是有效的。在均匀混合的情况下，最终尺寸关系给出了流行病的最终尺寸。然而，在异质混合的情况下，流行病的最终规模取决于人口中的混合情况。我们将研究这样一个问题，即在流行病的早期阶段从观察中获得哪些额外信息足以估计流行病的最终规模。这应该有助于选择一个最佳的治疗策略时，治疗包括在模型中。******在治疗可能导致疾病产生耐药菌株的模型中，提高治疗率可能导致更多的疾病病例，这是实验观察到的结果。一些分区模型似乎预测到了这一点（来自模拟，但尚未来自理论分析）。我们计划开发一个感染类型的年龄模型，为这种结果提供条件，并导致避免这种行为的策略。例如，延迟开始治疗可能会减少耐药性的产生，从而减少流行病的规模。******在霍乱中，感染既可通过直接接触传播，也可通过使用被感染者排出病原体污染的水传播。在许多空气传播疾病中，感染可能通过接触沉积在柜台、门把手或其他表面上的病原体而传播。这表明需要对疾病传播中接触的含义进行更深入的研究，从而导致对一般流行病模型中疾病接触传播术语的形式进行重新思考。******过去的大流行有时是一波一波的，可能是因为接触率随时间而变化，取决于温度和湿度，或者可能是季节性的，与学年有关。为了理解这种波，有必要研究具有时间依赖参数的模型，从周期模型开始，扩展到一般的非自治模型。

The age of infection epidemic model of Kermack and McKendrick (1927) was neglected for many years but has become a useful tool in studying epidemics. It allows a general compartmental structure including quarantine, isolation, and treatment and affords a way to compare effectiveness of different control strategies. However, it does not include such aspects as heterogeneity of mixing, indirect disease transmission through infection by contact with pathogens shed by infected individuals, and the development of drug resistant strains in populations including antiviral treatment of infected individuals. We plan to study extensions of the age of infection model to include these aspects.******If the initial exponential growth rate of an epidemic can be estimated experimentally and the mean infectivity as a function of age of infection is known, it is possible to estimate the basic reproduction number. This estimate is valid whether the mixing in the model is homogeneous or heterogeneous. In the case of homogeneous mixing, the final size relation gives the final size of the epidemic. In the case of heterogeneous mixing, however, the final size of the epidemic depends on the mixing in the population. We will study the question of what additional information from observation in the early stages of an epidemic would suffice to estimate the epidemic final size. This should aid in choosing an optimal treatment strategy when treatment is included in the model.******In models where treatment may lead to development of a drug-resistant strain of the disease, increasing the treatment rate may lead to more disease cases, an outcome that has been observed experimentally. Some compartmental models appear to predict this (from simulations, but not yet from theoretical analysis). We plan to develop a model of age of infection type giving conditions for such outcomes and leading to strategies that would avoid such behaviour. For example, a delay in the beginning of treatment might decrease the epidemic size by decreasing the development of resistance.******In cholera, infection may be transmitted either through direct contact or through use of water contaminated by shedding of pathogens by infected individuals. In many airborne diseases, infection may be transmitted through contact with pathogens that have been deposited on counters or door knobs or other surfaces. This suggests a need for a deeper examination of the meaning of contact in disease transmission, leading to a rethinking of the form of the disease contact transmission terms in general epidemic models.******Pandemics in the past have sometimes come in waves, possibly because contact rates may vary in time, depending on temperature and humidity, or may be seasonal with variations related to the school year. To understand such waves, it will be necessary to study models with time-dependent parameters, beginning with periodic models and extending to general non-autonomous models.