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。

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.