Modelling Infectious Diseases with discontinuities

随着最近的埃博拉疫情爆发，需要数学模型来了解世界各国之间和国家内部的传播至关重要。尽管我们在减少西方世界的疾病流行方面取得了一些成功，但许多疾病继续威胁着发展中国家，世界上超过50%的人口面临着一种或多种传染性感染的风险。近年来，西尼罗河病毒、SARS和猪流感等新出现的传染病威胁着西方，禽流感、MERS和其他传染病的潜在大流行迫在眉睫。我们可以使用的工具包括疫苗、药物、检疫和教育。这种干预的许多影响都是以短时间爆发或冲击的形式出现的。例如，在许多发展中国家，每年两次，为期1-2天的麻疹、脊髓灰质炎等全国免疫日。在此期间，数百万儿童立即接种疫苗。印度在一次全国免疫日为1.74亿儿童接种疫苗。在较小的范围内，艾滋病毒的抗逆转录病毒治疗包括服用药丸，其作用时间约为20分钟，大大短于药丸之间的时间。隔离通常涉及在短时间内重新安置受感染者，而教育可能涉及媒体信息的快速爆发。为了分析这些影响，我们将使用脉冲微分方程，脉冲扩展方程和Filippov系统开发数学模型。这些方法被用来分析短期，急剧的冲击，无论是在状态变量或其衍生物。通过利用短爆模型的力量，可以使用新的数学技术分析许多问题。这在处理生物、物理或其他真实世界模型的应用环境中非常有用，因为这些模型的结果精度非常重要。*

With the recent Ebola outbreak, the need for mathematical models to understand transmission between and within countries worldwide is critical. Although we have had some success in reducing disease prevalence in the Western world, many diseases continue to threaten developing nations, with over 50% of the world's population at risk for one or more transmissable infections. Emerging infections such as West Nile Virus, SARS and swine flu have threatened the West in recent years, with potential pandemics from avian influenza, MERS and others looming. Tools at our disposal include vaccines, drugs, quarantine and education.***Many of the effects of such interventions occur in short bursts or shocks. For example, National Immunization Days (NIDs) for measles, polio etc occur in many developing countries over 1-2 days, twice a year. During this time, millions of children are vaccinated at once. India vaccinates 174,000,000 children in a single NID. On a smaller scale, antiretroviral treatment for HIV involves taking pills, whose duration of action is about 20 minutes long, significantly shorter than the period of hours between pills. Quarantine often involves relocation of infected individuals occuring on a short timescale, whereas education can involve bursts of media information happening quickly.******To analyse these effects, we will develop mathematical models using impulsive differential equations, impulse extension equations and Filippov systems. These methods are used to analyse short, sharp shocks, either in the state variables or their derivatives. By harnessing the power of short-burst modelling, a great many problems can be analysed using novel mathematical techniques. This will be useful in an applied context when dealing with biological, physical or other real-word models where precision of the results are important.************