Modelling Infectious Diseases with discontinuities

随着最近埃博拉疫情的爆发，需要数学模型来了解世界各国之间和国家内部的传播。尽管我们在降低西方世界的疾病流行率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%的人口面临一种或多种可传播感染的风险。近年来，西尼罗河病毒、SARS和猪流感等新出现的感染威胁着西方，禽流感、中东呼吸综合征和其他疾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。*这种干预措施的许多效果都是在短时间内发生的。例如，许多发展中国家一年两次的麻疹、脊髓灰质炎等国家免疫接种日超过1-2天。在此期间，数百万儿童同时接种疫苗。印度在一个NID中为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药，其作用持续时间约为20分钟，明显短于两次服药之间的几个小时。隔离通常涉及在短时间尺度上发生的受感染人员的重新安置，而教育可能涉及快速发生的媒体信息突发。*为了分析这些影响，我们将使用脉冲微分方程、脉冲扩展方程和菲利波夫系统来建立数学模型。这些方法被用来分析状态变量或其导数中的短而剧烈的冲击。通过利用短脉冲建模的力量，可以使用新的数学技术来分析大量问题。在处理生物、物理或其他真实世界模型时，这将在应用环境中很有用，因为在这些模型中，结果的精度很重要。

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.************