Modelling infectious diseases with stochastic discontinuities

传染病是世界性的全球性问题。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50%以上的人口面临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。这种干预的许多效果都是在短时间内爆发或冲击的。例如，许多发展中国家的麻疹、脊髓灰质炎等国家免疫日超过12天，每年两次。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。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系统来建立数学模型。这些方法被用来分析短期、剧烈的冲击，无论是状态变量还是它们的导数。脉冲微分方程式建立在这样的假设之上，即通常很自然地假设系统中有足够短的扰动是瞬时发生的，因为与过程的持续时间相比，它们的长度可以忽略不计。脉冲扩展方程通过将脉冲微分方程扩展到包括其连续模拟来解决假设的有效性问题，以便将两者进行比较。Filippov系统是导数不连续的动力系统。菲利波夫体系容易强加一个经济门槛：当一种疾病的成本足够高时，就会采取行动。我们将使用这些公式来分析随机变化对支点的影响。过去，人们对随机微分方程进行了大量的研究，但很少有人研究随机性对不连续逼近的影响。例如，对于脉冲微分方程，脉冲的时间可以改变，跳跃的强度也可以改变。菲利波夫阈值的位置可能会发生变化，这可能会对位于阈值附近的真实均衡和虚拟均衡产生影响。通过利用短脉冲建模的强大功能，可以使用新的数学技术来分析大量问题。通过研究随机变化对阈值的影响，我们可以在数学和人类行为之间建立一个接口。在处理生物、物理或其他真实世界的模型时，这在应用环境中将是有用的，其中阈值很重要，但人类的行动可能会降低结果的可预测性。

Infectious diseases are a global problem worldwide. 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 transmissible 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. 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 12 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. 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. Impulsive differential equations are founded upon the assumption that it is often natural to assume that sufficiently short perturbations in the system occur instantaneously, since their length is negligible in comparison with the duration of the process. Impulse extension equations address the question of the validity of the assumption that the duration of short bursts can be ignored by extending the impulsive differential equation to include its continuous analogue, in order to compare the two. Filippov systems are dynamical systems with discontinuities in the derivatives. Filippov systems lend themselves to imposing an economic threshold: when the cost of a disease is sufficiently high, action will be instigated. We will use these formulations to analyse the effects of stochastic variations on the pivot points. Much work has been done on stochastic differential equations in the past; however, very little has been done on the effects of stochasticity on discontinuous approximations. E.g., for impulsive differential equations, the timing of the impulse may vary, as well as the strength of the jump. The location of Filippov thresholds may be subject to variation, which may have implications for both real and virtual equilibria that are located near the threshold. By harnessing the power of short-burst modelling, a great many problems can be analysed using novel mathematical techniques. By investigating the effect of stochastic variations on the threshold, we can develop an interface between mathematics and human behaviour. This will be useful in an applied context when dealing with biological, physical or other real-world models where thresholds are important, but the actions of humans may reduce the predictability of the outcome.