Modelling infectious diseases with stochastic discontinuities

具有随机不连续性的传染病建模

• 批准号: RGPIN-2020-05485
• 负责人: Smith, Robert
• 金额: $ 1.97万
• 依托单位: University of Ottawa
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=716588
• 关键词: Modelling; infectious; diseases; stochastic; discontinuities

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.

传染病是全球一个全球问题。尽管我们在降低西方世界的疾病患病率方面取得了一些成功，但许多疾病仍在威胁发展国家，世界上有50％以上人口有一种或多种可传播感染的风险。近年来，西尼罗河病毒，SARS和猪流感等新兴感染威胁着西方，并有可能来自鸟类流感，MERS和其他人迫在眉睫的大流行。这种干预措施的许多影响发生在短爆发或冲击中。例如，在许多发展中国家，每年两次发生麻疹，脊髓灰质炎等国家免疫日（NID）。在此期间，数以百万计的儿童立即接种疫苗。印度在一个NID中接种174,000,000名儿童。在较小的规模上，对HIV的抗逆转录病毒治疗涉及服用药丸，其作用持续时间约为20分钟，比药丸之间的小时时间短得多。为了分析这些效果，我们将使用冲动的微分方程，冲动扩展方程和Filippov系统开发数学模型。这些方法用于分析状态变量或其衍生物中的短而尖锐的冲击。冲动的微分方程是基于以下假设，即通常认为系统中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。脉冲扩展方程解决了以下假设的有效性问题，即可以通过将冲动的微分方程扩展到包含其连续类似物以比较两者，从而忽略了短突发的持续时间。 Filippov系统是动态系统，在衍生物中具有不连续性。 Filippov系统将自己施加经济阈值：当疾病的成本足够高时，就会煽动行动。我们将使用这些公式来分析随机变化对枢轴点的影响。过去，在随机微分方程上做了很多工作。但是，关于随机性对不连续近似的影响，几乎没有完成。例如，对于冲动的微分方程，冲动的时机可能会有所不同，并且跳跃的强度。 Filippov阈值的位置可能会受到变化的影响，这可能对位于阈值附近的真实和虚拟平衡都有影响。通过利用短期建模的力量，可以使用新颖的数学技术来分析许多问题。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发展一个接口。在处理阈值很重要的生物，物理或其他现实世界模型时，这将在应用的环境中很有用，但是人类的行为可能会降低结果的可预测性。