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: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=740615
• 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 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. 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.

传染病是世界范围内的全球性问题。尽管我们在减少西方世界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% 的人口面临一种或多种传染性感染的风险。近年来，西尼罗河病毒、非典和猪流感等新出现的传染病威胁着西方，禽流感、中东呼吸综合征等潜在大流行病迫在眉睫。此类干预措施的许多影响都是在短暂的爆发或冲击中发生的。例如，许多发展中国家每年举办两次针对麻疹、脊髓灰质炎等疾病的国家免疫日 (NID)，为期 1-2 天。在此期间，数百万儿童同时接种疫苗。印度在一次 NID 中为 1.74 亿儿童接种了疫苗。在较小范围内，艾滋病毒的抗逆转录病毒治疗涉及服用药物，其作用持续时间约为 20 分钟，明显短于服药之间的时间间隔。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。这些方法用于分析状态变量或其导数中的短期剧烈冲击。 脉冲微分方程建立在这样的假设之上：通常很自然地假设系统中的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假设的有效性问题，即通过扩展脉冲微分方程以包括其连续模拟，可以忽略短脉冲的持续时间，以便比较两者。 Filippov 系统是导数不连续的动力系统。菲利波夫系统有助于设定经济门槛：当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枢轴点的影响。过去，人们在随机微分方程方面做了很多工作；然而，关于随机性对不连续近似的影响却知之甚少。例如，对于脉冲微分方程，脉冲的时间以及跳跃的强度可能会变化。 Filippov 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平衡和虚拟平衡产生影响。 通过利用短突发建模的力量，可以使用新颖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，我们可以开发数学和人类行为之间的接口。当处理生物、物理或其他现实世界模型时，这在应用环境中非常有用，其中阈值很重要，但人类的行为可能会降低结果的可预测性。