Disease Spread at High Order

在过去的一年里，大多数公众已经意识到疾病传播的数学模型是有用的。它们使我们能够预测未来的疾病水平，并量化各种可能的干预策略的效果，例如封锁，社交距离和面部覆盖。此外，保持基本再生数（R 0）低于值1的重要性现在已被广泛理解。有各种不同的数学模型可用。它们在需要提供的信息量方面各不相同。在这个项目中，我们将在个人层面上建模，跟踪每个人的状态，因为疾病在人群中随机传播。在这种情况下，我们通常假设接触信息是可用的，或者可以估计-给定两个人，我们知道这对人是否接触（因此可能会传播感染）。在这种情况下，我们可以算出R 0的一个简单公式，从而确定在任何特定情况下疾病是否会迅速消亡。当我们以这种方式使用“成对”接触信息时，一个内在的假设是，我们被感染的机会与我们所拥有的受感染接触者的数量成正比。然而，这显然过于简单化了。例如，与四个受感染的同事共用一台复印机（在不知不觉中）的风险可能不会是与一个受感染的同事共用一台复印机的四倍，如果该物品在每次使用之间都被清洁的话。另一方面，如果存在病毒载量阈值，那么与四名受感染乘客共用一辆车的风险可能是与一名受感染乘客共用一辆车的风险的四倍多。此外，整个群体的规模可能会产生影响：在固定的教室里，可能会有一个学生人数的截止点，超过这个截止点，社交距离就不可行了。该提案旨在通过开发和分析直接处理群体而不仅仅是配对的数学模型来解决这些缺陷。从数学的角度来看，这使我们从图到超图，从线性到非线性的感染率。一些初步的工作表明，建立、分析和从以这种方式定义的模型中获得见解是可能的，但在这些模型变得真正有用之前，还有几个重要的步骤要做。该项目有三个主题：1。建模问题：通过搜索关于疾病传播的实验室和现实世界研究的越来越多的文献，我们将构建适当的数学方程，说明感染在不同群体环境中传播的方式;例如在教室、办公室、超市或酒吧。2.平均场模型及其分析：通过研究这些模型的简化版本，我们将推导出R 0和相关量的良好近似。3.精确模型分析：使用更复杂的数学技术，我们将证明关于完整模型的严格结果;例如，保证R 0的上限。总的来说，这个数学科学“小赠款”提案旨在建立一个基础建模和分析框架，由说明性计算机模拟支持，以解释人类在群体中互动的事实，而不仅仅是成对的。一旦这一阶段取得成功，进一步的跨学科和面向应用的后续工作将涉及（a）开发有效的大规模模拟算法，（B）利用真实的数据进行模型校准和模型选择，以及（c）大规模情景测试，以便使开发的工具对决策者和公共卫生专业人员有用。因此，从长远来看，有了真实的交互数据和校准良好的模型参数，我们将有工具来预测完全或部分封锁、不同程度的学校关闭、可变的社交距离、公共交通限制和其他行为干预的影响。

Over the past year most members of the general public have become aware that mathematical models of disease spread can be useful. They allow us to predict future disease levels, and also to quantify the effect of various possible intervention strategies, such as lockdown, social distancing and face-covering. Moreover, the importance of keeping the basic reproduction number (R0) below the value one is now widely understood. There are a variety of different mathematical models available. They vary in the amount of information that needs to be supplied. In this project we will be modelling at the individual level, keeping track of the status of each person as a disease randomly propagates through the population. In this setting, we typically assume that contact information is available, or can be estimated---given two people, we know whether this pair comes into contact (and hence may pass on an infection). In this case we can work out a simple formula for R0, and hence determine whether the disease will rapidly die out, in any particular circumstance. When we use "pairwise" contact information in this way, a built-in assumption is that our chance of becoming infected increases in direct proportion to the number of infected contacts that we have. However, this is clearly an oversimplification. For example, (unknowingly) sharing a photocopier with four infected colleagues may not be four times as risky as sharing it with one infected colleague, if the item is cleaned between each use. On the other hand, if there is a viral load threshold then sharing a car with four infected passengers may be more than four times as risky as sharing a car with one infected passenger. Moreover the overall group size may have an effect: in a fixed classroom, there may be a cutoff on the number of students beyond which social distancing is not feasible. This proposal aims to address these deficiencies by developing and analysing mathematical models that deal directly with *groups* of people, not just pairs. From a mathematical perspective, this takes us from graphs to hypergraphs and from linear to nonlinear infection rates. Some initial work has shown that it is possible to set up, analyse and gain insights from models defined in this way, but there are several important steps to take before these models can become really useful. The project has three main themes: 1. Modelling Issues: by searching the growing literature on laboratory and real-world studies of disease transmission, we will construct appropriate mathematical equations for the way that infection is transmitted in different group contexts; for example in classrooms, offices, supermarkets or pubs. 2. Mean-field Models and their Analysis: By studying simplified versions of these models, we will derive good approximations for R0 and related quantities. 3. Analysis of the Exact Model: Using more sophisticated mathematical techniques, we will prove rigorous results about the full model; for example, guaranteed upper bounds on R0.Overall, this mathematical sciences "small grant'' proposal seeks to build an underpinning modelling and analysis framework, backed up by illustrative computer simulations, to account for the fact that humans interact in groups, not just in pairs. Once this phase is successful, further interdisciplinary and application-oriented follow-on work will involve (a) development of effective large-scale simulation algorithms, (b) model calibration and model selection with real data, and (c) large-scale scenario testing, so that the tools developed can be made useful for policymakers and public health professionals. So, in the longer term, with realistic interaction data and well-calibrated model parameters, we would have tools to predict the effect of full or partial lockdown, different levels of school closures, variable social distancing, public transport restrictions, and other behavioural interventions.