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可能是有用的。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，如封锁、保持社交距离和蒙面。此外，保持基本繁殖数（R0）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上有所不同。在这个项目中，我们将在个人层面进行建模，跟踪每个人在疾病随机传播时的状态。在这种情况下，我们通常假设联系信息是可用的，或者可以估计——给定两个人，我们知道这对人是否接触过（因此可能会传播感染）。在这种情况下，我们可以计算出R0的一个简单公式，从而确定疾病在任何特定情况下是否会迅速消亡。当我们以这种方式使用“两两”联系信息时，一个固有的假设是，我们被感染的机会与我们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，（在不知情的情况下）与四个受感染的同事共用一台复印机，如果在每次使用之间进行清洁，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。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以上。此外，整体群体规模可能会产生影响：在一个固定的教室里，学生人数可能会有一个界限，超过这个界限，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”而不仅仅是成对的数学模型来解决这些缺陷。从数学的角度来看，这让我们从图到超图，从线性到非线性感染率。一些初步的工作已经表明，以这种方式定义的模型是可以建立、分析和获得洞察力的，但是在这些模型真正有用之前，还有几个重要的步骤需要采取。该项目有三个主要主题：1。建模问题：通过搜索越来越多的关于疾病传播的实验室和现实世界研究的文献，我们将为感染在不同群体背景下的传播方式构建适当的数学方程；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，我们将得到R0和相关数量的良好近似值。3. 精确模型的分析：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R0的上界。总的来说，这项数学科学“小额拨款”提案旨在建立一个基础的建模和分析框架，以说明性计算机模拟为后盾，以解释人类是群体互动的事实，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)使用真实数据进行模型校准和模型选择，以及(c)大规模情景测试，以便开发的工具能够对决策者和公共卫生专业人员有用。因此，从长远来看，有了现实的互动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措施的影响。

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.