Disease Spread at High Order

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.