An effective theory of Covid-19 propagation

To fight Covid-19, we need an understanding of how the virus propagates in a community. Right now, the workhorse engine in the literature and in the media are compartmental models: (S)usceptible (I)nfected (R)ecovered and its cousins. The most popular versions of these models start with a set of coupled ordinary differential equations which, when solved, generate paths for each compartment. For example, in the simple SIR model, the infection starts with a large susceptible population which diminishes as the infected population rises. The infected population eventually declines as the recovered population increases and eventually the infected population goes to zero as the outbreak ends. The differential equations govern the dynamics of each compartment, generating equations of motion for the population.

Covid-19 propagation as a gamma subordinated negative binomial branching process

SIR models work well when we are discussing large populations, when we are interested in population averages, and when the random nature of the transmission doesn’t particularly matter. The solution to the coupled set of differential equations is deterministic. But when an outbreak gets started or when we are interested in the dynamics in the small population limit, we need more than deterministic models. SIR compartmental model with its focus on averages is not enough when dealing with the very early stages of an outbreak – and it’s the early stages where we really want our mitigation strategies to be the most successful. We need a stochastic model of transmission to understand the early stages of an outbreak.

Together with my colleagues Jerome Levesque, and David Shaw, we built a branching model of Covid-19 propagation. The idea is that an infected person randomly infects other people over the course of the communicable period. That is, we model transmission by imagining that an infected person generates “offspring”, continuous in time, during the communicable period and that each “child” has the same statistical law for generate more “offspring”. The infections branch out from each infected person into a tree that makes up the infected community. So while on average an infected person will infect R0 other people (the basic reproduction number) during the communicable period, there are a range of possible outcomes. We could get lucky and an initially infected person might not spread the virus at all, or we could get unlucky and the initially infected person might become a super-spreader, all in a model with the same R0. In fact, even with R0>1, there can be a substantial probability that the outbreak will go extinct on its own, all depending on the statistical branching nature of transmission.

In some research communities, there is a tendency to use agent based models to capture the stochastic nature of an outbreak. Such models simulate the behaviour of many different individuals or “agents” in a population by assigning a complicated set of transmission rules to each person. In the quest for high fidelity, agent based models tend to have lots of parameters. While agent based approaches have merit, and they have enjoyed success in many fields, we feel that in this context these models are often too difficult to interpret, contain many layers of hidden assumptions, are extraordinarily difficult to calibrate to data while containing lots of identifiability issues, are easily susceptible to chaotic outputs, and obscure trade-off analysis for decision makers. In a decision making context we need a parsimonious model, one that gets the essentials correct and generates insight for trade-off analysis that decision makers can use. We need an effective theory of Covid-19 propagation in which we project all the complicated degrees of freedom of the real world down to a limited number of free parameters around which we can build statistical estimators.

The abstract of our paper:

We build a parsimonious Crump-Mode-Jagers continuous time branching process of Covid-19 propagation based on a negative binomial process subordinated by a gamma subordinator. By focusing on the stochastic nature of the process in small populations, our model provides decision making insight into mitigation strategies as an outbreak begins. Our model accommodates contact tracing and isolation, allowing for comparisons between different types of intervention. We emphasize a physical interpretation of the disease propagation throughout which affords analytical results for comparison to simulations. Our model provides a basis for decision makers to understand the likely trade-offs and consequences between alternative outbreak mitigation strategies particularly in office environments and confined work-spaces.

We focus on two parameters that decision makers can use to set policy: the average waiting time between infectious events from an infected individual, and the average number of people infected at an event. We fix the communicable period (distribution) from clinical data. Those two parameters go into the probabilistic model for branching the infection through the population. The decision maker can weigh trade-offs like restricting meeting sizes and interaction rates in the office while examining the extinction probabilities, growth rates, and size distributions for each choice.

You can find our paper here:

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