Mean-field theory is actually in the news a lot right now and influencing discussions at the highest level of government. It is used in many of the mathematical models in epidemiology involved in discussions about ``flattening the curve.'' An example is the paper from the Doherty Institute at the University of Melbourne that had a significant influence on Australian government policy.
In an article in The Guardian Australia, Kathryn Snow has given a nice discussion for a general audience of the role and limitations of modeling and there is a New York Times article comparing five different models for the spread of COVID-19.
Many of the models being used are generalisations of the simplest SIR model.
A basic introduction is found in a course at ETH-Zurich, from which the figures below are taken.
My UQ colleague, Zoltan Neufeld also gave a clear and helpful introduction in a seminar, he recently gave for the School of Mathematics and Physics at UQ. [Video is here].
The population is divided into three groups: susceptible (S), infected (I) and recovered/dead (R).
This leads to a set of three coupled non-linear differential equations
The model provides many qualitative insights.
A key parameter is the R0, the basic reproduction number, which for this model is.
The solution to these equations for different parameter values gives a feel for how quantities such as the fraction of the population that get infected and the duration of the epidemic depend on parameters such as R0. That is what underlies discussions about ``flattening the curve''.
The Doherty Institute paper is a generalisation of this model where the population is divided into 15 different groups (such as quarantined and non-quarantined individuals) and there are fifteen parameters and fifteen first-order differential equations.
But what are the assumptions in the SIR model?
Foremost, it is a mean-field theory.
It assumes that in the population there is homogeneous mixing. You could think of the system being like a gas or liquid, which is composed of a uniform mixture of particles of three different types: infected, susceptible, and recovered. They collide at random with one another, just like molecules in a fluid, and there is a fixed probability (collision cross-section) for the species of a particle to change after a collision.
Consider different ways this mixing assumption can break down in the real world.
1. The parameters in the model may be different for different people and for different communities.
I find it misleading that people say, whether in science papers or in newspaper articles, that the R0 for COVID-19 is 2.6 plus or minus 0.2. Surely, the value is context dependent and model dependent. Doesn't it depend on the number and type of contacts that people have? It should be different for the western suburbs of Brisbane, a slum in Bangkok, Beijing, and the Bronx in New York City.
If the model parameters are stochastic, i.e. drawn from a probability distribution, are there qualitative changes in behaviour.
2. The real world has spatial structure and discrete structure. The discrete structure is taken into account in agent-based models such as those in NetLogo, which has a nice simulation Virus.
I downloaded Netlogo last year and used it for a talk on emergence and international relations. It is very easy to use and has lots of cool programs. By varying parameters you can see a lot of cool phenomena.
3. Network effects
Netlogo has a nice program Virus on a network
Network models can include the role of super-spreaders: a few highly connected individuals who spread the virus.
There is a nice review
Epidemic processes in complex networks
Romualdo Pastor-Satorras, Claudio Castellano, Piet Van Mieghem, Alessandro Vespignani
Of particular interest (and concern) are scale-free networks, for which there is no epidemic threshold. In different words, regardless of the value of R0, an epidemic will always occur, and spread extremely rapidly.
Aside. The last author of the review is leading a group working on COVID-19 at Northeastern University, and featured in a New York Times article last week.
Why is a mean field theory needed at all? Can't the mechanics be described exactly? Not obvious to me whether the qualitative solid state physics approach is anything more than a gross simplification
ReplyDeleteIs this question about the mechanics of a single virus particle? The post is about modelling the spread of the virus in a human population.
DeleteHey Ross
ReplyDeleteI think that making better models of an epidemic is a lot of fun. More sophisticated graph theory sounds like a very worthwhile approach.
However, as a way to manage the initial outbreak of the novel coronavirus, I think it would have been exactly the wrong way to go. There are two related things that strike me about the published models.
Firstly, even these very simple models took too long to solve. I hope people didn't sit around waiting for an answer about the virus spreading from Asia to Australia: by the time that paper was written, it was clear that the US and UK were our real problem. Similarly, by the time people had calculated the effect of reducing the reproduction number by 30%, Australia had taken social distancing measures that were going to be an order of magnitude more effective than that.
Secondly, at any point in time, my latest Fermi estimates have been more useful than the latest official calculations.
Want to know R? Sum some geometric series, compare them to the reported ratio of imported cases to local infections, then relax.
Assume that doing the same things in the US as they did in Wuhan will lead the same outcome, multiply that by the number of Australians in the US, then be really worried when Brendan Murphy capitulates to the politicians and lets those people return and circulate in the community.
Did we take social distancing past the point of diminishing returns? New Zealand went significantly further, and their outcome is right in the middle of the distribution of Australian states. We were wiping out the virus on the timescale that people recover from it; there wasn't a lot of scope to do better.
The experts took weeks to catch up with estimates like that. As usual with Fermi estimates, the detailed calcuations gave the same answers.
Now that the urgency has passed, accurate forcasts would be useful. However, the urgent step is a qualitative one, imagining what questions need to be asked. Italy didn't fail because the coronavirus was hard to distinguish from the flu—once they asked the right question, they answered it very quickly. And if anyone in Singapore had cared about people from South Asia, estimating how the virus might spread through a population that lived 20 to a bedroom would have been pretty easy.