Playground Pox

Introduction

The “full” model

In order that some of the approximate models we later build have any chance of working, we’ll need some more kids. Let’s say we have a big playground with 1000 children.

A fast stochastic model: A homogeneous playground

The full model has to keep track of all the kids — in particular whether or not they are infectious in any given minute (second, actually, in practice) — so it’s rather slow.

SIR

The simplest Susceptible — Infectious — Recovered model (see my earlier article) is essentially identical to the fast stochastic model. The expected number of people transitioning from susceptible to infectious in a given time step in the fast stochastic model is the expected value of the binomial distribution describing the number of people transferring in a given time step. This is the number of uninfected kids multiplied by the probability each child gets infected, which in turn is (roughly) the number of infected kids multiplied by the average number of contacts per child per time step multiplied by the probability of transmission). A similar argument applies to the expected number of people transitioning from infectious to recovered.

Comparisons

We can compare SIR and the fast model with the full model if we set the infectious time distribution in the full model to exponential and set all the contact rates between children to a constant value. The results are shown below with the full model in the same colour as before, the fast model in blue and SIR in red.

Next time

The full model is specified by a description of the amount of contact each child has with every other child, so it’s incredibly versatile. In my next article in this series, I will play with different contact structures to demonstrate phenomena such as super-spreading.

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Graeme Keith

Graeme Keith

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Mathematical modelling for business and the business of mathematical modelling. See stochastic.dk/articles for a categorized list of all my articles on medium.