# The most used mathematical model for modelling COVID-19

## Harlan Ichikawa

The SIR model is the most basic deterministic model of infectious disease dynamics. Navier-Stokes is to fluid models what SIR is to epidemiology. I suspect a lot of technically minded individuals are pinging the wikipedia page, before turning around because they don’t have time. The point of this post is to describe the SIR model in a couple paragraphs. Let’s go.

## Variables

The variables of the model are

• $s(t)$: the portion of the population susceptible to infection at time $t$.
• $i(t)$: the portion of the population currently infected
• $r(t)$: the portion of the population recovered

Since each of these variables represents a portion of the population, each takes a value between $0$ and $1$. That the system is closed means

We are ignoring the fact that people die in this model.

## Dynamics

New infections are created by interactions between infected people and susceptible people. In the case of random interactions, the number of interactions is proportional to the product “$i \cdot s$”. Infections are destroyed as people recover, at some rate $\gamma > 0$. If the rate of infection is $\beta > 0$ then $di/dt = \beta i \cdot s - \gamma i$. You can write down all this logic in the dynamical equations

To study the dynamics, we can rescale time by the infection rate $\tau := \gamma t$ and thus study the ODE

where $R_0 = \beta / \gamma$. This is the “R-naught” that is being spoken of in any of those articles you might read that talk about “flattening the curve”. It’s also called the “basic reproduction number”. If $R_0 > 1$ we get an epidemic, if it’s less than $1$ we get exponential decay. In the case of covid-19, $R_0 \approx 2.5$ without any intervention. In the (unrealistic) case where everybody without exception practices social distancing $R_0 = 0$.

## What could happen?

What could happen, qualitatively is illustrated in this plot

We have $R_0 = 2.5$, and the plot suggests that at some point, 20 percent of people would be infected. Which is crazy. About 20% of that 20% (so 4%) will need hospitalization, and about a quarter of those in need of hospitalization will need ventilators (so 1%), which makes me feel a little sick to write. I’m not an epidemiologist, but my guess is that many of the projections that are being made by the experts are reasoning in a way that resembles what’s written here.

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