**Eric Busboom**.

The SIR infection model is a simple set of differential equations that describe how a disease can spread under very homogeneous, stable conditions. The model gets it’s name from the three “compartments” for people in diferent conditions:

- Susceptible: people who can be infected.
- Infected: people who are infected.
- Recovered or Removed: people who were infected, and can’t get infected again.

There are many good overviews of the model online, so in this brief analysis, we will just use the model to visuaize what happens when the $R$ parameter for an epidemic changes.

$R$ is the “effective reproduction number” of a disease, which describes the rate of increase of infection for curent conditions. It differs from the similar number, one most people are more familliar with from recent press, the basic reproduction number $R_0$, in that $R$ can change, but $R_0$ cannot.

The SIR model doesn’t use $R$ or $R_0$ directly; it uses an infection rate $\beta$ and recovery rate $\gamma$, and $R_0$ = $\frac{\beta N}{\gamma}$, where $N$ is the total population. In this simulation, we’ll use the SIR model to see what happens when a popuation the size of San Diego County gets one infection, with one case with constant $R_0$ and other cases where $R$ changes from $R_0$ to a lower value. We will change $R$ by changing $\beta$ and leaving $\gamma$ constant, which is a simple simulation of San Diego impementing a social distancing policy.

These three plots similate first a constant $R_0$ and then two social distancing policies, one that is implemented at day 7, continues for 6 weeks and drives $R$ down to .9, and a second that has similar parameters but is impemented a week earlier, at day 7. Note that the $I$ variable, infections, is the count of current infections; the green line for recovered people shows the number of people who have ever been infected. In this model, most people get infected eventually, but the third case, social distancing at the first week, has about 600K fewer total infections (recovered) than not implemeting the policy. For COVID-19 and a 1% to 2% fatality rate, the early implementation of the social distance policy would save 6,000 to 12,000 lives.

The orange line for $I$ shows the current number of infected people, is an indication of the strain on the hosptal system, and early implementation of the policy reduces the peak by 200,000 infections over the late policy, and 300,000 over no policy.

These simulations are very hard to match up to historic data, so the SIR model is not very useful for predicting actual numbers of infections, but since the dymanics are accurate, even if the numbers are not, it does a very good job of illustrating the likely effects of simple policy changes.