I have a question about the standard epidemic SIR model: $$\frac{dS}{dt}=-\frac{\beta IS}{N},$$ $$\frac{dI}{dt}=\frac{\beta IS}{N}-\gamma I,$$ $$\frac{dR}{dt}=\gamma I.$$
How do I get the values $\beta$ and $\gamma$?
My question arises because I want to perform a real simulation applied to the COVID case. I have already done some research and have not found anything useful.
I hope you can help me, thank you.
In the SIR model shown, the $\frac{\beta IS}{N}$ term corresponds to people getting sick, and $\gamma I$ represents people becoming no longer sick, either by recovering or passing. I'm going to assume that $N$ is the total initial population of your modeled area.
To find $\beta$, we use the fact that the rate of new cases is given by $\frac{\beta IS}{N}$. It's probably best to look at data near the beginning of the pandemic, since in that case very few people have been infected, so $S\approx N$ is a reasonable assumption. Then you can take the ratio between new cases and current cases (averaged over multiple days to reduce error) to find $\beta$.
To find $\gamma$, we can simply divide the number of new deaths/recoveries by the total infections for the day. Given what we know about recovery times (usually around 14 days), I would not expect this quantity to be consistent over the course of the pandemic, especially in the exponential growth portion at the beginning. I would suggest taking data from times the case count was plateauing or dropping.