I'm using the standard infection model on some data I am working with.
$ dS = -\beta SI $
$ dI = \beta SI - \gamma I $
$ dR = \gamma I $
Where $S$ is the number of susceptible subjects, $I$ is the infected, and $R$ is the recovered. I'm trying various methods for estimating the parameters $\beta$ and $\gamma$.
For any given discrete, fixed-width time period, I know the number of infected and the total population, which is fixed. One of the methods I have used to estimate the parameters is to feed the initial state into a differential equation solver in R and loop through several values for $\beta$ and $\gamma$ until they minimized the Mean Square Error.
I've been told it's possible to do a maximum likelihood estimate of the parameters, but am at a total loss as to how to begin doing this.
One idea that was presented to me involved using a normal curve and estimating the parameters of the distribution using the well-known maximum likelihood estimates for the parameters of normal distributions. My problem with this is that I'm dealing with the number of infected (or even proportion of infected) in a population, not with anything that follows the necessary assumptions of a probability distribution.
If I were to do this, I would need to introduce another parameter to shift the normal curve upward towards my data. By that I mean, if $f(t;\mu,\sigma)$ is the normal distribution, I would need another parameter $k>1$ such that
$$ I_t\approx kf(t;\mu,\sigma)$$
where $I_t$ is the number of infected (or the proportion, doesn't matter) at time $t$.
If I were to do this method, I believe I would estimate the parameters $\beta$ and $\gamma$ by:
- Use the maximum likelihood estimate of the normal(ish) curve using the above method.
- Use the least-squares estimate as in this question to fit the infection model against the normal curve instead of the actual data.
I'm not really sure what else to do, so I greatly appreciate any insight you can provide.