Least Squares Estimate of Infection Model Parameters

Question

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 $\gamma$.

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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.

To be thorough, I would like to do this using a least squares regression estimate of the parameters as well. Given that I know $I$, I can easily calculate $dI$ for each time period. However, I'm having trouble getting $S$ from my information.

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One of my attempts assumed that all the infected from time $t-1$ moved to $R$ at time $t$ and that all infected at time $t$ had come from $S$. Therefore, $S$ was simply reduced by the number of infected each time period. I understood going into it that this was a risky assumption and the results were quite disappointing.

Any tips you have on how to find $S$ at each time period are greatly appreciated.

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To be more clear, my goal is to do a regression on the equation

$$ dI = \beta (SI) + \gamma (-I) $$

to get the least squares estimates for $\beta$ and $\gamma$.

Answer 1

Based on your description, other appropriate forms for your model would include nonlinear growth or diffusion models. These are models like those used in the analysis of new products, tech innovations, movie ticket sales as well as the spread of diseases and contagions. The underlying process can range from simple, sigmoid-shaped Gompertz curves and loglets based on Fisher-Pry transformations to sophisticated Bose-Einstein processes. Good discussions of these issues can be found here:

http://phe.rockefeller.edu/current/areas/about/Diffusion%20of%20Social%20Phenomena

http://www.amazon.com/Hollywood-Economics-Uncertainty-Routledge-Contemporary/dp/0415312612/ref=sr_1_1?ie=UTF8&qid=1446043976&sr=8-1&keywords=hollywood+economics

Also and specifically with respect to the statistical issues and challenges involved in maximum likelihood estimation of the unknown parameters, Jack Weiss' Lecture Notes at UNC are an invaluable resource. He includes plenty of references to R modules as well:

http://www.unc.edu/courses/2008fall/ecol/563/001/docs/lectures/lecture27.htm

Answer 2

Since it was, as I feared, impossible to calculated $S_t$ directly, I will post my "best" solution to the problem.

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Using my initial method of looping through values for $\beta$ and $\gamma$, I came upon an estimate that minimized the Mean Square Error (MSE) with respect to $I_t$. Using these estimates for $\beta$ and $\gamma$ and the model in the question, I as able to estimate $S_t$ for any $t$.

Now that we have $S_t$ for all $t$, we can perform regression on

$$I_{t+1}-I_t=\beta(S_tI_t)+\gamma(-I_t)$$ $$\text{or}$$ $$I_{t+1}=\beta(S_tI_t)+(1-\gamma)I_t.$$

The big difference here is that this is minimizing the MSE with respect to the difference $I_{t+1}-I_t$ instead of $I_t$ as in the first method. I should note that this did not result in a "better" model, but it allowed me to introduce some measure of the error in the estimates.