I'm working with a SIR infection model, which is $$\begin{array}{rcl} \frac{dS}{dt} & = & -\beta IS\\ \frac{dI}{dt} & = & \beta IS-\gamma I\\ \frac{dR}{dt} & = & \gamma I \end{array}$$ in continuous time, where $S$, $I$, and $R$ are the proportion of Susceptible, Infected, and Recovered, respectively.
However, since I am working with fixed-width discrete-time data, I think it would be more appropriate to modify the equations accordingly. I know this is incorrect (based on getting a negative values for $\hat\gamma$ and $\hat\beta$, neither of which should be negative): $$\begin{array}{rcl} S_{t+1} & = & -\beta I_{t}S_{t}\\ I_{t+1} & = & \beta I_{t}S_{t}-\gamma I_{t}\\ R_{t+1} & = & \gamma I_{t} \end{array}$$
There is a model on Page 2 of this document which incorporates the population $N$, but i'm unsure of the reasoning behind this and doesn't appear to fix my problems in solving for the parameters.
Ultimately, I would like to estimate $\beta$ and $\gamma$ by doing regression on $$I_{t+1} = \beta (I_{t}S_{t})+\gamma (-I_{t})+U_{t} \text{, where }U_{t}\LARGE\text{~}\normalsize N(0,\sigma^2)$$ or whatever the discrete version is.
Edit: I feel as though I should note that I essentially used the discrete version above when solving for the parameters "manually" by looping through several values to get the minimized Mean Square Error. So, I would have expected to be able to just do the regression on the same equation for $I_{t+1}$. Apparently this is not the case as I am getting very different results with the regression.
You've missed some things. Remember a derivative is the small $\Delta$ limit of, for example, $(S(t+\Delta)-S(t))/\Delta$. So, your first differential equation is like $$ \frac{S(t+\Delta)-S(t)}{\Delta}=-\beta IS $$ or, where $\Delta$ is your time interval width, $$ S_{t+1}=S_t(1-\Delta\beta I_t). $$ You missed the $S(t)$ term on the right hand side. $S$ should be decreasing with time, and looking at the equation we see that this would be the case if $\beta$ is positive.
Also note that $S+I+R=N$ is constant, so you only need 2 differential equations, you can always find $R$ by doing $n-S-I$.