Hey /sci/, could you help me with a question of mine?

I am in the process of writing a program that models the spread of a disease on a lattice of agents where the agents move about randomly. Infected agents are able to transmit the disease to anyone within radius $\delta$ of them. The lattice itself is a square with area $l^2$ (the edges of the lattice wrap around on themselves). If a susceptible agent is within the "infective radius" of an infected agent, they will become infected with probability $\alpha$ per time step. The area of the entire "infected region" is $I_t \pi \delta^2$, where $I_t$ is the number of infected individuals at time $t$. The number of susceptible agents within this region is that area multiplied by the overall density of susceptible agents in the lattice, or $S_t/l^2$, where $S_t$ is the number of susceptible agents at time t. I should specify that agents are not uniformly distributed throughout the lattice. Infective regions are allowed to overlap, and they very often do.

Given all this, the difference in the susceptible agent population per time step would be $\Delta S = -I_tS_t \pi \delta^2 \alpha/l^2$. Is this correct? This formula doesn't match my sim. results, and I figured one of you may be able to help.