Logistic Model for Propagation of a Disease

Suppose $n$ is the number of turns which have been played in the disease game, and $I_n$ is the number of diseased individuals in the $n$th turn of the game. Then one may write

$$I_{n+1} = I_n + I_{new},$$

where $I_new$ is the number of individuals which are newly infected during turn $n$. A beginning model can be put together by assuming that the distribution of infective hexes and individuals creating them is random. If each infected individual `splats,' `sprays,' or `sneezes' into a zone covering $z$ hexes, and the board contains $H$ hexes, then an approximation for the total number of hexes which are infectious at turn $n$ is

$$I_n \times \frac{z}{H}.$$

The number of individuals which are still susceptible to being infected on turn $n$, $S_n$, is the total ($T$) less the number of currently infected individuals, that is

$$S_n = T - I_n.$$

Then the number of new infections can be approximated

$$I_{new} = S_n \times I_n \times \frac{z}{H}.$$

Putting this all together gives an initial, discrete logistic model for the propagation of disease:

$$I_{n+1} = I_n + I_n \times \frac{z}{H} \times (T - I_n).$$

In the case of the basic disease game, $T=50$, $z=7$ (six hexes surrounding each diseased individual + the hex they stand in), and $H=100$ for the hex-grids provided. This model can serve as a foundation to build other, more advanced models from.