I came up with an ill-defined problem that I'm not sure on how to approach: We have X€R^2 space and I(X)€{0,1}. If I(X)=1 X is infected, if I(X)=0 it's not. The probability of an X_0 infecting X is proportional to e^(-(X-X_0)^2). If at t=0 the circle X^2<1 is infected what's the average looking I(X,t)? What kind of maths deals with these problems? Rigor: An obvious way to approach this is a discrete way. Instead of X€R^2 use N^2. And instead of t€R use t€N and decrease the size of dt and dX. The weird part is N^2 is not isotropical like R^2 is. To give rigor to proportional to e^(-(X-X_0)^2), I could say that the time derivative of density in infected cells is proportional to the integral of e^(-(X-X_0)^2) or something of the sorts. I'll keep it short and just wait or suggestions.