Branching Process Models

We consider an exponentially growing cell populations modeled as a multi-type branching process in which type i cells are those with i ≥ 0 mutations and we let Zi(t) be the number of type i cells at time t. Type i cells give birth at rate ai and die at rate bi, where the growth rate λi = ai − bi > 0. Thinking of cancer we will usually restrict our attention to the case in which i → λi is increasing, but we will see an application later in the chapter with λ1 < λ0. To take care of mutations we suppose that individuals of type i in addition give birth at rate ui+1 to individuals of type i + 1. This is slightly different than the approach of having mutations with probability ui+1 at birth, which translates into a mutation rate of aiui+1, and this must be kept in mind when comparing results.