I have a biological process I want to study. We have a bacterial culture in which the probability of one individual getting some specific unwanted mutation is $p$. By a technical process we achieve a way of lowering this probability to $p^2$. This mutation is preferentially selected, so it is a matter of time that virtually all the population has it. We want to estimate how much better is our system with our technical development. For example, we could estimate the time gaining: if the time in which the size of the mutated subpopulation rises to (say) 10% is $T$ when the probability is $p$, which time $T'$ do we get with $p^2$?
I have thought of the following analysis (beware, I am no expert in this!):
We can model the independent, sequential, homogeneous in time bacterial mutations by a Poisson process. Let $N$ be the total population. In a fixed time $t$, if the mean number of mutations in $t$ is $m(t)$, then the probability that a given bacteria will mutate in time $t$ is $p(t)=m(t)/N$ (is this actually sound?). If we now change the time to $kt$, then $m(kt)=km(t)$ (by the properties of the Poisson distribution) and thus $p(kt)=kp(t)$.
Consider now the mean number of mutations for time $t$ corresponding to probability $p(t)^2$, $M(t)=p(t)^2N$. Since we want to estimate the time gain, we will find $k$ such that $M(kt)=m(t)$: $p(kt)^2N=p(t)N$ implies $k^2p^2(t)=p(t)$ and therefore $$k=p(t)^{-1/2}.$$ So if in usual conditions the mutated bacteria are a 1% of the total population in time $T$, then $p(T)=m(T)/N=10^{-2}$ and hence with our technical improvement we get a time of $10T$ to reach the same level of mutations, but the time to arrive at 50% of the population is just $\sqrt{2}$ times the usual time for this to happen.
So it seems that, although the improved culture starts to be "corrupted" way later, its rate of change is higher and arrives at a 100% of corruption at the same time as the usual one. Note that this seems to be at odds with the fact that in a Poisson process the mean is linear with time, but (I think) this apparent paradox is solved by noting that $M$ does not follow a Poisson process (precisely because of our $p^2(t)$ with a Poisson $p(t)$).
a) Is this analysis really correct? Would you do the estimation otherwise?
b) Which distribution studies the time intervals of a cumulative Poisson? How would we apply it to this problem?