Suppose we have a contact process on a finite lattice. I'm asked to give a heuristic argument for the fact that the extinction time for the contact process is exponential in the size of the lattice when it is in the supercritical phase, and logarithmic in the size of the lattice when it is subcritical. The supercritical phase means that on the infinite lattice, the infection never goes extinct almost surely.
I really don't know why this is the case. I get that the extinction time in the supercritical phase will grow rapidly, because if you make your lattice larger, the number of infected nodes grows. Because all these nodes are infected and they infect other nodes quickly (because of the supercritical phase), the other nodes will stay infected much longer. But is there any reason why this should be exponential (or logarithmic in the other case)?
See Liggett (1991), Stochastic Interacting Systems, part I, section 3, p.71. I'll provide some insight below in how to think about it. This should be helpful in understanding the difference in extinction time on the extreme ends of super- and sub-criticality. Of course, near criticality, the analysis becomes much harder.
Subcritical: In the subcritical case, the argument in Liggett essentially shows that the extinction time for the subcritical process on a lattice with $N$ sites is $\tau_d\approx \log N$.
Consider an extreme case where birth rates are nearly zero and all sites start off occupied. The births effectively can be ignored and we are just waiting for $N$ exponential death clocks to go off. The extinction time would be the maximum time of $N$ i.i.d. exponential random variables. The maximum of $N$ i.i.d. exponential random variables in the sum of the harmonic series up to $N$, which grows like $\log N$.
Supercritical: Similarly as above with the subcritical case, let the birth rate be one and the death rate be $\epsilon$ (close to zero). Assume that a birth will occur after one unit of time, then we want to know the probability that all $N$ exponential death clocks will go off before within that one unit of time. This probability is $e^{-\epsilon N}$.
Here is another way to look at it. Assume the supercritical contact process is stationary on the infinite lattice. There is going to be a positive average density of infected sites $-$ each site is occupied with probability $p$. If you were to randomly isolate a finite region with $N$ sites, the probability that you will isolate a region with no infected sites is $(1-p)^N$ (exponential in $N$). As you select larger and larger regions, the probability of not capturing any infected sites quickly goes to zero.
This isn't exactly what you are asking for, but it is one way to understand why the extinction time for the supercritical process on a finite lattice is exponential in the lattice size. We are simply waiting for the random event that the exponential 'death clocks' all happen to go off in series fast enough to overcome all the 'birth clocks'. It's a somewhat rare event since births are MUCH faster than deaths.