The logistic equation is another model at which the finite population size starts limiting the process. As the number of cases approaches the population size, the rate slows.

(%i8)  diff(f(x),x)=gamma*f(x)*(G-f(x));
                       d
(%o8)                  -- (f(x)) = (G - f(x)) f(x) gamma
                       dx

Here G is the total population size. As the number of infections reaches the population size, the rate of new infections decreases. So as the number of infections reaches 50% percent, the rate of growth will decrease by 50%. That’s what the logistic equation models. Basically, like a wildfire, the fire burns itself out.

So, a typical large metropolis has about six million people G = 6. When the number of infections, has reached 3 million, or 50%, the rate of increase with a simple logistic model will be 50% less.