The logistic equation

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));
(%o8)                  -- (f(x)) = (G - f(x)) f(x) gamma

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.

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