I'm sure this has been done to death, but I just like to work things out on my own.
Suppose there is a very large population, and a certain percentage pt are infected with some germ at time t.
Each period (day, week, whatever is appropriate for the disease), each person who is infected infects r0 other people randomly selected from the population. However, those who have already been infected are immune, and do not develop the disease.
So the percentage of people infected at time t is pt==pt-1*r0*(1-Pt-1), where Pt-1 is the cumulative percentage of people infected. So Pt==Pt-1+(Pt-1-Pt-2)*r0*(1-Pt-1).
It is probably possible to analyse this analytically, but I'm really tired, so I just pulled up a spreadsheet and did it numerically. Starting in all cases with an initial infection rate on day one of 0.001%, we have:
If r0=1.1, the pandemic reaches its peak on period 79, with about 0.44% of the population infected (that is, this is the highest rate of infection at one point in time). By this time, 9.3% of the population has been infected, but ultimately the cumulative infection rate will be 17.9%.
If r0=1.5, the pandemic reaches its peak on period 26, with about 7.24% of the population infected (that is, this is the highest rate of infection at one point in time). By this time, 37.8% of the population has been infected, but ultimately the cumulative infection rate will be 61.8%.
If r0=2.0, the pandemic reaches its peak on period 17, with about 19.5% of the population infected (that is, this is the highest rate of infection at one point in time). By this time, 63.6% of the population has been infected, but ultimately the cumulative infection rate will be 86.9%.
If r0=3.0, the pandemic reaches its peak on period 12, with about 40.6% of the population infected (that is, this is the highest rate of infection at one point in time). By this time, 104.4% of the population has been infected, but ultimately the cumulative infection rate will be 98.9%. The nonsensical numbers indicate a problem with the model - when r0 is high, a certain percentage of people will be infected by more than one already infected person, reducing the number of new infections somewhat.
Comments on the model/analysis? How to improve?