Covid-19 made me speculate of how to mathematically model the spread of a disease. So let's say there's some virus that has a basis reproduction number R0 = 3 (so one infected person does infect exactly 3 people (in the following week)). So we get the time (i.e. week)-dependent function of new infections, f(t) = 3^t, which means exponential growth.
Let's say someone who has been infected is immune forever. Then at some point in time there are so many immune people that the reproduction number decreases, e.g. only two people get infected because the third one is already immune.
1) Now my first question is, how to model our function f(t) in order to take this effect into account? And how to calculate the point in time where this effect shows up? I know about how to calculate the percentage p of people necessary for herd immunity (p = 1/(1-R0)), but this only says how many people have to be immune in order to have less than one new infection per current infection, it doesn't say anything about the crease of R from R0 = 3 to R<1.
2) I've found many graphs of simulated spreads on the internet, always depicted as the graph of a normal destribution (for example:
Is the graph the integral of the modified function we are looking for in 1), i.e. is the function in 1) the density function?
3) In such graph, where is the point of herd immunity? I guess it can't be at the peak of the graph, because i.e. for R = 3, we have p = 66% and that would mean that the area of the graph is equal to 132%, which doesn't make any sense...
I'm looking forward to your answers or to discussing my questions with you here.
You must use a SIR model, see e.g. https://en.wikipedia.org/wiki/Compartmental_models_in_epidemiology#The_SIR_model
And there are simulators on the internet, e.g. http://www.public.asu.edu/~hnesse/classes/sir.html
For the current outbreak, you can find data here: https://blogs.sas.com/content/sascom/2020/03/10/using-data-visualization-to-track-the-coronavirus-outbreak/ They leap a bit but describe the current situation quite well.