Does anybody know what is the mathematical relationship between a transmission rate and an epidemic peak in a finite population?
Now why is this a purely math problem? Because the "infection" can be reduced to anything: it would be the same thing as solving this problem: we have $50$ million balls, with only one red ball at the beginning and the rest are white. Each red ball can touch a total of $10$ other balls per unit of time, and can only touch other $10$ other balls during $2$ consecutive time units (so $20$ total), whether they're white or red, completely at random between the $50$ million balls. If a red ball touches a white ball, the white ball becomes red. If a red ball touches a red ball, nothing happens. Now the question is: what is the greatest number of red balls created per one unit of time at some point? (only the number of new red balls per unit of time counts). Very likely it's going to be when half of the balls are red, because it is at this time we have a max white balls to be touched by a max number of newly created red balls! Now the question becomes: what is the relationship between the number of red balls a newly created red ball can touch, and the peak? What if it was $15$ balls instead of $10$, how would the peak change?
Thank you very much if you can help! This question is very hard to find an answer to online.