In your link, look at this:
https://en.wikipedia.org/wiki/Mathematical_modelling_of_infe...
S = N - I - R, and lets used normalized ratios, so i = I/N, r = R/N, etc... so the dynamics of i are given by:
di/dt = beta (1 - i - r) i - gamma i
di/dt = (beta - gamma) i - beta (i^2 + ri)
At the beginning of the disease, the ratio of infected and recovered is very small, say 1e-5 for 100 cases in a population of 10 million. So initially the second term is 1e-10 whereas the first is 1e-5. The initial dynamics of a new disease are given by:
di/dt ~ (beta - gamma) i
exponential growth. It will start deviating from exponential growth once i becomes large enough. If 10% of the population have had the disease it will deviate from exponential growth by 10%. Once it gets to half the population being infected or recovered you start seeing a real deviation.
In reality this might never happen, because we are taking a lot of measures to get beta down. So really you have something like
di/dt ~ (beta(t) - gamma) i
where beta(t) will capture all the countermeasures people take. If the countermeasures are effective and manage to push beta(t) below gamma before a large chunk of the population is infected, we might never see non-exponential behaviour from the disease. It will have an exponentially growing phase, and then an exponentially shrinking phase.
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Here is just one paper I found with two seconds of google that looks at this, very recent, a bit basic:
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6962332/
But here is what it says in the abstract:
"The initial exponential growth rate of an epidemic is an important measure of the severeness of the epidemic, and is also closely related to the basic reproduction number."
"Classical compartmental transmission models assume exponential growth during the early phase of a well-mixed population"