COVID: How is the R coefficient calculated?
This article provides a good explanation.
Realistic models are much more complicated, but the basic ideas are:
$R_0$ is the basic reproductive number. If you put an infected individual in a population that is fully susceptible to the disease, then that individual is expected to infect $R_0$ people.
$R_t$ is the effective reproductive number (same as $R_e$). People include the subscript to distinguish $R_t$ from $R_0$ because $R$ is ambiguous. In a population $P$ at time $t$, an infected individual causes $R_t$ new infections on average. Therefore, given the susceptible population $S\subseteq P$ (i.e. the people who are not immune at time $t$), we have $R_t=R_0\frac{|S|}{|P|}$. Again, the actual $R_t$ is affected by many other factors, but if we only considered $R_0$ and the fraction of susceptible people in a population at a given time, then this is the interpretation of $R_t$.
Therefore, the growth ($R_t&gt;1$) or decay ($R_t&lt;1$) in the number of cases is exponential.

For live updates on COVID $R_t$ values in the US per state, you can check out: https://rt.live