= Effective reproduction number
{title2=$R_{\mathrm{eff}}(t)$}
An <effective reproduction number> describes transmission in the current population, accounting for available susceptibility and transmission conditions. The phrase is used with different time conventions, so a definition must specify whether it means an <instantaneous reproduction number> or a <case reproduction number>. The CDC uses a time-varying effective quantity $R_t$ to describe the direction of community transmission. https://www.cdc.gov/cfa-modeling-and-forecasting/modeling-handbook/mh-rt.html[CDC: time-varying effective reproduction number]{ref}.
For the closed, homogeneous <SIR model> with constant $\beta,\gamma$, the instantaneous convention gives
$$
R_{\mathrm{eff}}(t)=\frac{\beta}{\gamma}s(t)=R_0s(t),\qquad
\frac{di}{dt}=\gamma\bigl(R_{\mathrm{eff}}(t)-1\bigr)i(t).
$$
Thus the <infectious compartment> grows when $R_{\mathrm{eff}}>1$ and declines when $R_{\mathrm{eff}}<1$, provided $i>0$. At the <SIR infection peak> it equals one. This freezes the current susceptible <fraction>; it does not count a newly infected person's actual future secondary <infections> while $s$ continues to decrease. The distinction comes directly from the <SIR model equations> and the two time conventions above.
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