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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. ^[ref].

For the closed, homogeneous SIR model with constant $\beta ,\gamma$, the instantaneous convention givesThus 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.

An epidemic is disease occurrence above the expected level for a specified population, place and period. The comparison needs a baseline: an increase in incidence alone does not establish an epidemic, and cases need not be increasing throughout an outbreak. For example, if a hypothetical community normally records two cases per week, a fall from twenty cases to eight still leaves occurrence above that baseline.^[ref]

An epidemic curve plots cases against time, commonly using symptom onset. A common-source outbreak reflects shared exposure, whereas a propagated outbreak reflects transmission between hosts; these mechanisms can also combine.^[ref]

In a closed homogeneous SIR model, let $s$ and $i$ be susceptible and infectious fractions, with transmission coefficient $\beta$ and recovery rate $\gamma >0$. The equations giveFor $i>0$, the infectious compartment grows when the effective reproduction number ${\mathcal{R}}_{0}s$ exceeds one and shrinks below one. This is a dynamical growth threshold, distinct from comparison with an observed baseline. An epidemic invasion threshold concerns growth after a small introduction; a final size relation for an epidemic concerns total susceptible depletion. An endemic equilibrium instead has continuing transmission balanced by the population's other processes.

A propagated outbreak spreads through transmission from infected hosts to further hosts, producing successive infection generations. The transmission route can involve direct contact, a shared vehicle or a vector. Separated peaks in an epidemic curve can be suggestive, but overlapping transmission generations need not give visibly separated waves.^[ref]

For an elementary branching approximation, let $I_k$ count infections in generation $k$ and suppose each infectious host generates an average $R$ new infections while susceptibility remains effectively unchanged. Then $\mathbb{E}[I_{k+1}\mid I_k]=R{I}_k$ and $\mathbb{E}{I}_k=R^k{I}_0$. Starting from one infection with $R=2$ gives expected generation sizes $1,2,4,8$. This is a mean calculation, not a guarantee that every introduction survives: random transmission can produce extinction.

The generation interval controls the timing between infection generations. Depletion of susceptible hosts, changing contacts or interventions can change the effective reproduction number, invalidating the constant-$R$ approximation. A common-source outbreak can seed this mechanism, so distinguishing the initial source from later transmission is part of reconstructing the outbreak.