Generation-interval distribution

The generation-interval distribution gives the probability law of the generation interval. In discrete time, its probabilities $g_{\tau }$ form a nonnegative sequence summing to one.

The infectious disease renewal equation expresses current incidence as a convolution of past incidence with an infectivity profile, multiplied by a time-varying transmission level.

For incidence $I_t$ and discrete generation-interval probabilities $g_{\tau }$, the total infectiousness at time $t$ is ${\Lambda }_t={\sum }_{\tau \ge 1}{g}_{\tau }{I}_{t-\tau }$.

A time-varying reproduction number describes transmission at a specified calendar time while allowing transmission conditions to change during an epidemic.

The instantaneous reproduction number freezes the transmission conditions at calendar time $t$ and counts the expected secondary infections produced over a complete infectious lifetime under those conditions.

The case reproduction number is the expected number of secondary infections actually generated by a person infected at time $t$, allowing transmission conditions to change during that person's infectious lifetime. It follows an infection cohort through the subsequent conditions it experiences. The instantaneous reproduction number instead freezes conditions at the reference time. Cori and colleagues distinguish these two constructions in their framework for time-varying reproduction numbers. ^[ref].

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.