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The first subscript in ${\beta }_{t,\tau }$ is the calendar time $t$, and the second is the infection age $\tau$, the time elapsed since the source individual became infected. Thus ${\beta }_{t,\tau }$ is the rate at which an individual of infection age $\tau$ generates infections at time $t$. The corresponding discrete infectious disease renewal equation isup to a separately modelled term for imported infection. The upper limit may instead be a fixed maximal infectious age, with unavailable terms set to zero.

If incidence grows exponentially, ${\Delta }_t=C{e}^{\rho t}$, substitution in the infectious disease renewal equation gives the discrete Euler-Lotka equationFor $X\sim \mathrm{Geometric}(p)$ on $\{0,1,\dots \}$, $g_i=\mathbb{P}(X=i-1)=p(1-p{)}^{i-1}$. The finite geometric series therefore yieldsand consequently

Assume the infectivity profile is separable:Then $g_{\tau }$ is the discretized generation-interval distribution, $R_t$ is the instantaneous reproduction number, and the infectious disease renewal equation becomesHence $R_t={\Delta }_t/{\Lambda }_t$ whenever the total infectiousness ${\Lambda }_t$ is positive.