The renewal equation and the time-varying reproduction number R(t)

The renewal equation

New infections are made by old infections — scaled by R(t)

Infections are not independent. Each new case was infected by someone who is already infectious. The renewal equation writes this down directly: the expected number of new infections on day t is the reproduction number times a weighted sum of recent infections, where the weights are the generation interval g(s) — the distribution of times between an infector being infected and them infecting someone else:

I(t) = R(t) · Σ_s=1^∞ I(t − s) · g(s)

Read it piece by piece:

R(t) — the average number of secondary infections per infectious person at time t. It is allowed to change as susceptibility, behaviour, policy and seasonality change.
g(s) — the generation interval. It says how a person infected today spreads their onward infections across the coming days. We set g(0) = 0: nobody infects others on the very same day.
The sum is a convolution of the infection time series with itself — the renewal equation is the convolution from the previous session, but feeding back on the same series being generated.

The headline. Because g(s) sums to 1, on average each infectious person produces R(t) others. So R(t) > 1 → growth, R(t) < 1 → decline, and the epidemic peaks at the moment R(t) crosses 1. This is the single most useful fact about R(t), and the simulator below makes it visible.

This is more general than the basic reproduction number R₀ (secondary infections in a fully susceptible population) and includes the simple SIR model as a special case — there R(t) = R₀ S/N, falling as susceptibles are depleted.

The renewal step, as a self-convolution slide · multiply · sum

Today's cases = R × a weighted sum of recent cases

The renewal equation is the same convolution as turning infections into onsets — except the series convolves with itself: each day's cases are built from the recent days' cases, weighted by the generation interval g(s) and scaled by R. Step a day forward and watch its bar get assembled from the ones before it.

I(t) = R · Σ_{s ≥ 1} I(t − s) g(s)

incidence I(t) generation-interval weight g(s) the day being built

—

Reproduction number R: 1.30 Generation interval mean: 5.0 Day being built t: 20

Because the output feeds straight back into the input, a constant R compounds into exponential growth — and the generation interval sets how fast each turn of the sum (one "generation") comes round. This self-convolution is the engine behind every R(t) figure on this page.

A step in R, and the growth rate it produces the centrepiece

R says how many; the generation interval says how fast

Apply a step change to R(t) — from an initial R₀ down or up to a later R₁ on a chosen day (think of an intervention, with an optional smooth transition). The epidemic is seeded with a handful of infections, then the renewal recursion I(t) = R(t)·Σ_s I(t−s) g(s) runs forward for 120 days using the generation interval set right here.

Three panels share the day axis. The top is R(t), with a slate dashed line at R = 1. The middle is incidence I(t) (toggle a log axis to see the straight-line exponential phases). The bottom is the realised per-day growth rate r(t) = ln(I(t)/I(t−1)). The two purple dashed lines mark the equilibrium growth rates r₀ and r₁ that R₀ and R₁ imply through the Euler–Lotka relation. Watch the key thing: when R steps, r(t) does not jump — it glides from r₀ to r₁ over roughly one generation interval.

R(t) R = 1 / r = 0 incidence I(t) realised r(t) equilibrium r₀, r₁

Generation interval mean (days): 5.0 Generation interval SD (days): 2.0 Initial R₀: 2.0 Step on day: 40 Later R₁: 0.70 Transition width (days): 2 log incidence axis

equilibrium r before = —/day equilibrium r after = —/day doubling time after = — r reaches r₁ ~— days after the step

The key point. The realised growth rate lags the R step by about one generation interval — it cannot turn over instantly because today's infections were seeded by a whole cohort of past cases. Shorten the generation-interval mean and the same R step produces a larger |r| and a faster response; lengthen it and r is smaller and slower to settle. R fixes the destination; the generation interval sets the speed and the journey.

The R ↔ growth-rate map the link

One R, many growth rates — the generation interval decides which

R and the per-day growth rate r are two views of the same exponential phase, related through the generation interval by the Euler–Lotka equation. For a constant R the asymptotic growth rate solves

1R = Σ_s g(s) e^{−r s} ⇒ R = 1Σ_s g(s) e^{−r s} doubling time = ln 2r

The right-hand side decreases monotonically in r, so each R maps to exactly one growth rate (and back). The curve below plots that map for the current generation interval. The two dots are the operating points from the step above: (R₀, r₀) and (R₁, r₁). Now slide the generation-interval mean: the whole curve shifts, so the same R slides to a different r — and the same observed r implies a different R. R alone does not pin down the growth rate.

Euler–Lotka map for current GI map at a different GI operating points (R₀,r₀) & (R₁,r₁)

Generation interval mean (days): 5.0 Compare against GI mean (days): 3.0 Read off at R: 1.50

at R = —, current GI gives r = —/day — same R at compare GI gives r = —/day

Why the generation interval matters. R only counts how many people each case infects, not how fast. With a short generation interval those secondary infections arrive sooner, so the same R compounds more quickly: a higher growth rate r and a shorter doubling time. Slide the generation-interval mean down and watch the curve steepen and the operating points climb, even though R has not moved.

Takeaways

The renewal equation I(t) = R(t)·Σ_s I(t−s) g(s) generates incidence from the reproduction number and the generation interval — a convolution of the infection series with itself.
R(t) > 1 means growth, R(t) < 1 means decline, and the epidemic peaks where R(t) crosses 1 (give or take one generation's lag).
The generation interval sets the timescale: it links R to the exponential growth rate r and the doubling time. The same R grows faster with a shorter interval.
R(t) is rarely observed directly. In the course you instead estimate it: invert the renewal equation in Stan, place priors on R(t) (e.g. a geometric random walk for smoothness), and propagate uncertainty — see the session.
Because this version is deterministic, the equation gives the expectation of I(t); fitting to data adds observation noise (e.g. Poisson) on top.