Convolutions — from infections to observations

The idea in one equation

Population-level delays turn into a convolution of two time series

Work in discrete daily time. Let I(t) be the number of latent infections on day t, and let f(s) be the delay distribution — the probability that an infected person is observed (e.g. develops symptoms) exactly s days later. An infection on day t−s shows up on day t with probability f(s). Sum those contributions over every earlier day and you get the observed count C(t):

C(t) = Σ_s=0^∞ I(t − s) · f(s)

That single sum is the convolution, written C = I ∗ f. Read it as a recipe: for each output day t, walk back over the infection history, weight each past day by how likely its infections are to surface today, and add. Two things happen as a result, and they are the whole story of this page:

Shift. Because every contribution arrives late, the observed curve is pushed later in time — by roughly the mean of the delay distribution.
Smear. Because different individuals take different numbers of days, a sharp pulse of infections is spread out into a broader bump — by roughly the spread (standard deviation) of the delay.

A clean spike of infections never produces a clean spike of onsets. The delay distribution acts like a blur filter sliding along the time axis. The wider the delay, the later and flatter the curve you observe — which is exactly why nowcasting the "true" recent signal from delayed reports is hard.

Symbol	Meaning	In the figures
I(t)	latent incidence (infections) on day t	blue filled bars
f(s)	discretised delay distribution (a PMF over days)	teal kernel weights
C(t)	observed counts (onsets / reports) on day t	orange bold line

The delay, as a daily PMF f(s)

A continuous delay distribution, discretised into per-day probabilities

The incubation period is naturally a continuous distribution, but our counts are tallied by day. We therefore discretise it into a probability mass function f(s), one bar per whole day of delay. Here we use a lognormal delay with a chosen mean m and standard deviation s_sd. Match the moments, then take daily differences of the CDF:

σ² = log(1 + s_sd² / m²) μ = log(m) − σ²/2
F(x) = Φ( (log x − μ) / σ ) f(s) = F(s+1) − F(s)

The bars are then renormalised to sum to 1 over the days we keep (0 … S_max). Slide the mean and the spread and watch the shape: a small mean piles probability on the first day or two; a large spread flattens it into a long right tail. This f(s) is the kernel that does all the shifting and smearing in the next figures.

delay PMF f(s) mean delay m

Delay mean m (days): 5.0 Delay sd s_sd (days): 2.0

mode ≈ day — P(delay = 0) = — P(delay ≤ mean) = —

Convolution builder forward process

Feed an infection curve through the delay and watch the onsets appear

Here is the whole forward process at a glance. The blue filled bars are a latent infection curve I(t) — control its peak day and width, and add an optional second wave. The orange line is the resulting onset curve C(t) after convolving with the same delay PMF as above. Move any slider and watch the orange curve slide right (by about the delay mean) and flatten out (by about the delay spread).

infections I(t) onsets C(t) peak of I vs peak of C

Infection peak day: 14 Infection width: 3.5 Second wave size: 0.0 Delay mean m: 5.0 Delay sd s_sd: 2.0

peak of I ≈ day — peak of C ≈ day — shift ≈ — days

The shift between the two peaks tracks the mean of the delay; the broadening tracks its spread. Set the delay spread small and the orange curve keeps the shape of the blue one, just translated. Crank the spread up and even a tall, narrow outbreak is observed as a low, lazy hump.

Slide, multiply, sum how C(t) is built

The convolution sum, one output day at a time

This is the heart of it. To get the single observed value C(t*) on one output day t*, the convolution does three things:

Slide. Take the delay PMF, flip it, and lay it over the infection history ending at t*. The bar sitting on infection day τ carries the weight f(t* − τ) — recent infections get the small-delay weights, older ones the long-delay tail.
Multiply. On each day, multiply the infections by that weight: I(τ) · f(t* − τ). Those are the teal products.
Sum. Add the products. That single total is the height of the orange point raised at t*.

Drag t* across the axis — or press ▶ play — and watch the flipped kernel sweep left to right, its weighted products rise and fall, and their running sum trace out the entire onset curve, point by point.

infections I(τ) weighted product I(τ)·f(t*−τ) onsets C(t) so far current sum C(t*)

Output day t*: 10 Delay mean m: 5.0 Delay sd s_sd: 2.0

C(t*) = Σ I(τ)·f(t*−τ) = —

largest single contribution: day —

Notice that the biggest contribution to today's onsets usually comes not from today's infections but from those around m days ago — wherever the flipped kernel's peak lands on the infection curve. That lag is the shift you saw in the builder, now visible term by term.

Forward and back deconvolution

Why inference is the convolution run in reverse

Everything above is the forward direction: given infections and a delay, produce the observations. In practice we have the opposite problem — we see the onsets C(t) and want the infections I(t) that we never observed directly. Recovering I from C and f is deconvolution, and the course fits it as a Bayesian model.

It is genuinely harder than it looks. The smearing throws away fine detail: many slightly different infection curves convolve to almost the same onset curve, so the inverse is ill-posed and noisy. Sharp features in I become unrecoverable once blurred.
The delay anchors recent days. Because today's onsets mostly reflect infections from m days ago, the most recent infections are barely represented in the data yet — the deep reason nowcasts of the latest days are so uncertain.
Observation noise sits on top. The convolution is a smooth, deterministic operation; real counts are noisy. The session adds a Poisson layer, C^obs(t) ∼ Poisson(C(t)), so the model must separate genuine signal from sampling jitter.

The forward convolution is the easy, honest part of the model — a clear statement of how delays generate the data. The hard, interesting work is inverting it well: estimating the latent infection curve, with calibrated uncertainty, from the shifted and smeared signal you actually get to see.

	Forward (this page)	Inverse (inference)
You know	I(t) and f(s)	C(t) and f(s)
You want	C(t)	I(t)
Operation	convolution I ∗ f	deconvolution
Difficulty	one direct sum	ill-posed; needs priors & MCMC
Uncertainty	none (deterministic)	large, esp. for recent days