Renewal Window RT

Estimate rolling renewal-equation Rt as window incidence divided by lag-weighted infectiousness; no uncertainty interval or imported-case correction is inferred.

Description

Calculate rolling renewal-equation reproduction ratios from daily incidence and normalized generation-lag weights.

A renewal model relates new infections to the infectiousness contributed by earlier infections. A generation-interval weight describes how much a previous day’s incidence contributes at a particular lag. Summing past incidence times those weights gives an infectiousness term against which current incidence can be compared.^[1]

A window estimate combines incidence and infectiousness over several days, treating the reproduction number as constant within that window. It is a ratio of the two sums, rather than an average of separate daily ratios. The result depends on the lag distribution and how closely the input incidence represents infection times.

Renewal Window RT produces point estimates with zero-based window-end indexes. It uses supplied daily data and weights, without fitting the generation distribution, reconstructing infections, or adding a Bayesian prior.

When to calculate rolling renewal ratios

Use the tool for a transparent renewal calculation with a chosen daily generation-lag distribution and sufficient incidence history. Keep the origin date and weight definition with the results so indexes can be mapped back to dates.

Compare rolling windows using the same lag weights.
Inspect the effect of window length on a point estimate.
Check a ratio-of-sums calculation without an uncertainty model.

How to enter daily incidence and calculate window estimates

Enter nonnegative Incidence counts in chronological daily order, including zeros for days with zero incidence.
Enter Generation weights from lag 1 onward: the first weight multiplies the preceding day, the second the day before that, and so on.
Use nonnegative weights no greater than one that sum to one. Enter positive integer Window days.
Run the tool and pair each Reproduction estimate with its zero-based Window end index.

Λ_t = sum_j(w_j × I_(t−j−1)); Rt_window = sum_window(I_t)/sum_window(Λ_t)

Here j starts at zero. The implementation first uses t = number of supplied lag weights, ensuring every lag has an observed incidence value.

The first output index is lagCount + windowDays − 1. Subsequent estimates advance one day at a time using rolling sums. The tool accepts noninteger nonnegative incidence values, but requires enough observations for a full lag history and window. It checks weight normalization within a small tolerance and rejects excessively large convolutions.

json

{
  "input": {
    "incidenceCounts": [
      10,
      10,
      20,
      20
    ],
    "generationWeights": [
      1
    ],
    "windowDays": 2
  },
  "output": {
    "windowEndIndexes": [
      2,
      3
    ],
    "reproductionEstimates": [
      1.5,
      1.3333333333333333
    ]
  }
}

Every reported window must have positive infectiousness. A zero denominator causes an input error for the calculation, rather than an infinite estimate or an automatically skipped window.

Limitations of renewal point estimates

No reporting-delay correction, imported-case separation, smoothing, prior, or uncertainty interval is inferred.
The series is interpreted as daily. Missing days cannot simply be removed without changing the lag meaning.
Weights start at lag one and omit same-day transmission. Early days lacking the complete supplied lag history are not estimated.
Negative serial intervals cannot be used as lag weights. The relationship between onset intervals and generation intervals needs separate consideration.

Alternative or Complementary reproduction and timing analyses

A growth-rate conversion uses a different summary and generation-distribution assumption. If paired infection and onset data are available, compare the timing distributions before selecting renewal weights. Full inference should address the uncertainty and observation process omitted here.

References

A New Framework and Software to Estimate Time-Varying Reproduction Numbers During Epidemics — American Journal of Epidemiology; A. Cori and colleagues (2013)

Similar or alternative tools

Growth R0 Gamma Generation
Convert a supplied growth rate using a gamma generation-interval mean and shape.
Generation Serial Interval Pairs
Compare infection-to-infection and onset-to-onset intervals in aligned transmission pairs.