Signature
B = 2^(d / T_d)
| Inputs | Definition | Unit |
|---|---|---|
d | Delay between a case being reported and its death being reported | time, in the same unit as T_d |
T_d | Time taken for the number of cases to double during exponential growth | time, in the same unit as d, above zero |
B | Approximate factor by which the crude case fatality rate understates the case fatality rate | ratio, 1 when there is no delay |
|---|
Function
Case fatality rate estimation, outbreak bias and population mortality function
Maps counts of deaths, recoveries and cases to the case fatality rate (CFR), the proportion of cases meeting a case definition who die of the disease over a stated period, and relates it to the delay bias of crude estimates during exponential growth and to population mortality through the attack rate. The records follow the notation of the Case Fatality Rate article. Matching the period of a CFR to a model cycle uses HE-FM-TP-003, and the attack rate itself is HE-FM-ATR-001.
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Implementations
Excel
Understatement factor from delay and doubling time
Excel raises 2 to the ratio of the named cells holding the reporting delay and the doubling time, both in the same unit.
=2^(ReportDelay/DoublingTime)
Assumptions
Constant exponential growth of reported cases
Reported cases grow exponentially with a fixed doubling time over the delay. As growth slows or the epidemic declines, the factor no longer describes the bias.
One fixed delay from case report to death report
Every death is reported the same time after its case. Real delays vary from case to case, so the factor indicates the size of the bias rather than giving an exact correction.
Worked examples
Three-week delay with a two-week doubling time
In Lipsitch and colleagues' example a three-week delay and a two-week doubling time give 2 raised to the power 1.5, about 2.8, so the crude estimate is close to one third of the case fatality rate.
d = 3; T_d = 2; B = 2.83
Delay equal to one doubling time
When deaths are reported one doubling time after their cases, the crude estimate is about half the case fatality rate (computed here for illustration).
d = 2; T_d = 2; B = 2
Common errors
Delay and doubling time in different units
Entering the three-week delay as 21 days with a doubling time of 2 weeks gives 2 raised to the power 10.5, about 1,448, instead of about 2.8 (computed here for illustration). Both times are entered in days or both in weeks.
Sources
Delay heuristic for the crude case fatality rate in a growing epidemic
Lipsitch M, Donnelly CA, Fraser C, Blake IM, Cori A, Dorigatti I, et al. Potential biases in estimating absolute and relative case-fatality risks during outbreaks. PLoS Neglected Tropical Diseases. 2015;9(7):e0003846. Figure 1 legend: with a three-week reporting delay and a two-week doubling time the crude CFR is underestimated by a factor of about 2 to the power 3/2, or 2.8, the exponent being the number of doubling times between case and death reporting.
Canonical Identity
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