Functions & Formulae

Each applied formula has its own function page, with a signature, implementations, and tests.

Case fatality rate estimation, outbreak bias and population mortality function

CFR = f(D, N_c); CFR_r = g(D, R); B = h(d, T_d); M = AR * CFR

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.

  • Case fatality rate among confirmed cases

    CFR = D / N_c

    Divides the deaths from the disease among the counted cases by the number of cases that meet the case definition. The numerator is restricted to deaths among the cases in the denominator, so the CFR is a proportion rather than a true rate. It applies over the period in which those deaths were counted, which a precise definition states, for example a one-month CFR.

  • Case fatality rate among resolved cases

    CFR_r = D / (D + R)

    Restricts the calculation to detected cases whose outcome is known, dividing deaths by deaths plus recoveries. The WHO scientific brief on estimating mortality from COVID-19 sets it out as a simple response to the delay between becoming a case and dying, but the estimate is still biased when deaths and recoveries happen at different speeds.

  • Understatement factor of a crude case fatality rate during exponential growth

    B = 2^(d / T_d)

    Gives the approximate factor by which cumulative deaths divided by cumulative cases understates the case fatality rate when an epidemic grows exponentially and deaths are reported a fixed delay after cases. The exponent is the number of epidemic doubling times that pass between case reporting and death reporting, a heuristic given by Lipsitch and colleagues.

  • Population mortality from attack rate and case fatality rate

    M = AR * CFR

    Links the case fatality rate to the proportion of a closed population that dies of the disease over an outbreak. Deaths over the population equal cases over the population multiplied by deaths over cases, so the identity holds only when both measures share the case definition, the period and the population. The result is a cumulative incidence of death for the period, not a rate per person-year. The attack rate is computed with HE-FM-ATR-001.