Signature
T_c = T_E + T_I; R_exp = 1 + r * T_c; R_seir = (1 + r * T_E) * (1 + r * T_I); R_delta = exp(r * T_c)
| Inputs | Definition | Unit |
|---|---|---|
T_E | Mean time from infection until a person becomes infectious, 0 for an SIR infection | days |
T_I | Mean time a person stays infectious, 1/gamma in the SIR model | days |
r | Rate at which new infections grow in the early phase, log(2) divided by the doubling time | per day |
T_c | Mean time between infection of a case and infection of the people it infects, here the sum of the mean latent and infectious periods | days |
|---|---|---|
R_exp | Reproduction number implied by r when the generation interval is exponentially distributed with mean T_c | cases per case |
R_seir | Reproduction number implied by r when one exponential latent stage with mean T_E is followed by one exponential infectious stage with mean T_I | cases per case |
R_delta | Reproduction number implied by r when every generation interval equals T_c, the upper bound for that mean | cases per case |
Function
Basic reproduction number function from transmission parameters or a next-generation matrix
Maps the transmission parameters of an infection in a fully susceptible population, either a transmission rate and a recovery rate or a next-generation matrix of expected new cases by group, to the basic reproduction number R_0, a count of cases per case with no time unit. The same package derives the effective reproduction number once part of the population is immune, the herd immunity threshold with the critical vaccination coverage, and the reproduction number implied by an early exponential growth rate. R_0 estimated after a completed epidemic from its final attack rate is HE-FM-ATR-005, and the final attack rate and infections averted for a given R_0 and pre-epidemic coverage are computed in HE-CF-ATR-001, so neither is restated here. Vaccine effectiveness from attack rates (HE-FM-ATR-004) supplies E, the epidemic doubling time used in HE-FM-CFR-003 converts to the growth rate r, a per-cycle recovery probability for a cohort model follows the rate conversion of HE-FN-TP-001, and the economic treatment of herd protection as an external benefit is HE-FM-EXT-002. Notation follows the Basic Reproduction Number article.
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Implementations
Excel
Reproduction number from the growth rate in named cells
With the growth rate in GrowthRate and the mean latent and infectious periods in LatentPeriod and InfectiousPeriod, the first three cells return the exponential, two-stage and fixed-interval estimates, held in RepExp, RepTwoStage and RepDelta. The fourth converts a doubling time in DoublingTime to the growth rate.
=1+GrowthRate*(LatentPeriod+InfectiousPeriod); =(1+GrowthRate*LatentPeriod)*(1+GrowthRate*InfectiousPeriod); =EXP(GrowthRate*(LatentPeriod+InfectiousPeriod)); =LN(2)/DoublingTime
Assumptions
Early exponential growth in a fully susceptible population for the growth-rate relation
The growth rate is measured while almost everyone is susceptible and new infections grow at a steady exponential rate, so the result is R_0; measured later it is an effective reproduction number. Diekmann, Heesterbeek and Roberts prove for compartmental models that R_0 is above 1 if and only if r is above zero and equals 1 if and only if r is zero, so the sign of r fixes the side of the threshold whatever the shape of the generation interval.
Generation interval shape chosen to match the natural history
Wallinga and Lipsitch show that the different equations differ only in the assumed shape of the generation interval distribution, so the shape is chosen from data on the infection, such as an observed generation interval distribution. The two-stage form takes the latent and infectious periods as independent and exponentially distributed, with constant infectiousness during the infectious stage, and r, T_E and T_I share one time unit.
Worked examples
Article's growth rate of 0.20 per day with a 10-day exponential generation interval
The article's infection grows at 0.20 per day with a 10-day mean infectious period and no latent stage, so the exponential and two-stage forms both return R_0 of 3, while a fixed 10-day interval would imply about 7.39, the upper bound in step 1 of the article.
r = 0.2; T_E = 0; T_I = 10; T_c = 10; R_exp = 3; R_seir = 3; R_delta = 7.389
Growth rate of 0.20 per day with an illustrative four-day latent and six-day infectious period
Splitting the same 10-day mean generation interval into a 4-day latent and a 6-day infectious stage (illustrative values) raises the two-stage estimate to 3.96, above the exponential value of 3 and below the fixed-interval bound of about 7.39, because the generation interval is less dispersed.
r = 0.2; T_E = 4; T_I = 6; T_c = 10; R_exp = 3; R_seir = 3.96; R_delta = 7.389
Zero growth rate returns a reproduction number of one for every interval shape
With no growth every form returns 1, the threshold value, whatever the generation interval, a limiting case.
r = 0; T_E = 4; T_I = 6; T_c = 10; R_exp = 1; R_seir = 1; R_delta = 1
Common errors
Infectious period used as the generation interval when there is a latent stage
Ignoring the 4-day latent stage of the illustrative example and using R = 1 plus r T_I with the 6-day infectious period gives 2.2 instead of 3.96 at a growth rate of 0.2 per day, which would put the herd immunity threshold at about 0.545 instead of 0.747 (computed here for illustration).
Sources
Wallinga and Lipsitch on growth rates and reproduction numbers
Wallinga J, Lipsitch M. How generation intervals shape the relationship between growth rates and reproductive numbers. Proceedings of the Royal Society B. 2007;274(1609):599-604. Section 2(b), equation 2.9, R = 1/M(minus r) with M the moment generating function of the generation interval distribution; section 3(a), equation 3.1, R = 1 plus r/b for an exponential distribution with 1/b the mean generation interval, and equation 3.2, R = (1 plus r/b_1)(1 plus r/b_2) for exponential latent and infectious periods with mean T_c = 1/b_1 plus 1/b_2; section 3(c), equation 3.5, R = exp(r T_c) for a delta distribution; and the abstract, which states that the equations differ only in the assumed shape of the generation interval distribution and that equal generation intervals give an upper bound.
van den Driessche on R_0 from the early growth rate of the SIR model
van den Driessche P. Reproduction numbers of infectious disease models. Infectious Disease Modelling. 2017;2(3):288-303. Section 2.1, which gives the early growth I(t) approximately equal to I(0) exp[gamma(R_0 minus 1)t], and section 9.6, which states that if the exponential growth rate r of the initial phase is observed then R_0 = 1 plus r/gamma, a formula based on the simple SIR model with limited applicability.
Diekmann, Heesterbeek and Roberts on the sign link between R_0 and the growth rate
Diekmann O, Heesterbeek JA, Roberts MG. The construction of next-generation matrices for compartmental epidemic models. Journal of the Royal Society Interface. 2010;7(47):873-885. Abstract and appendix A, which prove for compartmental systems that R_0, defined as the dominant eigenvalue of the next-generation matrix, and the real-time growth rate r of the early phase are connected so that R_0 is above 1 if and only if r is above 0, and R_0 = 1 if and only if r = 0.
Canonical Identity
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