Signature
D = 1 / gamma; R_0 = beta * D
| Inputs | Definition | Unit |
|---|---|---|
gamma | Rate at which infectious people recover and stop transmitting, above zero | per day |
beta | Rate at which one infectious person causes new infections when everyone else is susceptible, the product p times c of the WHO guide | new infections per infectious person per day |
D | Mean time a person stays infectious, the inverse of the recovery rate | days, or the time unit of gamma |
|---|---|---|
R_0 | Expected number of new infections caused by one infectious person over the whole infectious period in a fully susceptible population | cases per case, no time unit |
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
SIR basic reproduction number from named rate cells
With the transmission rate in TransRate and the recovery rate in RecoveryRate, the first cell returns the mean infectious period, named InfectiousPeriod, and the second returns R_0 in a cell named BasicRepNumber. Names such as R0 or R1 are not used because Excel reads them as cell references.
=1/RecoveryRate; =TransRate*InfectiousPeriod
Assumptions
Closed homogeneously mixing SIR population behind beta over gamma
The population is closed over the outbreak, mixes homogeneously and gains lasting immunity after recovery, as in section 2.1 of van den Driessche. Van den Driessche writes transmission by mass action, which gives R_0 = beta S_0/gamma; with beta defined per infectious person in a fully susceptible population, as in the article, S_0 is absorbed into beta and the formula reduces to beta/gamma. Where births and natural deaths matter the denominator becomes d + gamma, the form given in the same section.
Constant recovery rate implying an exponential infectious period
A constant recovery rate gamma means the time infectious is exponentially distributed with mean 1/gamma, so D is a mean and individual durations vary. The probability of recovering within a model cycle of length t is 1 minus exp(minus gamma t) by the rate conversion of HE-FN-TP-001, not gamma times t. Beta and gamma are in the same time unit, so their ratio has none, and Delamater and colleagues describe R_0 as cases per case, not a rate.
Worked examples
Article's illustrative infection with beta of 0.30 and gamma of 0.10 per day
An infection that causes 0.30 new infections per infectious person per day and has a recovery rate of 0.10 per day has a mean infectious period of 10 days and R_0 of 3, the illustrative values of the article's worked example.
gamma = 0.10; beta = 0.30; D = 10; R_0 = 3
SIR basic reproduction number with an influenza-like infectious period of five days
Van den Driessche gives a mean infectious time of about 5 days for influenza, a recovery rate of 0.2 per day. With an illustrative transmission rate of 0.26 per day, R_0 is 1.3.
gamma = 0.2; beta = 0.26; D = 5; R_0 = 1.3
SIR basic reproduction number at the threshold when beta equals gamma
With transmission and recovery both at 0.25 per day each case replaces itself exactly once and R_0 is 1, the boundary at which van den Driessche's sharp threshold sits, a limiting case for checking an implementation.
gamma = 0.25; beta = 0.25; D = 4; R_0 = 1
Common errors
Infectious period of 10 days entered as the recovery rate
Typing 10 days into the recovery rate cell gives R_0 = 0.3/10 = 0.03 instead of 3 for the article's infection, which would show an epidemic-prone infection as one that dies out at once.
Same basic reproduction number read as the same speed of spread
R_0 carries no time unit. Two infections with R_0 of 3, one with a 10-day and one with a 3-day mean infectious period (beta of 0.30 and 1.0 per day), have early growth rates of 0.2 and about 0.667 per day by the SIR relation r = gamma(R_0 minus 1), and doubling times of about 3.47 and 1.04 days (computed here for illustration). Delamater and colleagues stress that R_0 does not show whether secondary cases appear within a day or months later.
Sources
van den Driessche on R_0 of the SIR model and the threshold at one
van den Driessche P. Reproduction numbers of infectious disease models. Infectious Disease Modelling. 2017;2(3):288-303. Section 2.1, which defines R_0 = beta S_0/gamma for the SIR model as the product of the transmission rate, the mean infectious time 1/gamma and S_0, notes that R_0 is independent of the fraction dying from the disease, states that R_0 = 1 acts as a sharp threshold between the disease dying out and causing an epidemic, and gives R_0 = beta S_0/(d + gamma) once births and natural deaths are included.
WHO guide on R_0 as transmission probability times contact rate times duration
World Health Organization. WHO guide for standardization of economic evaluations of immunization programmes, 2nd edition (WHO/IVB/19.10). Geneva, WHO, 2019. Section 6.1.1, which writes R_0 = pcD for a homogeneously mixing population without immigration or emigration, with p the probability of transmission following exposure, c the average number of new contacts per unit of time and D the duration of infectiousness, and states that the reproduction number depends on infectious disease characteristics (p and D) and population characteristics (c and x).
Delamater and colleagues on R_0 as cases per case
Delamater PL, Street EJ, Leslie TF, Yang YT, Jacobsen KH. Complexity of the basic reproduction number (R0). Emerging Infectious Diseases. 2019;25(1):1-4. The abstract, which states that R_0 is not a biological constant for a pathogen, a rate over time or a measure of disease severity, and the section Various Names for R0, which states that R_0 does not indicate whether new cases occur within 24 hours of the initial case or months later and is most accurately described in terms of cases per case.
Canonical Identity
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