Signature
R_t = R_0 * x_t
| Inputs | Definition | Unit |
|---|---|---|
R_0 | Basic reproduction number of the infection in a fully susceptible population | cases per case |
x_t | Proportion of the population that can still be infected, at least 0 and at most 1 | proportion |
R_t | Expected number of new infections caused by one infectious person when only the proportion x_t of the population is susceptible | 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
Effective reproduction number from R_0 and the susceptible share in named cells
With R_0 in BasicRepNumber and the susceptible proportion in SusceptibleShare, the first cell returns R_t, held in a cell named EffRepNumber. The second returns the susceptible proportion left by an all-or-nothing vaccine with coverage in Coverage and effectiveness in VaccineEff.
=BasicRepNumber*SusceptibleShare; =1-Coverage*VaccineEff
Assumptions
Homogeneous mixing and few infectious people for the effective reproduction number
Everyone meets everyone else at the same rate, and the infectious proportion is negligible apart from the one infectious person introduced, the conditions under which the WHO guide states R_t = R_0 x_t. In a structured population the susceptible share of each group scales the matching row of the next-generation matrix of HE-FM-BRN-005 instead.
Immune people fully protected for the effective reproduction number
People counted as immune cannot be infected or transmit, and people counted as susceptible are fully susceptible, as with an all-or-nothing vaccine. A leaky vaccine that partly reduces each vaccinated person's risk does not divide the population into immune and susceptible people, so x_t then has to be read as average relative susceptibility, and the later course of the epidemic differs from the all-or-nothing case.
Worked examples
Article's effective reproduction number at 50% coverage with a 90% effective vaccine
In the article's example, 50% coverage with a vaccine that fully protects 90% of those vaccinated leaves 55% of the population susceptible, so R_t is 3 times 0.55, or 1.65, and an epidemic can still occur.
R_0 = 3; x_t = 0.55; R_t = 1.65
Article's effective reproduction number at 80% coverage
At 80% coverage the protected share is 72% and the susceptible share 28%, so R_t is 0.84, below 1, and a major epidemic cannot occur in the article's dynamic model.
R_0 = 3; x_t = 0.28; R_t = 0.84
Effective reproduction number equal to R_0 in a fully susceptible population
With x_t of 1 the effective and basic reproduction numbers coincide, a limiting case.
R_0 = 3; x_t = 1; R_t = 3
Common errors
Current reproduction number quoted as the basic reproduction number
An estimate of R_t made after immunity has built up understates R_0. Reading the article's R_t of 1.65 as R_0 gives a herd immunity threshold of 1 minus 1/1.65, about 0.394, instead of 0.667 (computed here for illustration), so the coverage target would be set far too low.
Sources
WHO guide on the effective reproduction number and vaccination
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 defines the effective reproduction number R_t as the number of secondary cases a single infectious individual causes on average at time t and gives R_t = R_0 x_t, assuming a negligible proportion of the population is infectious, and section 6.2, which states that vaccination reduces the proportion of susceptibles x_t while R_0 remains constant.
Pitman and colleagues on the effective reproduction number
Pitman R, Fisman D, Zaric GS, Postma M, Kretzschmar M, Edmunds J, Brisson M. Dynamic transmission modeling. A report of the ISPOR-SMDM Modeling Good Research Practices Task Force-5. Value in Health. 2012;15(6):828-834. Section on the basic reproduction number, which defines R_0 as the average number of secondary infections generated by a typical case in a fully susceptible population and the effective reproduction number as R_0 multiplied by the susceptible fraction of the population s(t), with a value of 1 giving a threshold for invasion.
Canonical Identity
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