Signature
H = 1 - 1 / R_0; V_c = H / E
| Inputs | Definition | Unit |
|---|---|---|
R_0 | Basic reproduction number, above 1 for the threshold to be positive | cases per case |
E | Proportion of vaccinated people fully protected against infection, above zero and at most 1, for example from attack rates as in HE-FM-ATR-004 | proportion |
H | Proportion of the population that must be immune for each case to infect fewer than one other person on average | proportion |
|---|---|---|
V_c | Proportion of the population that must be vaccinated to reach H; a value above 1 means that vaccine alone cannot reach the target | proportion |
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
Herd immunity threshold and critical coverage from named cells
With R_0 in BasicRepNumber and effectiveness in VaccineEff, the first cell returns H, named HerdThreshold, and the second returns V_c, named CriticalCoverage. The third flags a target that a vaccine of that effectiveness cannot reach.
=1-1/BasicRepNumber; =HerdThreshold/VaccineEff; =IF(CriticalCoverage>1,"not reachable","reachable")
Assumptions
Homogeneous mixing and an R_0 above one for the herd immunity threshold
The formula assumes homogeneous mixing and R_0 above 1; for R_0 of 1 or less H is zero or negative and no vaccination is needed to prevent sustained spread. The WHO guide reports that crude estimates of R_0 and of the critical proportion are overestimates if transmission is greatest among young people and declines with age, and underestimates if transmission increases with age, so a structured model is used where mixing is uneven.
Lasting all-or-nothing protection given before exposure for the critical coverage
Each vaccinated person is either fully and lastingly protected, with probability E, or not protected at all, and is vaccinated before exposure. The WHO guide notes that the simple formula implicitly assumes vaccination at birth or as soon as maternal immunity has waned, and that actual coverage must exceed the critical effective immunisation level because protection is usually imperfect. E is effectiveness against infection, since protection against disease alone does not stop a vaccinated person from transmitting.
Worked examples
Article's herd immunity threshold and critical coverage for R_0 of 3 and a 90% effective vaccine
With R_0 of 3, two thirds of the population must be immune, and a vaccine that fully protects 90% of those vaccinated needs coverage of about 74.1%, as in step 2 of the article's worked example.
R_0 = 3; E = 0.9; H = 0.6667; V_c = 0.7407
Masterson and colleagues' critical coverage for R_0 of 4 and a 90% effective vaccine
Masterson and colleagues report that at R_0 of 4 the herd immunity threshold is 75% and that more than 80% of a population needs vaccinating with a vaccine of 90% effectiveness; the formula gives about 83.3%.
R_0 = 4; E = 0.9; H = 0.75; V_c = 0.8333
Measles critical proportion at the lower WHO estimate of R_0
With R_0 of 9.3, the lower end of the measles range in Table 7 of the WHO guide, and effectiveness set to 1 so that V_c is the critical effective immunisation level, both values are about 0.892, which the table rounds to 89%.
R_0 = 9.3; E = 1; H = 0.8925; V_c = 0.8925
Measles at the upper WHO estimate of R_0 with a 90% effective vaccine
At R_0 of 15.6 the threshold is about 0.936, the 94% of Table 7, and a vaccine that protects 90% of recipients would need coverage of about 104% (computed here for illustration), so that vaccine alone cannot reach the threshold in this simple model.
R_0 = 15.6; E = 0.9; H = 0.9359; V_c = 1.0399
Common errors
Herd immunity threshold used as the coverage target for an imperfect vaccine
Vaccinating 66.7% of the population with a 90% effective vaccine immunises 60%, leaving R_t of 3 times 0.4, or 1.2, so spread continues; the article's critical coverage is 74.1%.
Herd immunity threshold read as the share that will be infected
The threshold marks where incidence starts to fall, not where the epidemic ends. With no programme an SIR epidemic with R_0 of 3 infects about 94% of the population (the article's final size, computed as in HE-CF-ATR-001), well beyond the threshold of 66.7%, because infections continue while the epidemic declines.
Sources
WHO guide on the herd immunity threshold and critical effective immunisation
World Health Organization. WHO guide for standardization of economic evaluations of immunization programmes, 2nd edition (WHO/IVB/19.10). Geneva, WHO, 2019. Section 6.2, which gives the threshold susceptible proportion x* = 1/R_0, the herd immunity threshold H = 1 minus x* = 1 minus 1/R_0 and the critical effective immunisation level p_c of at least 1 minus 1/R_0, states that actual coverage should be greater because the protective efficacy of a vaccine is usually imperfect and it is not always given at birth, gives measles R_0 of 9.3 to 15.6 with a critical proportion of 89 to 94% in Table 7, and reports that crude estimates are overestimates or underestimates under age-dependent transmission.
van den Driessche on herd immunity with a perfect vaccine
van den Driessche P. Reproduction numbers of infectious disease models. Infectious Disease Modelling. 2017;2(3):288-303. Section 2.1, which states that if a perfect vaccine is given to a fraction p of the population the disease will not spread if (1 minus p) R_0 is below 1, that is if p exceeds 1 minus 1/R_0, giving herd immunity.
Pitman and colleagues on coverage above the herd immunity threshold
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 states that in a homogeneously mixing population, for herd immunity to occur, vaccine coverage V_c has to be greater than 1 minus 1/R_0.
Masterson and colleagues on critical coverage divided by vaccine effectiveness
Masterson SG, Lobel L, Carroll MW, Wass MN, Michaelis M. Herd immunity to Ebolaviruses is not a realistic target for current vaccination strategies. Frontiers in Immunology. 2018;9:1025. Materials and methods, equations 1 and 2, which compute the herd immunity threshold I_c = 1 minus 1/R_0 and the critical vaccination coverage V_c = I_c/E with E the vaccine effectiveness, citing Anderson 1992 and Fine, Eames and Heymann 2011 among others, and the abstract, which reports a threshold of 75% at R_0 of 4 and more than 80% coverage needed at an effectiveness of 90%.
Canonical Identity
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