Signature
pi = max(0, 1 - 1 / (R_0 * (1 - v))); MPB = L * pi; MEB = (v < 1 - 1 / R_0) * L / (R_0 * (1 - v))
| Inputs | Definition | Unit |
|---|---|---|
R_0 | Basic reproduction number of the infection, above 1 | cases per case |
v | Proportion of newborns vaccinated at birth at which the benefit of one more vaccination is read, at least 0 and below 1 | proportion |
L | Money value of the loss from one infection, the same for the family deciding and for society, above zero | currency per infection |
pi | Probability that an unvaccinated child is infected over its lifetime at coverage v, from HE-FM-VEXT-001 | probability |
|---|---|---|
MPB | Expected loss avoided by the vaccinated child, L times pi | currency per vaccination |
MEB | Loss avoided by unvaccinated children through one more vaccination, L/(R_0(1 minus v)) below the threshold and 0 at or above it | currency per vaccination |
Function
Endemic vaccination externality function from coverage at birth to infection risk, benefit split and voluntary uptake
Maps the basic reproduction number R_0 of an endemic infection, the share v of each birth cohort vaccinated at birth, the loss L per infection and the private cost c of one vaccination to the lifetime infection risk of an unvaccinated child and the infections per birth (HE-FM-VEXT-001), the private and external benefit of one more vaccination (HE-FM-VEXT-002) and the coverage that voluntary uptake reaches (HE-FM-VEXT-003), all in the long-run steady state of an SIR model with births and deaths. The marginal social benefit as marginal private plus marginal external benefit is HE-FM-EXT-001, the herd immunity threshold H = 1 minus 1/R_0 is HE-FM-BRN-003 and the effective reproduction number is HE-FM-BRN-002, so none of them is restated here; the Pigouvian vaccination subsidy with linear curves is HE-FM-EXT-002, with its bill and net welfare under HE-FN-PIGS-001. The article's social cost per 1,000 births, GBP 250,000 under voluntary uptake, GBP 212,500 under free provision and GBP 200,000 at the threshold, is 1,000 times the expected cost per birth at a chance node, HE-FM-CHN-001, with a vaccinated branch of probability v costing the full resource cost of a vaccination whoever pays it and an unvaccinated branch of probability 1 minus v costing the expected loss L times pi from HE-FM-VEXT-001, so it is not drafted as a formula. The lifetime chance lambda/(lambda + mu) of infection before death is the long-horizon limit of the competing-hazards record HE-FM-TP-004, the final size of a closed epidemic is HE-CF-ATR-001 and vaccine effectiveness from attack rates is HE-FM-ATR-004. Notation follows the Vaccination Externalities article.
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Implementations
Excel
Private and external benefit of one vaccination from named cells
With CoverageAtBirth, BasicRepNum and the loss per infection in LossPerInfection, and LifetimeInfRisk computed as in HE-IM-VEXT-001, the first formula returns MPB in a cell named PrivateBenefitVacc and the second returns MEB in a cell named ExternalBenefitVacc; the IF returns 0 at or above the threshold without evaluating the division. For the article's inputs 5, 0.75 and 500 the cells return 100 and 400.
=LossPerInfection*LifetimeInfRisk; =IF(CoverageAtBirth<1-1/BasicRepNum,LossPerInfection/(BasicRepNum*(1-CoverageAtBirth)),0)
Assumptions
Same loss per infection for the family and for society in the vaccination benefit split
L is one money value of an infection for the family deciding and for society, as in the article, and families are risk neutral, as Goodkin-Gold and colleagues assume. Where the family values the loss differently from society, MPB takes the family's value and MEB society's, and their sum is no longer L (derived).
Marginal benefits read at one steady-state coverage of a fully protective vaccine
MPB and MEB are the steady-state effect of one more vaccination per birth at coverage v of a fully protective vaccine, comparing steady states with the transition and discounting ignored, as in the article and in the limiting case without discounting that Goodkin-Gold and colleagues study; with an all-or-nothing vaccine of efficacy theta they scale all three benefits by theta. Boulier, Datta and Goldfarb report from a standard epidemiological model that the size of the externality is not necessarily monotonic in the number vaccinated, vaccine efficacy or infectiousness and that the marginal externality can exceed one case prevented among the unvaccinated, so the closed form here holds only for this endemic model.
Worked examples
Private and external benefit of a vaccination at the article's 60% coverage
At the article's voluntary coverage of 0.6, with R_0 of 5 and a loss of GBP 500 per infection, the unvaccinated risk is 0.5, so the vaccinated child captures GBP 250 and unvaccinated children receive the other GBP 250, which is 500/(5 × 0.4). The two add to the GBP 500 marginal social benefit the article compares with the GBP 250 cost. Illustrative inputs.
R_0 = 5; v = 0.6; L = 500; pi = 0.5; MPB = 250; MEB = 250
Private and external benefit of a vaccination at the article's 75% coverage
At 75% coverage the unvaccinated risk is 0.2, so the vaccinated child captures GBP 100, exactly the GBP 100 of time and travel the parent still bears under free provision, and unvaccinated children receive GBP 400, which is 500/(5 × 0.25). The external share has risen from 0.5 to 0.8, as in the article's interpretation. Illustrative inputs.
R_0 = 5; v = 0.75; L = 500; pi = 0.2; MPB = 100; MEB = 400
No private or external benefit from a vaccination at the herd immunity threshold
At the threshold coverage of 0.8 no infections remain, so one more vaccination prevents nothing and both benefits are 0, the reason the article's 200 vaccinations above the threshold under a universal requirement add GBP 50,000 of cost and no benefit. Illustrative inputs.
R_0 = 5; v = 0.8; L = 500; pi = 0; MPB = 0; MEB = 0
Common errors
External benefit of a vaccination treated as a fixed amount per dose
Carrying the GBP 250 external benefit found at the article's 60% coverage to the move at 75% coverage understates it by GBP 150, since it is GBP 400 there, and understates the subsidy that the marginal external benefit would justify. The external benefit per vaccination depends on the coverage at which it is read, as the article's misreadings list says.
External benefit counted as L times 1 minus pi at or above the threshold
Above the threshold pi is 0, so L times 1 minus pi returns the whole GBP 500 as external benefit, when one more vaccination prevents no infection and the marginal externality is zero, as Goodkin-Gold and colleagues state. Applied to the 200 vaccinations above the threshold in the article's universal requirement it credits GBP 100,000 of external benefit that does not exist.
Sources
Goodkin-Gold and colleagues on marginal private benefit, marginal social benefit and the marginal externality
Goodkin-Gold M, Kremer M, Snyder CM, Williams HL. Optimal vaccine subsidies for endemic and epidemic diseases. NBER Working Paper 28085. Cambridge, MA: National Bureau of Economic Research; 2020. Section 2.2, Consumer Demand, which gives the marginal private benefit of a risk-neutral consumer as theta H Phi_I(Q), with H the expected harm of an infection and Phi_I the lifetime infection probability, and zero at or above Q_0 (equation 16); and section 2.4, Normative Measures, which gives the marginal social benefit as theta H below Q_0 and zero at or above it (equation 21), defines the marginal externality as marginal social benefit less marginal private benefit, theta H/((1 minus theta Q/mu) R_0) below Q_0 and zero above it (equation 22), and states that for Q below Q_0 the marginal social benefit is a constant that does not vary with R_0.
Boulier, Datta and Goldfarb on how the vaccination externality varies
Boulier BL, Datta TS, Goldfarb RS. Vaccination externalities. The B.E. Journal of Economic Analysis & Policy. 2007;7(1). Abstract, which states that a standard epidemiological model is used to analyse how the magnitude of the externality varies with the number of vaccinations, vaccine efficacy and disease infectiousness, that the size of the externality is not necessarily monotonic in the number vaccinated, vaccine efficacy or disease infectiousness, and that the marginal externality of a vaccination can be greater than one case of illness prevented among the nonvaccinated.
Canonical Identity
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