Signature
pi = max(0, 1 - 1 / (R_0 * (1 - v))); n = (1 - v) * pi
| Inputs | Definition | Unit |
|---|---|---|
R_0 | Basic reproduction number of the infection, above 1, as in HE-FM-BRN-001 | cases per case |
v | Proportion of newborns given a fully protective vaccine at birth, at least 0 and below 1 | proportion |
pi | Probability that a child not vaccinated at birth is infected at some point in life once the endemic steady state is reached, at least 0 and at most 1 minus 1/R_0 | probability |
|---|---|---|
n | Expected lifetime infections per child born, the unvaccinated share times pi | infections per birth |
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
Endemic infection risk and infections per birth from named cells
With the share vaccinated at birth in a cell named CoverageAtBirth and the basic reproduction number in BasicRepNum, the first formula returns pi in a cell named LifetimeInfRisk and the second returns n in a cell named InfectionsPerBirth. MAX returns 0 at or above the threshold; CoverageAtBirth must be below 1, since a value of 1 makes the division return #DIV/0!. For the article's inputs 5 and 0.6 the cells return 0.5 and 0.2.
=MAX(0,1-1/(BasicRepNum*(1-CoverageAtBirth))); =(1-CoverageAtBirth)*LifetimeInfRisk
Assumptions
Endemic SIR steady state with births balancing deaths behind the infection risk
The infection follows the single-group SIR model used by Bauch and Earn, with births and deaths at the same rate mu, lifelong immunity after recovery and a constant population, and has reached the endemic steady state, at which coverage equals uptake; Goodkin-Gold and colleagues use a model of the same form with a vaccinated compartment and a non-fatal disease. Bauch and Earn note that the birth rate and the infectious period drop out of pi_p, so R_0 and v are the only inputs. The formula does not cover waning immunity, age-structured mixing, seasonal transmission or deaths from the infection.
Fully protective vaccination at birth and coverage below one for the endemic infection risk
Each vaccinated newborn is fully and lastingly protected, and no child is infected before vaccination, the simplification Bauch and Earn state. v is at least 0 and below 1, since at a coverage of 1 no child is unvaccinated and pi is undefined, and R_0 is above 1. At or above H = 1 minus 1/R_0 the formula returns 0, the case the appendix of Bauch and Earn states (pi_p = 0 for p at or above p_crit). For an all-or-nothing vaccine that protects a share theta of those vaccinated, Goodkin-Gold and colleagues write the immunised share theta Q/mu in place of v, so v is then entered as theta times coverage, with theta for example from HE-FM-ATR-004.
Worked examples
Article's endemic infection risk of an unvaccinated child at 60% coverage
The article's illustrative endemic infection has R_0 of 5, so H is 0.8 by HE-FM-BRN-003. At the voluntary coverage of 0.6 an unvaccinated child's lifetime risk is 1 minus 1/(5 × 0.4), or 0.5, and infections per birth are 0.4 × 0.5 = 0.2, that is 200 per 1,000 births, the same as the unvaccinated share 0.4 less the equilibrium susceptible share 0.2. The inputs are invented, as the article says.
R_0 = 5; v = 0.6; pi = 0.5; n = 0.2
Article's endemic infection risk of an unvaccinated child at 75% coverage under free provision
With free provision the article's coverage rises to 0.75. The unvaccinated child's risk falls to 1 minus 1/(5 × 0.25), or 0.2, and infections per birth to 0.25 × 0.2 = 0.05, the article's 50 infections per 1,000 births, equal to 0.8 less 0.75. Illustrative inputs.
R_0 = 5; v = 0.75; pi = 0.2; n = 0.05
Endemic infection risk of an unvaccinated child with no vaccination
With no vaccination every child is unvaccinated, the lifetime risk is 1 minus 1/5, or 0.8, and infections per birth are also 0.8, the article's 800 infections per 1,000 births in step 3. This risk with no vaccination is the pi_0 of Bauch and Earn. Illustrative inputs.
R_0 = 5; v = 0; pi = 0.8; n = 0.8
Endemic infection risk of an unvaccinated child at the herd immunity threshold
At the article's threshold coverage of 0.8 the expression 1 minus 1/(R_0(1 minus v)) reaches 0, so an unvaccinated child is not infected and no infections remain, the limiting case at which the infection dies out in the steady state. Any coverage above 0.8 also returns 0. Illustrative inputs.
R_0 = 5; v = 0.8; pi = 0; n = 0
Common errors
Epidemic final size used as the lifetime risk in an endemic infection
The final size of a closed epidemic without births, HE-CF-ATR-001, answers a different question. Applied to the article's R_0 of 5 with 0.4 of the population susceptible it returns a final attack rate of about 0.319 of the whole population, or about 0.797 of the unvaccinated, against the endemic lifetime risk of 0.5 at 60% coverage, so read as the risk of an unvaccinated child it overstates the private benefit of vaccination by about 59% and understates the external share.
Unvaccinated child's risk applied to every birth in the endemic infection count
pi is the risk of a child who is not vaccinated. Multiplying it by all births in place of the unvaccinated births gives 0.5 × 1,000 = 500 infections per 1,000 births at the article's 60% coverage in place of 200, two and a half times the true count, and inflates the loss under voluntary uptake from GBP 100,000 to GBP 250,000.
Sources
Bauch and Earn on the lifetime infection risk of an unvaccinated child in the endemic SIR model
Bauch CT, Earn DJ. Vaccination and the theory of games. Proceedings of the National Academy of Sciences of the United States of America. 2004;101(36):13391-13394. Section Incorporation of an Epidemic Model, which uses the SIR model with births and deaths at rate mu and vaccination of a proportion p of newborns, assuming that individuals are never infected before being vaccinated, gives the critical coverage p_crit = 1 minus 1/R_0 for R_0 above 1 (Eq. 12), the endemic state S-hat = 1 minus p_crit (Eq. 13) and I-hat = f/(1 + f)(p_crit minus p) (Eq. 14), and expresses the probability that an unvaccinated individual eventually becomes infected as the proportion of susceptible individuals becoming infected versus dying in any unit time, pi_p = 1 minus 1/(R_0(1 minus p)) (Eq. 15), noting that the parameter f does not appear in it; and the Appendix, which states that pi_p decreases with p until p reaches the eradication threshold and that pi_p = 0 for p at or above p_crit.
Goodkin-Gold and colleagues on the steady state and lifetime infection probability with vaccination of newborns
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 2.1, Epidemiological Model, which adds births and deaths at rate mu and immunisation of newborns to the SIR model and gives the steady-state susceptible share as the smaller of 1 minus theta Q/mu and 1/R_0 (equation 7) and the infected share mu/(alpha + mu) times (1 minus 1/R_0 minus theta Q/mu), floored at 0 (equation 8); and section 2.2, Consumer Demand, which treats infection and natural death as competing Poisson risks and gives the probability that a susceptible consumer is infected over the lifetime as beta I/(beta I + mu) (equation 14), equal to 1 minus 1/((1 minus theta Q/mu) R_0) below the eradication quantity Q_0 and 0 at or above it (equation 15).
Canonical Identity
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