Signature
r = c / L; v_vol = 1 - 1 / (R_0 * max(1 - r, 1 / R_0))
| Inputs | Definition | Unit |
|---|---|---|
c | Cost of one vaccination as the deciding family sees it, any price paid plus its time and travel, zero or above | currency per vaccination |
L | Money value the family puts on the loss from one infection of its child, above zero | currency per infection |
R_0 | Basic reproduction number of the infection, above 1 | cases per case |
r | c divided by L, the lifetime infection risk at which a family is indifferent about vaccinating, the relative risk r of Bauch and Earn | ratio, no unit |
|---|---|---|
v_vol | Share of births vaccinated when every family vaccinates only while its expected loss from infection exceeds its private cost; 0 when r is at least 1 minus 1/R_0 | proportion |
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
Voluntary vaccination coverage from named cells
With the family's private cost in a cell named PrivateCostVacc, the loss per infection in LossPerInfection and R_0 in BasicRepNum, the first formula returns r in a cell named CostLossRatio and the second returns v_vol in a cell named VoluntaryCoverage. For the article's inputs 250, 500 and 5 the cells return 0.5 and 0.6, and with 100 in place of 250 they return 0.2 and 0.75.
=PrivateCostVacc/LossPerInfection; =1-1/(BasicRepNum*MAX(1-CostLossRatio,1/BasicRepNum))
Assumptions
Families compare expected loss with private cost at the steady-state infection risk
Every family has the same information, the same L and c, and acts on the steady-state risk pi, as Bauch and Earn assume when all individuals are given the same information and use it in the same way to assess risks; Goodkin-Gold and colleagues take homogeneous, risk-neutral consumers in the steady state without discounting. At v_vol families are indifferent, so v_vol is the share that vaccinates rather than each family's choice. Bauch and Earn read the risks as perceived risks, so a change in perceived cost, such as during a vaccine scare, moves v_vol.
No voluntary vaccination once the cost ratio reaches the risk with no vaccination
c is zero or above and L above zero. Bauch and Earn show that the equilibrium is never to vaccinate when r is at least pi_0, the risk with no vaccination, which is 1 minus 1/R_0 by their Eq. 15 at p of 0 (derived); Goodkin-Gold and colleagues reach the same case when R_0 is at most 1/(1 minus c/(theta H)). The term max(1 minus r, 1/R_0) returns 0 in that case, and returns Eq. 17 of Bauch and Earn otherwise.
Worked examples
Voluntary coverage when parents bear the article's full GBP 250 cost
In the article's example each vaccination costs the parent GBP 250 and an infection costs GBP 500, so r is 0.5 and uptake settles where the unvaccinated risk is 0.5, at 1 minus 1/(5 × 0.5), a coverage of 0.6, twenty percentage points short of the 0.8 threshold. Illustrative inputs.
c = 250; L = 500; r = 0.5; R_0 = 5; v_vol = 0.6
Voluntary coverage under free provision with GBP 100 of time and travel
When the health service pays the GBP 150 for the vaccine and its delivery, parents still bear GBP 100 of time and travel, so r is 0.2 and coverage settles at 1 minus 1/(5 × 0.8), or 0.75, the article's step 4, five percentage points short of the threshold. Illustrative inputs.
c = 100; L = 500; r = 0.2; R_0 = 5; v_vol = 0.75
Voluntary coverage reaches the threshold only at zero private cost
With no private cost at all r is 0 and v_vol equals the threshold of 0.8, the limiting case in which an unvaccinated child's risk has to fall to zero before families stop vaccinating (derived). Any positive private cost leaves coverage below 0.8. Illustrative inputs.
c = 0; L = 500; r = 0; R_0 = 5; v_vol = 0.8
No voluntary vaccination when the cost ratio exceeds the risk with no vaccination
If the private cost were GBP 450 against a loss of GBP 500, r of 0.9 would exceed the risk with no vaccination of 0.8, so no family vaccinates, the pure never-vaccinate equilibrium of Bauch and Earn. Eq. 17 used alone would return minus 1. Illustrative inputs.
c = 450; L = 500; r = 0.9; R_0 = 5; v_vol = 0
Common errors
Resource cost used in place of the family's private cost in voluntary uptake
Free provision changes the cost the family sees, not the cost to society. Entering the full GBP 250 resource cost when the family bears only GBP 100 predicts coverage of 0.6 in place of 0.75, a shortfall from the threshold of 0.2 in place of 0.05, four times too large.
Bauch and Earn's interior equilibrium used when nobody vaccinates
Eq. 17 holds only when r is below 1 minus 1/R_0. With R_0 of 5, a cost ratio of 0.9 gives 1 minus 1/(5 × 0.1), or minus 1, and a cost ratio of 1.2 gives 2, neither a coverage; the equilibrium in both cases is 0, which the max term in the formula returns.
Sources
Bauch and Earn on the Nash equilibrium of the vaccination game
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 The Vaccination Game, subsections Description of Game and Characterization of Nash Equilibria, which give the payoff E(P, p) = minus r P minus pi_p(1 minus P) with relative risk r = r_v/r_i (Eq. 2) and state that if r is at least pi_0 the convergently stable Nash equilibrium is never vaccinate and otherwise vaccinate with a probability P* above 0 and below 1; section Incorporation of an Epidemic Model, which gives the mixed equilibrium obtained by solving r = pi_P* as P* = 1 minus 1/(R_0(1 minus r)) (Eq. 17); section Results and Discussion, which states that for any perceived relative risk r above 0 the expected uptake is below the eradication threshold and that parents can be expected to play a pure nonvaccinator strategy if r > pi_0, that is if r > 1 minus 1/R_0 (Eq. 18); and the Appendix, which shows that the Nash equilibrium is the unique solution of pi_P* = r when r < pi_0.
Goodkin-Gold and colleagues on steady-state vaccine demand and the no-purchase case
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 states that consumers buy the vaccine when its marginal private benefit exceeds the price, that when some but not all buy they must be indifferent so that the price equals the marginal private benefit, and gives the demand d(P) = (mu/theta)(1 minus 1/((1 minus P/(theta H)) R_0)) (equation 17) with its boundary cases (equation 18); and section 2.5, Perfectly Competitive Equilibrium, which sets the price at the cost c, bounds case LR2 by R_0 at most 1/(1 minus c~), with c~ = c/(theta H), a case in which the marginal private benefit is below c even if no other consumer buys and the vaccine quantity stays 0, and states that some but not all newborns buy in the next range of R_0, case LR3.
Canonical Identity
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