Vaccination Externalities

Vaccination Externalities: Herd Protection, Free-Riding and Programme Evaluation

Vaccination externalities arise because a vaccinated person is less likely to catch an infection and pass it on, so people who were never vaccinated also face a lower risk. In health economics this indirect protection is treated as a positive externality in consumption: voluntary uptake tends to stop short of the level that maximises social welfare, and an economic evaluation that ignores transmission can misstate the value of a vaccination programme. This page sets out the gap between private and social benefit, why free-riding grows near the herd immunity threshold, how subsidies, free provision and mandates respond, and how dynamic and static models capture indirect effects, with an illustrative worked example and the UK decision on HPV vaccination for boys.

Indirect protection as an external benefit

The World Health Organization guide on economic evaluation of immunisation programmes defines externalities as costs or benefits arising from an individual's production or consumption decision that indirectly affect the well-being of others, and herd immunity as the reduction in exposure of susceptible people to a pathogen through vaccination of other people. The person vaccinated gains direct protection, and the people they would otherwise have infected gain indirect protection.

Fine, Eames and Heymann note that "herd immunity" carries several meanings, but that the common implication is that immune people reduce the risk for susceptible people, sometimes called indirect protection or a herd effect. That indirect protection makes vaccination a case of an externality. The UK Cost-Effectiveness Methodology for Immunisation Programmes and Procurements (CEMIPP) group, reporting to the Department of Health in 2016, separated it from other possible benefits, such as peace of mind for vaccinated people and their families and a caring externality.

Private benefit, social benefit and under-vaccination

A person deciding whether to be vaccinated weighs their own expected gain against their own cost. The gain to others does not enter the decision, so the marginal social benefit of a vaccination exceeds the marginal private benefit, the pattern of a positive externality.

$$MSB = MPB + MEB$$

where $MSB$ is the marginal social benefit of one more vaccination, $MPB$ is the marginal private benefit to the person vaccinated and $MEB$ is the marginal external benefit to everyone else. Voluntary uptake settles where marginal private benefit equals the private cost, so when $MEB$ is positive, coverage falls below the level at which marginal social benefit equals marginal social cost.

The size of $MEB$ is not a fixed property of a vaccine. Boulier, Datta and Goldfarb, using a standard epidemiological model with parameters for influenza and mumps, found that the externality is not necessarily monotonic in the number vaccinated, vaccine efficacy or infectiousness, and that the marginal externality of one vaccination can exceed one case of illness prevented among the unvaccinated. The externality can also be partly negative. The ISPOR-SMDM dynamic transmission modelling task force notes that lower infection risk raises the average age at infection, which for many diseases raises complication risk and mortality, and that partial rubella vaccine coverage has been followed in several countries by paradoxical increases in congenital rubella.

Free-riding near the herd immunity threshold

The WHO guide gives the herd immunity threshold for a simple model as $H = 1 - 1/R_0$, where $R_0$ is the basic reproduction number, and notes that actual coverage should exceed it because vaccine protection is usually imperfect. In Bauch and Earn's model, an unvaccinated person's chance of infection falls as coverage rises and reaches zero at the eradication threshold.

That falling risk drives free-riding. Bauch and Earn modelled voluntary vaccination as a game in which parents weigh the perceived risk of vaccination against the risk from infection. They showed that for any positive perceived relative risk of vaccination, expected uptake at the Nash equilibrium is below the eradication threshold, so a disease cannot be eradicated through voluntary vaccination when individuals act in their own interest. Philipson's review of economic epidemiology adds that private demand for prevention is prevalence-elastic: as a disease disappears, so does the demand for vaccines, and initially successful programmes make eradication progressively harder.

Worked example: voluntary uptake stalls below the threshold

This example is illustrative; its numbers are invented. An endemic infection follows a simple SIR model with lifelong immunity, children are vaccinated at birth with a fully protective vaccine, and $R_0 = 5$, so the threshold is $H = 1 - 1/5 = 0.8$. Each infection causes a loss of GBP 500, valued the same by the family and by society. Each vaccination uses GBP 250 of resources: GBP 150 for the vaccine and its delivery and GBP 100 of the parent's time and travel. Costs are compared per 1,000 births at the long-run equilibrium, ignoring the transition and discounting.

While the infection persists, the long-run susceptible share settles at $\frac{1}{R_0}$, and the lifetime chance that an unvaccinated child is infected is

$$\pi(v) = \frac{\lambda}{\lambda + \mu} = 1 - \frac{1}{R_0 (1 - v)}$$

where $\pi(v)$ is that chance when a share $v$ of each birth cohort is vaccinated, $\lambda$ is the force of infection at equilibrium and $\mu$ is the birth and death rate. The second form follows from setting the inflow of unvaccinated births, $\mu(1 - v)$, equal to the outflow from the susceptible pool, $(\lambda + \mu)/R_0$. At or above the threshold, $\pi(v) = 0$.

Step 1. Voluntary uptake. A parent paying GBP 250 vaccinates only while the expected loss from infection, 500 × π(v), exceeds 250. Uptake settles where π(v) = 250 / 500 = 0.5, so 5 × (1 − v) = 2, 1 − v = 0.4 and coverage is 0.6.

Step 2. Infections and the external share. Infections per 1,000 births are 1,000 × 0.4 × 0.5 = 200. In this model infections per birth equal the unvaccinated share minus the equilibrium susceptible share, (1 − v) − 0.2, so each extra vaccination below the threshold prevents exactly one infection in total. The vaccinated child captures 0.5 of it, worth GBP 250, and the other 0.5, also worth GBP 250, goes to unvaccinated children. The marginal social benefit is GBP 500 against a cost of GBP 250.

Step 3. Social cost per 1,000 births. With no vaccination, 800 infections cost GBP 400,000. Under voluntary uptake, 600 vaccinations cost 600 × 250 = 150,000 and 200 infections cost 200 × 500 = 100,000, a total of GBP 250,000. At the threshold, 800 vaccinations cost GBP 200,000 and no infections remain. Since each vaccination below the threshold prevents an infection worth more than its cost, the threshold is the social optimum in this steady state, and voluntary uptake costs GBP 50,000 more per 1,000 births.

Step 4. Free provision. If the health service pays the GBP 150, parents still bear GBP 100 of time and travel. Uptake settles where π(v) = 100 / 500 = 0.2, so 5 × (1 − v) = 1.25, 1 − v = 0.25 and coverage is 0.75. Infections fall to 1,000 × (0.8 − 0.75) = 50, and the social cost is 750 × 250 = 187,500 plus 50 × 500 = 25,000, a total of GBP 212,500. Who pays the GBP 150 does not change the resource cost.

Interpretation. At 75% coverage an unvaccinated child's risk is 0.2, worth GBP 100, exactly the parent's remaining cost, so coverage stalls five percentage points short of the threshold while the remaining parents rely on other children's vaccination. The external share of each vaccination has risen from 0.5 to 1 − 0.2 = 0.8. Closing the gap by price would mean paying parents for their time, since the private benefit is zero at the threshold. A rule requiring every child to be vaccinated would cost 1,000 × 250 = GBP 250,000 in this model, GBP 50,000 more than at the threshold, because the 200 vaccinations above it prevent nothing; in this example that equals the cost of voluntary uptake.

Policy responses: subsidy, free provision and mandates

The corrective response to a positive externality is a subsidy equal to the marginal external benefit, the mirror image of the Pigouvian tax. Philipson argues that for vaccines such classic justifications of Pigouvian subsidies may be highly ineffective, because higher vaccination among people covered by a subsidy or a mandatory programme lowers the incentive for people outside it; in the extreme, total demand is inelastic to subsidies. He also argues that eradication is never Pareto optimal for the current population alone, because the benefit of reducing an almost extinct disease eventually falls below the cost of vaccinating more people.

Free provision removes the money price but leaves time, travel and perceived risk, so the worked example still stops short. Requirements set coverage directly rather than through price and raise their own questions of autonomy, exemptions and trust, covered under vaccination policy.

Capturing indirect effects in economic evaluation

The ISPOR-SMDM task force explains that static models hold the risk of infection constant, while in a dynamic transmission model the risk depends on the number of infectious people, so people not reached by a programme can benefit. It notes that many decision-analytic models, such as Markov models, ignore the indirect effects of averted infections, and cites a review in which only 11% of 208 cost-effectiveness studies of vaccination programmes used an approach that could include them.

The task force accepts a static model when the target groups are not epidemiologically important, when effects are almost entirely direct, as in vaccinating the elderly against influenza, or when the static result is already cost-effective and dynamic effects would add to it. When a static model shows a programme as unattractive or borderline, it recommends supplementary dynamic modelling. The WHO guide treats the static or dynamic choice as the main modelling decision for vaccination, and its flow chart prefers a dynamic model when a target group, such as children for airborne infections, is epidemiologically influential. The task force also warns that individually randomised trials underestimate effectiveness, because neither arm experiences a reduced force of infection.

The external share can be large. The US Centers for Disease Control and Prevention reported that in 2003 the 7-valent pneumococcal conjugate vaccine given to young children prevented more than twice as many cases of invasive pneumococcal disease through indirect effects on transmission as through direct protection. CEMIPP noted that herd immunity may give the incremental cost-effectiveness ratio a complex relationship with output, and that negative unintended consequences, such as a higher average age at infection, should be included in economic analyses of vaccination programmes.

HPV vaccination for boys in the UK

In 2008 the Joint Committee on Vaccination and Immunisation (JCVI) judged that high coverage in girls would provide herd protection to boys, so vaccinating boys was unlikely to be cost-effective. In its 2018 statement, the standard analysis at GBP 20,000 per QALY and a 3.5% discount rate, incremental on the girls' programme, indicated a threshold price of around zero pounds per dose. JCVI advised extending vaccination to boys only after supporting a 1.5% discount rate because of HPV's long natural history, and it noted that a combined girls' and boys' programme compared with no vaccination would be highly likely to be cost-effective. The 2018 statement again noted that the girls' programme would give boys substantial herd protection.

Boundaries and common misreadings

Herd immunity is the epidemiological phenomenon; the vaccination externality is its economic consequence, the part of a vaccination's value that accrues to people other than the decision-maker. Market failure is the broader account of why private choices fall short, and the perspective of an evaluation decides which benefits are counted. Several misreadings recur.

• The externality is a fixed amount per dose. It varies with coverage, efficacy and infectiousness.
• A static model is always conservative. The task force accepts a static model as a worst case only when herd immunity or age shifts cannot produce negative effects.
• Coverage should be pushed as high as possible. The economic target is where marginal social benefit no longer exceeds marginal social cost.

Sources

• Bauch CT, Earn DJD. Vaccination and the theory of games. Proceedings of the National Academy of Sciences of the USA. 2004;101(36):13391-13394. doi:10.1073/pnas.0403823101
• Boulier BL, Datta TS, Goldfarb RS. Vaccination externalities. The B.E. Journal of Economic Analysis & Policy. 2007;7(1). doi:10.2202/1935-1682.1487
• Centers for Disease Control and Prevention. Direct and indirect effects of routine vaccination of children with 7-valent pneumococcal conjugate vaccine on incidence of invasive pneumococcal disease, United States, 1998-2003. MMWR Morbidity and Mortality Weekly Report. 2005;54(36):893-897.
• Cost-Effectiveness Methodology for Immunisation Programmes and Procurements (CEMIPP) group. Review of Cost-Effectiveness Methodology for Immunisation Programmes and Procurements. Report presented to the Department of Health, 20 July 2016. London: GOV.UK; 2018.
• Fine P, Eames K, Heymann DL. "Herd immunity": a rough guide. Clinical Infectious Diseases. 2011;52(7):911-916. doi:10.1093/cid/cir007
• Joint Committee on Vaccination and Immunisation. JCVI Interim Statement on Extending HPV Vaccination to Adolescent Boys. London: GOV.UK; 19 July 2017.
• Joint Committee on Vaccination and Immunisation. Statement on HPV Vaccination. London: GOV.UK; July 2018.
• Philipson T. Economic epidemiology and infectious diseases. NBER Working Paper 7037. Cambridge, MA: National Bureau of Economic Research; 1999.
• Pitman R, Fisman D, Zaric GS, et al. Dynamic transmission modeling: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force-5. Value in Health. 2012;15(6):828-834. doi:10.1016/j.jval.2012.06.011
• World Health Organization. WHO Guide for Standardization of Economic Evaluations of Immunization Programmes. 2nd ed. WHO/IVB/19.10. Geneva: WHO; 2019.