Measured Games · Paper 01 of 07 · Epidemiology

The vaccination game

After C. T. Bauch and D. J. D. Earn (2004), building on P. E. M. Fine and J. A. Clarkson (1986)

Abstract

Voluntary vaccination is modelled as a population game in which the payoff to declining is not fixed but generated by a transmission model running underneath the decision. Because the risk of infection falls as coverage rises, the two options become equally attractive at a coverage level strictly below the elimination threshold, for any positive perceived cost of vaccination. The shortfall is a property of the equilibrium rather than a coordination failure that better information would remove.

Keywordsherd immunity · SIR final size · imitation dynamics · convergently stable equilibrium

1Setup

Setup

Each person decides whether to vaccinate. Vaccinating costs something certain and usually small: side effects, time, the worry itself. Declining costs nothing up front but carries a risk of infection. The whole model turns on one observation: that risk is not a fixed number. It is produced by an epidemic running underneath the decision, and the epidemic is shaped by how many other people vaccinated.

So the payoff to declining improves as coverage rises. When almost nobody vaccinates, declining is dangerous. When almost everybody does, declining is free. Somewhere in between the two options are worth the same, and that is where the population stops moving.

2The model

The model

# perceived relative cost of vaccination r = cost of vaccine / cost of infection # herd immunity threshold from the transmission model pcrit = 1 - 1/R0 # SIR final size at coverage p, susceptible fraction s = 1 - p z = s · ( 1 - e-R0·z ) z = fraction of population infected π(p) = z / s = 1 - e-R0·z risk to one unvaccinated person # payoffs vaccinate = -r certain decline = -1 with prob π(p), else 0 equilibrium at π(p*) = r

Because π falls to zero at p_crit and r is strictly positive, the crossing always happens strictly below the threshold. The gap is not a coordination failure. It is the solution.

Figure 1Imitation dynamics over 1,000 agents with a live SIR model

This figure needs JavaScript. The panels below are live simulations rather than images, so with scripting turned off they stay empty and the controls do nothing. Everything the figure demonstrates is also stated in the surrounding text: the model is given in full in section 2 and the result in section 3.

Coverage by season · imitation dynamics over 1,000 agents

Payoff to declining as a function of coverage · hover to read values

The epidemic underneath the decision, at the coverage currently observed

Coverage Herd immunity threshold Predicted equilibrium Cost of vaccinating Infectious Susceptible Recovered

Charts are focusable. With a chart focused, use the left and right arrow keys to read values along the horizontal axis.

Basic reproduction number R₀

Relative cost of vaccinating

Imitation rate

Starting coverage

Scenarios

Threshold p_crit—

Predicted p*—

Observed coverage—

Shortfall—

Infected this season—

Season0

Figure 1. Agents imitate: each season every agent compares its realised payoff against one other agent drawn at random and copies the better outcome with probability proportional to the difference. Coverage is an emergent count, not a target. Vaccine scare raises perceived cost sharply for twenty seasons, then releases it.The lower panel integrates the SIR system at whatever coverage the agents have currently reached, so the epidemic shrinking as coverage climbs is the coupling made visible. Hover the payoff curve to read the infection risk at any coverage level.

3Results

Results

Bauch and Earn (2004) proved that for any positive perceived cost there is a unique convergently stable Nash equilibrium coverage, and that it is strictly below the elimination threshold. Voluntary vaccination cannot eradicate a disease. The result holds across a broad class of transmission models rather than one particular formulation, which is what gives it force.

The mechanism is self-limiting in both directions. Push coverage up and declining becomes more attractive. Push it down and an outbreak makes vaccinating more attractive. The system has a floor and a ceiling, and the ceiling sits under the target. Run the simulator from a coverage of 1.0 and watch it fall; run it from zero and watch it climb to the same place.

A scare does not simply dent coverage once. It shifts perceived cost r, which moves the equilibrium itself, and recovery afterwards is slow because the disease has to return before the incentive does.

4Evidence

Evidence

Fine and Clarkson (1986) made the original argument that individual and public priorities diverge in vaccination policy, working from pertussis data for England and Wales. Bauch, Galvani and Earn (2003) applied the game to smallpox and found that under most plausible parameters individual self-interest supports substantially less pre-emptive vaccination than group interest does.

The sharpest test is Galvani, Reluga and Chapman (2007). Using contact and epidemiological data for the United States, they compared the individually optimal influenza vaccination strategy against the utilitarian optimum. The individually optimal strategy concentrates vaccination in the elderly, who carry the highest risk of influenza mortality. The utilitarian optimum concentrates it in children, who are responsible for most of the transmission. The long-standing recommendations of the Centers for Disease Control prioritised the elderly, which is to say they followed individual self-interest rather than the population optimum. That is the divergence the model predicts, showing up in a standing public health recommendation.

Bauch (2005) added imitation dynamics, the rule implemented in the simulator above. That version reproduces the oscillating coverage seen after real vaccine scares rather than a smooth approach to equilibrium.

5Limitations

Limitations

The model assumes a perfectly effective vaccine and homogeneous mixing. Both are wrong in ways that matter: imperfect efficacy raises the equilibrium shortfall further, and clustering of unvaccinated people means outbreaks can occur at coverage levels the aggregate model calls safe.

The larger problem is r. It is a perceived relative cost, not a measurable quantity, and it absorbs everything from side-effect rates to media coverage to how much someone trusts a health authority. The model predicts that coverage tracks r, which is close to unfalsifiable unless r is pinned down independently. The imitation version helps by generating r-driven dynamics with observable signatures, but it adds free parameters of its own.

Finally, the equilibrium result is about voluntary programmes. It says nothing about whether mandates work, only that the absence of one leaves a predictable gap.

References

Bauch, C. T. (2005). Imitation dynamics predict vaccinating behaviour. Proceedings of the Royal Society B: Biological Sciences, 272(1573), 1669–1675. https://doi.org/10.1098/rspb.2005.3153
Bauch, C. T., & Earn, D. J. D. (2004). Vaccination and the theory of games. Proceedings of the National Academy of Sciences, 101(36), 13391–13394. https://doi.org/10.1073/pnas.0403823101
Bauch, C. T., Galvani, A. P., & Earn, D. J. D. (2003). Group interest versus self-interest in smallpox vaccination policy. Proceedings of the National Academy of Sciences, 100(18), 10564–10567. https://doi.org/10.1073/pnas.1731324100
Fine, P. E. M., & Clarkson, J. A. (1986). Individual versus public priorities in the determination of optimal vaccination policies. American Journal of Epidemiology, 124(6), 1012–1020. https://doi.org/10.1093/oxfordjournals.aje.a114471
Galvani, A. P., Reluga, T. C., & Chapman, G. B. (2007). Long-standing influenza vaccination policy is in accord with individual self-interest but not with the utilitarian optimum. Proceedings of the National Academy of Sciences, 104(13), 5692–5697. https://doi.org/10.1073/pnas.0606774104
Kermack, W. O., & McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society A, 115(772), 700–721. https://doi.org/10.1098/rspa.1927.0118

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