Group interest versus self-interest in smallpox vaccination policy

Abstract

The recent threat of bioterrorism has fueled debate on smallpox vaccination policy for the United States. Certain policy proposals call for voluntary mass vaccination; however, if individuals decide whether to vaccinate according to self-interest, the level of herd immunity achieved may differ from what is best for the population as a whole. We present a synthesis of game theory and epidemic modeling that formalizes this conflict between self-interest and group interest and shows that voluntary vaccination is unlikely to reach the group-optimal level. This shortfall results in a substantial increase in expected mortality after an attack.

Fig. 1.

Sensitivity analysis histograms for the difference in vaccine coverage, Δp = p_gr - p_ind (A), and the relative increase in mortality at the individual equilibrium, ΔC/C = [C(p_ind) - C(p_gr)]/C(p_gr)(B). This analysis shows that, for a wide range of parameter values, self-interested decision making can cause coverage levels to fall short of the group-optimal levels, resulting in a significant increase in expected mortality. We analyzed 10⁵ realizations of the model by randomly selecting parameter values from the ranges listed in Table 1. All horizontal axis labels except “0” give upper limits of corresponding bins; lower limits are given by the left adjacent label. The bin corresponding to the “0” label contains outcomes that were precisely zero.

Fig. 2.

Vaccination coverage levels (solid line, individual equilibrium p_ind; dashed line, group optimum p_gr) at various values of basic reproductive ratio R₀ (A), postattack vaccination rate v (B), and attack risk r (C). The dependence of p_ind and p_gr on model parameters is qualitatively similar but quantitatively very different. (D-F) How the expected mortality C(p) varies with model parameters at both the individual equilibrium [C(p_ind)] and the group optimum [C(p_gr)]. Other parameter values were set to the estimated values in Table 1. Because the payoffs for delayer and vaccinator are equal at the mixed Nash equilibrium [E_del(p_ind) = E_vac = -d_v, Eqs. 1 and 4] and C(p) = -pE_vac - (1 - p)E_del(p) (Eq. 5), it follows that C(p_ind) = d_v whenever the Nash equilibrium p_ind is a mixed strategy, which explains why C(p_ind) = 10^-6 over wide parameter ranges in D-F.