Vaccination dynamics in well-mixed populations. The fractions (*a*) vaccinated and (*b*) infected are shown as functions of the relative cost of vaccination, *c*, for the intensity of selection *β* = 1 and 10. The lines are analytical predictions from deterministic equations (see the electronic supplementary material). The deviation between simulation and theory is largely due to stochasticity in disease transmission: holding vaccination constant at some level below the herd immunity threshold (1 − (1/*R*_{0}) = 0.6), simulated infection risk is smaller than the prediction in equation (2.1) (see electronic supplementary material, figure S1*b*). Individuals in the simulation respond to this decreased risk by vaccinating less than in the theory, which in turn leads to a larger epidemic versus the theory. Strong selection magnifies individuals' responses, producing larger deviations. For vaccination coverage above the theoretical herd immunity level, the deterministic approximation underestimates infection risk, leading to an opposite deviation at low *c*. Parameters: population size *N* = 5000, *R*_{0} = 2.5 (realized by setting *r* = 5/(6*N*) d^{−1} person^{−1} and *g* = ⅓ d^{−1}), number of infection seeds *I*_{0} = 5. (*a*) Simulations: open squares, *β* = 1; filled squares, *β* = 10; theory: dotted line, *β* = 1; solid line, *β* = 10. (*b*) Simulations: open inverted triangles, *β* = 1; filled inverted triangles, *β* = 10; theory: dotted line, *β* = 1; solid line, *β* = 10.

## PubMed Commons