Frank, S. A. 1998. Inducible defence and the social evolution of herd immunity. Proceedings of the Royal Society of London B 265:1911-1913.

Many organisms vary their level of investment in defensive characters. Protective traits may be induced upon exposure to predators or parasites. In a similar way, humans vaccinate in response to threatening epidemics. When most group members defend themselves, epidemics die out quickly because parasites cannot spread. A high level of group (herd) immunity is therefore beneficial to the group. There is, however, a well-known divergence between the optimum degree of induction for selfish individuals and the level of induction that maximises group benefit.

I develop two optimality models for the frequency of induction. The first model shows that higher relatedness favours more induction and a smaller difference between selfish and cooperative optima. The second model assumes variation in the vigour of individuals and therefore differences in the relative cost for induction. The model predicts that strong individuals induce more easily than weak individuals. Small differences in vigour cause a large divergence in the optimal levels of induction for strong and weak individuals.

The concept of genetic relatedness in an evolutionary model is analogous to correlated interests and correlated strategies in an economic model of human behaviour. The evolutionary models presented here therefore provide a basis for further study of human vaccination.