Evolutionary stability in continuous nonlinear public goods games

Smallpox was eradicated in the 1970s, but new outbreaks could be seeded by bioterrorism or accidental release. Substantial vaccine-induced morbidity and mortality make pre-emptive mass vaccination controversial, and if vaccination is voluntary, then there is a conflict between self- and group interests. This conflict can be framed as a tragedy of the commons, in which herd immunity plays the role of the commons, and free-riding (i.e. not vaccinating pre-emptively) is analogous to exploiting the commons. This game has been analysed previously for a particular post-outbreak vaccination scenario. We consider several post-outbreak vaccination scenarios and compare the expected increase in mortality that results from voluntary versus imposed vaccination. Below a threshold level of post-outbreak vaccination effort, expected mortality is independent of the level of response effort. A lag between an outbreak starting and a response being initiated increases the post-outbreak vaccination effort necessary to reduce mortality. For some post-outbreak vaccination scenarios, even modest response lags make it impractical to reduce mortality by increasing post-outbreak vaccination effort. In such situations, if decreasing the response lag is impossible, the only practical way to reduce mortality is to make the vaccine safer (greater post-outbreak vaccination effort leads only to fewer people vaccinating pre-emptively).