We use an agent-based compartmental (SIRV) model with birth and death processes. Agents inhabit a finite two-dimensional region; at birth an agent is assigned a fixed, random position and may choose to vaccinate or not. We assume that the agents will choose their vaccination strategy in order to minimize the expected cost that they incur, given by

Each agent estimates the likelihood of their becoming infected if not vaccinated by observing the state of all other agents within a given distance

As illustrated in the figure below we find that when contagion dynamics are local and there is an intermediate relative cost

Below are the mean and standard deviation of the nonzero vaccination levels at time 150, as a function of the size of the information neighborhood. Vaccination levels increase with information, while at the same time standard deviation decreases, reflecting the fact that more information leads to dynamics determined by population averages rather than local neighborhoods. There is qualitatively different behavior in vaccination as we move from an information neighborhood smaller than the interaction neighborhood (pink line) to larger. To the left very low levels of infection and vaccination can survive in the population.

It may seem counterintuitive that when the information neighborhood is large average vaccination levels are highest and yet the disease is more likely to survive. However, on closer inspection this makes sense. In the time course below we can observe that vaccination levels have an oscillatory dynamic, but when the knowledge neighborhood is small (left panel) the fall in vaccination level is slower than when the knowledge neighborhood is large (right panel). Furthermore, although maximum and minimum nonzero vaccination levels are lower with smaller knowledge neighborhood, there is a lower average and minimum ratio of infected and recovered agents to vaccinated agents. That is, there are more vaccinated agents per infected or recovered individual on average, and vaccination in response to local but small infections can be sustained. This can be understood by noting that there is a threshold number of infected plus recovered neighbors above which an agent will vaccinate and below which it will not. The larger the knowledge neighborhood, the larger this threshold number is. Hence, agents can vaccinate when there are lower average infection levels (which could be high using a local measure) only when they observe less of the population. This turns out to be crucial in containing the infection.