Herd immunity with spatial adaptation based on global prevalence information

During epidemic outbreaks, individuals often adapt their behaviour in response to global prevalence information, using spatial strategies such as reducing mobility or transmission range. We study the effects of such distance-based adaptive behaviors on epidemic dynamics in a spatially distributed population. Specifically, a fraction of agents modify their susceptibility and transmission range based on global prevalence. We analyse three adaptation scenarios: a constant adaptive fraction, a power-law dependence and a sigmoidal dependence on prevalence. In the spatially well-mixed limit, we derive critical thresholds of the adaptive fraction required to contain the outbreak. In the static spatial limit, we draw analogies to continuum percolation with polydisperse discs to estimate bounds on the critical adaptive fraction. Our results show that a linear response offers no advantage over constant adaptation, while a strongly superlinear response is necessary for effective containment. For the sigmoidal case, we identify conditions under which prevalence exhibits oscillations and peak prevalence showing a non-monotonic dependence on the width of the sigmoidal response—revealing an optimal parameter range that minimizes epidemic severity. Our work highlights the role of global-prevalence-driven spatial adaptation in shaping epidemic outcomes and offers a framework for understanding emergent spatial herd immunity in adaptive populations.