Background: A realistic description of the social processes leading to the increasing reluctance to various forms of vaccination is a very challenging task. This is due to the complexity of the psychological and social mechanisms determining the positioning of individuals and groups against vaccination and associated activities. Understanding the role played by social media and the Internet in the current spread of the anti-vaccination (AV) movement is of crucial importance. Methods: We present novel, long-term Big Data analyses of Internet activity connected with the AV movement for such different societies as the US and Poland. The datasets we analyzed cover multiyear periods preceding the COVID-19 pandemic, documenting the behavior of vaccine related Internet activity with high temporal resolution. To understand the empirical observations, in particular the mechanism driving the peaks of AV activity, we propose an Agent Based Model (ABM) of the AV movement. The model includes the interplay between multiple driving factors: contacts with medical practitioners and public vaccination campaigns, interpersonal communication, and the influence of the infosphere (social networks, WEB pages, user comments, etc.). The model takes into account the difference between the rational approach of the pro-vaccination information providers and the largely emotional appeal of anti-vaccination propaganda. Results: The datasets studied show the presence of short-lived, high intensity activity peaks, much higher than the low activity background. The peaks are seemingly random in size and time separation. Such behavior strongly suggests a nonlinear nature for the social interactions driving the AV movement instead of the slow, gradual growth typical of linear processes. The ABM simulations reproduce the observed temporal behavior of the AV interest very closely. For a range of parameters, the simulations result in a relatively small fraction of people refusing vaccination, but a slight change in critical parameters (such as willingness to post anti-vaccination information) may lead to a catastrophic breakdown of vaccination support in the model society, due to nonlinear feedback effects. The model allows the effectiveness of strategies combating the anti-vaccination movement to be studied. An increase in intensity of standard pro-vaccination communications by government agencies and medical personnel is found to have little effect. On the other hand, focused campaigns using the Internet and social media and copying the highly emotional and narrative-focused format used by the anti-vaccination activists can diminish the AV influence. Similar effects result from censoring and taking down anti-vaccination communications by social media platforms. The benefit of such tactics might, however, be offset by their social cost, for example, the increased polarization and potential to exploit it for political goals, or increased ‘persecution’ and ‘martyrdom’ tropes.