Misinformation is usually adjusted to fit distinct narratives and propagates rapidly through social networks. False beliefs, once adopted, are rarely corrected. Amidst the COVID-19 crisis, pandemic-deniers and people who oppose wearing face masks or quarantine have already been a substantial aspect of the development of the pandemic. With the vaccine for COVID-19, different anti-vaccine narratives are being created and are probably being adopted by large population groups with critical consequences. Assuming full adherence to vaccine administration, we use a diffusion model to analyse epidemic spreading and the impact of different vaccination strategies, measured with the average years of life lost, in three network topologies (a proximity, a scale-free and a small-world network). Then, using a similar diffusion model, we consider the spread of anti-vaccine views in the network, which are adopted based on a persuasiveness parameter of anti-vaccine views. Results show that even if anti-vaccine narratives have a small persuasiveness, a large part of the population will be rapidly exposed to them. Assuming that all individuals are equally likely to adopt anti-vaccine views after being exposed, more central nodes in the network, which are more exposed to these views, are more likely to adopt them. Comparing years of life lost, anti-vaccine views could have a significant cost not only on those who share them, since the core social benefits of a limited vaccination strategy (reduction of susceptible hosts, network disruptions and slowing the spread of the disease) are substantially shortened.