The dynamics of a spreading disease and individual behavioral changes are entangled processes that have to be addressed together in order to effectively manage an outbreak. Here, we relate individual risk perception to the adoption of a specific set of control measures, as obtained from an extensive large-scale survey performed via Facebook-involving more than 500,000 respondents from 64 countries-showing that there is a "one-to-one" relationship between perceived epidemic risk and compliance with a set of mitigation rules. We then develop a mathematical model for the spreading of a disease-sharing epidemiological features with COVID-19-that explicitly takes into account non-compliant individual behaviors and evaluates the impact of a population fraction of infectious risk-deniers on the epidemic dynamics. Our modeling study grounds on a wide set of structures, including both synthetic and more than 180 real-world contact patterns, to evaluate, in realistic scenarios, how network features typical of human interaction patterns impact the spread of a disease. In both synthetic and real contact patterns we find that epidemic spreading is hindered for decreasing population fractions of risk-denier individuals. From empirical contact patterns we demonstrate that connectivity heterogeneity and group structure significantly affect the peak of hospitalized population: higher modularity and heterogeneity of social contacts are linked to lower peaks at a fixed fraction of risk-denier individuals while, at the same time, such features increase the relative impact on hospitalizations with respect to the case where everyone correctly perceive the risks.