Proceedings of the 12th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2012 La Manga, Spain, July, 2–5, 2012. Fractional calculus and superdiffusion in epidemiology: shift of critical thresholds Urszula Skwara 1,3 , Jos´ e Martins 2 , Peyman Ghaffari 1,4 , Ma´ ıra Aguiar 1 , Jo˜ ao Boto 1 and Nico Stollenwerk 1 1 Centro de Matem´ atica e Aplica¸ c˜ oes Fundamentais, Universidade de Lisboa, Portugal 2 Departamento de Matem´ atica, Escola Superior de Tecnologia e Gest˜ ao, Instituto Polit´ ecnico de Leir´ ıa, Portugal 3 Institute of Mathematics, Maria Curie Sklodowska University, Poland 4 Complexity and Networks Group, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom emails: uskwara@o2.pl,urszula@ptmat.fc.ul.pt, jmmartins@estg.ipleiria.pt, peyman@ptmat.fc.ul.pt, maira@ptmat.fc.ul.pt, boto@ptmat.fc.ul.pt, nico@ptmat.fc.ul.pt Abstract Spatially extended stochastic processes in epidemiology lead to classical reaction- diffusion process, when infection spreads only locally. This notion can be generalized using fractional derivatives, especially fractional Laplacian operators, leading to L´ evy flights and sub- or super-diffusion. Especially super-diffusion is a more realistic mecha- nism of spreading epidemics than ordinary diffusion, hence fractional calculus has quite some relevance for epidemiological applications since it shifts the critical threshold be- tween extinction and presistence of the disease. Key words: fractional Laplace operator, critical thresholds, reinfection models, in- fluenza, dengue fever 1 Introduction Classical derivatives of integer order have been generalized historically in various ways to derivatives of fractional order [1, 2] and especially [3, 4] with more reference and results in c CMMSE ISBN: 978–84–615–5392–1