Fractional derivative in continuous-time Markov processes and applications to epidemics in networks

Continuous-time Markov processes are governed by the Chapman-Kolmogorov differential equation. We show that replacing the standard time derivative of the governing equation with a Caputo fractional derivative of order 0 <alpha < 1, leads to a fractional differential equation whose solution can describe the state probabilities of a class of non-Markovian stochastic processes. We show that the same state probabilities also solve a system of equations that describe semi-Markov processes in which the sojourn times follow a Mittag-Leffler distribution, contrasting the usual Markov processes with exponentially distributed sojourn times. We apply the fractional framework to the e-SIS epidemic process on any contact graph and we propose a microscopic epidemic description in which infection and curing events follow a Mittag-Leffler distribution and are not independent. We analytically prove that the description exactly solves the fractional extension of the Chapman-Kolmogorov differential equation, and we provide an extensive study of how the dependence between events strongly affects the dynamics of the spreading process. We conclude verifying the proposed framework with Monte Carlo simulations.