Simulating non-Markovian stochastic processes.

We present a simple and general framework to simulate statistically correct realizations of a system of non-Markovian discrete stochastic processes. We give the exact analytical solution and a practical and efficient algorithm like the Gillespie algorithm for Markovian processes, with the difference being that now the occurrence rates of the events depend on the time elapsed since the event last took place. We use our non-Markovian generalized Gillespie stochastic simulation methodology to investigate the effects of nonexponential interevent time distributions in the susceptible-infected-susceptible model of epidemic spreading. Strikingly, our results unveil the drastic effects that very subtle differences in the modeling of non-Markovian processes have on the global behavior of complex systems, with important implications for their understanding and prediction. We also assess our generalized Gillespie algorithm on a system of biochemical reactions with time delays. As compared to other existing methods, we find that the generalized Gillespie algorithm is the most general because it can be implemented very easily in cases (such as for delays coupled to the evolution of the system) in which other algorithms do not work or need adapted versions that are less efficient in computational terms.

[1]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[2]  D. Gillespie.  A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions , 1976 .

[3]  Paul Erdös,et al.  On random graphs, I , 1959 .

[4]  Alessandro Vespignani,et al.  Epidemic spreading in scale-free networks. , 2000, Physical review letters.

[5]  Jean-Pierre Eckmann,et al.  Entropy of dialogues creates coherent structures in e-mail traffic. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[6]  Igor A. Ushakov,et al.  Probabilistic Reliability Models , 2012 .

[7]  V. Eguíluz,et al.  Update rules and interevent time distributions: slow ordering versus no ordering in the voter model. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  L. Amaral,et al.  On Universality in Human Correspondence Activity , 2009, Science.

[9]  Walter L. Smith,et al.  ON THE SUPERPOSITION OF RENEWAL PROCESSES , 1954 .

[10]  Vicenç Méndez,et al.  Non-Markovian model for transport and reactions of particles in spiny dendrites. , 2008, Physical review letters.

[11]  S. Havlin,et al.  Scaling laws of human interaction activity , 2009, Proceedings of the National Academy of Sciences.

[12]  Mark Newman,et al.  Networks: An Introduction , 2010 .

[13]  Albert-László Barabási,et al.  The origin of bursts and heavy tails in human dynamics , 2005, Nature.

[14]  M. Barthelemy,et al.  Microdynamics in stationary complex networks , 2008, Proceedings of the National Academy of Sciences.

[15]  Tianhai Tian,et al.  Oscillatory Regulation of Hes1: Discrete Stochastic Delay Modelling and Simulation , 2006, PLoS Comput. Biol..

[16]  Adilson E. Motter,et al.  A Poissonian explanation for heavy tails in e-mail communication , 2008, Proceedings of the National Academy of Sciences.

[17]  D. Gillespie.  Exact Stochastic Simulation of Coupled Chemical Reactions , 1977 .

[18]  Michael A. Gibson,et al.  Efficient Exact Stochastic Simulation of Chemical Systems with Many Species and Many Channels , 2000 .

[19]  Offer Kella,et al.  Superposition of renewal processes and an application to multi-server queues , 2006 .

[20]  A. J. Hall.  Infectious diseases of humans: R. M. Anderson & R. M. May. Oxford etc.: Oxford University Press, 1991. viii + 757 pp. Price £50. ISBN 0-19-854599-1 , 1992 .

[21]  Alessandro Vespignani,et al.  Epidemic spreading in complex networks with degree correlations , 2003, cond-mat/0301149.

[22]  A. Barrat,et al.  Dynamical and bursty interactions in social networks. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  Jeffrey O. Kephart,et al.  Directed-graph epidemiological models of computer viruses , 1991, Proceedings. 1991 IEEE Computer Society Symposium on Research in Security and Privacy.

[24]  D. Gillespie.  Approximate accelerated stochastic simulation of chemically reacting systems , 2001 .

[25]  Alessandro Vespignani,et al.  Dynamical Processes on Complex Networks , 2008 .

[26]  Alain Barrat,et al.  Social network dynamics of face-to-face interactions , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  David F Anderson,et al.  A modified next reaction method for simulating chemical systems with time dependent propensities and delays. , 2007, The Journal of chemical physics.

[28]  H. D. Miller,et al.  The Theory Of Stochastic Processes , 1977, The Mathematical Gazette.

[29]  M. Newman.  Spread of epidemic disease on networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  R. Pastor-Satorras,et al.  Epidemic spreading in correlated complex networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[31]  D. Volfson,et al.  Delay-induced stochastic oscillations in gene regulation. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[32]  K. Sneppen,et al.  Sustained oscillations and time delays in gene expression of protein Hes1 , 2003, FEBS letters.

[33]  K. Goh,et al.  Spreading dynamics following bursty human activity patterns. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[34]  P. V. Mieghem,et al.  Non-Markovian Infection Spread Dramatically Alters the Susceptible-Infected-Susceptible Epidemic Threshold in Networks , 2013 .

[35]  Maria A. Kazandjieva,et al.  A high-resolution human contact network for infectious disease transmission , 2010, Proceedings of the National Academy of Sciences.

[36]  A. Barabasi,et al.  Impact of non-Poissonian activity patterns on spreading processes. , 2006, Physical review letters.

[37]  Tobias Galla,et al.  Stochastic processes with distributed delays: chemical Langevin equation and linear-noise approximation. , 2013, Physical review letters.

[38]  Alessandro Vespignani,et al.  Absence of epidemic threshold in scale-free networks with degree correlations. , 2002, Physical review letters.

[39]  Rui Zhu,et al.  Validation of an algorithm for delay stochastic simulation of transcription and translation in prokaryotic gene expression , 2006, Physical biology.

[40]  J. Griffiths.  The Theory of Stochastic Processes , 1967 .

[41]  Xiaodong Cai,et al.  Exact stochastic simulation of coupled chemical reactions with delays. , 2007, The Journal of chemical physics.