Dynamics of stochastic epidemics on heterogeneous networks

Epidemic models currently play a central role in our attempts to understand and control infectious diseases. Here, we derive a model for the diffusion limit of stochastic susceptible-infectious-removed (SIR) epidemic dynamics on a heterogeneous network. Using this, we consider analytically the early asymptotic exponential growth phase of such epidemics, showing how the higher order moments of the network degree distribution enter into the stochastic behaviour of the epidemic. We find that the first three moments of the network degree distribution are needed to specify the variance in disease prevalence fully, meaning that the skewness of the degree distribution affects the variance of the prevalence of infection. We compare these asymptotic results to simulation and find a close agreement for city-sized populations.

[1]  N. Ferguson,et al.  Planning for smallpox outbreaks , 2003, Nature.

[2]  H. Andersson,et al.  Stochastic Epidemic Models and Their Statistical Analysis , 2000 .

[3]  E. Volz.  SIR dynamics in random networks with heterogeneous connectivity , 2007, Journal of mathematical biology.

[4]  M. Keeling,et al.  The effects of local spatial structure on epidemiological invasions , 1999, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[5]  Frank Ball,et al.  Network epidemic models with two levels of mixing. , 2008, Mathematical biosciences.

[6]  Roy M. Anderson,et al.  The Transmission Dynamics of Human Immunodeficiency Virus (HIV) , 1988 .

[7]  Matt J Keeling,et al.  Modeling dynamic and network heterogeneities in the spread of sexually transmitted diseases , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[8]  P. Driessche,et al.  Effective degree network disease models , 2011, Journal of mathematical biology.

[9]  Joel C Miller,et al.  Edge-based compartmental modelling for infectious disease spread , 2011, Journal of The Royal Society Interface.

[10]  D. Sherrington.  Stochastic Processes in Physics and Chemistry , 1983 .

[11]  Joel E. Cohen,et al.  Infectious Diseases of Humans: Dynamics and Control , 1992 .

[12]  L. Meyers,et al.  Susceptible–infected–recovered epidemics in dynamic contact networks , 2007, Proceedings of the Royal Society B: Biological Sciences.

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

[14]  Andrew J Black,et al.  Stochastic fluctuations in the susceptible-infective-recovered model with distributed infectious periods. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  T. Kurtz.  Solutions of ordinary differential equations as limits of pure jump markov processes , 1970, Journal of Applied Probability.

[16]  Matt J. Keeling,et al.  Networks and the Epidemiology of Infectious Disease , 2010, Interdisciplinary perspectives on infectious diseases.

[17]  T. Kurtz.  Limit theorems for sequences of jump Markov processes approximating ordinary differential processes , 1971, Journal of Applied Probability.

[18]  R. May,et al.  The transmission dynamics of human immunodeficiency virus (HIV). , 1988, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

[19]  Rowland R Kao,et al.  The effect of contact heterogeneity and multiple routes of transmission on final epidemic size. , 2006, Mathematical biosciences.

[20]  C. Fraser,et al.  Transmission Dynamics of the Etiological Agent of SARS in Hong Kong: Impact of Public Health Interventions , 2003, Science.

[21]  B. Grenfell,et al.  The epidemiology of varicella–zoster virus infections: the influence of varicella on the prevalence of herpes zoster , 1992, Epidemiology and Infection.

[22]  R. May,et al.  Infection dynamics on scale-free networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[24]  M. Pascual,et al.  Stochastic amplification in epidemics , 2007, Journal of The Royal Society Interface.

[25]  L. Decreusefond,et al.  Large graph limit for an SIR process in random network with heterogeneous connectivity , 2010, 1007.3958.

[26]  Catherine H Mercer,et al.  Scale-Free Networks and Sexually Transmitted Diseases: A Description of Observed Patterns of Sexual Contacts in Britain and Zimbabwe , 2004, Sexually transmitted diseases.

[27]  J. Ross,et al.  A stochastic metapopulation model accounting for habitat dynamics , 2006, Journal of mathematical biology.

[28]  Christel Kamp,et al.  Untangling the Interplay between Epidemic Spread and Transmission Network Dynamics , 2009, PLoS Comput. Biol..

[29]  R. Durrett.  Random Graph Dynamics: References , 2006 .

[30]  Rob Deardon,et al.  Optimal reactive vaccination strategies for a foot-and-mouth outbreak in the UK , 2006, Nature.

[31]  David A. Rand,et al.  Correlation Equations and Pair Approximations for Spatial Ecologies , 1999 .

[32]  L. Meyers,et al.  When individual behaviour matters: homogeneous and network models in epidemiology , 2007, Journal of The Royal Society Interface.

[33]  M. Jit,et al.  Vaccination against pandemic influenza A/H1N1v in England: a real-time economic evaluation. , 2010, Vaccine.

[34]  Alessandro Vespignani,et al.  Epidemic dynamics and endemic states in complex networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[35]  M. Keeling,et al.  Integrating stochasticity and network structure into an epidemic model , 2008, Journal of The Royal Society Interface.

[36]  T. Rogers.  Maximum-entropy moment-closure for stochastic systems on networks , 2011, 1103.4980.

[37]  Alessandro Vespignani,et al.  The role of the airline transportation network in the prediction and predictability of global epidemics , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[38]  Joel C. Miller.  A note on a paper by Erik Volz: SIR dynamics in random networks , 2009, Journal of mathematical biology.

[39]  R. May,et al.  Infectious Diseases of Humans: Dynamics and Control , 1991, Annals of Internal Medicine.

[40]  Bruce A. Reed,et al.  A Critical Point for Random Graphs with a Given Degree Sequence , 1995, Random Struct. Algorithms.

[41]  R. Mikolajczyk,et al.  Social Contacts and Mixing Patterns Relevant to the Spread of Infectious Diseases , 2008, PLoS medicine.

[42]  O. Diekmann.  Mathematical Epidemiology of Infectious Diseases , 1996 .

[43]  M. Keeling,et al.  Modeling Infectious Diseases in Humans and Animals , 2007 .