SIR Dynamics with Vaccination in a Large Configuration Model

We consider an SIR model with vaccination strategy on a sparse configuration model random graph. We show the convergence of the system when the number of nodes grows and characterize the scaling limits. Then, we prove the existence of optimal controls for the limiting equations formulated in the framework of game theory, both in the centralized and decentralized setting. We show how the characteristics of the graph (degree distribution) influence the vaccination efficiency for optimal strategies, and we compute the limiting final size of the epidemic depending on the degree distribution of the graph and the parameters of infection, recovery and vaccination. We also present several simulations for two types of vaccination, showing how the optimal controls allow to decrease the number of infections and underlining the crucial role of the network characteristics in the propagation of the disease and the vaccination program.

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

[2]  I. I. Gikhman,et al.  The Theory of Stochastic Processes III , 1979 .

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

[4]  Acquaintance Vaccination in an Epidemic on a Random Graph with Specified Degree Distribution , 2013, Journal of Applied Probability.

[5]  Martin L. Puterman,et al.  Markov Decision Processes: Discrete Stochastic Dynamic Programming , 1994 .

[6]  Nicolas Fournier,et al.  A microscopic probabilistic description of a locally regulated population and macroscopic approximations , 2004, math/0503546.

[7]  Helena A Herrmann,et al.  Using network science to propose strategies for effectively dealing with pandemics: The COVID-19 example , 2020, medRxiv.

[8]  Bruno Gaujal,et al.  A MEAN FIELD GAME ANALYSIS OF SIR DYNAMICS WITH VACCINATION , 2020, Probability in the Engineering and Informational Sciences.

[9]  B. Gaujal,et al.  Discrete mean field games: Existence of equilibria and convergence , 2019, Journal of Dynamics & Games.

[10]  David R. Cox,et al.  The Theory of Stochastic Processes , 1967, The Mathematical Gazette.

[11]  Jingzhou Liu,et al.  The Impact of Imitation on Vaccination Behavior in Social Contact Networks , 2012, PLoS Comput. Biol..

[12]  Gesine Reinert,et al.  Approximating the epidemic curve , 2013, 1301.3288.

[13]  Yong Han Kang,et al.  Stability analysis and optimal vaccination of an SIR epidemic model , 2008, Biosyst..

[14]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

[15]  Aric Hagberg,et al.  Exploring Network Structure, Dynamics, and Function using NetworkX , 2008, Proceedings of the Python in Science Conference.

[16]  M. Yor,et al.  Continuous martingales and Brownian motion , 1990 .

[17]  Xuebin Chi,et al.  The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission , 2002 .

[18]  H E Stanley,et al.  Classes of small-world networks. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[19]  R. May,et al.  How Viruses Spread Among Computers and People , 2001, Science.

[20]  Piet Van Mieghem,et al.  Epidemic processes in complex networks , 2014, ArXiv.

[21]  L. Amaral,et al.  The web of human sexual contacts , 2001, Nature.

[22]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[23]  Stephen P. Borgatti,et al.  Centrality and network flow , 2005, Soc. Networks.

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

[25]  Daniel I. S. Rosenbloom,et al.  Imitation dynamics of vaccination behaviour on social networks , 2011, Proceedings of the Royal Society B: Biological Sciences.

[26]  S. Roelly-Coppoletta.  A criterion of convergence of measure-valued processes: Application to measure branching processes , 1986 .

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

[28]  Herbert W. Hethcote,et al.  Optimal vaccination schedules in a deterministic epidemic model , 1973 .

[29]  Svante Janson.  The Probability That a Random Multigraph is Simple , 2009, Comb. Probab. Comput..

[30]  Weimin Han,et al.  Convergence of the forward-backward sweep method in optimal control , 2012, Comput. Optim. Appl..

[31]  Joel C. Miller,et al.  Mathematics of Epidemics on Networks: From Exact to Approximate Models , 2017 .

[32]  Gabriel Turinici,et al.  Global optimal vaccination in the SIR model: properties of the value function and application to cost-effectiveness analysis. , 2015, Mathematical biosciences.

[33]  Viet Chi Tran.  Modèles particulaires stochastiques pour des problèmes d'évolution adaptative et pour l'approximation de solutions statistiques , 2006 .

[34]  P. Billingsley,et al.  Convergence of Probability Measures , 1970, The Mathematical Gazette.

[35]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .

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

[37]  Svante Janson,et al.  Law of large numbers for the SIR epidemic on a random graph with given degrees , 2013, Random Struct. Algorithms.

[38]  S. Janson,et al.  Graphs with specified degree distributions, simple epidemics, and local vaccination strategies , 2007, Advances in Applied Probability.

[39]  Rodrigo Castro,et al.  EB-DEVS: A Formal Framework for Modeling and Simulation of Emergent Behavior in Dynamic Complex Systems , 2021, J. Comput. Sci..

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

[41]  Timothy C. Reluga,et al.  Long-standing influenza vaccination policy is in accord with individual self-interest but not with the utilitarian optimum , 2007, Proceedings of the National Academy of Sciences.

[42]  Frank Ball,et al.  Acquaintance Vaccination in an Epidemic on a Random Graph with Specified Degree Distribution , 2013, J. Appl. Probab..

[43]  P. Holland,et al.  Transitivity in Structural Models of Small Groups , 1971 .

[44]  Joel C. Miller,et al.  A primer on the use of probability generating functions in infectious disease modeling , 2018, Infectious Disease Modelling.

[45]  Y. Moreno,et al.  Epidemic outbreaks in complex heterogeneous networks , 2001, cond-mat/0107267.

[46]  F. Brauer,et al.  Mathematical Models in Population Biology and Epidemiology , 2001 .

[47]  Sylvie Roelly‐ Coppoletta.  A criterion of convergence of measure‐valued processes: application to measure branching processes , 1986 .

[48]  Lauren Ancel Meyers,et al.  Erratic Flu Vaccination Emerges from Short-Sighted Behavior in Contact Networks , 2011, PLoS Comput. Biol..

[49]  Emmanuel Trélat,et al.  Contrôle optimal : théorie & applications , 2005 .

[50]  P. Lions,et al.  Ordinary differential equations, transport theory and Sobolev spaces , 1989 .

[51]  Jukka-Pekka Onnela,et al.  Efficient vaccination strategies for epidemic control using network information , 2018, Epidemics.

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

[53]  Luiz Henrique Alves Monteiro,et al.  An epidemic model to evaluate the homogeneous mixing assumption , 2014, Commun. Nonlinear Sci. Numer. Simul..

[54]  P. Bermolen,et al.  The jamming constant of uniform random graphs , 2013, 1310.8475.

[55]  Nuno Crokidakis,et al.  Dynamics of epidemic spreading with vaccination: Impact of social pressure and engagement , 2016, 1605.03914.

[56]  Jeong Han Kim,et al.  Poisson Cloning Model for Random Graphs , 2008, 0805.4133.

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

[58]  Alberto Bressan,et al.  Viscosity Solutions of Hamilton-Jacobi Equations and Optimal Control Problems , 2011 .

[59]  Reuven Cohen,et al.  Efficient immunization strategies for computer networks and populations. , 2002, Physical review letters.

[60]  F. Takeuchi,et al.  Effectiveness of realistic vaccination strategies for contact networks of various degree distributions. , 2006, Journal of theoretical biology.