Heterogeneous SIS model for directed networks and optimal immunization

We investigate the influence of a contact network structure over the spread of epidemics in an heterogeneous population. Basically the epidemics spreads over a directed weighted graph. We describe the epidemic process as a continuous-time individual-based susceptible–infected–suscepti-ble (SIS) model using a first-order mean-field approximation. First we consider a network without a specific topology, investigating the epidemic threshold and the stability properties of the system. Then we analyze the case of a community network, relying on the graph-theoretical notion of equitable partition, and using a lower-dimensional dynamical system in order to individuate the epidemic threshold. Moreover we prove that the positive steady-state of the original system, that appears above the threshold, can be computed by this lower-dimensional system. In the second part of the paper we treat the important issue of the infectious disease control. Taking into account the connectivity of the network, we provide a cost-optimal distribution of resources to prevent the disease from persisting indefinitely in the population; for a particular case of two-level immunization problem we report on the construction of a polynomial time complexity algorithm. 1. Introduction. Most studies refer to epidemic process with homogeneous infection (recovery) rate. However in many real situations, in social, biological and data communications networks, it is more appropriate to consider an heterogeneous setting than an homogeneous one [36]. A short overview on the literature considering heterogeneous populations can be found in [39, 27]. Our work strongly emphasises the influence on the population contact network. Our scope is to investigate the influence of a certain network topology on the epidemics spreading and to control it. For this purpose we consider a given and static graph (i.e. our set of nodes edges do not change in time), that can result from some random experiment, or as a man made architecture.

[1]  Piet Van Mieghem,et al.  Exact Markovian SIR and SIS epidemics on networks and an upper bound for the epidemic threshold , 2014, 1402.1731.

[2]  Ingemar Nåsell,et al.  Stochastic models of some endemic infections. , 2002, Mathematical biosciences.

[3]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[4]  Christos Faloutsos,et al.  Fractional Immunization in Networks , 2013, SDM.

[5]  F. Ball,et al.  Epidemics with two levels of mixing , 1997 .

[6]  F. Ball,et al.  Analysis of a stochastic SIR epidemic on a random network incorporating household structure. , 2010, Mathematical biosciences.

[7]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[8]  Donald F. Towsley,et al.  The effect of network topology on the spread of epidemics , 2005, Proceedings IEEE 24th Annual Joint Conference of the IEEE Computer and Communications Societies..

[9]  L. Massoulié,et al.  Epidemics and Rumours in Complex Networks: From microscopic to macroscopic dynamics , 2009 .

[10]  P Van Mieghem,et al.  Accuracy criterion for the mean-field approximation in susceptible-infected-susceptible epidemics on networks. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  P. Van Mieghem,et al.  Virus Spread in Networks , 2009, IEEE/ACM Transactions on Networking.

[12]  Delio Mugnolo.  Semigroup Methods for Evolution Equations on Networks , 2014 .

[13]  Bo Qu,et al.  SIS Epidemic Spreading with Heterogeneous Infection Rates , 2015, IEEE Transactions on Network Science and Engineering.

[14]  B. McKay,et al.  Feasibility conditions for the existence of walk-regular graphs , 1980 .

[15]  Kim-Chuan Toh,et al.  Solving semidefinite-quadratic-linear programs using SDPT3 , 2003, Math. Program..

[16]  I. Nåsell.  The quasi-stationary distribution of the closed endemic sis model , 1996, Advances in Applied Probability.

[17]  Yuan Lou,et al.  Asymptotic Profiles of the Steady States for an SIS Epidemic Patch Model , 2007, SIAM J. Appl. Math..

[18]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[19]  A. Schwenk.  COMPUTING THE CHARACTERISTIC POLYNOMIAL OF A GRAPH. , 1974 .

[20]  Frank Ball,et al.  Reproduction numbers for epidemic models with households and other social structures. I. Definition and calculation of R0 , 2012, Mathematical biosciences.

[21]  Otso Ovaskainen,et al.  Metapopulation theory for fragmented landscapes. , 2003, Theoretical population biology.

[22]  Piet Van Mieghem,et al.  Generalized Epidemic Mean-Field Model for Spreading Processes Over Multilayer Complex Networks , 2013, IEEE/ACM Transactions on Networking.

[23]  Naoki Masuda,et al.  Effects of diffusion rates on epidemic spreads in metapopulation networks , 2010, 1004.2391.

[24]  Piet Van Mieghem,et al.  Decay towards the overall-healthy state in SIS epidemics on networks , 2013, ArXiv.

[25]  Piet Van Mieghem,et al.  In-homogeneous Virus Spread in Networks , 2013, ArXiv.

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

[27]  V. Isham,et al.  Seven challenges for metapopulation models of epidemics, including households models. , 2015, Epidemics.

[28]  Frank Ball,et al.  Threshold behaviour and final outcome of an epidemic on a random network with household structure , 2008, Advances in Applied Probability.

[29]  Stefania Ottaviano,et al.  Epidemic outbreaks in two-scale community networks. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  Chinwendu Enyioha,et al.  Optimal vaccine allocation to control epidemic outbreaks in arbitrary networks , 2013, 52nd IEEE Conference on Decision and Control.

[31]  Matt J. Keeling,et al.  Calculation of Disease Dynamics in a Population of Households , 2010, PloS one.

[32]  I. Kiss,et al.  Exact epidemic models on graphs using graph-automorphism driven lumping , 2010, Journal of mathematical biology.

[33]  Frank Ball,et al.  A general model for stochastic SIR epidemics with two levels of mixing. , 2002, Mathematical biosciences.

[34]  Christian Kuehn,et al.  Heterogeneous population dynamics and scaling laws near epidemic outbreaks. , 2014, Mathematical biosciences and engineering : MBE.

[35]  Francesco De Pellegrini,et al.  Epidemic Outbreaks in Networks with Equitable or Almost-Equitable Partitions , 2014, SIAM J. Appl. Math..

[36]  J. Yorke,et al.  A Deterministic Model for Gonorrhea in a Nonhomogeneous Population , 1976 .