Heterogeneous population dynamics and scaling laws near epidemic outbreaks.

In this paper, we focus on the influence of heterogeneity and stochasticity of the population on the dynamical structure of a basic susceptible-infected-susceptible (SIS) model. First we prove that, upon a suitable mathematical reformulation of the basic reproduction number, the homogeneous system and the heterogeneous system exhibit a completely analogous global behaviour. Then we consider noise terms to incorporate the fluctuation effects and the random import of the disease into the population and analyse the influence of heterogeneity on warning signs for critical transitions (or tipping points). This theory shows that one may be able to anticipate whether a bifurcation point is close before it happens. We use numerical simulations of a stochastic fast-slow heterogeneous population SIS model and show various aspects of heterogeneity have crucial influences on the scaling laws that are used as early-warning signs for the homogeneous system. Thus, although the basic structural qualitative dynamical properties are the same for both systems, the quantitative features for epidemic prediction are expected to change and care has to be taken to interpret potential warning signs for disease outbreaks correctly.

[1]  V. Dakos,et al.  Detecting dynamical changes in nonlinear time series using locally linear state‐space models , 2012 .

[2]  E. Allen.  Derivation of stochastic partial differential equations for size- and age-structured populations , 2009, Journal of biological dynamics.

[3]  C. Kuehn.  Multiple Time Scale Dynamics , 2015 .

[4]  David G. Rand,et al.  the high Himalayas and Antarctica Phylogeography of microbial phototrophs in the dry valleys of Supplementary data tml , 2010 .

[5]  J. Hyman,et al.  Using mathematical models to understand the AIDS epidemic , 1988 .

[6]  Barbara Mayer,et al.  Mathematical Epidemiology Of Infectious Diseases Model Building Analysis And Interpretation , 2016 .

[7]  Christian Kuehn,et al.  Warning signs for wave speed transitions of noisy Fisher–KPP invasion fronts , 2012, Theoretical Ecology.

[8]  I. Longini,et al.  The critical vaccination fraction for heterogeneous epidemic models. , 2003, Mathematical biosciences.

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

[10]  David J. Marchette,et al.  Modeling computer virus prevalence with a susceptible-infected-susceptible model with reintroduction , 2004, Comput. Stat. Data Anal..

[11]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[12]  Martin A. Nowak,et al.  Infectious Disease Modeling of Social Contagion in Networks , 2010, PLoS Comput. Biol..

[13]  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.

[14]  Eduardo Massad,et al.  Modelling heterogeneities in individual frailties in epidemic models , 1999 .

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

[16]  Christian Kuehn,et al.  A Mathematical Framework for Critical Transitions: Normal Forms, Variance and Applications , 2011, J. Nonlinear Sci..

[17]  Stephen Schecter,et al.  Persistent unstable equilibria and closed orbits of a singularly perturbed equation , 1985 .

[18]  V. Veliov,et al.  On the effect of population heterogeneity on dynamics of epidemic diseases , 2005, Journal of mathematical biology.

[19]  Matt J. Keeling,et al.  Insights from unifying modern approximations to infections on networks , 2010, Journal of The Royal Society Interface.

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

[21]  M. Keeling,et al.  Networks and epidemic models , 2005, Journal of The Royal Society Interface.

[22]  Peter Szmolyan,et al.  Extending slow manifolds near transcritical and pitchfork singularities , 2001 .

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

[24]  C. Castillo-Chavez,et al.  Resource Consumption, Sustainability, and Cancer , 2015, Bulletin of mathematical biology.

[25]  P. Van Mieghem,et al.  Epidemics in networks with nodal self-infection and the epidemic threshold. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  Ira B Schwartz,et al.  Fluctuating epidemics on adaptive networks. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  A. Ives,et al.  Seeing the forest and the trees: multilevel models reveal both species and community patterns , 2012 .

[28]  M. Iannelli,et al.  Optimal Screening in Structured SIR Epidemics , 2012 .

[29]  R. Hickson,et al.  How population heterogeneity in susceptibility and infectivity influences epidemic dynamics. , 2014, Journal of theoretical biology.

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

[31]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[32]  N. Berglund,et al.  Geometric singular perturbation theory for stochastic differential equations , 2002, math/0204008.

[33]  Vladimir M. Veliov,et al.  Optimal control of heterogeneous systems: Basic theory , 2008 .

[34]  Herbert W. Hethcote,et al.  Mixing patterns between age groups in social networks , 2007, Soc. Networks.

[35]  Desmond J. Higham,et al.  An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations , 2001, SIAM Rev..

[36]  Meng Yang,et al.  A modified SIS model with an infective medium on complex networks and its global stability , 2011 .

[37]  A. Novozhilov.  On the spread of epidemics in a closed heterogeneous population. , 2008, Mathematical biosciences.

[38]  S. Busenberg,et al.  A general solution of the problem of mixing of subpopulations and its application to risk- and age-structured epidemic models for the spread of AIDS. , 1991, IMA journal of mathematics applied in medicine and biology.

[39]  S. Shreve,et al.  Stochastic differential equations , 1955, Mathematical Proceedings of the Cambridge Philosophical Society.

[40]  P. King.  Mathematical models in population biology and epidemiology [Book Reviews] , 2001, IEEE Engineering in Medicine and Biology Magazine.

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

[42]  D. Clancy,et al.  The effect of population heterogeneities upon spread of infection , 2013, Journal of mathematical biology.

[43]  Riccardo Zecchina,et al.  Containing epidemic outbreaks by message-passing techniques , 2013, ArXiv.

[44]  J. Yorke,et al.  Dynamics and Control of the Transmission of Gonorrhea , 1978, Sexually transmitted diseases.

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

[46]  Z. Duan,et al.  An SIS model with infective medium on complex networks , 2008 .

[47]  S. Carpenter,et al.  Early-warning signals for critical transitions , 2009, Nature.

[48]  Linda J S Allen,et al.  Probability of a Disease Outbreak in Stochastic Multipatch Epidemic Models , 2013, Bulletin of mathematical biology.

[49]  D. Williams.  STOCHASTIC DIFFERENTIAL EQUATIONS: THEORY AND APPLICATIONS , 1976 .

[50]  P. Hosseini,et al.  Seasonality and the dynamics of infectious diseases. , 2006, Ecology letters.

[51]  Christopher Jones,et al.  Geometric singular perturbation theory , 1995 .

[52]  O. Diekmann,et al.  Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation , 2000 .

[53]  O. Diekmann,et al.  On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations , 1990, Journal of mathematical biology.

[54]  Christian Kuehn,et al.  Early warning signs for saddle-escape transitions in complex networks , 2014, Scientific Reports.

[55]  Carlos Castillo-Chavez,et al.  Competitive Exclusion in Gonorrhea Models and Other Sexually Transmitted Diseases , 1996, SIAM J. Appl. Math..

[56]  M. Osterholm,et al.  Preparing for the next pandemic. , 2005, The New England journal of medicine.

[57]  J. O. Irwin,et al.  MATHEMATICAL EPIDEMIOLOGY , 1958 .

[58]  B. Øksendal.  Stochastic Differential Equations , 1985 .

[59]  J. Dushoff,et al.  Host heterogeneity and disease endemicity: a moment-based approach. , 1999, Theoretical population biology.

[60]  John M. Drake,et al.  Theory of early warning signals of disease emergenceand leading indicators of elimination , 2013, Theoretical Ecology.

[61]  Hisashi Inaba,et al.  Endemic threshold results in an age-duration-structured population model for HIV infection. , 2006, Mathematical biosciences.

[62]  G. Hek.  Geometric singular perturbation theory in biological practice , 2010 .

[63]  W. O. Kermack,et al.  A contribution to the mathematical theory of epidemics , 1927 .

[64]  Tom Britton,et al.  Stochastic epidemic models: a survey. , 2009, Mathematical biosciences.

[65]  O. Bjørnstad,et al.  DYNAMICS OF MEASLES EPIDEMICS: SCALING NOISE, DETERMINISM, AND PREDICTABILITY WITH THE TSIR MODEL , 2002 .

[66]  Kurt Wiesenfeld,et al.  Noisy precursors of nonlinear instabilities , 1985 .

[67]  Shweta Bansal,et al.  The dynamic nature of contact networks in infectious disease epidemiology , 2010, Journal of biological dynamics.

[68]  N. Berglund,et al.  Noise-Induced Phenomena in Slow-Fast Dynamical Systems: A Sample-Paths Approach , 2005 .

[69]  Thilo Gross,et al.  Epidemic dynamics on an adaptive network. , 2005, Physical review letters.

[70]  H. Hethcote.  Qualitative analyses of communicable disease models , 1976 .

[71]  S. Carpenter,et al.  Rising variance: a leading indicator of ecological transition. , 2006, Ecology letters.

[72]  Artem S. Novozhilov,et al.  Epidemiological Models With Parametric Heterogeneity : Deterministic Theory for Closed Populations , 2012, 1202.2448.

[73]  Vladimir M. Veliov,et al.  MAXIMUM PRINCIPLE FOR AGE AND DURATION STRUCTURED SYSTEMS: A TOOL FOR OPTIMAL PREVENTION AND TREATMENT OF HIV , 2004 .

[74]  R. Anderson,et al.  Sexually transmitted diseases and sexual behavior: insights from mathematical models. , 1996, The Journal of infectious diseases.