2016, 13(5): 1093-1118. doi: 10.3934/mbe.2016032

Heterogeneous population dynamics and scaling laws near epidemic outbreaks

1. 

ORCOS, Institute of Statistics and Mathematical Methods in Economics, Vienna University of Technology, Wiedner Hauptstrasse 8, A-1040 Vienna, Austria

2. 

Faculty of Mathematics, Technical University of Munich, Boltzmannstrasse 3, 85748 Garching, Germany

Received  August 2015 Revised  April 2016 Published  July 2016

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.
Citation: Andreas Widder, Christian Kuehn. Heterogeneous population dynamics and scaling laws near epidemic outbreaks. Mathematical Biosciences & Engineering, 2016, 13 (5) : 1093-1118. doi: 10.3934/mbe.2016032
References:
[1]

E. J. Allen, Derivation of stochastic partial differential equations for size-and age-structured populations,, Journal of Biological Dynamics, 3 (2009), 73. doi: 10.1080/17513750802162754. Google Scholar

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F. Altarelli, A. Braunstein, L. Dall'Asta, J. R. Wakeling and R. Zecchina, Containing epidemic outbreaks by message-passing techniques,, Physical Review X, 4 (2014). doi: 10.1103/PhysRevX.4.021024. Google Scholar

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S. Bansal, B. T. Grenfell and L. A. Meyers, When individual behaviour matters: Homogeneous and network models in epidemiology,, Journal of the Royal Society Interface, 4 (2007), 879. doi: 10.1098/rsif.2007.1100. Google Scholar

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S. Bansal, J. Read, B. Pourbohloul and L. A. Meyers, The dynamic nature of contact networks in infectious disease epidemiology,, Journal of Biological Dynamics, 4 (2010), 478. doi: 10.1080/17513758.2010.503376. Google Scholar

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N. Berglund and B. Gentz, Geometric singular perturbation theory for stochastic differential equations,, Journal of Differential Equations, 191 (2003), 1. doi: 10.1016/S0022-0396(03)00020-2. Google Scholar

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N. Berglund and B. Gentz, Noise-Induced Phenomena in Slow-Fast Dynamical Systems,, Springer, (2006). Google Scholar

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F. Brauer and C. Castillo-Chavez, Mathematical models in population biology and epidemiology,, Springer, (2012). doi: 10.1007/978-1-4614-1686-9. Google Scholar

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T. Britton, Stochastic epidemic models: A survey,, Mathematical Biosciences, 225 (2010), 24. doi: 10.1016/j.mbs.2010.01.006. Google Scholar

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S. Busenberg and C. Castillo-Chavez, 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,, Mathematical Medicine and Biology, 8 (1991), 1. doi: 10.1093/imammb/8.1.1. Google Scholar

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S. R. Carpenter and W. A. Brock, Rising variance: A leading indicator of ecological transition,, Ecology Letters, 9 (2006), 311. doi: 10.1111/j.1461-0248.2005.00877.x. Google Scholar

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C. Castillo-Chavez, W. Huang and J. Li, Competitive exclusion in gonorrhea models and other sexually transmitted diseases,, SIAM Journal on Applied Mathematics, 56 (1996), 494. doi: 10.1137/S003613999325419X. Google Scholar

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D. Clancy and C. J. Pearce, The effect of population heterogeneities upon spread of infection,, Journal of Mathematical Biology, 67 (2013), 963. doi: 10.1007/s00285-012-0578-x. Google Scholar

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F. A. B. Coutinho, E. Massad, L. F. Lopez and M. N. Burattini, Modelling heterogeneities in individual frailties in epidemic models,, Mathematical and Computer Modelling, 30 (1999), 97. doi: 10.1016/S0895-7177(99)00119-3. Google Scholar

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J. Dushoff, Host heterogeneity and disease endemicity: A moment-based approach,, Theoretical Population Biology, 56 (1999), 325. doi: 10.1006/tpbi.1999.1428. Google Scholar

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G. Feichtinger, T. Tsachev and V. M. Veliov, Maximum principle for age and duration structured systems: A tool for optimal prevention and treatment of HIV,, Mathematical Population Studies, 11 (2004), 3. doi: 10.1080/08898480490422301. Google Scholar

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[26]

T. Gross, C. J. Dommar D'Lima and B. Blasius, Epidemic dynamics on an adaptive network,, Physical Review Letters, 96 (2006). doi: 10.1103/PhysRevLett.96.208701. Google Scholar

[27]

G. Hek, Geometric singular perturbation theory in biological practice,, Journal of Mathematical Biology, 60 (2010), 347. doi: 10.1007/s00285-009-0266-7. Google Scholar

[28]

H. W. Hethcote, Qualitative analyses of communicable disease models,, Mathematical Biosciences, 28 (1976), 335. doi: 10.1016/0025-5564(76)90132-2. Google Scholar

[29]

R. I. Hickson and M. G. Roberts, How population heterogeneity in susceptibility and infectivity influences epidemic dynamics,, Journal of Theoretical Biology, 350 (2014), 70. doi: 10.1016/j.jtbi.2014.01.014. Google Scholar

[30]

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations,, SIAM Review, 43 (2001), 525. doi: 10.1137/S0036144500378302. Google Scholar

[31]

A. N. Hill and I. M. Longini, The critical vaccination fraction for heterogeneous epidemic models,, Mathematical Biosciences, 181 (2003), 85. doi: 10.1016/S0025-5564(02)00129-3. Google Scholar

[32]

A. Hill, D. G. Rand, M. A. Nowak and N. A. Christakis, Emotions as infectious diseases in a large social network: The SISa model,, Proceedings of the Royal Society B, 277 (2010), 3827. doi: 10.1098/rspb.2010.1217. Google Scholar

[33]

A. Hill, D. G. Rand, M. A. Nowak and N. A. Christakis, Infectious disease modeling of social contagion in networks,, PLoS Computational Biology, 6 (2010). doi: 10.1371/journal.pcbi.1000968. Google Scholar

[34]

T. House and M. J. Keeling, Insights from unifying modern approximations to infections on networks,, Journal of The Royal Society Interface, 8 (2011), 63. doi: 10.1098/rsif.2010.0179. Google Scholar

[35]

J. M. Hyman and E. A. Stanley, Using mathematical models to understand the AIDS epidemic,, Mathematical Biosciences, 90 (1988), 415. doi: 10.1016/0025-5564(88)90078-8. Google Scholar

[36]

H. Inaba, Endemic threshold results in an age-duration-structured population model for HIV infection,, Mathematical Biosciences, 201 (2006), 15. doi: 10.1016/j.mbs.2005.12.017. Google Scholar

[37]

A. R. Ives and V. Dakos, Detecting dynamical changes in nonlinear time series using locally linear state-space models,, Ecosphere, 3 (2012), 1. doi: 10.1890/ES11-00347.1. Google Scholar

[38]

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[39]

I. Kareva, M. Benjamin and C. Castillo-Chavez, Resource consumption, sustainability, and cancer,, Bulletin of Mathematical Biology, 77 (2015), 319. doi: 10.1007/s11538-014-9983-1. Google Scholar

[40]

M. J. Keeling and K. T. Eames, Networks and epidemic models,, Journal of the Royal Society Interface, 2 (2005), 295. doi: 10.1098/rsif.2005.0051. Google Scholar

[41]

M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals,, Princeton University Press, (2008). Google Scholar

[42]

J. O. Kephart and S. R. White, Directed-graph epidemiological models of computer viruses,, in Proceedings. 1991 IEEE Computer Society Symposium on Research in Security and Privacy, (1991), 343. doi: 10.1109/RISP.1991.130801. Google Scholar

[43]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics,, Proceedings of the Royal Society London A, 115 (1927), 700. Google Scholar

[44]

M. Krupa and P. Szmolyan, Extending slow manifolds near transcritical and pitchfork singularities,, Nonlinearity, 14 (2001), 1473. doi: 10.1088/0951-7715/14/6/304. Google Scholar

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C. Kuehn, A mathematical framework for critical transitions: Normal forms, variance and applications,, Journal of Nonlinear Science, 23 (2013), 457. doi: 10.1007/s00332-012-9158-x. Google Scholar

[46]

C. Kuehn, Multiple Time Scale Dynamics,, Springer, (2015). doi: 10.1007/978-3-319-12316-5. Google Scholar

[47]

C. Kuehn, Warning signs for wave speed transitions of noisy Fisher-KPP invasion fronts,, Theoretical Ecology, 6 (2013), 295. doi: 10.1007/s12080-013-0189-1. Google Scholar

[48]

C. Kuehn, G. Zschaler and T. Gross, Early warning signs for saddle-escape transitions in complex networks,, Scientific reports, 5 (2015). doi: 10.1038/srep13190. Google Scholar

[49]

G. E. Lahodny Jr and L. J. Allen, Probability of a disease outbreak in stochastic multipatch epidemic models,, Bulletin of Mathematical Biology, 75 (2013), 1157. doi: 10.1007/s11538-013-9848-z. Google Scholar

[50]

A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population,, Mathematical Biosciences, 28 (1976), 221. doi: 10.1016/0025-5564(76)90125-5. Google Scholar

[51]

Y. Moreno, R. Pastor-Satorras and A. Vespignani, Epidemic outbreaks in complex heterogeneous networks,, The European Physical Journal B-Condensed Matter and Complex Systems, 26 (2002), 521. doi: 10.1140/epjb/e20020122. Google Scholar

[52]

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[53]

A. S. Novozhilov, Epidemiological models with parametric heterogeneity: Deterministic theory for closed populations,, Mathematical Modelling of Natural Phenomena, 7 (2012), 147. doi: 10.1051/mmnp/20127310. Google Scholar

[54]

A. S. Novozhilov, On the spread of epidemics in a closed heterogeneous population,, Mathematical Biosciences, 215 (2008), 177. doi: 10.1016/j.mbs.2008.07.010. Google Scholar

[55]

S. M. O'Regan and J. M. Drake, Theory of early warning signals of disease emergence and leading indicators of elimination,, Theoretical Ecology, 6 (2013), 333. Google Scholar

[56]

M. T. Osterholm, Preparing for the next pandemic,, New England Journal of Medicine, 352 (2005), 1839. Google Scholar

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S. Schecter, Persistent unstable equilibria and closed orbits of a singularly perturbed equation,, Journal of Differential Equations, 60 (1985), 131. doi: 10.1016/0022-0396(85)90124-X. Google Scholar

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M. Scheffer, J. Bascompte, W. A. Brock, V. Brovkhin, S. R. Carpenter, V. Dakos, H. Held, E. H. van Nes, M. Rietkerk and G. Sugihara, Early-warning signals for critical transitions,, Nature, 461 (2009), 53. doi: 10.1038/nature08227. Google Scholar

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H. Shi, Z. Duan and G. Chen, An SIS model with infective medium on complex networks,, Physica A, 387 (2008), 2133. doi: 10.1016/j.physa.2007.11.048. Google Scholar

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V. M. Veliov, On the effect of population heterogeneity on dynamics of epidemic diseases,, Journal of Mathematical Biology, 51 (2005), 123. doi: 10.1007/s00285-004-0288-0. Google Scholar

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show all references

References:
[1]

E. J. Allen, Derivation of stochastic partial differential equations for size-and age-structured populations,, Journal of Biological Dynamics, 3 (2009), 73. doi: 10.1080/17513750802162754. Google Scholar

[2]

F. Altarelli, A. Braunstein, L. Dall'Asta, J. R. Wakeling and R. Zecchina, Containing epidemic outbreaks by message-passing techniques,, Physical Review X, 4 (2014). doi: 10.1103/PhysRevX.4.021024. Google Scholar

[3]

S. Altizer, A. Dobson, P. Hosseini, P. Hudson, M. Pascual and P. Rohani, Seasonality and the dynamics of infectious diseases,, Ecology Letters, 9 (2006), 467. doi: 10.1111/j.1461-0248.2005.00879.x. Google Scholar

[4]

B. Ainseba and M. Iannelli, Optimal screening in structured SIR epidemics,, Mathematical Modelling of Natural Phenomena, 7 (2012), 12. doi: 10.1051/mmnp/20127302. Google Scholar

[5]

L. Arnold, Stochastic Differential Equations: Theory and Applications,, Wiley, (1974). Google Scholar

[6]

S. Bansal, B. T. Grenfell and L. A. Meyers, When individual behaviour matters: Homogeneous and network models in epidemiology,, Journal of the Royal Society Interface, 4 (2007), 879. doi: 10.1098/rsif.2007.1100. Google Scholar

[7]

S. Bansal, J. Read, B. Pourbohloul and L. A. Meyers, The dynamic nature of contact networks in infectious disease epidemiology,, Journal of Biological Dynamics, 4 (2010), 478. doi: 10.1080/17513758.2010.503376. Google Scholar

[8]

N. Berglund and B. Gentz, Geometric singular perturbation theory for stochastic differential equations,, Journal of Differential Equations, 191 (2003), 1. doi: 10.1016/S0022-0396(03)00020-2. Google Scholar

[9]

N. Berglund and B. Gentz, Noise-Induced Phenomena in Slow-Fast Dynamical Systems,, Springer, (2006). Google Scholar

[10]

F. Brauer and C. Castillo-Chavez, Mathematical models in population biology and epidemiology,, Springer, (2012). doi: 10.1007/978-1-4614-1686-9. Google Scholar

[11]

F. Brauer, P. van den Driessche and J. Wu, Mathematical Epidemiology,, Vol. 1945, (1945). doi: 10.1007/978-3-540-78911-6. Google Scholar

[12]

T. Britton, Stochastic epidemic models: A survey,, Mathematical Biosciences, 225 (2010), 24. doi: 10.1016/j.mbs.2010.01.006. Google Scholar

[13]

S. Busenberg and C. Castillo-Chavez, 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,, Mathematical Medicine and Biology, 8 (1991), 1. doi: 10.1093/imammb/8.1.1. Google Scholar

[14]

S. R. Carpenter and W. A. Brock, Rising variance: A leading indicator of ecological transition,, Ecology Letters, 9 (2006), 311. doi: 10.1111/j.1461-0248.2005.00877.x. Google Scholar

[15]

C. Castillo-Chavez, W. Huang and J. Li, Competitive exclusion in gonorrhea models and other sexually transmitted diseases,, SIAM Journal on Applied Mathematics, 56 (1996), 494. doi: 10.1137/S003613999325419X. Google Scholar

[16]

D. Clancy and C. J. Pearce, The effect of population heterogeneities upon spread of infection,, Journal of Mathematical Biology, 67 (2013), 963. doi: 10.1007/s00285-012-0578-x. Google Scholar

[17]

F. A. B. Coutinho, E. Massad, L. F. Lopez and M. N. Burattini, Modelling heterogeneities in individual frailties in epidemic models,, Mathematical and Computer Modelling, 30 (1999), 97. doi: 10.1016/S0895-7177(99)00119-3. Google Scholar

[18]

S. Y. Del Valle, J. M. Hyman, H. W. Hethcote and S. G. Eubank, Mixing patterns between age groups in social networks,, Social Networks, 29 (2007), 539. Google Scholar

[19]

O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation,, Vol. 146. Wiley, (2000). Google Scholar

[20]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations,, Journal of Mathematical Biology, 28 (1990), 365. doi: 10.1007/BF00178324. Google Scholar

[21]

J. Dushoff, Host heterogeneity and disease endemicity: A moment-based approach,, Theoretical Population Biology, 56 (1999), 325. doi: 10.1006/tpbi.1999.1428. Google Scholar

[22]

G. Feichtinger, T. Tsachev and V. M. Veliov, Maximum principle for age and duration structured systems: A tool for optimal prevention and treatment of HIV,, Mathematical Population Studies, 11 (2004), 3. doi: 10.1080/08898480490422301. Google Scholar

[23]

C. Gardiner, Stochastic Methods,, 4th edition, (2009). Google Scholar

[24]

G. P. Garnett and R. M. Anderson, Sexually transmitted diseases and sexual behavior: Insights from mathematical models,, Journal of Infectious Diseases, 174 (1996), 150. doi: 10.1093/infdis/174.Supplement_2.S150. Google Scholar

[25]

B. T. Grenfell, O. N. Bjørnstad and B. F. Finkenstädt, Dynamics of measles epidemics: scaling noise, determinism, and predictability with the TSIR model,, Ecological Monographs, 72 (2002), 185. Google Scholar

[26]

T. Gross, C. J. Dommar D'Lima and B. Blasius, Epidemic dynamics on an adaptive network,, Physical Review Letters, 96 (2006). doi: 10.1103/PhysRevLett.96.208701. Google Scholar

[27]

G. Hek, Geometric singular perturbation theory in biological practice,, Journal of Mathematical Biology, 60 (2010), 347. doi: 10.1007/s00285-009-0266-7. Google Scholar

[28]

H. W. Hethcote, Qualitative analyses of communicable disease models,, Mathematical Biosciences, 28 (1976), 335. doi: 10.1016/0025-5564(76)90132-2. Google Scholar

[29]

R. I. Hickson and M. G. Roberts, How population heterogeneity in susceptibility and infectivity influences epidemic dynamics,, Journal of Theoretical Biology, 350 (2014), 70. doi: 10.1016/j.jtbi.2014.01.014. Google Scholar

[30]

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations,, SIAM Review, 43 (2001), 525. doi: 10.1137/S0036144500378302. Google Scholar

[31]

A. N. Hill and I. M. Longini, The critical vaccination fraction for heterogeneous epidemic models,, Mathematical Biosciences, 181 (2003), 85. doi: 10.1016/S0025-5564(02)00129-3. Google Scholar

[32]

A. Hill, D. G. Rand, M. A. Nowak and N. A. Christakis, Emotions as infectious diseases in a large social network: The SISa model,, Proceedings of the Royal Society B, 277 (2010), 3827. doi: 10.1098/rspb.2010.1217. Google Scholar

[33]

A. Hill, D. G. Rand, M. A. Nowak and N. A. Christakis, Infectious disease modeling of social contagion in networks,, PLoS Computational Biology, 6 (2010). doi: 10.1371/journal.pcbi.1000968. Google Scholar

[34]

T. House and M. J. Keeling, Insights from unifying modern approximations to infections on networks,, Journal of The Royal Society Interface, 8 (2011), 63. doi: 10.1098/rsif.2010.0179. Google Scholar

[35]

J. M. Hyman and E. A. Stanley, Using mathematical models to understand the AIDS epidemic,, Mathematical Biosciences, 90 (1988), 415. doi: 10.1016/0025-5564(88)90078-8. Google Scholar

[36]

H. Inaba, Endemic threshold results in an age-duration-structured population model for HIV infection,, Mathematical Biosciences, 201 (2006), 15. doi: 10.1016/j.mbs.2005.12.017. Google Scholar

[37]

A. R. Ives and V. Dakos, Detecting dynamical changes in nonlinear time series using locally linear state-space models,, Ecosphere, 3 (2012), 1. doi: 10.1890/ES11-00347.1. Google Scholar

[38]

C. K. R. T. Jones, Geometric singular perturbation theory,, in Dynamical Systems, 1609 (1995), 44. doi: 10.1007/BFb0095239. Google Scholar

[39]

I. Kareva, M. Benjamin and C. Castillo-Chavez, Resource consumption, sustainability, and cancer,, Bulletin of Mathematical Biology, 77 (2015), 319. doi: 10.1007/s11538-014-9983-1. Google Scholar

[40]

M. J. Keeling and K. T. Eames, Networks and epidemic models,, Journal of the Royal Society Interface, 2 (2005), 295. doi: 10.1098/rsif.2005.0051. Google Scholar

[41]

M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals,, Princeton University Press, (2008). Google Scholar

[42]

J. O. Kephart and S. R. White, Directed-graph epidemiological models of computer viruses,, in Proceedings. 1991 IEEE Computer Society Symposium on Research in Security and Privacy, (1991), 343. doi: 10.1109/RISP.1991.130801. Google Scholar

[43]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics,, Proceedings of the Royal Society London A, 115 (1927), 700. Google Scholar

[44]

M. Krupa and P. Szmolyan, Extending slow manifolds near transcritical and pitchfork singularities,, Nonlinearity, 14 (2001), 1473. doi: 10.1088/0951-7715/14/6/304. Google Scholar

[45]

C. Kuehn, A mathematical framework for critical transitions: Normal forms, variance and applications,, Journal of Nonlinear Science, 23 (2013), 457. doi: 10.1007/s00332-012-9158-x. Google Scholar

[46]

C. Kuehn, Multiple Time Scale Dynamics,, Springer, (2015). doi: 10.1007/978-3-319-12316-5. Google Scholar

[47]

C. Kuehn, Warning signs for wave speed transitions of noisy Fisher-KPP invasion fronts,, Theoretical Ecology, 6 (2013), 295. doi: 10.1007/s12080-013-0189-1. Google Scholar

[48]

C. Kuehn, G. Zschaler and T. Gross, Early warning signs for saddle-escape transitions in complex networks,, Scientific reports, 5 (2015). doi: 10.1038/srep13190. Google Scholar

[49]

G. E. Lahodny Jr and L. J. Allen, Probability of a disease outbreak in stochastic multipatch epidemic models,, Bulletin of Mathematical Biology, 75 (2013), 1157. doi: 10.1007/s11538-013-9848-z. Google Scholar

[50]

A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population,, Mathematical Biosciences, 28 (1976), 221. doi: 10.1016/0025-5564(76)90125-5. Google Scholar

[51]

Y. Moreno, R. Pastor-Satorras and A. Vespignani, Epidemic outbreaks in complex heterogeneous networks,, The European Physical Journal B-Condensed Matter and Complex Systems, 26 (2002), 521. doi: 10.1140/epjb/e20020122. Google Scholar

[52]

I. Nasell, The quasi-stationary distribution of the closed endemic SIS model,, Advances in Applied Probability, 28 (1996), 895. doi: 10.2307/1428186. Google Scholar

[53]

A. S. Novozhilov, Epidemiological models with parametric heterogeneity: Deterministic theory for closed populations,, Mathematical Modelling of Natural Phenomena, 7 (2012), 147. doi: 10.1051/mmnp/20127310. Google Scholar

[54]

A. S. Novozhilov, On the spread of epidemics in a closed heterogeneous population,, Mathematical Biosciences, 215 (2008), 177. doi: 10.1016/j.mbs.2008.07.010. Google Scholar

[55]

S. M. O'Regan and J. M. Drake, Theory of early warning signals of disease emergence and leading indicators of elimination,, Theoretical Ecology, 6 (2013), 333. Google Scholar

[56]

M. T. Osterholm, Preparing for the next pandemic,, New England Journal of Medicine, 352 (2005), 1839. Google Scholar

[57]

S. Schecter, Persistent unstable equilibria and closed orbits of a singularly perturbed equation,, Journal of Differential Equations, 60 (1985), 131. doi: 10.1016/0022-0396(85)90124-X. Google Scholar

[58]

M. Scheffer, J. Bascompte, W. A. Brock, V. Brovkhin, S. R. Carpenter, V. Dakos, H. Held, E. H. van Nes, M. Rietkerk and G. Sugihara, Early-warning signals for critical transitions,, Nature, 461 (2009), 53. doi: 10.1038/nature08227. Google Scholar

[59]

L. B. Shaw and I. B. Schwartz, Fluctuating epidemics on adaptive networks,, Physical Review E, 77 (2008). doi: 10.1103/PhysRevE.77.066101. Google Scholar

[60]

H. Shi, Z. Duan and G. Chen, An SIS model with infective medium on complex networks,, Physica A, 387 (2008), 2133. doi: 10.1016/j.physa.2007.11.048. Google Scholar

[61]

V. M. Veliov, On the effect of population heterogeneity on dynamics of epidemic diseases,, Journal of Mathematical Biology, 51 (2005), 123. doi: 10.1007/s00285-004-0288-0. Google Scholar

[62]

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