Heterogeneous population dynamics and scaling laws near epidemic outbreaks

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

Heterogeneous population dynamics and scaling laws near epidemic outbreaks

Andreas Widder ^1, and Christian Kuehn ^2,

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

Abstract
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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.

Keywords: warning signs, critical transition, transcritical bifurcation, heterogeneous population, stochastic perturbation, SIS-model, reproduction number., Epidemics, tipping point.

Mathematics Subject Classification: 34C60, 34D23, 37N25, 45J05, 91B69, 92B0.

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:

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[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), 021024. doi: 10.1103/PhysRevX.4.021024.
[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), 467-484. doi: 10.1111/j.1461-0248.2005.00879.x.
[4]	B. Ainseba and M. Iannelli, Optimal screening in structured SIR epidemics, Mathematical Modelling of Natural Phenomena, 7 (2012), 12-27. doi: 10.1051/mmnp/20127302.
[5]	L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, 1974.
[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), 879-891. doi: 10.1098/rsif.2007.1100.
[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), 478-489. doi: 10.1080/17513758.2010.503376.
[8]	N. Berglund and B. Gentz, Geometric singular perturbation theory for stochastic differential equations, Journal of Differential Equations, 191 (2003), 1-54. doi: 10.1016/S0022-0396(03)00020-2.
[9]	N. Berglund and B. Gentz, Noise-Induced Phenomena in Slow-Fast Dynamical Systems, Springer, 2006.
[10]	F. Brauer and C. Castillo-Chavez, Mathematical models in population biology and epidemiology, Springer, New York, 2012. doi: 10.1007/978-1-4614-1686-9.
[11]	F. Brauer, P. van den Driessche and J. Wu, Mathematical Epidemiology, Vol. 1945, Springer, 2008. doi: 10.1007/978-3-540-78911-6.
[12]	T. Britton, Stochastic epidemic models: A survey, Mathematical Biosciences, 225 (2010), 24-35. doi: 10.1016/j.mbs.2010.01.006.
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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), 208701. doi: 10.1103/PhysRevLett.96.208701.
[27]	G. Hek, Geometric singular perturbation theory in biological practice, Journal of Mathematical Biology, 60 (2010), 347-386. doi: 10.1007/s00285-009-0266-7.
[28]	H. W. Hethcote, Qualitative analyses of communicable disease models, Mathematical Biosciences, 28 (1976), 335-356. doi: 10.1016/0025-5564(76)90132-2.
[29]	R. I. Hickson and M. G. Roberts, How population heterogeneity in susceptibility and infectivity influences epidemic dynamics, Journal of Theoretical Biology, 350 (2014), 70-80. doi: 10.1016/j.jtbi.2014.01.014.
[30]	D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546. doi: 10.1137/S0036144500378302.
[31]	A. N. Hill and I. M. Longini, The critical vaccination fraction for heterogeneous epidemic models, Mathematical Biosciences, 181 (2003), 85-106. doi: 10.1016/S0025-5564(02)00129-3.
[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), 3827-3835. doi: 10.1098/rspb.2010.1217.
[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), e1000968, 15pp. doi: 10.1371/journal.pcbi.1000968.
[34]	T. House and M. J. Keeling, Insights from unifying modern approximations to infections on networks, Journal of The Royal Society Interface, 8 (2011), 63-73. doi: 10.1098/rsif.2010.0179.
[35]	J. M. Hyman and E. A. Stanley, Using mathematical models to understand the AIDS epidemic, Mathematical Biosciences, 90 (1988), 415-473. doi: 10.1016/0025-5564(88)90078-8.
[36]	H. Inaba, Endemic threshold results in an age-duration-structured population model for HIV infection, Mathematical Biosciences, 201 (2006), 15-47. doi: 10.1016/j.mbs.2005.12.017.
[37]	A. R. Ives and V. Dakos, Detecting dynamical changes in nonlinear time series using locally linear state-space models, Ecosphere, 3 (2012), 1-15. doi: 10.1890/ES11-00347.1.
[38]	C. K. R. T. Jones, Geometric singular perturbation theory, in Dynamical Systems, Springer, 1609 (1995), 44-118. doi: 10.1007/BFb0095239.
[39]	I. Kareva, M. Benjamin and C. Castillo-Chavez, Resource consumption, sustainability, and cancer, Bulletin of Mathematical Biology, 77 (2015), 319-338. doi: 10.1007/s11538-014-9983-1.
[40]	M. J. Keeling and K. T. Eames, Networks and epidemic models, Journal of the Royal Society Interface, 2 (2005), 295-307. doi: 10.1098/rsif.2005.0051.
[41]	M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, 2008.
[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, 343-359. doi: 10.1109/RISP.1991.130801.
[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), 700-721.
[44]	M. Krupa and P. Szmolyan, Extending slow manifolds near transcritical and pitchfork singularities, Nonlinearity, 14 (2001), 1473-1491. doi: 10.1088/0951-7715/14/6/304.
[45]	C. Kuehn, A mathematical framework for critical transitions: Normal forms, variance and applications, Journal of Nonlinear Science, 23 (2013), 457-510. doi: 10.1007/s00332-012-9158-x.
[46]	C. Kuehn, Multiple Time Scale Dynamics, Springer, 2015. doi: 10.1007/978-3-319-12316-5.
[47]	C. Kuehn, Warning signs for wave speed transitions of noisy Fisher-KPP invasion fronts, Theoretical Ecology, 6 (2013), 295-308. doi: 10.1007/s12080-013-0189-1.
[48]	C. Kuehn, G. Zschaler and T. Gross, Early warning signs for saddle-escape transitions in complex networks, Scientific reports, 5 (2015), art.nr. 13190. doi: 10.1038/srep13190.
[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), 1157-1180. doi: 10.1007/s11538-013-9848-z.
[50]	A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Mathematical Biosciences, 28 (1976), 221-236. doi: 10.1016/0025-5564(76)90125-5.
[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), 521-529. doi: 10.1140/epjb/e20020122.
[52]	I. Nasell, The quasi-stationary distribution of the closed endemic SIS model, Advances in Applied Probability, 28 (1996), 895-932. doi: 10.2307/1428186.
[53]	A. S. Novozhilov, Epidemiological models with parametric heterogeneity: Deterministic theory for closed populations, Mathematical Modelling of Natural Phenomena, 7 (2012), 147-167. doi: 10.1051/mmnp/20127310.
[54]	A. S. Novozhilov, On the spread of epidemics in a closed heterogeneous population, Mathematical Biosciences, 215 (2008), 177-185. doi: 10.1016/j.mbs.2008.07.010.
[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), 333-357.
[56]	M. T. Osterholm, Preparing for the next pandemic, New England Journal of Medicine, 352 (2005), 1839-1842.
[57]	S. Schecter, Persistent unstable equilibria and closed orbits of a singularly perturbed equation, Journal of Differential Equations, 60 (1985), 131-141. doi: 10.1016/0022-0396(85)90124-X.
[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), 53-59. doi: 10.1038/nature08227.
[59]	L. B. Shaw and I. B. Schwartz, Fluctuating epidemics on adaptive networks, Physical Review E, 77 (2008), 066101, 10pp. doi: 10.1103/PhysRevE.77.066101.
[60]	H. Shi, Z. Duan and G. Chen, An SIS model with infective medium on complex networks, Physica A, 387 (2008), 2133-2144. doi: 10.1016/j.physa.2007.11.048.
[61]	V. M. Veliov, On the effect of population heterogeneity on dynamics of epidemic diseases, Journal of Mathematical Biology, 51 (2005), 123-143. doi: 10.1007/s00285-004-0288-0.
[62]	V. M. Veliov, Optimal control of heterogeneous systems: Basic theory, Journal of Mathematical Analysis and Applications, 346 (2008), 227-242. doi: 10.1016/j.jmaa.2008.05.012.
[63]	K. Wiesenfeld, Noisy precursors of nonlinear instabilities, Journal of Statistical Physics, 38 (1985), 1071-1097. doi: 10.1007/BF01010430.
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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), 73-86. doi: 10.1080/17513750802162754.
[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), 021024. doi: 10.1103/PhysRevX.4.021024.
[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), 467-484. doi: 10.1111/j.1461-0248.2005.00879.x.
[4]	B. Ainseba and M. Iannelli, Optimal screening in structured SIR epidemics, Mathematical Modelling of Natural Phenomena, 7 (2012), 12-27. doi: 10.1051/mmnp/20127302.
[5]	L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, 1974.
[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), 879-891. doi: 10.1098/rsif.2007.1100.
[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), 478-489. doi: 10.1080/17513758.2010.503376.
[8]	N. Berglund and B. Gentz, Geometric singular perturbation theory for stochastic differential equations, Journal of Differential Equations, 191 (2003), 1-54. doi: 10.1016/S0022-0396(03)00020-2.
[9]	N. Berglund and B. Gentz, Noise-Induced Phenomena in Slow-Fast Dynamical Systems, Springer, 2006.
[10]	F. Brauer and C. Castillo-Chavez, Mathematical models in population biology and epidemiology, Springer, New York, 2012. doi: 10.1007/978-1-4614-1686-9.
[11]	F. Brauer, P. van den Driessche and J. Wu, Mathematical Epidemiology, Vol. 1945, Springer, 2008. doi: 10.1007/978-3-540-78911-6.
[12]	T. Britton, Stochastic epidemic models: A survey, Mathematical Biosciences, 225 (2010), 24-35. doi: 10.1016/j.mbs.2010.01.006.
[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), 1-29. doi: 10.1093/imammb/8.1.1.
[14]	S. R. Carpenter and W. A. Brock, Rising variance: A leading indicator of ecological transition, Ecology Letters, 9 (2006), 311-318. doi: 10.1111/j.1461-0248.2005.00877.x.
[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), 494-508. doi: 10.1137/S003613999325419X.
[16]	D. Clancy and C. J. Pearce, The effect of population heterogeneities upon spread of infection, Journal of Mathematical Biology, 67 (2013), 963-978. doi: 10.1007/s00285-012-0578-x.
[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), 97-115. doi: 10.1016/S0895-7177(99)00119-3.
[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), 539-554.
[19]	O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, Vol. 146. Wiley, Chichester, 2000.
[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), 365-382. doi: 10.1007/BF00178324.
[21]	J. Dushoff, Host heterogeneity and disease endemicity: A moment-based approach, Theoretical Population Biology, 56 (1999), 325-335. doi: 10.1006/tpbi.1999.1428.
[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), 3-28. doi: 10.1080/08898480490422301.
[23]	C. Gardiner, Stochastic Methods, 4th edition, Springer, Berlin Heidelberg, Germany, 2009.
[24]	G. P. Garnett and R. M. Anderson, Sexually transmitted diseases and sexual behavior: Insights from mathematical models, Journal of Infectious Diseases, 174 (1996), 150-161. doi: 10.1093/infdis/174.Supplement_2.S150.
[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), 185-202.
[26]	T. Gross, C. J. Dommar D'Lima and B. Blasius, Epidemic dynamics on an adaptive network, Physical Review Letters, 96 (2006), 208701. doi: 10.1103/PhysRevLett.96.208701.
[27]	G. Hek, Geometric singular perturbation theory in biological practice, Journal of Mathematical Biology, 60 (2010), 347-386. doi: 10.1007/s00285-009-0266-7.
[28]	H. W. Hethcote, Qualitative analyses of communicable disease models, Mathematical Biosciences, 28 (1976), 335-356. doi: 10.1016/0025-5564(76)90132-2.
[29]	R. I. Hickson and M. G. Roberts, How population heterogeneity in susceptibility and infectivity influences epidemic dynamics, Journal of Theoretical Biology, 350 (2014), 70-80. doi: 10.1016/j.jtbi.2014.01.014.
[30]	D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546. doi: 10.1137/S0036144500378302.
[31]	A. N. Hill and I. M. Longini, The critical vaccination fraction for heterogeneous epidemic models, Mathematical Biosciences, 181 (2003), 85-106. doi: 10.1016/S0025-5564(02)00129-3.
[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), 3827-3835. doi: 10.1098/rspb.2010.1217.
[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), e1000968, 15pp. doi: 10.1371/journal.pcbi.1000968.
[34]	T. House and M. J. Keeling, Insights from unifying modern approximations to infections on networks, Journal of The Royal Society Interface, 8 (2011), 63-73. doi: 10.1098/rsif.2010.0179.
[35]	J. M. Hyman and E. A. Stanley, Using mathematical models to understand the AIDS epidemic, Mathematical Biosciences, 90 (1988), 415-473. doi: 10.1016/0025-5564(88)90078-8.
[36]	H. Inaba, Endemic threshold results in an age-duration-structured population model for HIV infection, Mathematical Biosciences, 201 (2006), 15-47. doi: 10.1016/j.mbs.2005.12.017.
[37]	A. R. Ives and V. Dakos, Detecting dynamical changes in nonlinear time series using locally linear state-space models, Ecosphere, 3 (2012), 1-15. doi: 10.1890/ES11-00347.1.
[38]	C. K. R. T. Jones, Geometric singular perturbation theory, in Dynamical Systems, Springer, 1609 (1995), 44-118. doi: 10.1007/BFb0095239.
[39]	I. Kareva, M. Benjamin and C. Castillo-Chavez, Resource consumption, sustainability, and cancer, Bulletin of Mathematical Biology, 77 (2015), 319-338. doi: 10.1007/s11538-014-9983-1.
[40]	M. J. Keeling and K. T. Eames, Networks and epidemic models, Journal of the Royal Society Interface, 2 (2005), 295-307. doi: 10.1098/rsif.2005.0051.
[41]	M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, 2008.
[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, 343-359. doi: 10.1109/RISP.1991.130801.
[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), 700-721.
[44]	M. Krupa and P. Szmolyan, Extending slow manifolds near transcritical and pitchfork singularities, Nonlinearity, 14 (2001), 1473-1491. doi: 10.1088/0951-7715/14/6/304.
[45]	C. Kuehn, A mathematical framework for critical transitions: Normal forms, variance and applications, Journal of Nonlinear Science, 23 (2013), 457-510. doi: 10.1007/s00332-012-9158-x.
[46]	C. Kuehn, Multiple Time Scale Dynamics, Springer, 2015. doi: 10.1007/978-3-319-12316-5.
[47]	C. Kuehn, Warning signs for wave speed transitions of noisy Fisher-KPP invasion fronts, Theoretical Ecology, 6 (2013), 295-308. doi: 10.1007/s12080-013-0189-1.
[48]	C. Kuehn, G. Zschaler and T. Gross, Early warning signs for saddle-escape transitions in complex networks, Scientific reports, 5 (2015), art.nr. 13190. doi: 10.1038/srep13190.
[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), 1157-1180. doi: 10.1007/s11538-013-9848-z.
[50]	A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Mathematical Biosciences, 28 (1976), 221-236. doi: 10.1016/0025-5564(76)90125-5.
[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), 521-529. doi: 10.1140/epjb/e20020122.
[52]	I. Nasell, The quasi-stationary distribution of the closed endemic SIS model, Advances in Applied Probability, 28 (1996), 895-932. doi: 10.2307/1428186.
[53]	A. S. Novozhilov, Epidemiological models with parametric heterogeneity: Deterministic theory for closed populations, Mathematical Modelling of Natural Phenomena, 7 (2012), 147-167. doi: 10.1051/mmnp/20127310.
[54]	A. S. Novozhilov, On the spread of epidemics in a closed heterogeneous population, Mathematical Biosciences, 215 (2008), 177-185. doi: 10.1016/j.mbs.2008.07.010.
[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), 333-357.
[56]	M. T. Osterholm, Preparing for the next pandemic, New England Journal of Medicine, 352 (2005), 1839-1842.
[57]	S. Schecter, Persistent unstable equilibria and closed orbits of a singularly perturbed equation, Journal of Differential Equations, 60 (1985), 131-141. doi: 10.1016/0022-0396(85)90124-X.
[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), 53-59. doi: 10.1038/nature08227.
[59]	L. B. Shaw and I. B. Schwartz, Fluctuating epidemics on adaptive networks, Physical Review E, 77 (2008), 066101, 10pp. doi: 10.1103/PhysRevE.77.066101.
[60]	H. Shi, Z. Duan and G. Chen, An SIS model with infective medium on complex networks, Physica A, 387 (2008), 2133-2144. doi: 10.1016/j.physa.2007.11.048.
[61]	V. M. Veliov, On the effect of population heterogeneity on dynamics of epidemic diseases, Journal of Mathematical Biology, 51 (2005), 123-143. doi: 10.1007/s00285-004-0288-0.
[62]	V. M. Veliov, Optimal control of heterogeneous systems: Basic theory, Journal of Mathematical Analysis and Applications, 346 (2008), 227-242. doi: 10.1016/j.jmaa.2008.05.012.
[63]	K. Wiesenfeld, Noisy precursors of nonlinear instabilities, Journal of Statistical Physics, 38 (1985), 1071-1097. doi: 10.1007/BF01010430.
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