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A two-sex matrix population model to represent harem structure
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 |
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. |
[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. |
[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. |
[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. |
[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.
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.
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.
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, (2012).
doi: 10.1007/978-1-4614-1686-9. |
[11] |
F. Brauer, P. van den Driessche and J. Wu, Mathematical Epidemiology,, Vol. 1945, (1945).
doi: 10.1007/978-3-540-78911-6. |
[12] |
T. Britton, Stochastic epidemic models: A survey,, Mathematical Biosciences, 225 (2010), 24.
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.
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.
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.
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.
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.
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. Google Scholar |
[19] |
O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation,, Vol. 146. Wiley, (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.
doi: 10.1007/BF00178324. |
[21] |
J. Dushoff, Host heterogeneity and disease endemicity: A moment-based approach,, Theoretical Population Biology, 56 (1999), 325.
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.
doi: 10.1080/08898480490422301. |
[23] |
C. Gardiner, Stochastic Methods,, 4th edition, (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.
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. 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. |
[27] |
G. Hek, Geometric singular perturbation theory in biological practice,, Journal of Mathematical Biology, 60 (2010), 347.
doi: 10.1007/s00285-009-0266-7. |
[28] |
H. W. Hethcote, Qualitative analyses of communicable disease models,, Mathematical Biosciences, 28 (1976), 335.
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.
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.
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.
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.
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).
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.
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.
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.
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.
doi: 10.1890/ES11-00347.1. |
[38] |
C. K. R. T. Jones, Geometric singular perturbation theory,, in Dynamical Systems, 1609 (1995), 44.
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.
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.
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.
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. 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. |
[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. |
[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.
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).
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.
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.
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.
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.
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.
doi: 10.1051/mmnp/20127310. |
[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. |
[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. |
[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. |
[59] |
L. B. Shaw and I. B. Schwartz, Fluctuating epidemics on adaptive networks,, Physical Review E, 77 (2008).
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.
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.
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.
doi: 10.1016/j.jmaa.2008.05.012. |
[63] |
K. Wiesenfeld, Noisy precursors of nonlinear instabilities,, Journal of Statistical Physics, 38 (1985), 1071.
doi: 10.1007/BF01010430. |
[64] |
J. C. Wierman and D. J. Marchette, Modeling computer virus prevalence with a susceptible-infected-susceptible model with reintroduction,, Computational statistics & data analysis, 45 (2004), 3.
doi: 10.1016/S0167-9473(03)00113-0. |
[65] |
M. Yang, G. Chen and X. Fu, A modified SIS model with an infective medium on complex networks and its global stability,, Physica A, 390 (2011), 2408.
doi: 10.1016/j.physa.2011.02.007. |
[66] |
J. A. Yorke, H. W. Hethcote and A. Nold, Dynamics and control of the transmission of gonorrhea,, Sexually transmitted diseases, 5 (1978), 51.
doi: 10.1097/00007435-197804000-00003. |
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. |
[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. |
[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. |
[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. |
[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.
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.
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.
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, (2012).
doi: 10.1007/978-1-4614-1686-9. |
[11] |
F. Brauer, P. van den Driessche and J. Wu, Mathematical Epidemiology,, Vol. 1945, (1945).
doi: 10.1007/978-3-540-78911-6. |
[12] |
T. Britton, Stochastic epidemic models: A survey,, Mathematical Biosciences, 225 (2010), 24.
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.
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.
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.
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.
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.
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. Google Scholar |
[19] |
O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation,, Vol. 146. Wiley, (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.
doi: 10.1007/BF00178324. |
[21] |
J. Dushoff, Host heterogeneity and disease endemicity: A moment-based approach,, Theoretical Population Biology, 56 (1999), 325.
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.
doi: 10.1080/08898480490422301. |
[23] |
C. Gardiner, Stochastic Methods,, 4th edition, (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.
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. 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. |
[27] |
G. Hek, Geometric singular perturbation theory in biological practice,, Journal of Mathematical Biology, 60 (2010), 347.
doi: 10.1007/s00285-009-0266-7. |
[28] |
H. W. Hethcote, Qualitative analyses of communicable disease models,, Mathematical Biosciences, 28 (1976), 335.
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.
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.
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.
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.
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).
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.
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.
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.
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.
doi: 10.1890/ES11-00347.1. |
[38] |
C. K. R. T. Jones, Geometric singular perturbation theory,, in Dynamical Systems, 1609 (1995), 44.
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.
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.
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.
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. 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. |
[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. |
[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.
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).
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.
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.
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.
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.
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.
doi: 10.1051/mmnp/20127310. |
[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. |
[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. |
[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. |
[59] |
L. B. Shaw and I. B. Schwartz, Fluctuating epidemics on adaptive networks,, Physical Review E, 77 (2008).
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.
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.
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.
doi: 10.1016/j.jmaa.2008.05.012. |
[63] |
K. Wiesenfeld, Noisy precursors of nonlinear instabilities,, Journal of Statistical Physics, 38 (1985), 1071.
doi: 10.1007/BF01010430. |
[64] |
J. C. Wierman and D. J. Marchette, Modeling computer virus prevalence with a susceptible-infected-susceptible model with reintroduction,, Computational statistics & data analysis, 45 (2004), 3.
doi: 10.1016/S0167-9473(03)00113-0. |
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