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Epidemic models for complex networks with demographics
1. | Complex Systems Research Center, Shanxi University, Taiyuan, Shanxi 030051, China |
2. | LAMPS and CDM, Department of Mathematics and Statistics, York University, Toronto, ON, M3J1P3, Canada |
References:
[1] |
R. M. Anderson and R. M. May, Infectious Diseases of Humans, Oxford University Press, Oxford, 1992. |
[2] |
A.-L. Barabasi and R. Albert, Emergence of scaling in random networks, Science, 286 (1999), 509-511.
doi: 10.1126/science.286.5439.509. |
[3] |
M. Barthelemy, A. Barrat, R. Pastor-Satorras and A. Vespignani, Dynamical patterns of epidemic outbreaks in complex heterogeneous networks, Journal of Theoretical Biology, 235 (2005), 275-288.
doi: 10.1016/j.jtbi.2005.01.011. |
[4] |
E. Ben-Naim and P. L. Krapivsky, Addition-deletion networks, J. Phys. A: Math. Theor., 40 (2007), 8607-8619.
doi: 10.1088/1751-8113/40/30/001. |
[5] |
M. Boguna, R. Pastor-Satorras and A. Vespignani, Epidemic spreading in complex networks with degree correlations, e-print cond-mat/0301149, (2003). |
[6] |
S. Busenberg and P. van den Driessche, Analysis of a disease transmission model in a population with varying size, J. Math. Biol., 28 (1990), 257-270.
doi: 10.1007/BF00178776. |
[7] |
C. Castillo-Chavez and H. R. Thieme, Asymptotically autonomous epidemic models, in Mathematical Population Dynamics: Analysis of Heterogeneity (eds. O. Arino, D. Axelrod, M. Kimmel and M. Langlais), Theory of Epidemics, 1, Wuerz, Winnipeg, 1993, 33-50. |
[8] |
K. Emrah, Explicit formula for the inverse of a tridiagonal matrix by backward continued fractions, Applied Mathematics and Computation, 197 (2008), 345-357.
doi: 10.1016/j.amc.2007.07.046. |
[9] |
L. Q. Gao and H. W. Hethcote, Disease transmission models with density-dependent demographics, J. Math. Biol., 30 (1992), 717-731.
doi: 10.1007/BF00173265. |
[10] |
L. Hufnagel, D. Brockmann and T. Geisel, Forecast and control of epidemics in a globalized world, Proc. Natl. Acad. Sci. U.S.A., 101 (2004), 15124.
doi: 10.1073/pnas.0308344101. |
[11] |
Y. Jin and W. Wang, The effect of population dispersal on the spread of a disease, J. Math. Anal. Appl., 308 (2005), 343-364.
doi: 10.1016/j.jmaa.2005.01.034. |
[12] |
J. Joo and J. L. Lebowitz, Behavior of susceptible-infected-susceptible epidemics on heterogeneous networks with saturation, Phys. Rev. E, 69 (2004), 066105.
doi: 10.1103/PhysRevE.69.066105. |
[13] |
M. J. Keeling and K. T. D. Eames, Networks and epidemic models, J. R. Soc. Interface, 2 (2005), 295-307.
doi: 10.1098/rsif.2005.0051. |
[14] |
M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, 2007. |
[15] |
W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. A, 115 (1927), 700-711.
doi: 10.1098/rspa.1927.0118. |
[16] |
I. Z. Kiss, D. M. Green and R. R. Kao, Heterogeneity and multiple of transmission on final epidemic size, Mathematical Biosciences, 203 (2006), 124-136.
doi: 10.1016/j.mbs.2006.03.002. |
[17] |
I. Z. Kiss, P. L. Simon and R. R. Kao, A contact-network-based formulation of a preferential mixing model, Bulletin of Mathematical Biology, 71 (2009), 888-905.
doi: 10.1007/s11538-008-9386-2. |
[18] |
J. Lindquist, J. Ma, P. van den Driessche and F. H. Willeboords, Network evolution by different rewiring schemes, Physica D, 238 (2009), 370-378.
doi: 10.1016/j.physd.2008.10.016. |
[19] |
Z. Ma and J. Li, Dynamical Modeling and Anaylsis of Epidemics, World Scientific, 2009. |
[20] |
R. M. May and A. L. Lloyd, Infection dynamics on scale-free networks, Phys. Rev. E, 64 (2001), 066112.
doi: 10.1103/PhysRevE.64.066112. |
[21] |
Y. Moreno, R. Pastor-Satorras and A. Vespignani, Epidemic outbreaks in complex heterogeneous networks, Eur. Phys. J. B, 26 (2002), 521-529.
doi: 10.1140/epjb/e20020122. |
[22] |
R. Olinky and L. Stone, Unexpected epidemic thresholds in heterogeneous networks: The role of disease transmission, Phys. Rev. E, 70 (2004), 030902.
doi: 10.1103/PhysRevE.70.030902. |
[23] |
R. Pastor-Satorras and A. Vespignani, Epidemic dynamics and endemic states in complex networks, Phys. Rev. E, 63 (2001), 066117.
doi: 10.1103/PhysRevE.63.066117. |
[24] |
R. Pastor-Satorras and A. Vespignani, Epidemic spreading in scale-free networks, Phys. Rev. Let., 86 (2001), 3200.
doi: 10.1103/PhysRevLett.86.3200. |
[25] |
M. G. Roberta, An SEI model with density-dependent demographics and epidemiology, IMA Journal of Mathematics Applied in Medicine & Biology, 13 (1996), 245-257.
doi: 10.1093/imammb13.4.245. |
[26] |
L. B. Shaw and I. B. Schwartz, Fluctuating epidemics on adaptive networks, Phys. Rev. E, 77 (2008), 066101.
doi: 10.1103/PhysRevE.77.066101. |
[27] |
H. L. Smith, On the asymptotic behavior of a class of deterministic models of cooperating species, SIAM J. Appl. Math., 46 (1986), 368-375.
doi: 10.1137/0146025. |
[28] |
H. R. Thieme, Asymptotically autonomous differential equations in the plane, Rocky Mountain J. Math., 24 (1994), 351-380.
doi: 10.1216/rmjm/1181072470. |
[29] |
P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.
doi: 10.1016/S0025-5564(02)00108-6. |
[30] |
L. Wang and G. Z. Dai, Global stability of virus spreading in complex heterogeneous networks, Siam J. Appl. Math., 68 (2008), 1495-1502.
doi: 10.1137/070694582. |
[31] |
W. Wang and X.-Q. Zhao, An epidemic model in a patchy environment, Mathematical Biosciences, 190 (2004), 97-112.
doi: 10.1016/j.mbs.2002.11.001. |
[32] |
X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003.
doi: 10.1007/978-0-387-21761-1. |
[33] |
X.-Q. Zhao and Z.-J. Jing, Global asymptotic behavior in some cooperative systems of functional differential equations, Canad. Appl. Math. Quart., 4 (1996), 421-444. |
show all references
References:
[1] |
R. M. Anderson and R. M. May, Infectious Diseases of Humans, Oxford University Press, Oxford, 1992. |
[2] |
A.-L. Barabasi and R. Albert, Emergence of scaling in random networks, Science, 286 (1999), 509-511.
doi: 10.1126/science.286.5439.509. |
[3] |
M. Barthelemy, A. Barrat, R. Pastor-Satorras and A. Vespignani, Dynamical patterns of epidemic outbreaks in complex heterogeneous networks, Journal of Theoretical Biology, 235 (2005), 275-288.
doi: 10.1016/j.jtbi.2005.01.011. |
[4] |
E. Ben-Naim and P. L. Krapivsky, Addition-deletion networks, J. Phys. A: Math. Theor., 40 (2007), 8607-8619.
doi: 10.1088/1751-8113/40/30/001. |
[5] |
M. Boguna, R. Pastor-Satorras and A. Vespignani, Epidemic spreading in complex networks with degree correlations, e-print cond-mat/0301149, (2003). |
[6] |
S. Busenberg and P. van den Driessche, Analysis of a disease transmission model in a population with varying size, J. Math. Biol., 28 (1990), 257-270.
doi: 10.1007/BF00178776. |
[7] |
C. Castillo-Chavez and H. R. Thieme, Asymptotically autonomous epidemic models, in Mathematical Population Dynamics: Analysis of Heterogeneity (eds. O. Arino, D. Axelrod, M. Kimmel and M. Langlais), Theory of Epidemics, 1, Wuerz, Winnipeg, 1993, 33-50. |
[8] |
K. Emrah, Explicit formula for the inverse of a tridiagonal matrix by backward continued fractions, Applied Mathematics and Computation, 197 (2008), 345-357.
doi: 10.1016/j.amc.2007.07.046. |
[9] |
L. Q. Gao and H. W. Hethcote, Disease transmission models with density-dependent demographics, J. Math. Biol., 30 (1992), 717-731.
doi: 10.1007/BF00173265. |
[10] |
L. Hufnagel, D. Brockmann and T. Geisel, Forecast and control of epidemics in a globalized world, Proc. Natl. Acad. Sci. U.S.A., 101 (2004), 15124.
doi: 10.1073/pnas.0308344101. |
[11] |
Y. Jin and W. Wang, The effect of population dispersal on the spread of a disease, J. Math. Anal. Appl., 308 (2005), 343-364.
doi: 10.1016/j.jmaa.2005.01.034. |
[12] |
J. Joo and J. L. Lebowitz, Behavior of susceptible-infected-susceptible epidemics on heterogeneous networks with saturation, Phys. Rev. E, 69 (2004), 066105.
doi: 10.1103/PhysRevE.69.066105. |
[13] |
M. J. Keeling and K. T. D. Eames, Networks and epidemic models, J. R. Soc. Interface, 2 (2005), 295-307.
doi: 10.1098/rsif.2005.0051. |
[14] |
M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, 2007. |
[15] |
W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. A, 115 (1927), 700-711.
doi: 10.1098/rspa.1927.0118. |
[16] |
I. Z. Kiss, D. M. Green and R. R. Kao, Heterogeneity and multiple of transmission on final epidemic size, Mathematical Biosciences, 203 (2006), 124-136.
doi: 10.1016/j.mbs.2006.03.002. |
[17] |
I. Z. Kiss, P. L. Simon and R. R. Kao, A contact-network-based formulation of a preferential mixing model, Bulletin of Mathematical Biology, 71 (2009), 888-905.
doi: 10.1007/s11538-008-9386-2. |
[18] |
J. Lindquist, J. Ma, P. van den Driessche and F. H. Willeboords, Network evolution by different rewiring schemes, Physica D, 238 (2009), 370-378.
doi: 10.1016/j.physd.2008.10.016. |
[19] |
Z. Ma and J. Li, Dynamical Modeling and Anaylsis of Epidemics, World Scientific, 2009. |
[20] |
R. M. May and A. L. Lloyd, Infection dynamics on scale-free networks, Phys. Rev. E, 64 (2001), 066112.
doi: 10.1103/PhysRevE.64.066112. |
[21] |
Y. Moreno, R. Pastor-Satorras and A. Vespignani, Epidemic outbreaks in complex heterogeneous networks, Eur. Phys. J. B, 26 (2002), 521-529.
doi: 10.1140/epjb/e20020122. |
[22] |
R. Olinky and L. Stone, Unexpected epidemic thresholds in heterogeneous networks: The role of disease transmission, Phys. Rev. E, 70 (2004), 030902.
doi: 10.1103/PhysRevE.70.030902. |
[23] |
R. Pastor-Satorras and A. Vespignani, Epidemic dynamics and endemic states in complex networks, Phys. Rev. E, 63 (2001), 066117.
doi: 10.1103/PhysRevE.63.066117. |
[24] |
R. Pastor-Satorras and A. Vespignani, Epidemic spreading in scale-free networks, Phys. Rev. Let., 86 (2001), 3200.
doi: 10.1103/PhysRevLett.86.3200. |
[25] |
M. G. Roberta, An SEI model with density-dependent demographics and epidemiology, IMA Journal of Mathematics Applied in Medicine & Biology, 13 (1996), 245-257.
doi: 10.1093/imammb13.4.245. |
[26] |
L. B. Shaw and I. B. Schwartz, Fluctuating epidemics on adaptive networks, Phys. Rev. E, 77 (2008), 066101.
doi: 10.1103/PhysRevE.77.066101. |
[27] |
H. L. Smith, On the asymptotic behavior of a class of deterministic models of cooperating species, SIAM J. Appl. Math., 46 (1986), 368-375.
doi: 10.1137/0146025. |
[28] |
H. R. Thieme, Asymptotically autonomous differential equations in the plane, Rocky Mountain J. Math., 24 (1994), 351-380.
doi: 10.1216/rmjm/1181072470. |
[29] |
P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.
doi: 10.1016/S0025-5564(02)00108-6. |
[30] |
L. Wang and G. Z. Dai, Global stability of virus spreading in complex heterogeneous networks, Siam J. Appl. Math., 68 (2008), 1495-1502.
doi: 10.1137/070694582. |
[31] |
W. Wang and X.-Q. Zhao, An epidemic model in a patchy environment, Mathematical Biosciences, 190 (2004), 97-112.
doi: 10.1016/j.mbs.2002.11.001. |
[32] |
X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003.
doi: 10.1007/978-0-387-21761-1. |
[33] |
X.-Q. Zhao and Z.-J. Jing, Global asymptotic behavior in some cooperative systems of functional differential equations, Canad. Appl. Math. Quart., 4 (1996), 421-444. |
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