American Institute of Mathematical Sciences

Advanced
2014, 11(6): 1295-1317. doi: 10.3934/mbe.2014.11.1295

Epidemic models for complex networks with demographics

Zhen Jin 1, , Guiquan Sun 1, and  Huaiping Zhu 2,

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

Received  March 2014 Revised  June 2014 Published  September 2014

In this paper, we propose and study network epidemic models with demographics for disease transmission. We obtain the formula of the basic reproduction number $R_{0}$ of infection for an SIS model with births or recruitment and death rate. We prove that if $R_{0}\leq1$, infection-free equilibrium of SIS model is globally asymptotically stable; if $R_{0}>1$, there exists a unique endemic equilibrium which is globally asymptotically stable. It is also found that demographics has great effect on basic reproduction number $R_{0}$. Furthermore, the degree distribution of population varies with time before it reaches the stationary state.
Keywords: complex networks, demographics, Epidemic models, basic reproduction number, global stability..
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Zhen Jin, Guiquan Sun, Huaiping Zhu. Epidemic models for complex networks with demographics. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1295-1317. doi: 10.3934/mbe.2014.11.1295
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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