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July  2015, 20(5): 1393-1404. doi: 10.3934/dcdsb.2015.20.1393

The dynamics of an HBV epidemic model on complex heterogeneous networks

Meihong Qiao 1, , Anping Liu 1, and  Qing Tang 1,

1. 

School of Mathematics & Physics, China University of Geoscience, Wuhan 430074, Hubei Province, China, China, China

Received  April 2014 Revised  January 2015 Published  May 2015

In this paper, an HBV epidemic model on complex heterogeneous networks is proposed. Theoretical analysis of the HBV spreading dynamics is presented via mean-field approximation. Stabilities of the disease-free equilibrium and the endemic equilibrium are studied. The theoretical results reveal that disease propagation is impacted by the heterogeneous connectivity patterns and the underlying network structures.
Keywords: mathematical model, Heterogeneous network, equilibrium., HBV, global stability.
Mathematics Subject Classification: Primary: 32G3.
Citation: Meihong Qiao, Anping Liu, Qing Tang. The dynamics of an HBV epidemic model on complex heterogeneous networks. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1393-1404. doi: 10.3934/dcdsb.2015.20.1393
References:
[1]

H. W. Hethcote, The mathematics of infectious diseases,, SIAM Rev, 42 (2000), 599.  doi: 10.1137/S0036144500371907.  Google Scholar

[2]

M. H. Qiao, H. Qi and Y. C. Chen, Qualitative analysis of hepatitis B virus infection model with impulsive vaccination and time delay,, Acta Mathematica Scientia, 31 (2011), 1020.  doi: 10.1016/S0252-9602(11)60294-4.  Google Scholar

[3]

M. H. Qiao, A. P. Liu and U. Fory's, Qualitative analysis of the SICR epidemic model with impulsive vaccinations,, Math. Meth. Appl. Sci., 36 (2013), 695.  doi: 10.1002/mma.2620.  Google Scholar

[4]

M. H. Qiao, A. P. Liu and U. Fory's, The dynamics of a time delayed epidemic model on a population with birth pulse,, Applied Mathematics and Computation, 252 (2015), 166.  doi: 10.1016/j.amc.2014.12.022.  Google Scholar

[5]

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

[6]

X. Fu, M. Small, D. M. Walker and H. Zhang, Epidemic dynamics on scale-free networks with piecewise linear infectivity and immunization,, Phys Rev E, 77 (2008).  doi: 10.1103/PhysRevE.77.036113.  Google Scholar

[7]

J. Joo and J. L. Lebowitz, Behavior of susceptible-infected-susceptible epidemics on heterogeneous networks with saturation,, Phys Rev E, 69 (2004).  doi: 10.1103/PhysRevE.69.066105.  Google Scholar

[8]

Z. Liu and B. Hu, Epidemic spreading in community networks,, Europhys Lett, 72 (2005), 315.  doi: 10.1209/epl/i2004-10550-5.  Google Scholar

[9]

R. Olinky and L. Stone, Unexpected epidemic thresholds in heterogeneous networks: The role of disease transmission,, Phys Rev E, 70 (2004).  doi: 10.1103/PhysRevE.70.030902.  Google Scholar

[10]

R. Pastor-Satorras and A. Vespignani, Epidemic dynamics in scale-free networks,, Phys Rev Lett, 86 (2001).   Google Scholar

[11]

A. L. Barabási and R. Albert, Emergence of scaling in random networks,, Science, 286 (1999), 509.  doi: 10.1126/science.286.5439.509.  Google Scholar

[12]

Y. Moreno, R. Pastor-Satorras and A. Vespignani, Epidemic outbreaks in complex heterogeneous networks,, Eur Phys J. B., 26 (2002), 521.  doi: 10.1140/epjb/e20020122.  Google Scholar

[13]

L. Wang and G. Z. Dai. Global, stability of virus spreading in complex heterogeneous networks,, SIAM J. Appl. Math., 68 (2008), 1495.  doi: 10.1137/070694582.  Google Scholar

[14]

J. Liu and T. Zhang, Epidemic spreading of an SEIRS model in scale-free networks,, Commun Nonlinear Sci Numer Simul, 16 (2011), 3375.  doi: 10.1016/j.cnsns.2010.11.019.  Google Scholar

show all references

References:
[1]

H. W. Hethcote, The mathematics of infectious diseases,, SIAM Rev, 42 (2000), 599.  doi: 10.1137/S0036144500371907.  Google Scholar

[2]

M. H. Qiao, H. Qi and Y. C. Chen, Qualitative analysis of hepatitis B virus infection model with impulsive vaccination and time delay,, Acta Mathematica Scientia, 31 (2011), 1020.  doi: 10.1016/S0252-9602(11)60294-4.  Google Scholar

[3]

M. H. Qiao, A. P. Liu and U. Fory's, Qualitative analysis of the SICR epidemic model with impulsive vaccinations,, Math. Meth. Appl. Sci., 36 (2013), 695.  doi: 10.1002/mma.2620.  Google Scholar

[4]

M. H. Qiao, A. P. Liu and U. Fory's, The dynamics of a time delayed epidemic model on a population with birth pulse,, Applied Mathematics and Computation, 252 (2015), 166.  doi: 10.1016/j.amc.2014.12.022.  Google Scholar

[5]

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

[6]

X. Fu, M. Small, D. M. Walker and H. Zhang, Epidemic dynamics on scale-free networks with piecewise linear infectivity and immunization,, Phys Rev E, 77 (2008).  doi: 10.1103/PhysRevE.77.036113.  Google Scholar

[7]

J. Joo and J. L. Lebowitz, Behavior of susceptible-infected-susceptible epidemics on heterogeneous networks with saturation,, Phys Rev E, 69 (2004).  doi: 10.1103/PhysRevE.69.066105.  Google Scholar

[8]

Z. Liu and B. Hu, Epidemic spreading in community networks,, Europhys Lett, 72 (2005), 315.  doi: 10.1209/epl/i2004-10550-5.  Google Scholar

[9]

R. Olinky and L. Stone, Unexpected epidemic thresholds in heterogeneous networks: The role of disease transmission,, Phys Rev E, 70 (2004).  doi: 10.1103/PhysRevE.70.030902.  Google Scholar

[10]

R. Pastor-Satorras and A. Vespignani, Epidemic dynamics in scale-free networks,, Phys Rev Lett, 86 (2001).   Google Scholar

[11]

A. L. Barabási and R. Albert, Emergence of scaling in random networks,, Science, 286 (1999), 509.  doi: 10.1126/science.286.5439.509.  Google Scholar

[12]

Y. Moreno, R. Pastor-Satorras and A. Vespignani, Epidemic outbreaks in complex heterogeneous networks,, Eur Phys J. B., 26 (2002), 521.  doi: 10.1140/epjb/e20020122.  Google Scholar

[13]

L. Wang and G. Z. Dai. Global, stability of virus spreading in complex heterogeneous networks,, SIAM J. Appl. Math., 68 (2008), 1495.  doi: 10.1137/070694582.  Google Scholar

[14]

J. Liu and T. Zhang, Epidemic spreading of an SEIRS model in scale-free networks,, Commun Nonlinear Sci Numer Simul, 16 (2011), 3375.  doi: 10.1016/j.cnsns.2010.11.019.  Google Scholar

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