2015, 12(3): 415-429. doi: 10.3934/mbe.2015.12.415

An SIRS model with differential susceptibility and infectivity on uncorrelated networks

1. 

Department of Mathematics, North University of China, Taiyuan, Shanxi 030051, China

2. 

Department of Applied Mathematics, Yuncheng University, Yuncheng, Shanxi 044000, China

Received  July 2013 Revised  November 2014 Published  January 2015

We propose and study a model for sexually transmitted infections on uncorrelated networks, where both differential susceptibility and infectivity are considered. We first establish the spreading threshold, which exists even in the infinite networks. Moreover, it is possible to have backward bifurcation. Then for bounded hard-cutoff networks, the stability of the disease-free equilibrium and the permanence of infection are analyzed. Finally, the effects of two immunization strategies are compared. It turns out that, generally, the targeted immunization is better than the proportional immunization.
Citation: Maoxing Liu, Yuming Chen. An SIRS model with differential susceptibility and infectivity on uncorrelated networks. Mathematical Biosciences & Engineering, 2015, 12 (3) : 415-429. doi: 10.3934/mbe.2015.12.415
References:
[1]

M. Artzrouni, Transmission probabilities and reproductive numbers for sexually transmitted infections with variable infectivity: Application to the spread of HIV between low- and high-activity populations,, Mathematical Population Studies, 16 (2009), 266.  doi: 10.1080/08898480903251538.  Google Scholar

[2]

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

[3]

B. Bonzi, A. A. Fall, A. Iggidr and G. Sallet, Stability of differential susceptibility and infectivity epidemic models,, J. Math. Biol., 62 (2011), 39.  doi: 10.1007/s00285-010-0327-y.  Google Scholar

[4]

S. N. Dorogovtsev, A. V. Goltsev and J. F. F. Mendes, Critical phenomena in complex networks,, Rev. Mod. Phys., 80 (2008), 1275.  doi: 10.1103/RevModPhys.80.1275.  Google Scholar

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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

[6]

W.-P. Guo, X. Li and X.-F. Wang, Epidemics and immunization on Euclidean distance preferred small-world networks,, Physica A, 380 (2007), 684.  doi: 10.1016/j.physa.2007.03.007.  Google Scholar

[7]

J. M. Hyman and J. Li, Differential susceptibility epidemic models,, J. Math. Biol., 50 (2005), 626.  doi: 10.1007/s00285-004-0301-7.  Google Scholar

[8]

J. M. Hyman and J. Li, Differential susceptibility and infectivity epidemic models,, Math. Biosci. Engrg., 3 (2006), 89.  doi: 10.3934/mbe.2006.3.89.  Google Scholar

[9]

J. M. Hyman and J. Li, Epidemic models with differential susceptibility and staged progression and their dynamics,, Math. Biosci. Engrg., 6 (2009), 321.  doi: 10.3934/mbe.2009.6.321.  Google Scholar

[10]

A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population,, Math. Biosci., 28 (1976), 221.  doi: 10.1016/0025-5564(76)90125-5.  Google Scholar

[11]

P. De Leenheer and H. L. Smith, Virus dynamics: A global analysis,, SIAM J. Appl. Math., 63 (2003), 1313.  doi: 10.1137/S0036139902406905.  Google Scholar

[12]

F. Liljeros, C. R. Edling, L. A. N. Amaral, H. E. Stanley and Y. Åberg, The web of human sexual contacts,, Nature, 411 (2001), 907.  doi: 10.1038/35082140.  Google Scholar

[13]

J. Lou and T. Ruggeri, The dynamics of spreading and immune strategies of sexually transmitted diseases on scale-free network,, J. Math. Anal. Appl., 365 (2010), 210.  doi: 10.1016/j.jmaa.2009.10.044.  Google Scholar

[14]

Z. Ma, J. Liu and J. Li, Stability analysis for differential infectivity epidemic models,, Nonlinear Anal.: RWA, 4 (2003), 841.  doi: 10.1016/S1468-1218(03)00019-1.  Google Scholar

[15]

J. D. May, Mathematical Biology I: An Introduction,, 3rd Ed., (2002).   Google Scholar

[16]

R. M. May and A. L. Lloyd, Infection dynamics on scale-free networks,, Phys. Rev. E, 64 (2001).  doi: 10.1103/PhysRevE.64.066112.  Google Scholar

[17]

M. E. J. Newman, The structure and function of complex networks,, SIAM Rev., 45 (2003), 167.  doi: 10.1137/S003614450342480.  Google Scholar

[18]

R. Pastor-Satorras and A. Vespignani, Epidemic spreading in scale-free networks,, Phys. Rev. Lett., 86 (2001), 3200.  doi: 10.1103/PhysRevLett.86.3200.  Google Scholar

[19]

R. Pastor-Satorras and A. Vespignani, Epidemic dynamics in finite size scale-free networks,, Phys. Rev. E, 65 (2002).  doi: 10.1103/PhysRevE.65.035108.  Google Scholar

[20]

R. Pastor-Satorras and A. Vespignani, Immunization of complex networks,, Phys. Rev. E, 65 (2002).  doi: 10.1103/PhysRevE.65.036104.  Google Scholar

[21]

W. J. Reed, A stochastic model for the spread of a sexually transmitted disease which results in a scale-free network,, Math. Biosci., 201 (2006), 3.  doi: 10.1016/j.mbs.2005.12.016.  Google Scholar

[22]

A. Schneeberger, C. H. Mercer, S. A. J. Gregson, N. M. Ferguson, C. A. Nyamukapa, R. M. Anderson, A. M. Johmson and G. P. Garnett, Scale-free networks and sexually transmitted diseases: A description of observed patterns of sexual contacts in Britain and Zimbabwe,, Sex. Transm. Dis., 31 (2004), 380.  doi: 10.1097/00007435-200406000-00012.  Google Scholar

[23]

A. Smed-Sörensen et al., Differential susceptibility to human immunodeficiency virus type 1 infection of myeloid and Plasmacytoid dendritic cells,, J. Virology, 79 (2005), 8861.   Google Scholar

[24]

N. Sugimine and K. Aihara, Stability of an equilibrium state in a multiinfectious-type SIS model on a truncated network,, Artif. Life Robotics, 11 (2007), 157.  doi: 10.1007/s10015-007-0421-4.  Google Scholar

[25]

H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model),, SIAM J. Math. Anal., 24 (1993), 407.  doi: 10.1137/0524026.  Google Scholar

[26]

R. Yang et al., Epidemic spreading on heterogeneous networks with identical infectivity,, Phys. Lett. A, 364 (2007), 189.   Google Scholar

[27]

H. Zhang, L. Chen and J. J. Nieto, A delayed epidemic model with stage-structure and pulses for pest management strategy,, Nonlinear Anal. Real World Appl., 9 (2008), 1714.  doi: 10.1016/j.nonrwa.2007.05.004.  Google Scholar

[28]

Z. Zhang and J. Peng, A SIRS epidemic model with infection-age dependence,, J. Math. Anal. Appl., 331 (2007), 1396.  doi: 10.1016/j.jmaa.2006.09.061.  Google Scholar

[29]

Z. Zhang, J. Peng and J. Zhang, Analysis of a bacteria-immunity model with delay quorum sensing,, J. Math. Anal. Appl., 340 (2008), 102.  doi: 10.1016/j.jmaa.2007.08.027.  Google Scholar

[30]

S. Zou, J. Wu and Y. Chen, Multiple epidemic waves in delayed susceptible-infected-recovered models on complex networks,, Phys. Rev. E, 83 (2011).  doi: 10.1103/PhysRevE.83.056121.  Google Scholar

show all references

References:
[1]

M. Artzrouni, Transmission probabilities and reproductive numbers for sexually transmitted infections with variable infectivity: Application to the spread of HIV between low- and high-activity populations,, Mathematical Population Studies, 16 (2009), 266.  doi: 10.1080/08898480903251538.  Google Scholar

[2]

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

[3]

B. Bonzi, A. A. Fall, A. Iggidr and G. Sallet, Stability of differential susceptibility and infectivity epidemic models,, J. Math. Biol., 62 (2011), 39.  doi: 10.1007/s00285-010-0327-y.  Google Scholar

[4]

S. N. Dorogovtsev, A. V. Goltsev and J. F. F. Mendes, Critical phenomena in complex networks,, Rev. Mod. Phys., 80 (2008), 1275.  doi: 10.1103/RevModPhys.80.1275.  Google Scholar

[5]

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

[6]

W.-P. Guo, X. Li and X.-F. Wang, Epidemics and immunization on Euclidean distance preferred small-world networks,, Physica A, 380 (2007), 684.  doi: 10.1016/j.physa.2007.03.007.  Google Scholar

[7]

J. M. Hyman and J. Li, Differential susceptibility epidemic models,, J. Math. Biol., 50 (2005), 626.  doi: 10.1007/s00285-004-0301-7.  Google Scholar

[8]

J. M. Hyman and J. Li, Differential susceptibility and infectivity epidemic models,, Math. Biosci. Engrg., 3 (2006), 89.  doi: 10.3934/mbe.2006.3.89.  Google Scholar

[9]

J. M. Hyman and J. Li, Epidemic models with differential susceptibility and staged progression and their dynamics,, Math. Biosci. Engrg., 6 (2009), 321.  doi: 10.3934/mbe.2009.6.321.  Google Scholar

[10]

A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population,, Math. Biosci., 28 (1976), 221.  doi: 10.1016/0025-5564(76)90125-5.  Google Scholar

[11]

P. De Leenheer and H. L. Smith, Virus dynamics: A global analysis,, SIAM J. Appl. Math., 63 (2003), 1313.  doi: 10.1137/S0036139902406905.  Google Scholar

[12]

F. Liljeros, C. R. Edling, L. A. N. Amaral, H. E. Stanley and Y. Åberg, The web of human sexual contacts,, Nature, 411 (2001), 907.  doi: 10.1038/35082140.  Google Scholar

[13]

J. Lou and T. Ruggeri, The dynamics of spreading and immune strategies of sexually transmitted diseases on scale-free network,, J. Math. Anal. Appl., 365 (2010), 210.  doi: 10.1016/j.jmaa.2009.10.044.  Google Scholar

[14]

Z. Ma, J. Liu and J. Li, Stability analysis for differential infectivity epidemic models,, Nonlinear Anal.: RWA, 4 (2003), 841.  doi: 10.1016/S1468-1218(03)00019-1.  Google Scholar

[15]

J. D. May, Mathematical Biology I: An Introduction,, 3rd Ed., (2002).   Google Scholar

[16]

R. M. May and A. L. Lloyd, Infection dynamics on scale-free networks,, Phys. Rev. E, 64 (2001).  doi: 10.1103/PhysRevE.64.066112.  Google Scholar

[17]

M. E. J. Newman, The structure and function of complex networks,, SIAM Rev., 45 (2003), 167.  doi: 10.1137/S003614450342480.  Google Scholar

[18]

R. Pastor-Satorras and A. Vespignani, Epidemic spreading in scale-free networks,, Phys. Rev. Lett., 86 (2001), 3200.  doi: 10.1103/PhysRevLett.86.3200.  Google Scholar

[19]

R. Pastor-Satorras and A. Vespignani, Epidemic dynamics in finite size scale-free networks,, Phys. Rev. E, 65 (2002).  doi: 10.1103/PhysRevE.65.035108.  Google Scholar

[20]

R. Pastor-Satorras and A. Vespignani, Immunization of complex networks,, Phys. Rev. E, 65 (2002).  doi: 10.1103/PhysRevE.65.036104.  Google Scholar

[21]

W. J. Reed, A stochastic model for the spread of a sexually transmitted disease which results in a scale-free network,, Math. Biosci., 201 (2006), 3.  doi: 10.1016/j.mbs.2005.12.016.  Google Scholar

[22]

A. Schneeberger, C. H. Mercer, S. A. J. Gregson, N. M. Ferguson, C. A. Nyamukapa, R. M. Anderson, A. M. Johmson and G. P. Garnett, Scale-free networks and sexually transmitted diseases: A description of observed patterns of sexual contacts in Britain and Zimbabwe,, Sex. Transm. Dis., 31 (2004), 380.  doi: 10.1097/00007435-200406000-00012.  Google Scholar

[23]

A. Smed-Sörensen et al., Differential susceptibility to human immunodeficiency virus type 1 infection of myeloid and Plasmacytoid dendritic cells,, J. Virology, 79 (2005), 8861.   Google Scholar

[24]

N. Sugimine and K. Aihara, Stability of an equilibrium state in a multiinfectious-type SIS model on a truncated network,, Artif. Life Robotics, 11 (2007), 157.  doi: 10.1007/s10015-007-0421-4.  Google Scholar

[25]

H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model),, SIAM J. Math. Anal., 24 (1993), 407.  doi: 10.1137/0524026.  Google Scholar

[26]

R. Yang et al., Epidemic spreading on heterogeneous networks with identical infectivity,, Phys. Lett. A, 364 (2007), 189.   Google Scholar

[27]

H. Zhang, L. Chen and J. J. Nieto, A delayed epidemic model with stage-structure and pulses for pest management strategy,, Nonlinear Anal. Real World Appl., 9 (2008), 1714.  doi: 10.1016/j.nonrwa.2007.05.004.  Google Scholar

[28]

Z. Zhang and J. Peng, A SIRS epidemic model with infection-age dependence,, J. Math. Anal. Appl., 331 (2007), 1396.  doi: 10.1016/j.jmaa.2006.09.061.  Google Scholar

[29]

Z. Zhang, J. Peng and J. Zhang, Analysis of a bacteria-immunity model with delay quorum sensing,, J. Math. Anal. Appl., 340 (2008), 102.  doi: 10.1016/j.jmaa.2007.08.027.  Google Scholar

[30]

S. Zou, J. Wu and Y. Chen, Multiple epidemic waves in delayed susceptible-infected-recovered models on complex networks,, Phys. Rev. E, 83 (2011).  doi: 10.1103/PhysRevE.83.056121.  Google Scholar

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