# American Institute of Mathematical Sciences

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), 266-287. doi: 10.1080/08898480903251538. [2] A.-L. Barabási and R. Alberta, Emergence of scaling in random networks, Science, 286 (1999), 509-512. doi: 10.1126/science.286.5439.509. [3] B. Bonzi, A. A. Fall, A. Iggidr and G. Sallet, Stability of differential susceptibility and infectivity epidemic models, J. Math. Biol., 62 (2011), 39-64. doi: 10.1007/s00285-010-0327-y. [4] S. N. Dorogovtsev, A. V. Goltsev and J. F. F. Mendes, Critical phenomena in complex networks, Rev. Mod. Phys., 80 (2008), 1275-1335. doi: 10.1103/RevModPhys.80.1275. [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), 036113, 8pp. doi: 10.1103/PhysRevE.77.036113. [6] W.-P. Guo, X. Li and X.-F. Wang, Epidemics and immunization on Euclidean distance preferred small-world networks, Physica A, 380 (2007), 684-690. doi: 10.1016/j.physa.2007.03.007. [7] J. M. Hyman and J. Li, Differential susceptibility epidemic models, J. Math. Biol., 50 (2005), 626-644. doi: 10.1007/s00285-004-0301-7. [8] J. M. Hyman and J. Li, Differential susceptibility and infectivity epidemic models, Math. Biosci. Engrg., 3 (2006), 89-100. doi: 10.3934/mbe.2006.3.89. [9] J. M. Hyman and J. Li, Epidemic models with differential susceptibility and staged progression and their dynamics, Math. Biosci. Engrg., 6 (2009), 321-332. doi: 10.3934/mbe.2009.6.321. [10] A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci., 28 (1976), 221-236. doi: 10.1016/0025-5564(76)90125-5. [11] P. De Leenheer and H. L. Smith, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63 (2003), 1313-1327. doi: 10.1137/S0036139902406905. [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), 907-908. doi: 10.1038/35082140. [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), 210-219. doi: 10.1016/j.jmaa.2009.10.044. [14] Z. Ma, J. Liu and J. Li, Stability analysis for differential infectivity epidemic models, Nonlinear Anal.: RWA, 4 (2003), 841-856. doi: 10.1016/S1468-1218(03)00019-1. [15] J. D. May, Mathematical Biology I: An Introduction, 3rd Ed., Springer-Verlag, New York, 2002. [16] 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. [17] M. E. J. Newman, The structure and function of complex networks, SIAM Rev., 45 (2003), 167-256. doi: 10.1137/S003614450342480. [18] R. Pastor-Satorras and A. Vespignani, Epidemic spreading in scale-free networks, Phys. Rev. Lett., 86 (2001), 3200-3203. doi: 10.1103/PhysRevLett.86.3200. [19] R. Pastor-Satorras and A. Vespignani, Epidemic dynamics in finite size scale-free networks, Phys. Rev. E, 65 (2002), 035108. doi: 10.1103/PhysRevE.65.035108. [20] R. Pastor-Satorras and A. Vespignani, Immunization of complex networks, Phys. Rev. E, 65 (2002), 036104. doi: 10.1103/PhysRevE.65.036104. [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), 3-14. doi: 10.1016/j.mbs.2005.12.016. [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), 380-387. doi: 10.1097/00007435-200406000-00012. [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), 8861-8869. [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), 157-161. doi: 10.1007/s10015-007-0421-4. [25] H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407-435. doi: 10.1137/0524026. [26] R. Yang et al., Epidemic spreading on heterogeneous networks with identical infectivity, Phys. Lett. A, 364 (2007), 189-193. [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), 1714-1726. doi: 10.1016/j.nonrwa.2007.05.004. [28] Z. Zhang and J. Peng, A SIRS epidemic model with infection-age dependence, J. Math. Anal. Appl., 331 (2007), 1396-1414. doi: 10.1016/j.jmaa.2006.09.061. [29] Z. Zhang, J. Peng and J. Zhang, Analysis of a bacteria-immunity model with delay quorum sensing, J. Math. Anal. Appl., 340 (2008), 102-115. doi: 10.1016/j.jmaa.2007.08.027. [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), 056121. doi: 10.1103/PhysRevE.83.056121.

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##### 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), 266-287. doi: 10.1080/08898480903251538. [2] A.-L. Barabási and R. Alberta, Emergence of scaling in random networks, Science, 286 (1999), 509-512. doi: 10.1126/science.286.5439.509. [3] B. Bonzi, A. A. Fall, A. Iggidr and G. Sallet, Stability of differential susceptibility and infectivity epidemic models, J. Math. Biol., 62 (2011), 39-64. doi: 10.1007/s00285-010-0327-y. [4] S. N. Dorogovtsev, A. V. Goltsev and J. F. F. Mendes, Critical phenomena in complex networks, Rev. Mod. Phys., 80 (2008), 1275-1335. doi: 10.1103/RevModPhys.80.1275. [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), 036113, 8pp. doi: 10.1103/PhysRevE.77.036113. [6] W.-P. Guo, X. Li and X.-F. Wang, Epidemics and immunization on Euclidean distance preferred small-world networks, Physica A, 380 (2007), 684-690. doi: 10.1016/j.physa.2007.03.007. [7] J. M. Hyman and J. Li, Differential susceptibility epidemic models, J. Math. Biol., 50 (2005), 626-644. doi: 10.1007/s00285-004-0301-7. [8] J. M. Hyman and J. Li, Differential susceptibility and infectivity epidemic models, Math. Biosci. Engrg., 3 (2006), 89-100. doi: 10.3934/mbe.2006.3.89. [9] J. M. Hyman and J. Li, Epidemic models with differential susceptibility and staged progression and their dynamics, Math. Biosci. Engrg., 6 (2009), 321-332. doi: 10.3934/mbe.2009.6.321. [10] A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci., 28 (1976), 221-236. doi: 10.1016/0025-5564(76)90125-5. [11] P. De Leenheer and H. L. Smith, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63 (2003), 1313-1327. doi: 10.1137/S0036139902406905. [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), 907-908. doi: 10.1038/35082140. [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), 210-219. doi: 10.1016/j.jmaa.2009.10.044. [14] Z. Ma, J. Liu and J. Li, Stability analysis for differential infectivity epidemic models, Nonlinear Anal.: RWA, 4 (2003), 841-856. doi: 10.1016/S1468-1218(03)00019-1. [15] J. D. May, Mathematical Biology I: An Introduction, 3rd Ed., Springer-Verlag, New York, 2002. [16] 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. [17] M. E. J. Newman, The structure and function of complex networks, SIAM Rev., 45 (2003), 167-256. doi: 10.1137/S003614450342480. [18] R. Pastor-Satorras and A. Vespignani, Epidemic spreading in scale-free networks, Phys. Rev. Lett., 86 (2001), 3200-3203. doi: 10.1103/PhysRevLett.86.3200. [19] R. Pastor-Satorras and A. Vespignani, Epidemic dynamics in finite size scale-free networks, Phys. Rev. E, 65 (2002), 035108. doi: 10.1103/PhysRevE.65.035108. [20] R. Pastor-Satorras and A. Vespignani, Immunization of complex networks, Phys. Rev. E, 65 (2002), 036104. doi: 10.1103/PhysRevE.65.036104. [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), 3-14. doi: 10.1016/j.mbs.2005.12.016. [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), 380-387. doi: 10.1097/00007435-200406000-00012. [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), 8861-8869. [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), 157-161. doi: 10.1007/s10015-007-0421-4. [25] H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407-435. doi: 10.1137/0524026. [26] R. Yang et al., Epidemic spreading on heterogeneous networks with identical infectivity, Phys. Lett. A, 364 (2007), 189-193. [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), 1714-1726. doi: 10.1016/j.nonrwa.2007.05.004. [28] Z. Zhang and J. Peng, A SIRS epidemic model with infection-age dependence, J. Math. Anal. Appl., 331 (2007), 1396-1414. doi: 10.1016/j.jmaa.2006.09.061. [29] Z. Zhang, J. Peng and J. Zhang, Analysis of a bacteria-immunity model with delay quorum sensing, J. Math. Anal. Appl., 340 (2008), 102-115. doi: 10.1016/j.jmaa.2007.08.027. [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), 056121. doi: 10.1103/PhysRevE.83.056121.
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