
Mathematical Biosciences and Engineering, 2019, 16(5): 57295749. doi: 10.3934/mbe.2019286.
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Global dynamics of an SIRS model with demographics and transfer from infectious to susceptible on heterogeneous networks
1 School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, Hunan, 410114, P. R. China
2 College of Arts and Science, National University of Defense Technology, Changsha, Hunan, 410073, P.R. China
3 Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science and Technology, Changsha, Hunan, 410114, P. R. China
Received: , Accepted: , Published:
Special Issues: Differential Equations in Mathematical Biology
Keywords: SIRS model; heterogeneous network; basic reproduction number; global dynamics; immunization strategy
Citation: Haijun Hu, Xupu Yuan, Lihong Huang, Chuangxia Huang. Global dynamics of an SIRS model with demographics and transfer from infectious to susceptible on heterogeneous networks. Mathematical Biosciences and Engineering, 2019, 16(5): 57295749. doi: 10.3934/mbe.2019286
References:
 1. WHO Ebola Response Team, Ebola virus disease in west Africathe first 9 months of the epidemic and forward projections, N. Engl. J. Med., 371 (2014), 1481–1495.
 2. S. Watts, SARS: a case study in emerging infections, Soc. Hist. Med., 18 (2005), 498–500.
 3. R. Xu and Z. Ma, Stability of a delayed SIRS epidemic model with a nonlinear incidence rate, Chaos Solitons Fractals, 41 (2009), 2319–2325.
 4. J. M. Epstein, Modelling to contain pandemics, Nature, 460 (2009), 687–689.
 5. J. Chen, An SIRS epidemic model, Appl. Math. J. Chinese Univ., 19 (2004), 101–108.
 6. T. Li, F. Zhang, H. Liu, et al., Threshold dynamics of an SIRS model with nonlinear incidence rate and transfer from infectious to susceptible, Appl. Math. Lett., 70 (2017), 52–57.
 7. C. Huang, H. Zhang, J. Cao, et al., Stability and Hopf bifurcation of a delayed preypredator model with disease in the predator, Int. J. Bifurcat. Chaos, (2019), in press.
 8. M. Martcheva, Introduction to Mathematical Epidemiology, Springer, New York, 2015.
 9. W. O. Kermack and A.G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 115 (1927), 700–721.
 10. W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics. II. The problem of endemicity, Proc. R. Soc. Lond. A, 138 (1932), 55–83.
 11. W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics. III. Further studies of the problem of endemicity, Proc. R. Soc. Lond. A, 141 (1933), 94–122.
 12. H. W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci., 28 (1976), 335–356.
 13. S. Riley, C. Fraser and C. A. Donnelly, Transmission dynamics of the etiological agent of SARS in Hong Kong: impact of public health interventions, Science, 300 (2003), 1961–1966.
 14. M. Small and C. K. Tse, Small world and scale free model of transmission of SARS, Int. J. Bifurcat. Chaos, 15 (2005), 1745–1755.
 15. G. Zhu, G. Chen and X. Fu, Effects of active links on epidemic transmission over social networks, Phys. A, 468 (2017), 614–621.
 16. M. E. J. Newman, The structure and function of complex networks, SIAM Rev., 45 (2003), 167–256.
 17. X. Chu, Z. Zhang, J. Guan, et al., Epidemic spreading with nonlinear infectivity in weighted scalefree networks, Phys. A, 390 (2011), 471–481.
 18. H. Han, A. Ma and Z. Huang, An improved SIRS epidemic model on complex network, Int. Conf. Comput. Intell. Softw. Eng. IEEE, (2009), 1–5.
 19. R. Olinky and L. Stone, Unexpected epidemic thresholds in heterogeneous networks: The role of disease transmission, Phy. Rev. E, 70 (2004), 030902.
 20. R. PastorSatorras and A. Vespignani, Epidemic spreading in scalefree networks, Phys. Rev. Lett., 86 (2001), 3200–3213.
 21. L. Wang and G. Dai, Global stability of virus spreading in complex heterogeneous networks, SIAM J. Appl. Math., 68 (2008), 1495–1502.
 22. R. Yang, B. Wang, J. Ren, et al., Epidemic spreading on heterogeneous networks with identical infectivity, Phys. Lett. A, 364 (2007), 189–193.
 23. H. Zhang and X. Fu, Spreading of epidemics on scalefree networks with nonlinear infectivity, Nonlinear Anal., 70 (2009), 3273–327.
 24. X. Zhang, J. Wu, P. Zhao, et al., Epidemic spreading on a complex network with partial immunization, Soft Comput. 22 (2017), 1–9.
 25. C. H. Li, C. C. Tsai and S. Y. Yang, Analysis of epidemic spreading of an SIRS model in complex heterogeneous networks, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 1042–1054.
 26. C. Huang, J. Cao, F. Wen, et al., Stability analysis of SIR model with distributed delay on complex networks, PloS One, 11 (2016), e0158813.
 27. J. Huo and H. Y. Zhao, Dynamical analysis of a fractional SIR model with birth and death on heterogeneous complex networks, Phys. A, 448 (2016), 41–56.
 28. Z. Jin, G. Sun and H. Zhu, Epidemic models for complex networks with demographics, Math. Biosci. Eng., 11 (2014), 1295–1317.
 29. J. Liu, Y. Tang and Z. R. Yang, The spread of disease with birth and death on networks, J. Stat. Mech. Theory Exp., 8 (2004), P08008.
 30. Y. Wang, J. Cao, A. Alsaedi, et al., The spreading dynamics of sexually transmitted diseases with birth and death on heterogeneous networks, J. Stat. Mech. Theor. Exp., 2 (2017), 023502.
 31. P. Van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48.
 32. H. Yang, The basic reproduction number obtained from Jacobian and next generation matricesA case study of dengue transmission modelling, Biosystems, 126 (2014), 52–75.
 33. F. Chen, On a nonlinear nonautonomous predatorprey model with diffusion and distributed delay, J. Comput. Appl. Math., 180 (2005), 33–49.
 34. H. R. Thieme, Persistence under relaxed pointdissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407–435.
 35. H. L. Smith and P. De Leenheer, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63 (2003), 1313–1327.
 36. J. P. LaSalle, The stability of dynamical systems, SIAM, Philadelphia, 1976.
 37. R. Cohen, S. Havlin and D. BenAvraham, Effecient immunization strategies for computer networks and populations, Phys. Rev. Lett., 91 (2003), 247901.
 38. X. Fu, M. Small, D. M. Walker, et al., Epidemic dynamics on scalefree networks with piecewise linear infectivity and immunization, Phys. Rev. E, 77 (2008), 036113.
 39. F. Nian and X. Wang, Efficient immunization strategies on complex networks, J. Theor. Biol., 264 (2010), 77–83.
 40. R. Pastorsatorras and A.Vespignani, Immunization of complex networks, Phys. Rev. E, 65 (2002), 036104.
 41. D. S. Callaway, M. E. J. Newman, S. H. Strogatz, et al., Network robustness and fragility: percolation on random graphs, Phys. Rev. Lett., 85 (2000), 5468–5471.
 42. A. L. Barabasi and R. Albert, Emergence of scaling in random networks, Science, 286 (1999), 509–512.
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