Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

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

Special Issues: Differential Equations in Mathematical Biology

In this paper, by taking full consideration of demographics, transfer from infectious to sus-ceptible and contact heterogeneity of the individuals, we construct an improved Susceptible-Infected-Removed-Susceptible (SIRS) epidemic model on complex heterogeneous networks. Using the next generation matrix method, we obtain the basic reproduction number $\mathcal{R}_0$ which is a critical value and used to measure the dynamics of epidemic diseases. More specifically, if $\mathcal{R}_0$ < 1, then the disease-free equilibrium is globally asymptotically stable; if $\mathcal{R}_0$ > 1, then there exists a unique endemic equilib-rium and the permanence of the disease is shown in detail. By constructing an appropriate Lyapunov function, the global stability of the endemic equilibrium is proved as well under some conditions. Moreover, the effects of three major immunization strategies are investigated. Finally, some numerical simulations are carried out to demonstrate the correctness and validness of the theoretical results.
  Article Metrics

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): 5729-5749. doi: 10.3934/mbe.2019286


  • 1. WHO Ebola Response Team, Ebola virus disease in west Africa-the 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 prey-predator 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 scale-free 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. Pastor-Satorras and A. Vespignani, Epidemic spreading in scale-free 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 scale-free 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 immu-nization, 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 sub-threshold endemic equi-libria 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 matrices-A case study of dengue transmission modelling, Biosystems, 126 (2014), 52–75.
  • 33. F. Chen, On a nonlinear nonautonomous predator-prey model with diffusion and distributed delay, J. Comput. Appl. Math., 180 (2005), 33–49.
  • 34. H. R. Thieme, Persistence under relaxed point-dissipativity (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. Ben-Avraham, Effecient immunization strategies for computer net-works and populations, Phys. Rev. Lett., 91 (2003), 247901.
  • 38. X. Fu, M. Small, D. M. Walker, et al., Epidemic dynamics on scale-free 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. Pastor-satorras 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: perco-lation 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.


This article has been cited by

  • 1. Yi Wang, Jinde Cao, Gang Huang, Further dynamic analysis for a network sexually transmitted disease model with birth and death, Applied Mathematics and Computation, 2019, 363, 124635, 10.1016/j.amc.2019.124635
  • 2. Xin Yang, Shigang Wen, Zhifeng Liu, Cai Li, Chuangxia Huang, Dynamic Properties of Foreign Exchange Complex Network, Mathematics, 2019, 7, 9, 832, 10.3390/math7090832

Reader Comments

your name: *   your email: *  

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved