Export file:


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


  • Citation Only
  • Citation and Abstract

Bifurcation analysis of a pair-wise epidemic model on adaptive networks

1 School of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006, Shanxi, China
2 School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi, China
3 Complex Systems Research Center, Shanxi University, Taiyuan 030006, Shanxi, China

Special Issues: Transmission dynamics in infectious diseases

The topological structures of complex networks have been playing an important role on the epidemic spreading. There has been several studies of pairwise epidemic models on adaptive networks with Poisson distribution, all of which have shown that the rewiring behaviors can lead to complex dynamics numerically or analytically. However, the triples approximation formula under Poisson distribution overlooked the degree of center node of triples which has dramatic effects on the structures. Therefore in this paper, through a new moment closure incorporating the effect of center node’s degree, we study how the topological structures of adaptive networks influences epidemic dynamics. The SIS pairwise epidemic model is first closed by the new triple approximation formula, then we transform the model into an equivalent nondimensionalized three dimensional system. By the qualitative theory and the stability theory of ordinary differential equations, the basic reproduction number R0 of the model is obtained, the existence and stabilities of the equilibria are analyzed. Moreover, we prove that the model exhibits transcritical forward bifurcation, backward bifurcation, saddle-node bifurcation and Hopf bifurcation using the methods of bifurcation theory. In addition, by a numerical example, the normal form of Hopf bifurcation and the first Lyapunov coecient are derived, which show that a stable limit cycle can bifurcate from the endemic equilibrium with larger epidemicity. Our study show that the adaptive behavior can lead to rich dynamics on epidemic transmission, including oscillation and bistability. Finally the numerical simulations which is consistent with the analytical results above are given.
  Article Metrics


1. F. Brauer and C. Castillo-Chavez, Mathematical models in population biology and epidemiology, New York: Springer, 2001.

2. R. Pastor-Satorras and A. Vespignani, Epidemic dynamics and endemic states in complex networks, Physic. Rev. E, 63 (2001), 649–667.

3. R. Pastor-Satorras and A. Vespignani, Epidemic dynamics in finite size scale-free networks, Physic. Rev. E, 65 (2002), 035108.

4. M. Boguñá and R. Pastor-Satorras, Epidemic spreading in correlated complex networks, Physic. Rev. E, 66 (2002), 047104.

5. P. V. Mieghem, J. Omic and R. Kooij, Virus spread in networks, IEEE ACM T. Network, 17 (2009), 1–14.

6. L. X. Yang, M. Draief and X. F. Yang, Heterogeneous virus propagation in networks: a theoretical study, Math. Method Appl. Sci , 40 (2017), 1396–1413.

7. L. X. Yang, X. F. Yang and Y. Wu, The impact of patch forwarding on the prevalence of computer virus: A theoretical assessment approach, Appl. Math. Model, 43 (2017), 110–125.

8. L. X. Yang, X. F. Yang and Y. Y. Tang, A Bi-Virus Competing Spreading Model with Generic Infection Rates, IEEE Transact. Netw. Sci. Eng., 5 (2018), 2–13.

9. X. Zhang, G. Q. Sun, Y. X. Zhu, et al., Epidemic dynamics on semi-directed complex networks, Math. Biosci., 246 (2013), 242–251.

10. J. Y. Yang, Y. M. Chen and F. Xu, Efect of infection age on an SIS epidemic model on complex networks, J. Math. Biol., 73 (2016), 1227–1249.

11. X. L. Peng, Z. Q. Zhang, J. Y. Yang, et al., An SIS epidemic model with vaccination in a dynamical contact network of mobile individuals with heterogeneous spatial constraints, Commun. Nonlinear Sci., 73 (2019), 52–73.

12. G. Q. Sun, C. H. Wang, L. L. Chang, et al., Efects of feedback regulation on vegetation patterns in semi-arid environments, Appl. Math. Model., 61 (2018), 200–215.

13. G. Q. Sun, M. Jusup, Z. Jin, et al., Pattern transitions in spatial epidemics: Mechanisms and emergent properties, Phys. Life Rev., 19 (2016), 43–73.

14. M. J. Keeling, D. A. Rand and A. J. Morris, Correlation models for childhood epidemics, P. Roy. Soc. B-Biol. Sci., 264 (1385), 1149–1156.

15. M. J. Keeling, The efects of local spatial structure on epidemiological invasions, Proc. R. Soc. Lond. B., 266 (1999), 859–867.

16. K. T. D. Eames and M. J. Keeling, Monogamous networks and the spread of sexually transmitted diseases, Math. Biosci., 189 (2004), 115–130.

17. Y.Wang, J. D. Cao, A. Alsaedi, et al., Edge-based SEIR dynamics with or without infectious force in latent period on random networks, Commun. Nonlinear Sci., 45 (2017), 35–54.

18. Y. Wang, J. D. Cao, X. D. Li, et al., Edge-based epidemic dynamics with multiple routes of transmission on random networks, Nonlinear Dynam., 91 (2018), 403–420.

19. Y.Wang, J. D. Cao and M. Q. Li, et al., Global behavior of a two-stage contact process on complex networks, J. Franklin I., In Press.

20. J. Y. Yang and F. Xu, The coumputational approach for the basic reproduction number of epidemic models on complex networks, IEEE Access, 2019, 10.1109/ACESS2019.2898639.

21. L. Li, C. H. Wang, S. F. Wang, et al., Hemorrhagic fever with renal syndrome in China: Mechanisms on two distinct annual peaks and control measures, Int. J. Biomath., 11 (2018), 1850030.

22. L. Li, J. Zhang, C. Liu, et al., Analysis of transmission dynamics for Zika virus on networks, Appl. Math. Comput., 347 (2019), 566–577.

23. D. J.Watts and S. H. Strogatz, Collective dynamics of 'small-world' networks, Nature, 393 (1998), 440–442.

24. A. L. Barabási and R. Albert, Emergence of scaling in random networks, Science, 286 (1999), 509–512.

25. R. M. May and A. L. Lloyd, Infection dynamics on scale-free networks, Physic. Rev. E, 64 (2001), 066112.

26. V. M. Eguiluz and K. Klemm, Epidemic threshold in structured scale-free networks, Physic. Rev. Lett., 89 (2002), 108701.

27. M. Boguna, R. Pastor-Satorras, A. Vespignani, Absence of epidemic threshold in scale-free networks with degree correlations, Physic. Rev. Lett., 90 (2003), 028701.

28. H. Zhang, P. Shu, Z.Wang, et al., Preferential imitation can invalidate targeted subsidy policies on seasonal-influenza diseases, Appl. Math. Comput., 294 (2017), 332–342.

29. T. Gross, C. J. D. D'Lima and B. Blasius, Epidemic dynamics on an adaptive network, Physic. Rev. Lett., 96 (2006), 208701.

30. T. Gross and B. Blasius, Adaptive coevolutionary networks: a review, J. R. Soc. Inter., 5 (2008), 259–271.

31. X. Zhang, C. Shan, Z. Jin, et al., Complex dynamics of epidemic models on adaptive networks, J. Difer. Equa., 266 (2019), 803–832.

32. L. B. Shaw and I. B. Schwartz, Fluctuating epidemics on adaptive networks, Physic. Rev. E, 77 (2008), 066101.

33. D. H. Zanette and S. Risau-Gusman, Infection Spreading in a Population with Evolving Contacts, J. Biol. Phys., 34 (2008), 135–148.

34. Y. Schwarzkopf, A. Ráos and D. Mukamel, Epidemic spreading in evolving networks, Physic. Rev. E, 82 (2010), 036112.

35. T. Rogers, W. Cliford-Brown, C. Mills, et al., Stochastic oscillations of adaptive networks: application to epidemic modelling, J. Stat. Mech. Theory. E, 8 (2012), 1–15.

36. D. Juher, J. Ripoll and J. Saldaña, Outbreak analysis of an SIS epidemic model with rewiring, J. Math. Biol., 67 (2013), 411–432.

37. J. Zhou, G. Xiao, S. A. Cheong, et al., Epidemic reemergence in adaptive complex networks, Physic. Rev. E, 85 (2012), 036107.

38. A. Szabó-Solticzky, L. Berthouze, I. Z. Kiss, et al., Oscillating epidemics in a dynamic network model: stochastic and mean-field analysis, J. Math. Biol., 72 (2016), 1153–1176.

39. S.J. Fan, A new extracting formula and a new distinguishing means on the one variable cubic equation, Nat. Sci. J. Hainan Teacheres Coll., 2 (1989), 91–98 (in Chinese).

40. S. N. Chow and J. K. Hale, Methods of bifurcation theory, Springer Science & Business Media, 2012.

41. L. Perko, Diferential equations and dynamical systems, Springer Science & Business Media, 2013.

© 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

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved