Skip to main content

Synchronization and Self-organization in Complex Networks for a Tuberculosis Model


In this work, we propose and analyze the dynamics of a complex network built with non identical instances of a tuberculosis (TB) epidemiological model, for which we prove the existence of non-negative and bounded global solutions. A two nodes network is analyzed where the nodes represent the TB epidemiological situation of the countries Angola and Portugal. We analyze the effect of different coupling and intensity of migratory movements between the two countries and explore the effect of seasonal migrations. For a random complex network setting, we show that it is possible to reach a synchronization state by increasing the coupling strength and test the influence of the topology in the dynamics of the complex network. All the results are analyzed through numerical simulations where the given algorithms are implemented with the python 3.5 language, in a Debian/GNU-Linux environment.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7


  1. Angola is in the 30 high TB burden countries. of WHO’s Global tuberculosis report 2018 [15].


  1. Aziz-Alaoui, M.A.: Synchronization of chaos. Encycl. Math. Phys. 5, 213–226 (2006)

    Article  Google Scholar 

  2. Belykh, I., Hasler, M., Lauret, M., Nijmeijer, H.: Synchronization and graph topology. Int. J. Bifurc. Chaos 11, 3423–3433 (2005)

    MathSciNet  Article  Google Scholar 

  3. Cantin, G.: Non identical coupled networks with a geographical model for human behaviors during catastrophic events. Int. J. Bifurc. Chaos 27(14), 1750213 (2017)

    MathSciNet  Article  Google Scholar 

  4. Haken, H.: Information and Self-organization: A Macroscopic Approach to Complex Systems. Springer, New York (2006)

    MATH  Google Scholar 

  5. Hethcote, H.W., van den Driessche, P.: An SIS epidemic model with variable population size and a delay. J. Math. Biol. 34, 177–194 (1995)

    MathSciNet  Article  Google Scholar 

  6. Garner-Purkis, A., Hine, P., Gamage, A., Pererac, S., Gulliford, M.C.: Tuberculosis screening for prospective migrants to high-income countries: systematic review of policies. Public Health 168, 142–147 (2019)

    Article  Google Scholar 

  7. Kahoui, M.E., Otto, A.: Stability of disease free equilibria in epidemiological models. Math. Comput. Sci. 2, 517–533 (2009)

    MathSciNet  Article  Google Scholar 

  8. Li, X., Mou, C., Niu, W., et al.: Stability analysis for discrete biological models using algebraic methods. Math. Comput. Sci. 5, 247–262 (2011)

    MathSciNet  Article  Google Scholar 

  9. Lopes, J.S., Rodrigues, P., Pinho, S.T.R., Andrade, R.F.S., Duarte, R., Gomes, M.G.M.: Interpreting measures of tuberculosis transmission: a case study on the Portuguese population. BMC Infect. Dis. 14, 340 (2014)

    Article  Google Scholar 

  10. Pareek, M., et al.: The impact of migration on tuberculosis epidemiology and control in high-income countries: a review. BMC Med. 14, 48 (2016)

    Article  Google Scholar 

  11. Pecora, L.M., Carroll, T.L.: Master stability functions for synchronized coupled systems. Phys. Rev. Lett. 80(10), 2109–2112 (1998)

    Article  Google Scholar 

  12. Prigogine, I., Nicolis, G.: Self Organization in Non-equilibrium Systems. Wiley, New York (1977)

    MATH  Google Scholar 

  13. Rocha, E.M., Silva, C.J., Torres, D.F.M.: The effect of immigrant communities coming from higher incidence tuberculosis regions to a host country. Ric. Mat. 67(1), 89–112 (2018)

    MathSciNet  Article  Google Scholar 

  14. Smith, H.L., Thieme, H.R.: Dynamical Systems and Population Persistence. Graduate Studies in Mathematics. American Mathematical Soc., Providence (2011)

    MATH  Google Scholar 

  15. WHO: Global Tuberculosis Report 2018 (2018)

  16. Tuberculosis Fact sheets, WHO, 18 September (2018)

  17. Accessed 3 July (2019)

  18. Accessed 3 July (2019)

Download references


This work is partially supported by The Center for Research and Development in Mathematics and Applications (CIDMA) through the Portuguese Foundation for Science and Technology (FCT—Fundação para a Ciência e a Tecnologia), references UIDB/04106/2020 and UIDP/04106/2020. Silva is also supported by national funds (OE), through FCT, I.P., in the scope of the framework contract foreseen in the numbers 4, 5 and 6 of the article 23, of the Decree-Law 57/2016, of August 29, changed by Law 57/2017, of July 19. Silva is also grateful to the support of the COST Action CA16227—Investigation and Mathematical Analysis of Avant-garde Disease Control via Mosquito Nano-Tech-Repellents (WG2 on ?Structured Models & Optimal Control?).

Author information

Authors and Affiliations


Corresponding author

Correspondence to Cristiana J. Silva.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Silva, C.J., Cantin, G. Synchronization and Self-organization in Complex Networks for a Tuberculosis Model. Math.Comput.Sci. 15, 107–120 (2021).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Complex network
  • Graph topology
  • Python
  • Tuberculosis

Mathematics Subject Classification

  • 34A34
  • 34C60
  • 92B05