An Integrable SIS Model on Time Scales

  • Martin BohnerEmail author
  • Sabrina Streipert
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 312)


In this work, we generalize the dynamic model introduced in Bohner and Streipert (Pliska Stud. Math. 26:11–28, 2016, [5]) in the context of epidemiology. This model exhibits many similarities to the continuous susceptible-infected-susceptible model and is therefore of particular interest to formulate a generalization of a continuous model on time scales. In this work, we extend the results in Bohner and Streipert (Pliska Stud. Math. 26:11–28, 2016, [5]) for time-dependent coefficients rather than constant parameters and derive an explicit solution. We further discuss the stability of periodic solutions for the corresponding discrete model with periodic coefficients. We conclude the analysis of the SIS model by considering time-dependent vital dynamics and derive its explicit solution on a general time scale.


Time scales Dynamic equations Difference equations Epidemiology Periodic solution Stability 


  1. 1.
    Allen, L.J.S.: Some discrete-time SI, SIR, and SIS epidemic models. Math. Biosci. 124, 83–105 (1994)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bohner, M.: Some oscillation criteria for first order delay dynamic equations. Far East J. Appl. Math. 18, 289–304 (2005)Google Scholar
  3. 3.
    Bohner, M., Peterson, A.: Dynamic Equations on Time Scales. Birkhäuser, Boston (2001)CrossRefGoogle Scholar
  4. 4.
    Bohner, M., Peterson, A.: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston (2003)Google Scholar
  5. 5.
    Bohner, M., Streipert, S.: The SIS-model on time scales. Pliska Stud. Math. 26, 11–28 (2016)Google Scholar
  6. 6.
    Bohner, M., Streipert, S., Torres, D.F.M.: Exact solution to a dynamic SIR model. Nonlinear Anal. Hybrid Syst. 32, 228–238 (2019)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Kandhway, K., Kuri, J.: How to run a campaign: optimal control of SIS and SIR information epidemics. Appl. Math. Comput. 231, 79–92 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Keeling, M.J., Rohani, P.: Modeling Infectious Diseases in Humans and Animals, p. xiii. Princeton University Press, Princeton (2008)Google Scholar
  9. 9.
    Mkhatshwa, T., Mummert, A.: Modeling super-spreading events for infectious diseases: case study SARS. IAENG Int. J. Appl. Math. 41, (2011)Google Scholar
  10. 10.
    Nucci, M.C., Leach, P.G.L.: An integrable SIS model. J. Math. Anal. Appl. 290, 506–518 (2004)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Rodrigues, H.S., Fonseca, M.J.: Viral marketing as epidemiological model. In: Proceedings of the 15th International Conference on Computational and Mathematical Methods in Science and Engineering, pp. 946–955 (2015)Google Scholar
  12. 12.
    Wang, J., Wang, Y.Q.: SIR rumor spreading model with network medium in complex social networks. Chinese J. Phys. 53, Art. ID 020702, 21 (2015)Google Scholar

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Missouri University of Science and TechnologyRollaUSA
  2. 2.Centre for Applications in Natural Resource MathematicsUniversity of QueenslandSt LuciaAustralia

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