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Periodic Incidence in a Discrete-Time SIS Epidemic Model

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Book cover Mathematical Methods and Models in Biomedicine

Abstract

Mathematical models have continued to increase our understanding of the spread of infectious diseases and their control in both humans and animals. In most infectious diseases, the incidence coefficient or contact rate (the rate of new infections) plays a key role in ensuring that the model gives a reasonable qualitative description of the real disease dynamics. To accurately gauge the impact of infectious diseases prevention efforts, it is important to understand the relation between disease transmission and the host population dynamics. In [8–11], Castillo-Chavez and Yakubu introduced a framework for studying infectious disease dynamics in strongly fluctuating populations. In their model framework, Castillo-Chavez and Yakubu assumed that the host demographics is governed by the Ricker model and the contact rate is constant. However, periodicity in infectious disease incidence is known to occur in chickenpox, measles, pertussis, gonorrhea, mumps, influenza, and other infectious diseases.

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References

  1. Alexander, M.E., Moghadas, S.M.: Periodicity in an epidemic model with a generalized nonlinear incidence. Math. Biosci. 189, 75–96 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Allen, L.: Some discrete-time SI, SIR and SIS models. Math. Biosci. 124, 83–105 (1994)

    MATH  Google Scholar 

  3. Allen, L., Burgin, A.M.: Comparison of deterministic and stochastic SIS and SIR models in discrete-time. Math. Biosci. 163, 1–33 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Anderson, R.M., May, R.M.: Infectious Diseases of Humans: Dynamics and Control. Oxford University, Oxford (1992)

    Google Scholar 

  5. Bailey, N.T.J.: The simple stochastic epidemic: a complete solution in terms of known functions. Biometrika 50, 235–240 (1963)

    MathSciNet  MATH  Google Scholar 

  6. Best, J., Castillo-Chavez, C., Yakubu, A.-A.: Hierarchical competition in discrete time models with dispersal. Fields Institute Communications 36, 59–86 (2003)

    MathSciNet  Google Scholar 

  7. Beverton, R.J.H., Holt, S.J.: On the dynamics of exploited fish populations, vol. 2, p. 19. H.M. Stationery Off., London, Fish. Invest. (1957)

    Google Scholar 

  8. Castillo-Chavez, C., Yakubu, A.: Discrete-time S-I-S models with complex dynamics. Nonlin. Anal. TMA 47, 4753–4762 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Castillo-Chavez, C., Yakubu, A.: Dispersal, disease and life-history evolution. Math. Biosc. 173, 35–53 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Castillo-Chavez, C., Yakubu, A.: Epidemics on attractors. Contem. Math. 284, 23–42 (2001)

    MathSciNet  Google Scholar 

  11. Castillo-Chavez, C., Yakubu, A.A.: Intraspecific competition, dispersal and disease dynamics in discrete-time patchy environments. In: Castillo-Chavez, C., Blower, S., van den Driessche, P., Kirschner, D., Yakubu, A.-A. (eds.) Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction to Models, Methods and Theory. Springer, New York (2002)

    Chapter  Google Scholar 

  12. Cooke, K.L., Yorke, J.A.: Some equations modelling growth processes of gonorrhea epidemics. Math. Biosc. 16, 75–101 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cull, P.: Local and global stability for population models. Biol. Cybern. 54, 141–149 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  14. Elaydi, S.N.: Discrete Chaos. Chapman Hall/CRC, Boca Raton, FL (2000)

    MATH  Google Scholar 

  15. Elaydi, S.N., Yakubu, A.-A.: Global stability of cycles: Lotka-Volterra competition model With stocking. J. Diff. Equat. Appl. 8, 537–549 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ewald, P.W.: Evolution of Infectious Disease. Oxford University, Oxford, NY (1994)

    Google Scholar 

  17. Franke, J.E., Yakubu, A.-A.:Discrete-time SIS epidemic model in a seasonal environment. SIAM J. Appl. Math. 66, 1563–1587 (2006)

    Google Scholar 

  18. Franke, J.E., Yakubu, A.-A.: Periodically forced discrete-time SIS epidemic with disease-induced mortality. Math. Biosci. Eng. 8, 385–408 (2011)

    Article  MathSciNet  Google Scholar 

  19. Hadeler, K.P., van den Driessche, P.: Backward bifurcation in epidemic control. Math. Biosc. 146, 15–35 (1997)

    Article  MATH  Google Scholar 

  20. Hassell, M.P.: The dynamics of competition and predation. Studies in Biol, vol. 72. The Camelot Press Ltd, Southampton (1976)

    Google Scholar 

  21. Hassell, M.P., Lawton, J.H., May, R.M.: Patterns of dynamical behavior in single species populations. J. Anim. Ecol. 45, 471–486 (1976)

    Article  Google Scholar 

  22. Hethcote, H.W., van den Driessche, P.: Some epidemiological models with nonlinear incidence. J. Math. Biol. 29, 271–287 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  23. May, R.M.: Simple mathematical models with very complicated dynamics. Nature 261, 459–469 (1977)

    Article  Google Scholar 

  24. May, R.M.: Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton (1974)

    Google Scholar 

  25. May, R.M., Oster, G.F.: Bifurcations and dynamic complexity in simple ecological models. Am. Nat. 110, 573–579 (1976)

    Article  Google Scholar 

  26. McClusky, C.C., Muldowney, J.C.: Bendixson-Dulac criteria for difference equations. J. Dyn. Diff. Equat. 10, 567–575 (1998)

    Article  Google Scholar 

  27. Nicholson, A.J.: Compensatory reactions of populations to stresses, and their evolutionary significance. Aust. J. Zool. 2, 1–65 (1954)

    Article  Google Scholar 

  28. van den Driessche, P., Watmough, J.: A simple SIS epidemic model with a backward bifurcation. J. Math. Biol. 40, 525–540 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  29. Yakubu, A.-A.: Introduction to discrete-time epidemic models. DIMACS Ser. Discrete Math. Theor. Comput. Sci. 75, 83–109 (2010)

    MathSciNet  Google Scholar 

  30. Yakubu, A.-A., Ziyadi, N.: Discrete-time exploited fish epidemic models. Afrika Mathematika, 22, 177–199 (2011)

    Article  MathSciNet  Google Scholar 

  31. Yorke, J.A., London, W.P.: Recurrent outbreaks of measles, chickenpox and mumps II. Am. J. Epidemiol. 98, 453–468 (1973)

    Google Scholar 

  32. Zhao, X.-Q.: Asymptotic behavior for asymptotically periodic semiflows with applications. Comm. Appl. Nonl. Anal. 3, 43–66 (1996)

    MATH  Google Scholar 

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Acknowledgements

This research has been partially supported by the National Marine Fisheries Service, Northeast Fisheries Science Center (Woods Hole, MA 02543), Department of Homeland Security, DIMACS and CCICADA of Rutgers University, Mathematical Biosciences Institute of the Ohio State University and National Science Foundation under grants DMS 0931642 and 0832782.

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Authors and Affiliations

  1. Department of Mathematics, Morgan State University, Baltimore, MD, 21251, USA

    Najat Ziyadi

  2. Department of Mathematics, Howard University, Washington, DC, 20059, USA

    Abdul-Aziz Yakubu

Authors
  1. Najat Ziyadi
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  2. Abdul-Aziz Yakubu
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Correspondence to Najat Ziyadi .

Editor information

Editors and Affiliations

  1. , Mathematics and Statistics, Southern Illinois University Edwardsvill, Edwardsville, 62026, Illinois, USA

    Urszula Ledzewicz

  2. , Electrical and Systems Engineering, Washington University, Brookings Drive 1, Saint Louis, 63130, Missouri, USA

    Heinz Schättler

  3. Mathematical Biosciences Institute, Ohio State University, W. 18th Avenue 231, Columbus, 43210-1292, Ohio, USA

    Avner Friedman

  4. Tel Aviv University, Ramat Aviv, Tel Aviv, 69978, Israel

    Eugene Kashdan

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Ziyadi, N., Yakubu, AA. (2013). Periodic Incidence in a Discrete-Time SIS Epidemic Model. In: Ledzewicz, U., Schättler, H., Friedman, A., Kashdan, E. (eds) Mathematical Methods and Models in Biomedicine. Lecture Notes on Mathematical Modelling in the Life Sciences. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4178-6_15

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