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Modeling the Stochastic Dynamics of Influenza Epidemics with Vaccination Control, and the Maximum Likelihood Estimation of Model Parameters

  • Divine WandukuEmail author
  • C. Newman
  • O. Jegede
  • B. Oluyede
Chapter
  • 258 Downloads
Part of the Forum for Interdisciplinary Mathematics book series (FFIM)

Abstract

This study presents a family of stochastic models for the dynamics of influenza in a closed human population. We consider treatment for the disease in the form of vaccination and incorporate the periods of effectiveness of the vaccine and infectiousness for the individuals in the population. Our model is a SVIR model, with trinomial transition probabilities, where all individuals who recover from the disease acquire permanent natural immunity against the strain of the disease. A special SVIR model in the stochastic family based on correlated vaccination and infection probabilities at any instant is presented. The methods of maximum likelihood and expectation–maximization algorithm are applied to find estimates for the parameters of the chain. Moreover, estimators for some special epidemiological control parameters, such as the basic reproduction number, are computed. A numerical simulation example is presented to find the MLE of the parameters of the model.

Keywords

Influenza epidemics Chain binomial model MLE method EM algorithm Basic reproduction number 

Notes

Acknowledgements

This work was completed during the graduate studies of Cameron Newman, Omotomilola Jegede and Mymuna Monem in the department of Mathematical Sciences of Georgia Southern University (GSU) in 2018–2019 academic year, supervised by Dr. Wanduku. Mymuna Monem was part of the general group discussions.

References

  1. 1.
    CDC Estimating Seasonal Influenza-Associated Deaths. https://www.cdc.gov/flu/about/disease/us_flu-related_deaths.htm
  2. 2.
    D. Iuliano, K.Roguski, H. Chang, D. Muscatello, R. Palekar, S. Tempia et al., Estimates of global seasonal influenza-associated respiratory mortality: a modelling study. Lancet 391(10127), 1285–1300, 31 Mar 2018Google Scholar
  3. 3.
  4. 4.
    CDC  Flu  symptoms  and  complications. https://www.cdc.gov/flu/consumer/symptoms.htm/
  5. 5.
    CDC Types of Influenza Viruses. https://www.cdc.gov/flu/about/viruses/types.htm
  6. 6.
  7. 7.
    CDC Different types of flu vaccines. https://www.cdc.gov/flu/vaccines/index.htm
  8. 8.
    H.C. Tuckwell, R,J. Williams, Some properties of a simple stochastic epidemic model of SIR type. Math. Biosci. 208, 76–97 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    D. Wanduku, Threshold conditions for a family of epidemic dynamic models for malaria with distributed delays in a non-random environment, Int. J. Biomath. 11(6) 1850085 (46 pages) (2018).  https://doi.org/10.1142/S1793524518500857MathSciNetCrossRefGoogle Scholar
  10. 10.
    P. Witbooi, G.E. Muller, G.J. Van Schalkwyk, Vaccination control in a stochastic SVIR epidemic model. Comput. Math. Methods Med. 2015, Article ID 271654, 9 pages (2015).  https://doi.org/10.1155/2015/271654MathSciNetCrossRefGoogle Scholar
  11. 11.
    D. Wanduku, Complete global analysis of a two-scale network SIRS epidemic dynamic model with distributed delay and random perturbation. Appl. Math. Comput. 294, 49–76 (2017)MathSciNetzbMATHGoogle Scholar
  12. 12.
    D. Wanduku, G.S. Ladde, Global properties of a two-scale network stochastic delayed human epidemic dynamic model. Nonlinear Anal. Real World Appl. 13, 794–816 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    F. Etbaigha, A. Willms, Z. Poljak, An SEIR model of influenza A virus infection and reinfection within a farrow-to-finish swine farm. PLOS ONE 13(9), e0202493.  https://doi.org/10.1371/journal.pone.0202493CrossRefGoogle Scholar
  14. 14.
    M.E. Alexander, C. Bowman, S.M. Moghadas et al., Vaccination model for transmission dynamics of influenza. SIAM J. Appl. Dyn. Syst. 3, 503–524 (2004)MathSciNetCrossRefGoogle Scholar
  15. 15.
    D. Wanduku, The stochastic extinction and stability conditions for nonlinear malaria epidemics. Math. Biosci. Eng. 16, 3771–3806 (2019)MathSciNetCrossRefGoogle Scholar
  16. 16.
    A. Scherer, A.R. McLean, Mathematical models of vaccination. Brit. Med. Bull. 62, 187–199 (2002)CrossRefGoogle Scholar
  17. 17.
    H.S. Rodrigues, M.T.T. Monteiro, D.F.M. Torres, Vaccination models and optimal control strategies to dengue. Math. Biosci. 247, 1–12 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    A. Lloyd, Introduction to Epidemiological Modeling: Basic Models and Their Properties 23 Jan. 2017Google Scholar
  19. 19.
    M. Li, J.R. Graef, L. Wang, J. Karsai, Global dynamics of a SEIR model with varying total population size. Math. Biosci. 160, 191–213 (1999)MathSciNetCrossRefGoogle Scholar
  20. 20.
    C.M. Kribs-Zaleta, J.X. Velasco-Hernández, A simple vaccination model with multiple endemic states. J. Math. Biosci. 164, 183–201 (2000)CrossRefGoogle Scholar
  21. 21.
    D. Wanduku, G.S. Ladde, Fundamental properties of a two-scale network stochastic human epidemic dynamic model. Neural Parallel Sci. Comput. 19, 229–270 (2011)MathSciNetzbMATHGoogle Scholar
  22. 22.
    M. Ferrante, E. Ferraris, C. Rovira, On a stochastic epidemic SEIHR model and its diffusion approximation. TEST 25, 482 (2016).  https://doi.org/10.1007/s11749-015-0465-zMathSciNetCrossRefGoogle Scholar
  23. 23.
    R. Yaesoubi, T. Cohen, Generalized Markov models of infectious disease spread: a novel framework for developing dynamic health policies. Eur. J. Oper. Res. 215(3) (2011)Google Scholar
  24. 24.
    M. Greenwood, On the statistical measure of infectiousness. J. Hyg. Camb. 31, 336 (1931)CrossRefGoogle Scholar
  25. 25.
    H. Abbey, An examination of the Reed-Frost theory of epidemics. Hum. Biol. 24, 201 (1952)Google Scholar
  26. 26.
    J. Jacquez, A note on chain binomial models of epidemic spread: what is wrong with the Reed-Frost Model. Mathem. Biosci. 87(1), 73–82 (1987)MathSciNetCrossRefGoogle Scholar
  27. 27.
    J. Gani, D. Jerwood, Markov chain methods in chain binomial epidemic models. Biometrics 27 (1971)CrossRefGoogle Scholar
  28. 28.
    J. Lin, V. Andreasen, S.A. Levin, Dynamics of influenza A; the linear three strain model. Math. Biosci. 162, 33–51 (1999)MathSciNetCrossRefGoogle Scholar
  29. 29.
    M. Ajelli, P. Poletti, A. Melegaro S. Merler, The role of different social contexts in shaping influenza transmission during the 2009 pandemic. Sci. Rep. 4, Article number: 7218 (2014)Google Scholar
  30. 30.
    L. Allen, An introduction to stochastic epidemic models, in Mathematical Epidemiology ed. by Brauer F., van den Driessche P., Wu J. Lecture Notes in Mathematics, vol. 1945 (Springer, Berlin, Heidelberg), pp. 81–130CrossRefGoogle Scholar
  31. 31.
    M. Gupta, Y. Chen, Theory and use of the EM Algorithm. Found. Trends Sig. Process. 4(3) (2010)Google Scholar
  32. 32.
    J. Bilmes, A gentle tutorial of the EM algorithm and it’s Application to parameter estimation for Gausian mixture and hidden Markov models. Int. Comput. Sci. Inst. (1998)Google Scholar
  33. 33.
    G. Casella, R. Berger, Statistical Inference, 2 edn. (Duxbury, 2002)Google Scholar
  34. 34.
    K. Dietz, The estimation of the basic reproduction number for infectious diseases. Stat Methods Med Res. 2(1), 23–41 (1993)CrossRefGoogle Scholar
  35. 35.
    J. Jones, Notes on  \(R_0\) (Stanford University, Department of Anthropological Sciences, 2007)Google Scholar
  36. 36.
    P. Holme, N. Masuda, The basic reproduction number as a predictor for epidemic outbreaks in temporal networks. PLoS ONE 10(3) (2015)CrossRefGoogle Scholar
  37. 37.
    E. Vergu, H. Busson, P. Ezanno, Impact of the infection period distribution on the epidemic spread in a metapopulation model. PLoS ONE 5(2) (2010)CrossRefGoogle Scholar
  38. 38.
  39. 39.
  40. 40.
    US census Bureau, 2018 National and State Population Estimates. https://www.census.gov/newsroom/press-kits/2018/pop-estimates-national-state.html

Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  • Divine Wanduku
    • 1
    Email author
  • C. Newman
    • 1
  • O. Jegede
    • 1
  • B. Oluyede
    • 1
  1. 1.Department of Mathematical SciencesGeorgia Southern UniversityStatesboroUSA

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