A Disease Outbreak Prediction Model Using Bayesian Inference: A Case of Influenza

Document Type: Original Article

Authors

Information Technology Department, Faculty of Industrial Engineering, Tarbiat Modares University, Tehran, Iran

Abstract

Introduction: One major problem in analyzing epidemic data is the lack of data and high dependency among the available data, which is due to the fact that the epidemic process is not directly observable.
Methods: One method for epidemic data analysis to estimate the desired epidemic parameters, such as disease transmission rate and recovery rate, is data intensification. In this method, unknown quantities are considered as additional parameters of the model and are extracted using other parameters. The Markov Chain Monte Carlo algorithm is extensively used in this field.
Results: The current study presents a Bayesian statistical analysis of influenza outbreak data using Markov Chain Monte Carlo data intensification that is independent of probability approximation and provides a wider range of results than previous studies. A method for estimating the epidemic parameters has been presented in a way that the problem of uncertainty regarding the modeling of dynamic biological systems can be solved. The proposed method is then applied to fit an SIR-like flu transmission model to data from 19 years leading up to the seventh week of the 2017 incidence of influenza.
Conclusion: The proposed method showed an improvement in estimating the values of all the parameters considered in the study. The results of this study showed that the distributions are significant and the error ranges are real.

Keywords


  1. Anderson RM, May RM. Infectious diseases of humans: Dynamics and control. Oxford: Oxford University Press; 1991.
  2. Anderson RM. The role of mathematical models in the study of HIV transmission and the epidemiology of AIDS. J Acquir Immune Defic Syndr. 1988;1(3):241-256.
  3. Ferguson NM, Keeling MJ, Edmunds WJ, et al. Planning for smallpox outbreaks. Nature. 2003;425(6959):681-685. doi:10.1038/nature02007.
  4. Ngwa GA, Shu WS. A mathematical model for endemic malaria with variable human and mosquito populations. Math Comput Model. 2000;32(7-8):747-763. doi:10.1016/S0895-7177(00)00169-2.
  5. Fraser C, Donnelly CA, Cauchemez S, et al. Pandemic potential of a strain of influenza A (H1N1): early findings. Science. 2009;324(5934):1557-1561. doi:10.1126/science.1176062.
  6. Gibson GJ, Renshaw E. Estimating parameters in stochastic compartmental models using Markov chain methods. IMA J Math Appl Med Biol. 1998;15(1):19-40. doi:10.1093/imammb/15.1.19.
  7. Hethcote HW. The mathematics of infectious diseases. SIAM Rev Soc Ind Appl Math. 2000;42(4):599-653. doi:10.1137/S0036144500371907.
  8. Dietz K. Transmission and control of arbovirus diseases. In: Ludwig D, Cooke KL, eds. Epidemiology. Philadelphia: SIAM; 1975:104- 121.
  9. Just W, Callender HL. Differential equation models of disease transmission. March 2015. http://shorturl.at/dBZ36.
  10. Hamer W. Epidemic disease in England. Lancet. 1906;(1):733-739.
  11. Ross R. The Prevention of Malaria. 2nd ed. London: Murray; 1981.
  12. Bailey NT. The Mathematical Theory of Infectious Diseases. 2nd ed. New York: Hafner; 1975.
  13. Dietz K. Epidemics and rumours: A survey. J R Stat Soc Ser A. 1967;130(4):505-528. doi:10.2307/2982521.
  14. Dietz K. The first epidemic model: a historical note on PD En’ko. Aust J Stat. 1988;30A(1):56-65. doi:10.1111/j.1467-842X.1988.tb00464.x.
  15. Kermack WO, McKendrick AG. A contribution to the mathematical theory of epidemics. Proc R Soc Lond A. 1927;115(772):700-721. doi:10.1098/rspa.1927.0118.
  16. Becker N. The uses of epidemic models. Biometrics. 1979;35(1):295-305. doi:10.2307/2529951.
  17. Castillo-Chavez C. Mathematical and statistical approaches to AIDS epidemiology. Berlin: Springer-Verlag; 1989. doi:10.1007/978-3-642-93454-4.
  18. Dietz K. Density-dependence in parasite transmission dynamics. Parasitol Today. 1988;4(4):91-97. doi:10.1016/0169-4758(88)90034-8.
  19. Dietz K, Schenzle D. Mathematical models for infectious disease statistics In: Atkinson AC, Fienberg SE, eds. A Celebration of Statistics. New York: Springer-Verlag; 1985:167-204. doi:10.1007/978-1-4613-8560-8_8.
  20. Hethcote H.W. A thousand and one epidemic models. In: Levin SA, ed. Frontiers in mathematical biology. New York: Springer- Verlag; 1994:504-515. doi:10.1007/978-3-642-50124-1_29.
  21. Hethcote HW, Levin SA. Periodicity in epidemiological models. In: Levin SA, Hallam TG, Gross LJ, eds. Applied mathematical ecology. New York: Springer-Verlag; 1989:193-211. doi:10.1007/978-3-642-61317-3_8.
  22. Hethcote HW, Stech P. Periodicity and stability in epidemic models: A survey. In: Busenberg SN, Cooke KL, eds. Differential equations and applications in ecology, epidemics, and population problems. New York: Academic Press; 1981:65-82. doi:10.1016/B978-0-12-148360-9.50011-1.
  23. Brooks S, Gelman A, Jones G, Meng XL, editors. Handbook of markov chain monte carlo. CRC press; 2011 May 10. doi:10.1201/b10905.
  24. Wickwire K. Mathematical models for the control of pests and infectious diseases: a survey. Theor Popul Biol. 1977;11(2):182-238. doi:10.1016/0040-5809(77)90025-9.
  25. Dempster AP. A generalization of Bayesian inference. In: Yager RR, Liu L, eds. Classic works of the Dempster-Shafer theory of belief functions. Berlin: Springer; 2008. doi:10.1007/978-3-540-44792-4_1.
  26. Greenland S. Generalized conjugate priors for Bayesian analysis of risk and survival regressions. Biometrics. 2003;59(1):92-99. doi:10.1111/1541-0420.00011.
  27. Greenland S. Bayesian perspectives for epidemiological research: I. Foundations and basic methods. Int J Epidemiol. 2006;35(3):765-775. doi:10.1093/ije/dyi312.
  28. Yang B. Stochastic dynamics of an SEIS epidemic model. Adv Differ Equ. 2016;2016(1):226. doi:10.1186/s13662-016-0914-3.
  29. Nsoesie EO, Beckman RJ, Marathe MV. Sensitivity analysis of an individual-based model for simulation of influenza epidemics. PLoS One. 2012;7(10):e45414. doi:10.1371/journal.pone.0045414.
  30. Halloran ME, Ferguson NM, Eubank S, et al. Modeling targeted layered containment of an influenza pandemic in the United States. Proc Natl Acad Sci U S A. 2008;105(12):4639-4644. doi:10.1073/pnas.0706849105.
  31. Germann TC, Kadau K, Longini IM, Jr., Macken CA. Mitigation strategies for pandemic influenza in the United States. Proc Natl Acad Sci U S A. 2006;103(15):5935-5940. doi:10.1073/pnas.0601266103.
  32. Brooks LC, Farrow DC, Hyun S, Tibshirani RJ, Rosenfeld R. Flexible Modeling of Epidemics with an Empirical Bayes Framework. PLoS Comput Biol. 2015;11(8):e1004382. doi:10.1371/journal.pcbi.1004382.
  33. Kar TK, Batabyal A. Stability analysis and optimal control of an SIR epidemic model with vaccination. Biosystems. 2011;104(2- 3):127-135. doi:10.1016/j.biosystems.2011.02.001.
  34. Zaman G, Kang YH, Jung IH. Optimal treatment of an SIR epidemic model with time delay. Biosystems. 2009;98(1):43-50. doi:10.1016/j.biosystems.2009.05.006.