Journal of the Korean Data and Information Science Society

The Korean Data and Information Science Society (한국데이터정보과학회)
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An estimation method of probability of infection using Reed - Frost model

Reed - Frost 모형을 이용한 전염병 감염 확률 추정

  • Eom, Eunjin (Department of Statistics, Daegu University) ;
  • Hwang, Jinseub (Department of Computer Science and Statistics, Daegu University) ;
  • Choi, Boseung (Department of Applied Statistics, Korea University)
  • 엄은진 (대구대학교 대학원 통계학과) ;
  • 황진섭 (대구대학교 전산통계학과) ;
  • 최보승 (고려대학교 응용통계학과)
  • Received : 2016.12.30
  • Accepted : 2017.01.12
  • Published : 2017.01.31
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Abstract

SIR model (Kermack and McKendrik, 1927) is one of the most popular method to explain the spread of disease, In order to construct SIR model, we need to estimate transition rate parameter and recovery rate parameter. If we don't have any information of the two rate parameters, we should estimate using observed whole trajectory of pandemic of disease. Thus, with restricted observed data, we can't estimate rate parameters. In this research, we introduced Reed-Frost model (Andersson and Britton, 2000) to calculate the probability of infection in the early stage of pandemic with the restriction of data. When we have an initial number of susceptible and infected, and a final number of infected, we can apply Reed - Frost model and we can get the probability of infection. We applied the Reed - Frost model to the Vibrio cholerae pandemic data from Republic of the Cameroon and calculated the probability of infection at the early stage. We also construct SIR model using the result of Reed - Frost model.

질병의 확산 과정을 설명하기 위한 모형으로 가장 대표적인 방법은 Kermack과 McKendrick (1927)에 의해 제안된 SIR (susceptible - infectious - recovered) 모형이다. SIR 모형을 구축하기 위해서는 질병의 감염률 (transition rate)과 회복률 (recovery rate)이 주어져 있거나 질병의 전체 확산 과정이 데이터로 주어진 경우 추정을 통하여 구할 수 있다. 하지만 데이터가 제한적으로 관찰된 경우 직접적인 감염률와 회복률의 계산이 불가능 하다. 본 연구에서는 관찰된 자료가 가지는 한계점을 고려하여 질병의 초기 확산과정에서 질병 감염 확률을 추정하기 위하여 리드-프로스트 (Reed-Frost) 모형 (Andersson과 Britton, 2000)을 적용하였다. 리드-프로스트 모형은 질병의 최초 감염자 수, 최종 감염자 수, 그리고 최초 감염대상자의 수가 주어졌을 때 이를 통하여 감염 확률을 추정하기 위한 모형이다. 본 연구에서는 서아프리카의 카메룬 공화국에서 조사된 역학 조사 자료를 이용하여 콜레라의 초기 감염 확률을 추정하였다. 그리고 추정된 결과를 이용하여 다시 SIR 모형에 적용하여 질병의 확산 경로에 대한 예측을 수행하였다. 예측 결과 조사 지역의 주민 가운데 50% 이상이 감염될 것으로 예측되었으며 질병의 전파는 약 한달 정도 지속될 것으로 예측 되었다.

Keywords

References

  1. Abbey, H. (1952). An examination of the Reed-Frost theory of epidemics. Human Biology, 24, 201-233.
  2. Andersson, H. and Britton, T. (2000). Stochastic epidemic models and their statistical analysis, Springer, New York.
  3. Bernoulli, D. and Blower, S. (2004). An attempt at a new analysis of the mortality caused by smallpow and of the advantages of inoculation to prevent it. Reviews in Medical Virology, 14, 275-288. https://doi.org/10.1002/rmv.443
  4. Brauer, F. and Castillo-Chavez, C. (2001). Mathematical models in population biology and epidemiology, Springer, New York.
  5. Choi, B. and Rempala, G. A. (2012). Inference for discretely observed stochastic kinetic networks with applications to epidemic modeling. Biostatistics, 13, 153-165. https://doi.org/10.1093/biostatistics/kxr019
  6. Daley, D. J. and Gany, J. (2005). Epidemic Modeling: An Introduction, Cambridge University Press, New York.
  7. Deijfen, M. (2011). Epidemics and vaccination on weighted graphs. Mathematical biosciences, 232, 57-65. https://doi.org/10.1016/j.mbs.2011.04.003
  8. Eisenberg, M. C., Robertson, S. L. and Tien, J. H. (2012). Identifiability and estimation of multiple transmission pathways in cholera and waterborne disease. Journal of Theoretical Biology, 324, 84-102.
  9. Gillespie, D. T. (1977). Exact stochastic simulation of coupled chemical reactions. The Journal of Physical Chemistry, 81, 2340-2361. https://doi.org/10.1021/j100540a008
  10. Hoops, S., Sahle, S., Gauges, R., Lee, C., Pahle, J., Simus, N., Singhal, M., Xu, L., Mendes, P. and Kummer, U. (2006). COPASI-a complex pathway simulator. Bioinformatics, 22, 3067-3074. https://doi.org/10.1093/bioinformatics/btl485
  11. Hwang, N. A., Jeong, B. Y., Lim, Y. C. and Park, J. S. (2007). Diseases data analysis using sir nonlinear regression model. Journal of The Korean Data Analysis Society, 9, 49-59.
  12. Kermack, W. O. and McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society A: Mathematical Physical and Engineering Sciences, 115, 700-721. https://doi.org/10.1098/rspa.1927.0118
  13. Kim, C. S. (2010). Development and evaluation of influenza confinement strategy using mathematical estimating model, Final report, Korea Centers for Disease Control & Prevention, Osong, Korea.
  14. Lee, J. H., Murshed, M. S. and Park, J. S. (2009). Estimation of infection distribution and prevalence number of Tsutsugamushi fever in Korea. Journal of the Korean Data & Information Science Society, 20, 149-158.
  15. Lim, H. (2007). Contributing factors of infectous waterborne and foodborne outbreaks in Korea. Journal of the Korean Medical Association, 50, 582-591. https://doi.org/10.5124/jkma.2007.50.7.582
  16. Lim, Y., Do, M. and Choi, B. (2016). A construction of susceptible - infected - removed model using Korean MERS pandemic data. Journal of the Korean Data Analysis Society, 18, 105-115.
  17. Mendes, P., Hoops, S. and Gauges, R. (2009). Computational modeling of bochemical networks using COPASI. Methods in Molecular Biology-Clifton then Totowa, 500, 17-60.
  18. Neurirth, E., Arganbright, D. (2004). The active modeler: Mathematical modeling with Microsoft Excel, Tnomson Brooks/Cde, Belmont.
  19. Profitos, M., Mouhaman, A., Lee, S., Garabed, R., Moritz, M., Piperata, B., Tien, J., Bisesi, M. and Lee, J. (2014). Muddying the Waters: A new area of concern for drinking water contamination in Cameroon. International Journal of Environmental Research and Public Health, 11, 12454-12526. https://doi.org/10.3390/ijerph111212454
  20. Ryu, S. and Choi, B. (2015). Development of epidemic model using the stochastic method. Journal of the Korean Data & Information Science Society, 26, 301-312. https://doi.org/10.7465/jkdi.2015.26.2.301
  21. Schaber, J. (2012). Easy parameter identifiability analysis with COPASI. Bio systems, 110, 183-185. https://doi.org/10.1016/j.biosystems.2012.09.003
  22. Seo, M. and Choi, B. (2015). An estimation method for stochastic epidemic model. Journal of the Korean Data Analysis Society, 17, 1247-1259.
  23. Trottier, H. and Philippe, P. (2001). Deterministic modeling of infectious diseases: Theory and methods. The Internet Journal of Infectious Disease, 1.