Bulletin of Mathematical Biology

, Volume 75, Issue 7, pp 1031–1050

On the Exact Measure of Disease Spread in Stochastic Epidemic Models

Original Article

Abstract

The basic reproduction number, R0, is probably the most important quantity in epidemiology. It is used to measure the transmission potential during the initial phase of an epidemic. In this paper, we are specifically concerned with the quantification of the spread of a disease modeled by a Markov chain. Due to the occurrence of repeated contacts taking place between a typical infective individual and other individuals already infected before, R0 overestimates the average number of secondary infections. We present two alternative measures, namely, the exact reproduction number, Re0, and the population transmission number, Rp, that overcome this difficulty and provide valuable insight. The applicability of Re0 and Rp to control of disease spread is also examined.

Keywords

Disease spread Basic reproduction number Stochastic epidemic Vaccination coverage 

References

  1. Allen, L. J. S. (2003). An introduction to stochastic processes with applications to biology. Englewood Cliffs: Prentice-Hall. MATHGoogle Scholar
  2. Andersson, H., & Britton, T. (2000). Springer lecture notes in statistics: Vol. 151. Stochastic epidemic models and their statistical analysis. New York: Springer. MATHCrossRefGoogle Scholar
  3. Andreasen, V. (2011). The final size of an epidemic and its relation to the basic reproduction number. Bull. Math. Biol., 73, 2305–2321. MathSciNetCrossRefGoogle Scholar
  4. Artalejo, J. R., & Lopez-Herrero, M. J. (2011). The SIS and SIR stochastic epidemic models: a maximum entropy approach. Theor. Popul. Biol., 80, 256–264. CrossRefGoogle Scholar
  5. Artalejo, J. R., Economou, A., & Lopez-Herrero, M. J. (2010). On the number of recovered individuals in the SIS and SIR stochastic epidemic models. Math. Biosci., 228, 45–55. MathSciNetMATHCrossRefGoogle Scholar
  6. Artalejo, J. R., Economou, A., & Lopez-Herrero, M. J. (2012). Stochastic epidemic models revisited: analysis of some continuous performance measures. J. Biol. Dyn., 6, 189–211. MathSciNetCrossRefGoogle Scholar
  7. Bacaër, N., & Gomes, M. G. M. (2009). On the final size of epidemics with seasonality. Bull. Math. Biol., 71, 1954–1966. MathSciNetMATHCrossRefGoogle Scholar
  8. Bailey, N. T. J. (1975). The mathematical theory of infectious diseases and its applications. London: Charles Griffin & Company Ltd. MATHGoogle Scholar
  9. Ball, F., & Nåsell, I. (1994). The shape of the size distribution of an epidemic in a finite population. Math. Biosci., 123, 167–181. MATHCrossRefGoogle Scholar
  10. Böckh, R. (1886). Statistisches Fahrbuch der Stadt Berlin, Zwölfter Jahrgang. Statistik des Jahres (pp. 30–31). Berlin: P. Stankiewicz. Google Scholar
  11. Britton, T. (2010). Stochastic epidemic models: a survey. Math. Biosci., 225, 24–35. MathSciNetMATHCrossRefGoogle Scholar
  12. Ciarlet, P. G. (1989). Introduction to numerical linear algebra and optimisation. Cambridge: Cambridge University Press. Google Scholar
  13. Cross, P. C., Lloyd-Smith, J. O., Johnson, P. L. F., & Getz, W. M. (2005). Duelling time scales of host movement and disease recovery determine invasion of disease in structured populations. Ecol. Lett., 8, 587–595. CrossRefGoogle Scholar
  14. de Koejier, A. A., Diekmann, O., & de Jong, M. C. M. (2008). Calculating the extinction of a reactivating virus, in particular bovine herpes virus. Math. Biosci., 212, 111–131. MathSciNetCrossRefGoogle Scholar
  15. Diekmann, O., & Heesterbeek, J. A. P. (2000). Wiley series in mathematical and computational biology. Mathematical epidemiology of infectious diseases: model building, analysis and interpretation. Chichester: Wiley. Google Scholar
  16. Diekmann, O., Heesterbeek, J. A. P., & Metz, J. A. J. (1990). On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations. J. Math. Biol., 28, 365–382. MathSciNetMATHCrossRefGoogle Scholar
  17. Diekmann, O., de Jong, M. C. M., & Metz, J. A. J. (1998). A deterministic epidemic model taking account of repeated contacts between the same individuals. J. Appl. Probab., 35, 448–462. MathSciNetMATHCrossRefGoogle Scholar
  18. Diekmann, O., Heesterbeek, H., & Britton, T. (2013). Mathematical tools for understanding infectious disease dynamics. Princeton: Princeton University Press. MATHGoogle Scholar
  19. Forrester, M., & Pettitt, A. N. (2005). Use of stochastic epidemic modeling to quantify transmission rates of colonization with methicillin-resistant Staphylococcus aureus in an intensive care unit. Infect. Control Hosp. Epidemiol., 26, 598–606. CrossRefGoogle Scholar
  20. Green, D. M., Kiss, I. Z., & Zao, R. R. (2006). Parametrization of individual-based models: comparisons with deterministic mean-field models. J. Theor. Biol., 239, 289–297. CrossRefGoogle Scholar
  21. Heesterbeek, J. A. P., & Dietz, K. (1996). The concept of R 0 in epidemic theory. Stat. Neerl., 50, 89–110. MathSciNetMATHCrossRefGoogle Scholar
  22. Heffernan, J. M., Smith, R. J., & Wahl, I. M. (2005). Perspectives on the basic reproductive ratio. J. R. Soc. Interface, 2, 281–293. CrossRefGoogle Scholar
  23. Hethcote, H. W. (2000). The mathematics of infectious diseases. SIAM Rev., 42, 599–653. MathSciNetMATHCrossRefGoogle Scholar
  24. Hotta, L. K. (2010). Bayesian melding estimation of a stochastic SEIR model. Math. Popul. Stud., 17, 101–111. MathSciNetCrossRefGoogle Scholar
  25. Keeling, M. J., & Grenfell, B. T. (2000). Individual-based perspectives on R 0. J. Theor. Biol., 203, 51–61. CrossRefGoogle Scholar
  26. Keeling, M. J., & Rohani, P. (2008). Modeling infectious diseases in humans and animals. Princeton: Princeton University Press. MATHGoogle Scholar
  27. Keeling, M. J., & Ross, J. V. (2008). On methods for studying stochastic disease dynamics. J. R. Soc. Interface, 5, 171–181. CrossRefGoogle Scholar
  28. Kulkarni, V. G. (1995). Modeling and analysis of stochastic systems. Boca Raton: Chapman and Hall. MATHGoogle Scholar
  29. Li, J., Blakeley, D., & Smith, R. J. (2011). The failure of R 0. Comput. Math. Methods Med., 2011, 527610. MathSciNetCrossRefGoogle Scholar
  30. Ma, J. L., & Earn, D. J. D. (2006). Generality of the final size formula for an epidemic of a newly invading infectious disease. Bull. Math. Biol., 68, 679–802. MathSciNetCrossRefGoogle Scholar
  31. Nåsell, I. (2001). Extinction and quasi-stationarity in the Verhulst logistic model. J. Theor. Biol., 211, 11–27. CrossRefGoogle Scholar
  32. Nåsell, I. (2011). Springer lecture notes in mathematics: Vol. 2022. Extinction and quasi-stationarity in the stochastic logistic SIS model. Berlin: Springer. MATHCrossRefGoogle Scholar
  33. Nisbet, R. M., & Gurney, W. S. C. (2003). Modelling fluctuating populations. Caldwell: Blackburn Press. Google Scholar
  34. Norden, R. H. (1982). On the distribution of the time to extinction in the stochastic logistic population model. Adv. Appl. Probab., 14, 687–708. MathSciNetMATHCrossRefGoogle Scholar
  35. Orsel, K., Bouma, A., Dekker, A., Stegeman, J. A., & de Jong, M. C. M. (2009). Foot and mouth disease virus transmission during the incubation period of the disease in piglets, lambs, calves, and dairy cows. Prev. Vet. Med., 88, 158–163. CrossRefGoogle Scholar
  36. Pellis, L., Ball, F., & Trapman, P. (2012). Reproduction numbers for epidemic models with households and other social structures. I. Definition and calculation of R 0. Math. Biosci., 235, 85–97. MathSciNetMATHCrossRefGoogle Scholar
  37. Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. (2007). Numerical recipes: the art of scientific computing. New York: Cambridge University Press. Google Scholar
  38. Roberts, M. G. (2007). The pluses and minuses of R 0. J. R. Soc. Interface, 4, 946–961. Google Scholar
  39. Roberts, M. G. (2012). Epidemic models with uncertainty in the reproduction number. J. Math. Biol. doi:10.1007/s00285-012-0540-y. Google Scholar
  40. Stone, P., Wilkinson-Herbots, H., & Isham, V. (2008). A stochastic model for head lice infections. J. Math. Biol., 56, 743–763. MathSciNetMATHCrossRefGoogle Scholar
  41. van den Driessche, P., & Watmough, J. (2002). Reproduction number and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci., 180, 29–48. MathSciNetMATHCrossRefGoogle Scholar
  42. Wang, J., Wang, L., Magal, P., Wang, Y., Zhuo, J., Lu, X., & Ruan, S. (2011). Modelling the transmission dynamics of methicillin-resistant Staphylococcus aureus in Beijing Tongren hospital. J. Hosp. Infect., 79, 302–308. CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2013

Authors and Affiliations

  1. 1.Department of Statistics and Operations Research, Faculty of MathematicsComplutense University of MadridMadridSpain
  2. 2.School of StatisticsComplutense University of MadridMadridSpain

Personalised recommendations