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Behaviors of a Disease Outbreak During the Initial Phase and the Branching Process Approximation

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Quantitative Methods for Investigating Infectious Disease Outbreaks

Part of the book series: Texts in Applied Mathematics ((TAM,volume 70))

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Abstract

We consider that at the beginning, t = 0, there is no disease. We call the system at this condition the disease-free equilibrium. We assume that the entire population is susceptible. The size of the susceptible population is denoted by m.

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References

  • Allen, L. J. (2010). An introduction to stochastic processes with applications to biology. Boca Raton, FL: CRC Press.

    Google Scholar 

  • Anderson, D., & Watson, R. (1980). On the spread of a disease with gamma distributed latent and infectious periods. Biometrika, 67(1), 191–198.

    Article  Google Scholar 

  • Anderson, R. M., & May, R. M. (1991) Infectious diseases of humans, dynamics and control. Oxford: Oxford University Press.

    Google Scholar 

  • Arita, I., Shafa, E., & Kader, A. (1970). Role of hospital in smallpox outbreak in Kuwait. American Journal of Public Health and the Nations Health, 60(10), 1960–1966.

    Article  Google Scholar 

  • Assiri, A., McGeer, A., Perl, T. M., Price, C. S., Al Rabeeah, A. A., Cummings, D. A., et al. (2013). Hospital outbreak of Middle East respiratory syndrome coronavirus. New England Journal of Medicine, 369(5), 407–416.

    Article  Google Scholar 

  • Bacaër, N., & Abdurahman, X. (2008). Resonance of the epidemic threshold in a periodic environment. Journal of Mathematical Biology, 57, 649–673.

    Article  MathSciNet  Google Scholar 

  • Banks, R. B. (1994). Growth and diffusion phenomena: Mathematical frameworks and applications. Berlin: Springer.

    Book  Google Scholar 

  • Bartlett, M. S. (1955). An introduction to stochastic processes. London: Cambridge University Press.

    MATH  Google Scholar 

  • Barttlet, M. S. (1961). Stochastic population models in ecology and epidemiology. London: Methuen and Co. Ltd.

    Google Scholar 

  • Becker, N. G. (1989). Analysis of infectious disease data. London: Chapman and Hall/CRC.

    Google Scholar 

  • Bhattacharya, R. N., & Waymire, E. C. (1990). Stochastic processes with applications. New York, NY: Wiley.

    MATH  Google Scholar 

  • Borel, É. (1942). Sur l’emploi du théorème de Bernoulli pour faciliter le calcul d’une infinité de coefficients. Application au problème de l’attente à un guichet. Comptes rendus de l’Académie des Sciences, 214, 452–456.

    MathSciNet  MATH  Google Scholar 

  • Brauer, F. (2006). Some simple epidemic models. Mathematical Biosciences and Engineering, 3, 1–15.

    Article  MathSciNet  Google Scholar 

  • Castillo-Chavez C., Feng, Z., & Huang, W. (2002). On the computation Ro and its role on global stability. In C. Castillo-Chavez, P. van den Driessche, D. Kirschner, & A.-A. Yakubu (Eds.), Mathematical approaches for emerging and reemerging infectious diseases: An introduction, IMA (Vol. 125, pp. 229–250). Berlin: Springer.

    Chapter  Google Scholar 

  • Chowell, G., Hincapie-Palacio, D., Ospina, J., Pell, B., Tariq, A., Dahal, S., et al. (2016). Using phenomenological models to characterize transmissibility and forecast patterns and final burden of Zika epidemics. PLOS Currents Outbreaks, 8. https://doi.org/10.1371/currents.outbreaks.f14b2217c902f453d9320a43a35b9583

  • Chowell, G., Viboud, C., Hyman, J. M., & Simonsen, L. (2015). The Western Africa ebola virus disease epidemic exhibits both global exponential and local polynomial growth rates. PLoS Currents, 7. https://doi.org/10.1371/currents.outbreaks.8b55f4bad99ac5c5db3663e916803261

  • Corless, R. M., Gonnet, G. H., Hare, D. G. E., Jeffery, D. J., & Knuth, D. E. (1996). On the Lambert W function. Advances in Computational Mathematics, 5, 329–359.

    Article  MathSciNet  Google Scholar 

  • Daley, D. J., & Gani, J. (1999). Epidemic modelling, an introduction. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Diekmann, O., & Heesterbeek, J. A. P. (2000). Mathematical epidemiology of infectious diseases: Model building, analysis and interpretation. Mathematical and computational biology (Vol. 5). Chichester: Wiley.

    MATH  Google Scholar 

  • Dietz, K. (1995). Some problems in the theory of infectious diseases transmission and control. In D. Mollison (Ed.), Epidemic models: Their structure and relation to data (pp. 3–16). Cambridge: Cambridge University Press.

    Google Scholar 

  • Dion, J. P. (1975). Estimation of the variance of a branching process. The Annuals of Statistics, 3(5), 1183–1187.

    Article  MathSciNet  Google Scholar 

  • Feller, W. (1966). An introduction to probability theory and its applications. New York, NY: Wiley.

    MATH  Google Scholar 

  • Fenner, F., Henderson, D. A., Arita, I., Jezĕk, Z., & Ladnyi, I. D. (1988). Smallpox and its eradication. Geneva: World Health Organization.

    Google Scholar 

  • Goh, K. T., Cutter, J., Heng, B. H., Ma, S., Koh, B. K., Kwok, C., et al. (2006). Epidemiology and control of SARS in Singapore. Annals-Academy of Medicine Singapore, 35(5), 301.

    Google Scholar 

  • Haccou, P., Jagers, P., & Vatutin, V. (2005). Branching processes: Variation, growth, and extinction of populations. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Harris, T. (1948). Branching processes. Annals of Mathematical Statistics, 19, 474–494.

    Article  MathSciNet  Google Scholar 

  • Harris, T. (1963). The Theory of Branching Processes. Berlin: Springer.

    Book  Google Scholar 

  • Heyde, C. C. (1974). On estimating the variance of the offspring distribution in a simple branching process. Advances in Applied Probability, 6(3), 421–433.

    Article  MathSciNet  Google Scholar 

  • Hope Simpson, R. E. (1948). The period of transmission in certain epidemic diseases: An observational method for its discovery. Lancet, 2, 755–760.

    Article  Google Scholar 

  • Jagers, P. (1975). Branching processes with biological applications. London: Wiley.

    MATH  Google Scholar 

  • Karlin, S., & Taylor, H. M. (1975). A first course in stochastic processes (2nd ed.). Cambridge, MA: Academic Press.

    MATH  Google Scholar 

  • Kendall, D. (1956). Deterministic and stochastic epidemics in closed populations. In Proceedings of Fifth Berkeley Symposium on Mathematical Statistics and Probability (Vol. 4, pp. 149–165). Berkeley, CA: University of California Press.

    Google Scholar 

  • Marshall, A. W., & Olkin, I. (2007). Life distributions, structure of nonparametric, semiparametric and parametric families. New York, NY: Springer.

    MATH  Google Scholar 

  • Martin-Löf, A. (1988). The final size of a nearly critical epidemic, and the first passage time of a Wienner process to a parabolic barrier. Journal of Applied Probability, 35, 671–682.

    Article  Google Scholar 

  • Mode, C. J., & Sleeman, C. K. (2000). Stochastic processes in epidemiology, HIV/AIDS, other infectious diseases and computers. Singapore: World Scientific.

    Book  Google Scholar 

  • Nåsell, I. (1995). The threshold concept in stochastic and endemic models. In D. Mollison (Ed.), Epidemic models: Their structure and relation to data (pp. 71–83). Cambridge: Cambridge University Press.

    MATH  Google Scholar 

  • Nishiura, H., Yan, P., Sleeman, C. K., & Mode, C. J. (2012). Estimating the transmission potential of supercritical processes based on the final size distribution of minor outbreaks. Journal of Theoretical Biology, 294, 48–55.

    Article  MathSciNet  Google Scholar 

  • Roberts, G. M., & Heesterbeek, J. A. P. (2007). Model-consistent estimation of the basic reproduction number from the incidence of an emerging infection. Journal of Mathematical Biology, 55, 803–816.

    Article  MathSciNet  Google Scholar 

  • Smirnova, A., deCamp, L., & Chowell, G. (2017). Forecasting epidemics through nonparametric estimation of time-dependent transmission rates using the SEIR model. Bulletin of Mathematical Biology. https://doi.org/10.1007/s11538-017-0284-3

  • Varia, M., Wilson, S., Sarwal, S., McGeer, A., Gournis, E., & Galanis, E. (2003). Investigation of a nosocomial outbreak of severe acute respiratory syndrome (SARS) in Toronto, Canada. Canadian Medical Association Journal, 169(4), 285–292.

    Google Scholar 

  • Viboud, C., Bjornstad, O. N., Smith, D. L., Simonsen, L., Miller, M. A., & Grenfell, B. T. (2006). Synchrony, waves, and spatial hierarchies in the spread of influenza. Science, 312(5772), 447–451.

    Article  Google Scholar 

  • Viboud, C., Simonsen, L., Chowell, G. (2016). A generalized-growth model to characterize the early ascending phase of infectious disease outbreaks. Epidemics 15, 27–37.

    Article  Google Scholar 

  • Waugh, W. A. O’N. (1958). Conditioned Markov processes. Biometrika, 45(1–2), 241–249.

    Article  MathSciNet  Google Scholar 

  • White, L. F., & Pagano, M. (2008). A likelihood-based method for real-time estimation of the serial interval and reproductive number of an epidemic. Statistics in Medicine, 27, 2999–3016.

    Article  MathSciNet  Google Scholar 

  • Yan, P. (2008a). Distribution theory, stochastic processes and infectious disease modelling. In F. Brauer, P. van den Driessche, & J. Wu (Eds.), Mathematical epidemiology. Lecture notes in mathematics (Vol. 1945). Berlin: Springer.

    Google Scholar 

  • Yan, P. (2008b). Separate roles of the latent and infectious periods in shaping the relation between the basic reproduction number and the intrinsic growth rate of infectious disease outbreaks. Journal of Theoretical Biology, 251, 238–252.

    Article  MathSciNet  Google Scholar 

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Yan, P., Chowell, G. (2019). Behaviors of a Disease Outbreak During the Initial Phase and the Branching Process Approximation. In: Quantitative Methods for Investigating Infectious Disease Outbreaks. Texts in Applied Mathematics, vol 70. Springer, Cham. https://doi.org/10.1007/978-3-030-21923-9_4

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