Abstract
We start with simple models that describe the dynamics of disease transmission over time t in a constant population of size m and investigate the long-term epidemic dynamics as t →∞. In these simple models, we assume there is no replacement of susceptible individuals due to demographic input of susceptible newborns. The population is partitioned into compartments, with at least one compartment representing the prevalence of individuals who are susceptible to infection and at least one compartment representing the prevalence of individuals who are infectious (at time t).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Allen, L. J. (2010). An introduction to stochastic processes with applications to biology. Boca Raton, FL: CRC Press.
Allen, L. J. (2017). A primer on stochastic epidemic models: Formulation, numerical simulation, and analysis. Infectious Disease Modelling, 2(2), 128–142.
Anderson, D., & Watson, R. (1980). On the spread of a disease with gamma distributed latent and infectious periods. Biometrika, 67(1), 191–198.
Andersson, H., & Djehiche, B. (1998). A threshold limit theorem for the stochastic logistic epidemic. Journal of Applied Probability, 35(3), 662–670.
Bailey, N. T. J. (1975). The mathematical theory of infectious diseases and its applications (2nd ed.). London: The Griffin & Company Ltd.
Ball, F. G., Britton, T., & Neal, P. (2016). On expected durations of birth-death processes, with applications to branching processes and SIS epidemics. Journal of Applied Probability, 53, 203–215.
Brauer, F. (2008). Compartmental models in epidemiology. In F. Brauer, P. van den Driessche, & J. Wu (Eds.), Mathematical epidemiology (Chapter 2). Berlin: Springer.
Brauer, F., van den Driessche, P., & Wu, J. (Eds.). (2008). Mathematical epidemiology. Berlin: Springer.
Castillo-Chávez, C., Blower, S., van den Driessche, P., Kirschner D., & Yakubu, A. A. (2000). Mathematical approaches for emerging and reemerging infectious diseases. New York, NY: Springer.
Clancy, D. (2018). Precise estimates of persistence time for SIS infections in heterogeneous populations. Bulletin of Mathematical Biology, 80(11), 2871–2896. https://dol.org/10.1007/s11538-018-0491-6
Clancy, D., & Mendy, S. T. (2011). Approximating the quasi-stationary distribution of the SIS model for endemic infection. Methodology and Computing in Applied Probability, 12(3). https://doi.org/10.1007/s11009-010-9177-8
Cox, D. R. (2006). Principles of statistical inference. Cambridge: Cambridge Press.
Deakin, M. A. B. (1975). A standard form for the Kermack-McKendrick epidemic equations. Bulletin of Mathematical Biology, 37, 91–95.
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.
Erdös, P., & Rényi, A. (1961). On the evolution of random graphs. Bulletin of the International Statistical Institute, 38, 343–347.
Feng, Z., Xu, D., & Zhao, H. (2007). Epidemiological models with non-exponentially distributed disease stages and applications to disease control. Bulletin of Mathematical Biology. https://doi.org/10.1007/s11538-006-9174-9
Hernádez-Suárez, C. M., & Castillo-Chavex, C. (1999). A basic result on the integral for birth-death Markov processes. Mathematical Biosciences, 161, 95–104.
Hethcote, H. W., & van den Driessche, P. (1991). Some epidemiological models with nonlinear incidence. Journal of Mathematical Biology, 29, 271.
Isham, V. (1991). Assessing the variability of stochastic epidemics. Mathematical Biosciences, 107, 209–224.
Isham, V. (2005). Stochastic models for epidemics. In A. C. Davison, Y. Dodge, & N. Wermuth (Eds.), Celebrating statistics: papers in honour of Sir David Cox on his 80 th birthday. Oxford statistical science series (Chapter 1, Vol. 33). Oxford: Oxford University Press.
Karlin, S., & Taylor, H. M. (1975). A first course in stochastic processes (2nd ed.). Cambridge, MA: Academic Press.
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.
Kermack, W. O., & McKendrick, A. G. (1927). Contributions to the mathematical theory of epidemics, part I. Proceedings of the Royal Society London A, 115, 700–721.
Krishnarajah, I., Cook, A., Marion, G., & Gibson, G. (2005). Novel moment closure approximations in stochastic epidemics. Bulletin of Mathematical Biology, 67, 855–873.
Lefèvre, C., & Picard, P. (1995). Collective epidemic processes: A general modelling approach to the final outcome of SIR infectious diseases. In D. Mollison (Ed.), Epidemic models: Their structure and relation to data (pp. 53–70). Cambridge: Cambridge University Press.
Li, M. Y., Muldowney, J. S., & van den Driessche, P. (1999). Global stability of SEIRS models in epidemiology. Canadian Applied Mathematics Quarterly, 7(4), 409–425.
Ludwig, D. (1975). Final size distributions for epidemics. Mathematical Biosciences, 23, 33–46.
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.
Nåsell, I. (2002). Stochastic models of some endemic infections. Mathematical Biosciences, 179, 1–19.
Nåsell, I. (2003). Moment closure and the stochastic logistic model. Theoretical Population Biology, 63(2), 159–168.
Pinto, A., Martins, J., & Stollenwerk, N. (2009). The higher moments dynamic on SIS model. In T. E. Simos, et al. (Eds.), Numerical Analysis and Applied Mathematics, AIP Conference Proceedings (Vol. 1168, pp. 1527–1530). College Park, MD: AIP.
Ross, S. M. (1996). Stochastic processes (2nd ed.). New York, NY: Wiley.
Scalia-Tomba, G. (1985). Asymptotic final size distribution for some chain binomial processes. Advances in Applied Probability, 17, 477–495.
von Bahr, B., & Martin-Löf, A. (1980). Threshold limit theorems for some epidemic processes. Advances in Applied Probability, 12, 319–349.
Wearing, H. J., Rohani, P., & Keeling, M. J. (2005). Appropriate models from the management of infectious diseases. PLoS Medicine, 7, 621–627.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Crown
About this chapter
Cite this chapter
Yan, P., Chowell, G. (2019). Beyond the Initial Phase: Compartment Models for Disease Transmission. 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_5
Download citation
DOI: https://doi.org/10.1007/978-3-030-21923-9_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-21922-2
Online ISBN: 978-3-030-21923-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)