Abstract
W.O. Kermack and A.G. McKendrick introduced in their fundamental paper, A Contribution to the Mathematical Theory of Epidemics, published in 1927, a deterministic model that captured the qualitative dynamic behavior of single infectious disease outbreaks. A Kermack–McKendrick discrete-time general framework, motivated by the emergence of a multitude of models used to forecast the dynamics of epidemics, is introduced in this manuscript. Results that allow us to measure quantitatively the role of classical and general distributions on disease dynamics are presented. The case of the geometric distribution is used to evaluate the impact of waiting-time distributions on epidemiological processes or public health interventions. In short, the geometric distribution is used to set up the baseline or null epidemiological model used to test the relevance of realistic stage-period distribution on the dynamics of single epidemic outbreaks. A final size relationship involving the control reproduction number, a function of transmission parameters and the means of distributions used to model disease or intervention control measures, is computed. Model results and simulations highlight the inconsistencies in forecasting that emerge from the use of specific parametric distributions. Examples, using the geometric, Poisson and binomial distributions, are used to highlight the impact of the choices made in quantifying the risk posed by single outbreaks and the relative importance of various control measures.
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References
Allen, L. J. S., & van den Driessche, P. (2008). The basic reproduction number in some discrete-time epidemic models. J. Differ. Equ. Appl., 14(10–11), 1127–1147.
Brauer, F. (2008). Age-of-infection and the final size relation. Math. Biosci. Eng., 5(4), 681–690.
Brauer, F., & Castillo-Chavez, C. (2012). Mathematical models in population biology and epidemiology (2nd ed.). Berlin: Springer.
Brauer, F., Feng, Z., & Castillo-Chavez, C. (2010). Discrete epidemic models. Math. Biosci., 7, 1–15.
Casella, G., & Berger, R. L. (2001) Duxbury advance series: Vol. 70. Statistical inference (2nd ed.).
Castillo-Chávez, C. (1989). Number lecture notes in biomathematics: Vol. 83. Mathematical and statistical approaches to AIDS epidemiology. Berlin: Springer.
Chowell, G., Castillo-Chavez, C., Fenimore, P. W., Kribs-Zaleta, C. M., Arriola, L., & Hyman, J. M. (2004). Model parameters and outbreak control for SARS. Emerg. Infect. Dis., 10(7), 1258–1263.
Chowell, G., Fenimore, P. W., Castillo-Garsow, M. A., & Castillo-Chavez, C. (2003). SARS outbreaks in Ontario, Hong Kong and Singapore: the role of diagnosis and isolation as a control mechanism. J. Theor. Biol., 224(1), 1–8.
Diekmann, O., & Heesterbeek, J. A. P. (2000). Mathematical epidemiology of infectious diseases: model building, analysis and interpretation. New York: Wiley.
Dietz, K., & Heesterbeek, J. (2002). Daniel Bernoulli’s epidemiological model revisited. Math. Biosci., 180, 1–21.
Feng, Z., Xu, D., & Zhao, H. (2007). Epidemiological models with non-exponentially distributed disease stages and applications to disease control. Bull. Math. Biol., 69, 1511–1536.
Gumel, A., Ruan, S., Day, T., Watmough, J., van den Driessche, P., Brauer, F., Gabrielson, D., Bowman, C., Alexander, M., Ardal, S., Wu, J., & Sahai, B. (2004). Modeling strategies for controlling SARS outbreaks based on Toronto, Hong Kong, Singapore and Beijing experience. Proc. R. Soc. Lond., 271, 2223–2232.
Kermack, W. O., & McKendrick, A. G. (1927). Contributions to the mathematical theory of epidemics. I. Proc. R. Soc. Lond. Ser. A, 115, 700–721. Reprinted in Bull. Math. Biol. 53, 33–55 (1991).
Kermack, W. O., & McKendrick, A. G. (1932). Contributions to the mathematical theory of epidemics. II. The problem of endemicity. Proc. R. Soc. Lond. Ser. A, 138(834), 55–83.
Kermack, W. O., & McKendrick, A. G. (1933). Contributions to the mathematical theory of epidemics. III. Further studies of the problem of endemicity. Proc. R. Soc. Lond. Ser. A, 141(843), 94–122. Reprinted in Bull. Math. Biol. 53, 89–118 (1991).
Ludwig, D. (1975). Final size distribution for epidemics. Math. Biosci., 23(1), 33–46.
Ma, J., & Earn, D. J. (2006). Generality of the final size formula for an epidemic of a newly invading infectious disease. Bull. Math. Biol., 68(3), 679–702.
Pellis, L., Ferguson, N. M., & Fraser, C. F. (2008). The relationship between real-time and discrete-generation models of epidemic spread. Math. Biosci., 216(1), 63–70.
Ross, R. (1911). The prevention of malaria (2nd ed.). New York: Dutton.
Yang, C. K., & Brauer, F. (2008). Calculation of \(\mathcal{R}_{0}\) for age-of-infection models. Math. Biosci. Eng., 5(3), 585–599.
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We thank the reviewers for their helpful comments and suggestions. This research is supported in part by the NSF grant DMS-1022758 and by grant number 1R01GM100471-01 from the National Institute of General MedicalSciences (NIGMS) at the National Institutes of Health.
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Hernandez-Ceron, N., Feng, Z. & Castillo-Chavez, C. Discrete Epidemic Models with Arbitrary Stage Distributions and Applications to Disease Control. Bull Math Biol 75, 1716–1746 (2013). https://doi.org/10.1007/s11538-013-9866-x
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DOI: https://doi.org/10.1007/s11538-013-9866-x