Abstract
The main focus of this book is to address phenomenological questions regarding the spread of infectious diseases at the population level. Examples of such questions include:
-
1.
If one or a few infected individuals are “seeded” in a large and completely susceptible population, will it only lead to a handful of infected individuals and the (small) outbreak burns out; or will it lead to an “explosive” (large) outbreak that results in a significant proportion of the population infected?
-
(a)
If the outcome is the former, what is the expected total number of infected individuals and what is the expected time to extinction?
-
(b)
If the outcome is the latter, how fast will it grow?
-
(a)
-
2.
In a large outbreak, can we predict the peak burden of the disease and the timing of the peak? How about the long-term outcomes? Will it simply go away after a single wave or a few repeated waves, or will it settle down at some equilibrium state and the epidemic becomes endemic?
-
3.
What about the effects of control measures, such as public health interventions including quarantine, isolation, or pharmaceutical treatments and vaccination?
This is a preview of subscription content, log in via an institution to check access.
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.
Arnold, B. C. (1983). Pareto distributions. Fairland, MD: International Co-operative Publishing House.
Bacchetti, P. B., & Moss, A. R. (1989). Incubation period of AIDS in San Francisco, Nature, 338, 251–253.
Bellman, R. E., & Roth, R. S. (1984). The Laplace transform. Singapore: World Scientific.
Brauer, F. (2008). Compartmental models in epidemiology. In F. Brauer, P. van den Driessche, & J. Wu (Eds.), Mathematical epidemiology (Chapter 2). Berlin: Springer.
Brookmeyer, R., & Goedert, J. J. (1989). Censoring in an epidemic with an application to hemophilia-associated AIDS. Biometrics, 45, 325–335.
Cox, D. R., & Oaks, D. (1987). Analysis of survival data. New York, NY: Champan and Hall/CRC.
Devroye, L. (1992). Random variate generation for the digamma and trigamma distributions. Journal of Statistical Computation and Simulation, 43(3–4), 197–216.
Dubey, S. D. (1968). A compound Weibull distribution. Naval Research Logistics Quarterly, 15, 179–188.
Dubey, S. D. (1969). A new derivation of the logistic distribution. Naval Research Logistics Quarterly, 16, 37–40.
Gauvreau, K., Degruttola, V., & Pagano, M. (1994). The effect of covariates on the induction time of AIDS using improved imputation of exact seroconversion times. Statistics in Medicine, 13, 2021–2130.
Gleser, L. J. (1989). The gamma distribution as a mixture of exponential distributions. The American Statistician, 43, 115–117.
Goedert, J. J., Kessler, C. M., Aledort, L. M., Biggar, R. J., Andes, W. A., White, G. C., et al. (1989). A perspective study of human immunodeficiency virus type 1 infection and the development of AIDS in subjects with hemophilia. The New England Journal of Medicine, 321, 1141–1148.
Hesselager, O., Wang, S., & Willmot, G. E. (1998). Exponential and scale mixtures and equilibrium distributions. Scandinavian Actuarial Journal, 2, 125–142.
Hessol, N. A., Lifson, A. R., O’Malley, P. M., Doll, L. S., Jaffe, H. W., Rutherford, G. W. (1989). Prevalence, incidence, and progression of human immunodeficiency virus infection in homosexual and bisexual men in hepatitis B vaccine trials, 1978–1988. American Journal of Epidemiology, 130, 1167–1175.
Hougaard, P. (1984). Life table methods for heterogenous populations. Distributions describing the heterogeneity. Biometrika, 71, 75–83.
Italian Seroconversion Study. (1992). Disease progression and early predictors of AIDS in HIV-seroconverted injecting drug users. AIDS, 6, 421–426.
Johnson, N., Kotz, S., & Kemp, A. W. (1993). Univariate discrete distributions (2nd ed.). New York, NY: Wiley.
Kalbfleisch, J. D., & Prentice, R. L. (2002). Statistical analysis for failure time data (2nd ed.). New York, NY: Wiley.
Karlis, D., & Xekalaki, E. (2005). Mixed Poisson distributions. International Statistical Review, 73(1), 35–58.
Klar, B. (2002). A note on the L-class of life distributions. Journal of Applied Probability, 39, 11–19.
Lawless, J. F. (2003). Statistical models and methods for lifetime data (2nd ed.). New York, NY: Wiley.
Li, H., Han, D., Hou, Y., Chen, H., & Chen, Z. (2015). Statistical inference methods for two crossing survival curves: A comparison of methods. PLoS One, 10(1), e0116774. https://doi.org/10.1371/journal.pone.0116774
Lui, K. J., Darrow, W. W., & Ruthford, G. W. (1988). A model-based estimate of the mean incubation period for AIDS in homosexual men. Science, 240, 1333–1335.
Marshall, A. W., & Olkin, I. (2007). Life distributions, structure of nonparametric, semiparametric and parametric families. New York, NY: Springer.
Meeker, W. Q., & Escobar, L. A. (1998). Statistical methods for reliability data. Wiley Series in probability and statistics. New York, NY: Wiley.
Nishiura, H. (2007). Early efforts in modeling the incubation period of infectious diseases with an acute course of illness. Emerging Themes in Epidemiology, 4, 2. https://doi.org/10.1186/1742-7622-4-2
Sartwell, P. E. (1966). The incubation period and the dynamics of infectious disease. American Journal of Epidemiology, 83(2), 204–216.
Shaked, M., & Shanthikumar, J. G. (2007). Stochastic orders. New York, NY: Springer.
Smith, H. (2011). Distributed delay equations and the linear chain trick. In An introduction to delay differential equations with applications to the life sciences. Texts in applied mathematics (Vol. 57). New York, NY: Springer.
Stoyan, D. (1983). Comparison models for queues and other stochastic models. New York, NY: Wiley.
Walker, S. & Stephens, D. (1999). A multivariate family of distributions on (0, ∞)p. Biometrika, 86(3), 703–709.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Crown
About this chapter
Cite this chapter
Yan, P., Chowell, G. (2019). Shapes of Hazard Functions and Lifetime Distributions. 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_2
Download citation
DOI: https://doi.org/10.1007/978-3-030-21923-9_2
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)