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 這篇論文提出一個分析傳染病資料的新模型與演算法。利用點過程（point process）並且考慮連續型的解釋變數（continuous explanatory variable）以及更廣泛的感染函數（infectivity function）去建構出家庭中每一個人被感染與被移除（removal）的條件機率。我們定義兩個記數過程（counting process），各自代表著在一個家庭中每一個人何時被感染與何時被移除。這些發生的條件機率可以用來描述傳染病的擴散速度；同時，這些條件機率也受到一些個人的特徵與我們設計的函數所影響。我們利用貝氏分析（Bayesian inference）裡常用的馬可夫鏈蒙地卡羅（Markov Chain Monte Carlo）演算法發展出一種特別的演算法¬，並且用它分析傳染病的特性與人的特徵對傳染病的影響；包括分析模擬的結果以及分析真實資料的結果。
 This paper proposes a point process model for infectious disease data that take into consideration continuous explanatory variables regarding infectivity, susceptibility to infection and removal rate and allow parametric family of baseline infectivity functions.For each individual in a closed community, we define two counting processes; one jumps when this individual gets infected and the other jumps when this individual gets removed. The intensities of these counting processes are used to describe the spread of the infectious disease. These intensities have one component describing the way that individual covariates may affect infectivity, susceptibility to infection or removal; these intensities also have a baseline infectivity function, belonging to a parametric family of functions. Customized MCMC algorithms are developed for Bayesian inference based on removal times and covariates of each individual. Simulation studies and analysis of real infectious data are provided to illustrate the numerical performance of the methods.
 Contents1 Introduction 12 A point process model for infectious disease data 52.1 The model ......................................... 52.2 Identifiability of the parameters ................. 82.3 The likelihood .................................... 113 Bayesian inference based on removal times 133.1 Inference when the epidemic is over ............... 133.2 Inference during the epidemic ..................... 174 Simulation studies 234.1 Generating data for point processes ............... 234.2 Final size of the epidemics ....................... 244.3 Performance of the algorithms ..................... 265 Application to smallpox data 316 Measuring the missing information 337 Discussion 36References 38
 Aalen, O. O. (1980) A model for non-parametric regression analysisof counting processes. Lecture Notes in Statist, 2, 1-25, Springer,New York.Addy, C. L., Longini, I. M. and Haber, M. (1991) A generalizedstochastic model for the analysis of infectious disease final sizedata. Biometrics, 47, 961-974.Andersson, H. and Britton, T. (2000) Stochastic Epidemic Models andTheir Statistical Analysis. Springer Lecture Notes in Statistics,New York.Bailey, N. T. J. (1975) The Mathematical Theory of InfectiousDiseases and its Applications. Griffin, London.Bailey, N. T. J. and Thomas, A. S. (1971) The estimation ofparameters from population data on the general stochastic epidemic.Theor. Pop. Biol., 2, 53-70.Baker, R. D. and Stevens, R. H. (1995) A random-effects model foranalysis of infectious disease final-state data. Biometrics, 51,956-968.Ball, F. (1986) A unified approch to the distribution of total sizeand total area under the trajectory of infectives in epidemicmodels. Adv. Appl. Prob., 18, 289-310.Ball, F., Mollison, D. and Scalia-Tomba, G. (1997) Epidemics withtwo levels of mixing. Ann. Appl. Prob., 7, 46-89.Becker, N. G. (1989) Analysis of Infectious Disease Data.} Chapman &Hall, London.Becker, N. G. and Britton, T. (1999) Statistical studies ofinfectious disease incidence. J. R. Statist. Soc. B, 61, 287-307.Becker, N. G. and Hopper, J. L. (1983) The infectiousness of adisease in a community of households. Biometrika}, 70, 29-39.Becker, N. G. and Yip, P. (1989) Analysis of variation in aninfection rate. Australian Journal of Statistics, 31, 42-52.Bremaud, P. (1981) Point Processes and Queues, Martingale Dynamics.Springer-Verlag, New York.Britton, T. (1998) Estimation in multitype epidemics. J. R. Statist.Soc. B, 60, 663-679.Brooks, S. P. and Gelman, A. (1997) General methods for monitoringconvergence of iterative simulation. Journal of Computational andGraphical Statistics, 7, 434-455.Demiris, N. and O''Neill, P. D. (2005) Bayesian inference forepidemics with two levels of mixing. Scand. J. Statist., 32,265-280.Green, P. J. (1995) Reversible jump Markov chain Monte Carlocomputation and Bayesian model determination. Biometrika, 82, Part4, 711-732.Hethcote, H. W. and Van Ark, J. W. (1987) Epidemiological models forheterogeneous populations: proportionate mixing, parameterestimation and immunization programs. Math. Biosci., 84, 85-118.Longini, I. M. and Koopman, J. S. (1982) Household and communitytransmission parameters from final distributions of infections inhouseholds. Biometrics, 38, 115-126.Nicolae, D. L., Meng, X.-L. and Kong, A. (2008) Quantifying thefraction of missing information for hypothesis testing instatistical and genetic studies. Statistical Science, To appear.O''Neill, P. D. and Becker, N. G. (2001) Inference for an epidemicwhen susceptibility varies. Biostatistics, 2, 99-108.O''Neill, P. D. and Roberts, G. O. (1999) Bayesian inference forpartially observed stochastic epidemics. J. R. Statist. Soc. A, 162,121-129.Rhodes, P. H., Halloran, M. E. and Longini, I. M. (1996) Countingprocess models for infectious disease data: distinguishing exposureto infection from susceptibility. J. R. Statist. Soc. B, 58,751-762.Robert, C. P. and Casella, G. (2004) Monte Carlo Statistical Methods(2nd Edition). Springer-Verlag, New York.Thompson, D. and Foege, W.(1968) Faith Tabernacle smallpox epidemic.AbakaJiki, Nigeria. Geneva, Switzerland: World HealthOrganization. (WHO/SE/68.3) (http://whqlibdoc.who.int/smallpox/SE\_68.3.pdf).
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