| Summary: | 博士 === 國立臺灣大學 === 流行病學與預防醫學研究所 === 103 === Background
Deterministic models are conducive to estimate a very important indicator for assessing the spread of infectious disease such as epidemic, endemic, and extinction, namely, the basic reproductive number (R0). However, when small or moderate population size and the question of the probability of the extinction of infectious disease in question are involved the deterministic model is therefore not adequate. Furthermore, it may not be adequate when minor outbreak occurred if the R0 is less than 1. The alternative is the application of stochastic model.
Of these stochastic models, the branching process is one of considerations because it can be easily applied to estimating both the extinction probability and R0. In spite of these two advantages, because we often have the total number of infected individuals for a given period of time and generations usually overlap each other in reality that enables the branching processes difficult to estimate R0. The continuous-time Markov process embodied with birth-death process may be more appropriate.
Objectives
The objectives of my thesis are to develop various types of stochastic models for estimating R0 and the extinct probability of infectious disease by demonstrating the two examples of SARS and pulmonary TB. Specific aims are to
(1)apply the deterministic compartmental model to data on SARS poliomyelitis for estimating R0;
(2)develop branching process and birth-death process to SARS dataset to estimate both R0 and the extinct probability and also extend the simple branching process to mortal branching process for measles;
(3)apply the Becker’s SIR model to the data of TB for estimating latent period and incubation period;
(4)develop a novel three-state Markov model embodied with birth-death process to assess the effect of covariates (such as IGRA) on infection rate and conversion rate using Bayesian MCMC method and to further apply birth-death process to estimate extinct probability and the expected time to reach final size.
Materials and Methods
Generating Data by simulations
We simulated a branching process with 9 generations of data for a given offspring distribution under various values of R0=2, 1.5, 1.1 and 0.9 for calculating the extinction probability. We simulated a birth-death process with given birth rate, death rate and different initial infected cases. We calculate the mean and variance of arrival time from the initial state.
Empirical Data
. SARS
The thesis used 346 confirmed cases with SRAS from November 2002 to July 2003 in Taiwan obtained from Taiwan CDC and also 22,520,776 population of Taiwan at the beginning of 2003. This thesis also made use of total 56 infected with SARS in a hospital in Singapore from Mar. 26 to Apr. 15, 2003. Only 3 generation of offspring was noted after outbreak investigation.
Mycobacterium tuberculosis
The outbreaks of TB in the Long-term Care Facility
The data on outbreak of TB in the LTCF provide empirical data for estimating the unobserved incubation period and latent period before onset of symptoms.
Data for estimating parameters of TB natural course
Various datasets were used including a total of 2,420 TB cases with age ≥ 30 enrolled in our cohort study from 2009 to 2011 (surveillance system for TB from 2009 to 2011 in Changhua County), a total of 22,510 TB contacts with age ≥ 30 enrolled in our cohort study from 2005 to 2011 (B contact registry database from 2005 to 2011 in Changhua County), a matched case-control study for risk factors of TB from 2012 to 2014 in Changhua County, and a IGRA survey for general population from 2011 to 2013 in Changhua County
Model Specification and Statistical Analysis
Three types of stochastic processes were applied and proposed. We first applied branching process and birth-death process to estimate R0, extinct probability and the expected time to reach final size for SARS epidemics. We then applied the Becker’s SIR model to estimate unobserved incubation period (including latent period) to the outbreak of TB to estimate its R0 and extinct probability. The novel three-state Markov process embodied with birth-death process was develop to assess the effect of IGRA on the transition from susceptible to LTBI and the conversion from LTBI to TB with Bayesian MCMC method.
Results
Part I Simulation
Branching Process
The results of estimating R0 on the generating data of a branching process with six generations for a given offspring distribution (such as Poisson, Binomial, and Negative Binomial distributions) are presented. The estimated R0 were consistent with the nonparametric or parametric method with different distributions. However, the variances were heterogeneous by different methods.
Pure birth process
The simulated results of 1000 simulations for pure birth process assuming λ=0.5 compared with the true results estimated the exact equation for E(Ta). It is very interesting to note that the simulated curve with mean value was still deviant from the curve obtained from the exact formula. However, when n0 became larger, the simulated curve with mean value was close to the true curve obtained from the formula with larger n0 but deviant from the formula with smaller n0. When λ was enlarged to 3, the results were not changed at all.
Part II Estimation of R0 for the outbreak of SARS in Taiwan
The estimated R0 was 0.9971 (0.5090~1.4852) by using branching process given 16~22 generations assuming the incubation of 5 or 7 days. The estimated extinction probability was 0.9912 under the assumption of Poisson distribution.
Using Borel-Tanner distribution under the assumption of R<1, the stimulated R0 was from 0.9790 (0.8437 ~ 1.1143) to 1.0134 (0.8535 ~ 1.1733). The estimated extinction probability was 0.9709 ~ 0.9989.
As linear birth-death process did not fit well with data apply instead general birth death process to fit the observed cumulated SARS data. The estimated birth rates were 0.57 (< 55 day of outbreak), 11.45 (the 55th ~ 80th day of outbreak) and 1.413 (after the 80th day of outbreak). The expected time to reach final size a (Ta) were 54.97(10.09), 80.00 (10.41) and 112.01 (11.47) days for T32 , T300 and T346, respectively.
Part III Natural Course of TB
Outbreak of TB
The latent period was estimated about 223.6 days [λ=0.0045 (2.17*10-6) ] and the infectious period before symptoms onset was estimated about 55.9 days [ β=0.0179 (3.47*10-5)]. Hence, the incubation period was about 279.5 days. According to our estimation of latent period, there were at least two generations and at most 3 generations. R0 was bounded between 0.9739 and 0.9796 in this cluster. The extinction probability was almost certain.
The effect of IGRA on the occurrence of TB with a case-control study
Using a match-case-control study, the estimated odds ratios in multivariable logistic regression mode for positive QFT-GIT after further adjustment for positive TST was 2.47 (95% CI: 1.72-3.54). After further considering the interaction term in the model, the odds ratio of QFT-GIT for subjects with positive TST was estimated as 4.28 (95% CI: 1.16-15.76) whereas the odds ratio of QFT-GIT for subjects with negative TST was estimated as 1.15 (95% CI: 0.66-2.00).
The effect of IGRA on the infection rate and conversion rate with multi-state Markov model
The overall estimated infection rate (per person-years) and conversion rate (per year) were 0.0168 (95% CI: 0.0157-0.0180) and 0.0113 (95% CI: 0.0098-0.0129). The infection rate was higher for the young age group (30-44 years old) and male sex. Those with positive IGRA were 1.60 (RR=1.59, 95% CI:1.39-1.84) times likely to be susceptible to LTBI compared with negative IGRA. In contrast to the effect of age on infection rate, the older the subject was, the higher the conversion rate. Males still had higher conversion rate than females. Those with positive IGRA were two times (RR=2.12, 95% CI:1.57-2.85) likely to surface to TB compared with negative IGRA.
After taking the effect of age and sex on both infection rate and conversion rate into account, subjects with positive QFT-GIT still had higher risk of being infected and converting to tuberculosis with estimated RR being 1.71 (95% CI: 1.49-2.00) and 1.58 (95% CI: 1.15-2.17), respectively.
Application of birth-and-death process with the parameters obtained from three-state Markov model found one initial case may take about 61 days to have 10 of final size and 87 days to have 30 of final size without considering covariates. The young people, male and positive IGRA tended to spread quickly. The male aged less than 45 years with positive results of IGRA took only one week to reach final size given five initial cases. It should be noted that an increase in initial size reduced the time to reach the expected final size. When initial size was larger than five the extinct probability of TB was very unlikely.
Conclusion
There are five major conclusions on the practical findings reached as follows.
1.While evaluating SARS in the two regions, the estimation of R0 given 3~8 generations was between 1 and 1.5, and the estimated extinct probability was almost certain using branching process in Singarepore. The SAS outbreak yielded 0.99 of R0 using branching process in Taiwan. The estimated extinct probability was 0.99. The similar findings were noted by using the mortal branching process with Borel-Tanner distribution.
2.Estimate unobserved incubation period with approximately 9 months, including seven months of latent period and two months of infectious period before onset of symptoms given data from an outbreak of TB occuring even among subjects with negative TST result after undergoing TB screening. Surveillance of the elderly people even with a negative TST after TB screening is still necessary given a long latent period if the outbreak of TB in a long-term care facility is to be controlled.
3.This is the first study to assess the effect of IGRA on the occurrence of TB by conducting a case-control study making allowance for demographic characteristics and induration size of TST.
4.This is the first study to assess the effects of age, gender, and IGRA on infection from susceptible to LTBI and also the conversion from LTBI to TB in the natural course of TB. The young age was at increased risk for being LTBI but the old age enhanced the risk of conversion from LTBI to TB. Male had higher risk for being infected as LTBI and also the conversion from LTBI to TB. The elevated IGRA plays a significant role not in the infection rate (from free of LTBI (susceptible) to LTBI) but also in the conversion rate after adjusting for age and gender.
5.The application of infection rate (birth rate) and conversion rate (death rate) gives the time expected to reach number of LTBI of final size and the extinct probability by various combinations of age, gender, and the results of IGRA. Subjects with positive IGRA results had shorter expected time to reach final size than those with negative result.
This thesis has also contributed to developing the methodological part related to infectious disease consisting of three summary points:
1.Provide several statistical simulated methods for simulating various R0 with branching process and also birth-and-death process so as to estimate the extinct probability and the expected time to reach final size.
2.Demonstrate how to apply the Becker’s SIR model in conjunction with branching process to estimate incubation period and latent period for the surveillance of TB.
3.Develop a continuous-time Markov process embodied with birth-and-death process in conjunction with a novel case-cohort design data given the known sampling fraction to assess how covariates such as IGRA affect the infection rate and the conversion rate framed with a three-state Markov process. The further application of birth-and-death process used in the simulation of SARS process can compute the extinct probability and the expected time to reach final size, both of which provide a new insight into the golden period for the formulation of policy for the containment of infectious disease in question.
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