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  • 學位論文

隨機過程在肺結核及嚴重急性呼吸道症候群傳染病之運用

Stochastic Processes for SARS and TB Infectious Diseases

指導教授 : 陳秀熙
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摘要


研究背景 決定論模型(deterministic model)有助於推導非常重要的傳染病指標,用以評估傳染病的散播,如大規模的流行,地方性的流行等,此指標就是傳染病的閾值-基礎再生數(R0)。然而,決定論模型在面對中小型人口及傳播傳染病滅絕(extinction)機率的估計是有疑問的,此時使用決定論模型是不適當。此外,當基礎再生數小於1,卻仍發生小規模群突發時,使用決定論模型也是不適合的。此時,應用隨機模型便是其替代的方法。 在這些隨機模型中,分支過程(branching process) 是其中被考慮的隨機模型之一。因為它對於傳染病滅絕(extinction)的機率及基礎再生數(R0)進行評估是較容易的。儘管有上述的優點,但因為我們常常得到的資料是在一段期間中,受到感染的人數及受感染的各不同子代間,時常相互重疊,以至於造成使用分支過程估計基礎再生數的困難。因此,連續時間馬可夫過程(continue time Markov process)中的生死過程(birth-death process)會是適合的方法。 研究目的 本論文的研究目標是在發展一系列隨機模型,利用嚴重急性呼吸道症候群及結核病兩個應用實例,估計基礎再生數及傳染病的滅絶機率。具體研究目的為: (1)應用決定性間隔模型,估計嚴重急性呼吸道症候群之基礎再生數。 (2)以分支過程及生死過程,估計嚴重急性呼吸道症候群之基礎再生數及滅絶機率,並進一步擴展分支過程至mortal 分支過程。 (3)應用Becker的SIR模型,估計無法觀察之結核病從感染至發生症狀之潛伏期(incubation)及從感染至可傳染期之潛伏期(latent)。 (4)發展三階段馬可夫模型結合生死過程,並以貝氏蒙地卡羅-馬可夫鏈(MCMC; Markov chain Monte Carlo)方法,了解IGRA(Interferon-γ release assays)於潛伏結核病感染率及結核病轉移率在不同疾病進程上所扮演的角色,且進一步運用生死過程計算潛伏結核病感染數及滅絶機率。 研究材料及方法 模擬資料 本論文以分支過程給定2, 1.5, 1.1, 0.9等不同基礎再生數下,模擬9代的資料且計算滅絶機率。給定出生率、死亡率及不同初始感染個案數以生死過程模擬資料,且計算從初始個案到達最後狀態的平均時間及其變異數。 臨床及社區資料 嚴重急性呼吸道症候群 本研究利用台灣地區2002年11月至2003年7月期間,罹患嚴重急性呼吸道症候群共346位確定個案,以及新加坡2003年3月26日至4月15日共56位在醫院感染嚴重急性呼吸道症候群之個案資料進行研究。 結核菌 長期照護機構結核病群突發流行資料 利用長期照護機構之結核病群突發資料,進行結核菌從受感染至發生症狀之潛伏期(incubation)及受感染至發生可傳染之潛伏期(latent)之估計。 結核病自然病史估計資料 利用彰化縣結核病2009至2011年監視系統共2,420位30歲以上結核病個案資料,與2005至2011年接觸者登記個案系統共22,510位30歲以上結核病接觸個案資料,以及2012至2014年結核病危險因子病例對照配對研究,加上2011至2013年一般族群IGRA調查資料進行分析。 模式發展及統計分析 本論文提出三種隨機過程,首先以分支過程及生死過程估計嚴重急性呼吸道症候群之基礎再生數、滅絶機率與擴散至最終感染數的時間。應用Becker的SIR模型估計無法觀察之結核菌從受感染至發生症狀之潛伏期(incubation)及受感染至發生可傳染之潛伏期(latent),再以貝氏蒙地卡羅-馬可夫鏈方法,導入三階段馬可夫模型並結合生死過程,估計IGRA於潛伏結核病感染率及結核病轉移率在不同疾病進程之影響。 研究結果 1.模擬 分支過程 由假定布瓦松、二項及負二項等不同子代分佈,透過分支過程模擬6代的資料來估計基礎再生數。在不同分佈條件下,基礎再生數分別以無母數及母數方法進行估計,不同方法所估計之基礎再生數結果一致。然而不同方法下其變異數是具異質性。 純出生過程 在λ=0.5的假設下,經1000次的純出生過程(pure birth process)模擬並與精確公式所得之估計結果進行比較。經模擬的曲線與得自精確公式之結果曲線相異。然而當初始數越大,則經模擬的曲線就越接近精確公式所得結果曲線。而當λ大到3時,結果並没有改變太多。 2. 嚴重急性呼吸道症候群流行之基礎再生數估計 利用分支過程,在16-22代及5-7天的感染潛伏期假設下,基礎再生數為0.9971 (0.5090~1.4852),在布瓦松分佈的假設下,滅絶機率為0.9912。在Borel-Tanner分佈且基礎再生數小於1的條件下,基礎再生數介於0.9790 (0.8437 ~ 1.1143)與 1.0134 (0.8535 ~ 1.1733)之間,滅絶機率為0.9709 ~ 0.9989。 由於線性生死過程無法最適配觀測資料,我們採以一般生死過程對觀察到的累積性個案資料進行配適。出生率觀察為0.57 (於流行期小於55天),11.45 (於流行期介於55天至80天),以及1.413(於流行期超過80天)。預期達到最終感染數a的時間(Ta)為:在T32、T300以及T346分別為54.97(10.09)天、80.00 (10.41)天以及112.01 (11.47)天。 3. 結核病自然病史 結核病群突發 結核病受感染至發生可傳染之潛伏期(latent)經估計為223.6天[λ=0.0045 (2.17*10-6) ],而症狀發生前之感染期經估計為55.9天[ β=0.0179 (3.47*10-5)]。因此從受感染至發生症狀之潛伏期(incubation)約為279.6天。而依據潛伏期的估計,感染至少2代至多3代。基礎再生數的範圍介於0.9739 及 0.9796間。 IGRA對結核病發生的危險性 利用病例對照配對研究,在調整TST後,QFT-GIT陽性對結核病發生的危險性為2.47 (95% CI: 1.72-3.54),若廻歸模式考量交互作用, TST陽性者,QFT-GIT陽性對結核病發生的危險性為4.28 (95% CI: 1.16-15.76), TST陰性者,QFT-GIT陽性對結核病發生的危險性為1.15 (95% CI: 0.66-2.00)。 IGRA在潛伏結核病感染率及結核病轉移率不同疾病進程之影響 整體結核病感染率(每人年)及轉移率(每人年自潛伏結核病感染轉移至結核病)經估計分別為0.0168 (95% CI: 0.0157-0.0180)及 0.0113 (95% CI: 0.0098-0.0129)。感染率表現在年輕族群(30-44歲)及男性都較高。那些陽性IGRA檢測值者,相較於陰性IGRA者,有1.6倍 (RR=1.59, 95% CI:1.39-1.84)的危險性,較易成為潛伏結核病感染個案。相對地,年老族群有較高轉移率,但男性轉移率仍較女性為高。陽性IGRA檢測者,相較於陰性IGRA檢測者有約2倍 (RR=2.12, 95% CI:1.57-2.85)的危險性較易轉移至結核病。經年齡及性別的調整後,QFT-GIT陽性者在結核病感染及轉移率之危險比分別為1.71 (95% CI: 1.49-2.00) 及 1.58 (95% CI: 1.15-2.17)。 應用三階段馬可夫模式估計所得到參數於生死過程發現,在不考量共變數因子下,一個初始個案要擴散到10個個案約花61天,而要擴散到30個個案約花87天。年輕族群、男性及QFT-GIT陽性擴散愈迅速。年齡小於45歲且QFT-GIT陽性的男性,若在初始個案為5位的狀況下,擴散到最終為10位個案約需1週的時間。值得注意的是,若增加初始個案,則要達到預期的擴散個案數,所花費時間會越短。而初始個案若超過5位,則結核病滅絕的機率則幾乎不可能。 結論 本論文在結果的發現上可歸納出5個主要結論如下: 1.當評估新加坡及台灣兩個地區嚴重急性呼吸道症候群的基礎再生數時,在新加坡發生的嚴重急性呼吸道症候群3至8代的感染資料,經評估發現基礎再生數介於1至1.5之間,利用分支過程可幾乎確定必定滅絶。而透過分支過程剖析台灣地區嚴重急性呼吸道症候群流行,其基礎再生數為0.99,滅絶機率為0.99。Borel-Tanner分佈之分支過程也有相同的發現。 2.估計結核菌從感染到發生症狀之潛伏期約9個月,其中從感染至可傳染之潛伏期約7個月,在症狀發生前之可傳染期約2個月。結核病群突發時,進行TST篩檢監視,追蹤TST陰性個案後來也發生結核病。所以針對長照機構的TST陰性年老族群,仍需進行監視,以期在結核病群突發時能獲得控制。 3.本論文是在考量人口學特性及TST檢測結果下,針對IGRA對結核病發生之影響所進行的第一個病例對照研究。 4.本論文也是第一個針對評估年齡、性別及IGRA在潛伏結核病感染(LTBI)及結核病轉移不同疾病進程影響之結核病自然病史研究。年輕族群有較高潛伏結核病感染率,而年老族群有較高轉移率,男性在潛伏結核病感染率及轉移率均較女性高。考量年齡及性別因素後,IGRA在潛伏結核病感染率及轉移率上均扮演重要角色。 5.運用生死過程,在不同年齡、性別、IGRA狀態下,可在給定潛伏結核病感染率(出生率)及轉移率(死亡率)下,計算到達潛伏結核病感染數目之預期時間及滅絶機率。 本論文在傳染病相關的方法學發展有三項貢獻綜述如下: 1.提出多項統計模擬方法包含以分支過程模擬基礎再生數,或以生死過程估計滅絶機率及傳染病擴散時間。 2.示範如何應用Becker的SIR模型結合分支過程估計潛伏期及感染潛伏期以監視結核病。 3.以嶄新病例世代設計加上連續時間馬可夫模型,並結合生死過程以了解IGRA在潛伏結核病感染率及結核病轉移率之不同疾病進程上所扮演的角色。更進一步運用生死過程模擬SARS,以計算其滅絶機率及到達最後感染人數之預期時間。兩種方法對於在傳染病之防治政策問題上,能提供相當助益。

並列摘要


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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