珮倫弗羅柏氏定理及拉普拉斯轉換應用在多階段疾病參數及高階動差之估計

Abstract

多階段統計模型經常運用於回答族群動態轉變之關鍵問題，諸如運用於感染症模型中回答「傳染病在一開始有人感染後，是否會造成流行? 」與「在多久之後會爆發流行? 」；亦或是運用於慢性疾病時，回答「疾病將以多快之速率由早期進展至晚期?」。上述之問題與兩個主要統計量：感染症之基礎再生數(R0)與慢性疾病之平均滯留期(mean sojourn time, MST)密切相關。然而對於多階段模型建構時，由於疾病進展特性，其轉移機率矩陣常為非負矩陣且需運用摺機之方式進行推導，如此一來將使得計算複雜化。再者，多階段模型動差之推導，特別是高階動差，常常會面臨相當困難的運算。上述在多階段模型應用上諸多困難引致本人發展珮倫弗羅柏氏定理(Perron-Frobenius Theory)配合拉普拉斯變換(Laplace Transform)之方法以得到對於上述應用之簡易過程。雖然過去曾有一些統計方法被提出，但卻尚未有系統性的方法結合珮倫弗羅柏氏定理與拉普拉斯變換以解決上述之問題。 本論文之目的旨在 (1) 演示如何應用珮倫弗羅柏氏定理於包含易感期-感染期-恢復期模型(SIR model)之感染症多階段模型結合拉普拉斯變換得到基礎再生數之第一階動差以及高階動差; (2) 運用拉普拉斯變換於癌症三階段與五階段模型，簡化其轉移機率推導之摺機過程; (3) 以拉普拉斯變換推導得致三階段與五階段模型參數之第一階動差與高階動差; (4) 發展對於運用於三階模型與五階段模型中以拉普拉斯變換概似參數之期望值-最大化演算法(expectation-maximization, EM)。 本論文將演示兩類之應用實例，其一為感染症多階段模型之運用(台灣流行性感冒實例與不同國家之伊波拉爆發流行實例)。其二為乳癌由無病狀態、臨床症前期狀態以及臨床期狀態之三階段進展模型以及加入淋巴結轉移與腫瘤大小考量之五階段模型實例以及大腸直腸癌實例資料運用。 對於R0在流行性感冒與伊波拉爆發流行實例中，本文將比較利用拉普拉斯轉換所得到之第一階動差與傳統方法所得者以及運用拉普拉斯方法估算實例中無法由傳統方法得到之R0值高階動差。對於乳癌與大腸直腸癌實證資料之運用上文中將結合拉普拉斯變換以及期望值-最大化演算法得到轉移參數之估計。 本論文以應用實例演示所發展之方法可對於多階段感染症模型之R0以及多階段慢性疾病之MST中之不確定性加以考量。文中所發展之拉普拉斯變換方法與期望值-最大化演算法則解決了以傳統概似函數估計多階模型參數時需要收集轉移狀態發生之觀測時間之資料需求。本論文所發展之方法可運用於包含感染症與慢性疾病之多階段模型，以得到模型參數之高階動差、分佈函數以及其瞬時變化率。

Multistate statistical models are often used for dealing with the cardinal questions for infectious disease such as “whether and when the epidemic will occur after the introduction of infectives?” and also for chronic disease such as “how soon the disease will progress from early status to advanced one?”. Both questions are related to two main parameters, basic reproductive number for infectious disease and mean sojourn time for the progression of cancer. However, the derivation of transition kernels are often involved in non-negative matrix and also convolution form implicated in multistate disease process, which renders the statistical computation complex. Moreover, the derivation of moment, particularly higher order, is often hampered by intractable computation. These characteristics motivate me to propose Perron-Frobenius theory for dealing with non-negative matrix and apply Laplace transform to render statistical computation feasible. In spite of several statistical approaches proposed before, a systematic approach has been barely addressed. The aims of this thesis are there to (1) demonstrate how to apply Perron-Frobenius theory to multi-state model such as susceptible-infected-recovery model from which the first moment of basic reproductive number (R0) and its higher moments using Laplace transformation model can be derived; (2) to develop Laplace transformation of transition probabilities with convolution form for the widely used three-state and five-state stochastic process cancer ; (3) to estimate first moment and higher moments of the parameters implicated in three-state and five-state disease process with Laplace transformation;(4) to develop the estimation procedure for Laplace transformed likelihood with E-M algorithm for three-state and five-state model. Two applications were demonstrated, including the basic reproductive number used in infectious disease process (influenza epidemic in Taiwan and the epidemic of Ebola virus in different countries). The second is applied to three-state and five-state Markov model for the progression of breast cancer from free of breast cancer, preclinical detectable phase, and clinical phase with the consideration of lymph node invasion and tumour size as the advance and early state of preclinical detectable phase and clinical phase. My thesis compared the results of first moment of basic reproductive number in the outbreaks of influenza and Ebola using our proposed method in comparison with those based on the conventional methods and also demonstrated their second and high order moments, which cannot be reckoned by the conventional method. It illustrates how to estimate parameters based on Laplace transformed likelihood in conjunction with EM algorithm while applied to empirical data on breast cancer and colorectal cancer. The application of the proposed method is of assistance to elucidate the uncertainty of basic reproductive number and sojourns time in modelling infectious disease and cancer based on multistate disease process. The proposed Laplace transformed likelihood function to estimate parameters can solve the requirement of cumbersome computation and dispense with detailed time-stamped history data used for traditional likelihood function while the multistate disease process is implicated. The proposed approach can be applied to a number of multistate models pertaining to infectious and chronic disease for the derivation of high order moments, distribution function, and instantaneous change of transition of parameters.