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Continuous-time linear birth-death-immigration (BDI) processes are frequentlyused in ecology and epidemiology to model stochastic dynamics of the populationof interest. In clinical settings, multiple birth-death processes can describedisease trajectories of individual patients, allowing for estimation of theeffects of individual covariates on the birth and death rates of the process.Such estimation is usually accomplished by analyzing patient data collected atunevenly spaced time points, referred to as panel data in the biostatisticsliterature. Fitting linear BDI processes to panel data is a nontrivialoptimization problem because birth and death rates can be functions of manyparameters related to the covariates of interest. We propose a novelexpectation--maximization (EM) algorithm for fitting linear BDI models withcovariates to panel data. We derive a closed-form expression for the jointgenerating function of some of the BDI process statistics and use thisgenerating function to reduce the E-step of the EM algorithm, as well ascalculation of the Fisher information, to one-dimensional integration. Thisanalytical technique yields a computationally efficient and robust optimizationalgorithm that we implemented in an open-source R package. We apply our methodto DNA fingerprinting of Mycobacterium tuberculosis, the causative agent oftuberculosis, to study intrapatient time evolution of IS6110 copy number, agenetic marker frequently used during estimation of epidemiological clusters ofMycobacterium tuberculosis infections. Our analysis reveals previouslyundocumented differences in IS6110 birth-death rates among three major lineagesof Mycobacterium tuberculosis, which has important implications forepidemiologists that use IS6110 for DNA fingerprinting of Mycobacteriumtuberculosis.

Fitting birth-death processes to panel data with applications to bacterial DNA fingerprinting