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Continuous-time birth-death-shift (BDS) processes are frequently used instochastic modeling, with many applications in ecology and epidemiology. Inparticular, such processes can model evolutionary dynamics of transposableelements - important genetic markers in molecular epidemiology. Estimation ofthe effects of individual covariates on the birth, death, and shift rates ofthe process can be accomplished by analyzing patient data, but inferring theserates in a discretely and unevenly observed setting presents computationalchallenges. We propose a mutli-type branching process approximation to BDSprocesses and develop a corresponding expectation maximization (EM) algorithm,where we use spectral techniques to reduce calculation of expected sufficientstatistics to low dimensional integration. These techniques yield an efficientand robust optimization routine for inferring the rates of the BDS process, andapply more broadly to multi-type branching processes where rates can depend onmany covariates. After rigorously testing our methodology in simulationstudies, we apply our method to study intrapatient time evolution of IS6110transposable element, a frequently used element during estimation ofepidemiological clusters of Mycobacterium tuberculosis infections.

Likelihood-Based Inference for Discretely Observed Birth-Death-Shift Processes, with Applications to Evolution of Mobile Genetic Elements