Researchers struggle with likelihood-based inference from count data that arise continuously in time but we only intermittently observe them.  A major shortcoming lies in our inability to integrate most underlying stochastic processes generating the data over all possible realizations between observations.  Since these processes are ubiquitous across the natural, physical and social sciences as generative models, solutions should promote the use of statistical inference in many real-world problems.  One seemingly trivial example is a stochastic compartmental model tracking the count of susceptible, infectious and removed people during the spread of an infectious disease.  For over 90 years, many have believed the transition probabilities of this SIR model remain beyond reach.  However, applying a novel re-parameterization, integral transforms and other tools from numerical analysis shows that we can compute the transition probabilities in merely quadratic complexity in terms of the observed change in population size.  Other stochastic processes for modest numbers of outcomes, such as those employed to model molecular sequence evolution, yield well to advancing computing technology, such as many-core parallelization.  Examples in this talk stem from the dynamics of influenza across the global and the 2014-2015 West African ebola outbreak.

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