Stochastic processes and randomness are vital features of mathematical modeling in biology.Unfortunately analytical results are rarely available for even moderately complexstochastic processes leaving simulation and numerical techniques the main avenues of attack.We begin this work by exploring coupling bounds for birth-death processes, a fundamentaltype of stochastic process that describes how populations of individuals change overtime. By forming a coupling between a truncated version of the process and the originalunbounded version, we are able to compute both moments and transition probabilities forthe true process within an acceptable error bound. Second, we present an algorithm designframework for Interacting Particle Systems (IPSs). These are complex stochastic processeswith wide application to spatial phenomenon across many scientific disciplines. Here we describea method for efficiently sorting particles into classes based off of their type and spatialconfiguration in such a fashion that reduces the spatial simulation to that of a non-spatialwell-mixed process, albeit with a more complicated update step. This also allows us to applya large suite of well-developed stochastic simulation algorithms to IPSs with little additionalcoding cost. Third, we return to numerical methods, this time for multi-type branchingprocesses applied to gene therapy. We derive a series of ordinary differential equations thatgovern the evolution of the probability generating function and provide a straightforwardnumerical inversion approach to obtain marginalized probability distributions for probabilisticquantities of interest. We provide examples of our techniques applied to lentiviral genetherapy and the associated risk of oncogenesis in transplanted hematopoietic stem cell lines.Finally, we conclude with a chapter on future directions, both related to the previous threechapters as well as projects not previously addressed in this work
