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Stereotactic body radiation therapy (SBRT) is characterized by delivering a high amount of dose in a short period of time. In SBRT the dose is delivered using open fields (e.g., beam's-eye-view) known as 'apertures'. Mathematical methods can be used for optimizing treatment planning for delivery of sufficient dose to the cancerous cells while keeping the dose to surrounding organs at risk (OARs) minimal. Two important elements of a treatment plan are quality and delivery time. Quality of a plan is measured based on the target coverage and dose to OARs. Delivery time heavily depends on the number of beams used in the plan as the setup times for different beam directions constitute a large portion of the delivery time. Therefore the ideal plan, in which all potential beams can be used, will be associated with a long impractical delivery time. We use the dose to OARs in the ideal plan to find the plan with the minimum number of beams which is guaranteed to be epsilon-optimal (i.e., a predetermined maximum deviation from the ideal plan is guaranteed). Since the treatment plan optimization is inherently a multi-criteria-optimization problem, the planner can navigate the ideal dose distribution Pareto surface and select a plan of desired target coverage versus OARs sparing, and then use the proposed technique to reduce the number of beams while guaranteeing epsilon-optimality. We use mixed integer programming (MIP) for optimization. To reduce the computation time for the resultant MIP, we use two heuristics: a beam elimination scheme and a family of heuristic cuts, known as 'neighbor cuts', based on the concept of 'adjacent beams'. We show the effectiveness of the proposed technique on two clinical cases, a liver and a lung case. Based on our technique we propose an algorithm for fast generation of epsilon-optimal plans.

We consider the susceptible-infective (SI) epidemiological model, a variant of the Kermack–McKendrick models, and let the contact rate be a function of the number of infectives, an indicator of disease spread during the course of the epidemic. We represent the resultant model as a continuous-time Markov chain. The result is a pure death (or birth) process with state-dependent rates, for which we find the probability distribution of the associated Markov chain by solving the Kolmogorov forward equations. This model is used to find the analytic solution of the SI model as well as the distribution of the epidemic duration. We use the maximum likelihood method to estimate contact rates based on observations of inter-infection time intervals. We compare the stochastic model to the corresponding deterministic models through a numerical experiment within a typical household. We also incorporate different intervention policies for vaccination, antiviral prophylaxis, isolation, and treatment considering both full and partial adherence to interventions among individuals.

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