| Abstract | For many epidemiological and biomedical models, the dynamics can be described by a system of ordinary differential equations, that is controlled by an independent function of time known as the control function. Optimal control is the problem of finding the best control function for a given objective. For example, in an epidemic disease model, the control function might be the percentage of susceptible individuals being vaccinated per unit time. An optimal control problem for this situation is: Find the vaccination rate that minimizes both the number of infectious persons and the overall cost. Direct solution methods treat optimal control as a global optimization problem. A global search is performed, for the control function that minimizes the objective functional. Increasingly, Differential Evolution (DE) is being recognized as a powerful global optimizer for multimodal optimal control problems. However, like other evolutionary algorithms, it operates on discrete n-dimensional vectors, and becomes computationally unmanageable for large values of n. Evolutionary direct methods thus require a technique to represent control functions with a small number of real-valued parameters. This is known as Control Vector Parameterization (CVP). To date, solutions for epidemiological and biomedical models have used piecewise constant or piecewise linear parameterization. These have obvious limitations for approximating arbitrary functions. We introduce a new CVP, using Bezier curves, which can accurately represent continuous control functions, using only a few parameters. The new technique is shown to be robust and efficient when paired with DE, providing global, near optimal, continuous solutions, with reasonable computational cost. The effectiveness and versatility of the method is demonstrated by application to a range of models, including public health strategies for epidemics, and drug administration schedules for HIV and cancer. |
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