Sensitivity Functions for Delay Differential Equation Models

ROBBINS, DANIELLE. Sensitivity Functions for Delay Differential Equation Models. (Under the direction of H.T. Banks.) Delay differential equations are useful to model various biological, sociological, and physical processes in which there are hysteretic or memory effects. Nicholas Minorsky played a great role in establishing the use of these type of models for physical processes. From his work, physical processes like ship control systems are modeled using delay differential equations with delayed damping or delayed restoring force. G.E. Hutchinson also saw the importance of using delay systems to model ecological models. Hutchinson’s equation, also known as the delay-logistic equation, is used to model population growth of a species. For these biological and physical processes modeled using delay differential equations there are generally associated data sets. This data is used to estimate parameters within the model to gain the best predictive model for the process. When performing estimation procedures, parameter identifiability issues may occur resulting in unfavorable estimates. There also may not be enough data or repeated information in the data which will again produce unfavorable estimates. Sensitivity analysis improves the estimation process as traditional sensitivity functions can determine which parameters can be estimated and those that should be fixed. Generalized sensitivity functions will determine which regions in the data help estimate specific parameters. Thus using both type of sensitivity functions should lead to optimal parameter estimates. We will derive and compute traditional sensitivity functions for the delay logistic model, delayed damping and restoring force harmonic oscillator models, as well as a sociological model for the behavior of alcoholics. We will also prove the existence of a solution for the derived sensitivity equation with respect to the delay. We will use the computed traditional sensitivity functions, to compute generalized sensitivity functions and illustrate the effect of a delay on generalized sensitivity functions (which provide insight on sensitivity of estimated parameters to data). We compare the numerical approximations of the generalized sensitivity functions for the delay-logistic equations to the equations without delay. From the traditional sensitivity functions and generalized sensitivity functions we simulate ideal data sets to obtain optimal estimates for the delay parameter τ via the inverse problem. © Copyright 2011 by Danielle Robbins All Rights Reserved Sensitivity Functions for Delay Differential Equation Models

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