Efficient Methods for Parameter Estimation of Ordinary and Partial Differential Equation Models of Viral Hepatitis Kinetics

Alexander Churkin, Stephanie Lewkiewicz, Vladimir Reinharz, Harel Dahari and Danny Barash
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Alexander Churkin: Department of Software Engineering, Sami Shamoon College of Engineering, Beer-Sheva 8410501, Israel
Stephanie Lewkiewicz: Department of Mathematics, University of California at Los Angeles, Los Angeles, CA 90095, USA
Vladimir Reinharz: Department of Computer Science, Université du Québec à Montréal, Montreal, QC H3C 3P8, Canada
Harel Dahari: Program for Experimental and Theoretical Modeling, Division of Hepatology, Department of Medicine, Stritch School of Medicine, Loyola University Medical Center, Maywoood, IL 60153, USA
Danny Barash: Department of Computer Science, Ben-Gurion University, Beer-Sheva 8410501, Israel

Mathematics, 2020, vol. 8, issue 9, 1-30

Abstract: Parameter estimation in mathematical models that are based on differential equations is known to be of fundamental importance. For sophisticated models such as age-structured models that simulate biological agents, parameter estimation that addresses all cases of data points available presents a formidable challenge and efficiency considerations need to be employed in order for the method to become practical. In the case of age-structured models of viral hepatitis dynamics under antiviral treatment that deal with partial differential equations, a fully numerical parameter estimation method was developed that does not require an analytical approximation of the solution to the multiscale model equations, avoiding the necessity to derive the long-term approximation for each model. However, the method is considerably slow because of precision problems in estimating derivatives with respect to the parameters near their boundary values, making it almost impractical for general use. In order to overcome this limitation, two steps have been taken that significantly reduce the running time by orders of magnitude and thereby lead to a practical method. First, constrained optimization is used, letting the user add constraints relating to the boundary values of each parameter before the method is executed. Second, optimization is performed by derivative-free methods, eliminating the need to evaluate expensive numerical derivative approximations. The newly efficient methods that were developed as a result of the above approach are described for hepatitis C virus kinetic models during antiviral therapy. Illustrations are provided using a user-friendly simulator that incorporates the efficient methods for both the ordinary and partial differential equation models.

Keywords: parameter estimation; constrained optimization; derivative free optimization; multiscale models; differential equations; viral hepatitis (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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