A Robust and Efficient Numerical Method for RNA-Mediated Viral Dynamics

Reinharz, Vladimir; Churkin, Alexander; Dahari, Harel; Barash, Danny

doi:10.3389/fams.2017.00020

Cited by 6 publications

(12 citation statements)

References 55 publications

(111 reference statements)

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“…Then, in Appendix B , for each type of model, some examples are described. The results are presented using a newer (efficient) version of the user-friendly simulator that we have initially developed in [ 55 , 57 ] for both biphasic and multiscale models. We start from the biphasic model in Appendix B and end with the multiscale model in Appendix C .…”

Section: Resultsmentioning

confidence: 99%

“…It works directly on the multiscale model equations, preparing them in advance for the optimization procedure by taking their derivatives with respect to the parameters, in contrast to solving them first by an analytical approximation or performing the method of lines as a first step. For the solution of the model equations, the Rosenbrock method described in [ 55 ] is employed, as was shown to be advantageous in comparison to other solution schemes in [ 56 ]. For the constrained optimization procedure, as a departure from [ 57 ], either the Gauss–Newton or COBYLA are employed in full (not as a canned method) such that the user has access to the source code at each point in the procedure.…”

Section: Discussionmentioning

confidence: 99%

“…For the numerical solution of the multiscale model equations, properties such as approximation, stability, and convergence were discussed in [ 56 ] and numerical robustness was discussed in [ 55 , 56 ]. Future work should expand towards the advanced treatment of properties as covered in [ 63 , 64 ].…”

Section: Methodsmentioning

confidence: 99%

“…Extending to the multiscale model for HCV dynamics, we perform parameter estimation on the model taken from [ 48 ]. As previously described, in our simulator, we solved the model equations using the Rosenbrock method [ 55 ] and performed parameter estimation after preparing the derivative equations using a full implementation of our developed Gauss–Newton (LSF) and COBYLA methods that are suitable to our application domain without reverting to canned methods.…”

Section: Figure A1mentioning

confidence: 99%

“…Unlike the construction of numerical schemes in other applications, for example in the nonlinear diffusion of digital images [ 52 – 54 ] where accuracy can be limited, herein it is advisable to construct a stable and efficient scheme that belongs to the Runge–Kutta family with a higher accuracy than in nonlinear diffusion. Our numerical solution strategy was outlined in [ 55 – 57 ] and herein we continue [ 57 ] by providing an efficient parameter estimation method that follows this strategy.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2020

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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.

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Section: Resultsmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: Figure A1mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

et al. 2020

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A Parameter Estimation Method for Multiscale Models of Hepatitis C Virus Dynamics

et al. 2019

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Numerical schemes for solving and optimizing multiscale models with age of hepatitis C virus dynamics

Reinharz

Dahari

Barash

2018

Mathematical Biosciences

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Age-structured PDE models have been developed to study viral infection and treatment. However, they are notoriously difficult to solve. Here, we investigate the numerical solutions of an age-based multiscale model of hepatitis C virus (HCV) dynamics during antiviral therapy and compare them with an analytical approximation, namely its long-term approximation. First, starting from a simple yet flexible numerical solution that also considers an integral approximated over previous iterations, we show that the long-term approximation is an underestimate of the PDE model solution as expected since some infection events are being ignored. We then argue for the importance of having a numerical solution that takes into account previous iterations for the associated integral, making problematic the use of canned solvers. Second, we demonstrate that the governing differential equations are stiff and the stability of the numerical scheme should be considered. Third, we show that considerable gain in efficiency can be achieved by using adaptive stepsize methods over fixed stepsize methods for simulating realistic scenarios when solving multiscale models numerically. Finally, we compare between several numerical schemes for the solution of the equations and demonstrate the use of a numerical optimization scheme for the parameter estimation performed directly from the equations.

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A Robust and Efficient Numerical Method for RNA-Mediated Viral Dynamics

Cited by 6 publications

References 55 publications

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

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

A Parameter Estimation Method for Multiscale Models of Hepatitis C Virus Dynamics

Numerical schemes for solving and optimizing multiscale models with age of hepatitis C virus dynamics

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