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

An ODEs multiscale model with cell proliferation for hepatitis C virus infection treated with direct acting antiviral agents

In a recent study, a mathematically identical ODE model is derived from a multiscale PDE model of hepatitis C virus infection, which helps to overcome the limitations of the PDE model in the analysis. Here, an extended proposed model is formulated for this transformed ODE model by including the hepatocyte proliferation of both uninfected and infected cells. Unlike the transformed model, the proposed model can predict the triphasic viral decline and the virus level after therapy cessation without oscillations. Numerical simulations are performed to investigate the effect of hepatocyte proliferation and therapy with direct-acting antivirals agents (DAAs). The basic reproduction number is obtained, the equilibrium points are specified, and their stability is analysed. A bifurcation analysis is performed to specify the bifurcation points and to study the effect of varying system parameters. Various viral load profiles generated by the model are confirmed to fit with reported data in the literature.

Advances in Parameter Estimation and Learning from Data for Mathematical Models of Hepatitis C Viral Kinetics

Mathematical models, some of which incorporate both intracellular and extracellular hepatitis C viral kinetics, have been advanced in recent years for studying HCV–host dynamics, antivirals mode of action, and their efficacy. The standard ordinary differential equation (ODE) hepatitis C virus (HCV) kinetic model keeps track of uninfected cells, infected cells, and free virus. In multiscale models, a fourth partial differential equation (PDE) accounts for the intracellular viral RNA (vRNA) kinetics in an infected cell. The PDE multiscale model is substantially more difficult to solve compared to the standard ODE model, with governing differential equations that are stiff. In previous contributions, we developed and implemented stable and efficient numerical methods for the multiscale model for both the solution of the model equations and parameter estimation. In this contribution, we perform sensitivity analysis on model parameters to gain insight into important properties and to ensure our numerical methods can be safely used for HCV viral dynamic simulations. Furthermore, we generate in-silico patients using the multiscale models to perform machine learning from the data, which enables us to remove HCV measurements on certain days and still be able to estimate meaningful observations with a sufficiently small error.

HCVMultiscaleFit: A Simulator For Parameter Estimation in Multiscale Models Of Hepatitis C Virus Dynamics

Callibration 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 or fitting (callibration) that solves all cases of data points available presents a formidable challenge, as efficiency considerations need to be employed in order for the method to become practical. In the case of multiscale models of hepatitis C virus dynamics that deal with partial differential equations (PDEs), 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 derviative approximations. These steps that were successful in significantly speeding up a highly non-efficient approach, rendering it practical, can also be adapted to multiscale models of other viruses and other sophisticated differential equation models. The newly efficient methods that were developed as a result of the above approach are described. Illustrations are provided using a user-friendly simulator that incorporates the efficient methods for multiscale models. We provide a simulator called HCVMultiscaleFit with a Graphical User Interface that applies these methods and is useful to perform parameter estimation for simulating viral dynamics during antiviral treatment.

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

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.

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

Mathematical models that are based on differential equations require detailed knowledge about the parameters that are included in the equations. Some of the parameters can be measured experimentally while others need to be estimated. When the models become more sophisticated, such as in the case of multiscale models of hepatitis C virus dynamics that deal with partial differential equations (PDEs), several strategies can be tried. It is possible to use parameter estimation on an analytical approximation of the solution to the multiscale model equations, namely the long-term approximation, but this limits the scope of the parameter estimation method used and a long-term approximation needs to be derived for each model. It is possible to transform the PDE multiscale model to a system of ODEs, but this has an effect on the model parameters themselves and the transformation can become problematic for some models. Finally, it is possible to use numerical solutions for the multiscale model and then use canned methods for the parameter estimation, but the latter is making the user dependent on a black box without having full control over the method. The strategy developed here is to start by working directly on the multiscale model equations for preparing them toward the parameter estimation method that is fully coded and controlled by the user. It can also be adapted to multiscale models of other viruses. The new method is described, and illustrations are provided using a user-friendly simulator that incorporates the method.

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

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.