Dan Stanescu

Random coefficient differential equation models for bacterial growth

In the mathematical modeling of population growth, and in particular of bacterial growth, parameters are either measured directly or determined by curve fitting. These parameters have large variability that depends on the experimental method and its inherent error, on differences in the actual population sample size used, as well as other factors that are difficult to account for. In this work the parameters that appear in the Monod kinetics growth model are considered random variables with specified distributions. A stochastic spectral representation of the parameters is used, together with the polynomial chaos method, to obtain a system of differential equations, which is integrated numerically to obtain the evolution of the mean and higher-order moments with respect to time.

Epidemic models with random coefficients

Mathematical models are very important in epidemiology. Many of the models are given by differential equations and most consider that the parameters are deterministic variables. But in practice, these parameters have large variability that depends on the measurement method and its inherent error, on differences in the actual population sample size used, as well as other factors that are difficult to account for. In this paper the parameters that appear in SIR and SIRS epidemic model are considered random variables with specified distributions. A stochastic spectral representation of the parameters is used, together with the polynomial chaos method, to obtain a system of differential equations, which is integrated numerically to obtain the evolution of the mean and higher-order moments with respect to time.

Parameter estimation using polynomial chaos and maximum likelihood

When modelling biological processes, there are always errors, uncertainties and variations present. In this paper, we consider the coefficients in the mathematical model to be random variables, whose distribution and moments are unknown a priori, and need to be determined by comparison with experimental data. A stochastic spectral representation of the parameters and the solution stochastic process is used, based on polynomial chaoses. The polynomial chaos representation generates a system of equations of the same type as the original model. The inverse problem of finding the parameters is reduced to establishing the best-fit values of the random variables that represent them, and this is done using maximum likelihood estimation. In particular, in modelling biofilm growth, there are variations, measurement errors and uncertainties in the processes. The biofilm growth model is given by a parabolic differential equation, so the polynomial chaos formulation generates a system of partial differential equations. Examples are presented.

Biofilm growth on medical implants with randomness

Biofilms are colonies of bacteria that attach to surfaces by producing extracellular polymer substances. They may cause serious infections in humans and animals, and also cause problems in hydraulic machinery. In this paper we model the growth of a biofilm established on a medical implant. We assume that the biofilms growth is given by a logistic reaction term with the growth rate being a random variable with a given distribution. This way we take into account the variability in the bacterial populations, and the measurement and experimental errors. The diffusion coefficient of the microbes is also taken to be random. A stochastic spectral representation of the parameters and the unknown stochastic process is used, together with the polynomial chaos method, to obtain a system of partial differential equations, which is integrated numerically to obtain the evolution of the mean and higher-order moments with respect to time. Some examples are presented.

High-order W-methods

Implicit methods are the natural choice for solving stiff systems of ODEs. Rosenbrock methods are a class of linear implicit methods for solving such stiff systems of ODEs. In the Rosenbrock methods the exact Jacobian must be evaluated at every step. These evaluations can make the computations costly. By contrast, W-methods use occasional calculations of the Jacobian matrix. This makes the W-methods popular among the class of linear implicit methods for numerical solution of stiff ODEs. However, the design of high-order W-methods is not easy, because as the order of the W-methods increases, the number of order conditions of the W-methods increases very fast. In this paper, we describe an approach to constructing high-order W-methods.

Development of a Spectral Element DNS/LES Method for Turbulent Flow Simulations

This paper describes a turbulent flow simulation method, which is based on combination of spectral element and large eddy simulation (LES) technique. The robust, high-order discontinuous Galerkin (DG) spectral element method for large-eddy simulation of compressible flows allows for arbitrary order of accuracy and has excellent stability properties. A local spectral discretization in terms of Legendre polynomials is used on each element of the (possibly unstructured) mesh, which allows for high-accurate simulations of turbulent flows. Discontinuities across the interfaces of the elements are resolved using a Riemann solver. An isoparametric representation of the geometry is implemented, with boundaries of the domain discretized to the same order of accuracy as the solution, and explicit low-storage Runge-Kutta methods are used for time integration. Large eddy simulation has proven to be a valuable technique for the calculation of turbulent flows. An element based filtering technique is used in conjunction with the standard Smagorinsky eddy viscosity model to estimate the effect of sub-grid scales stresses in this paper. The recently developed nonlinear model [1] will also be added in the future. The final aim of this project is to use the LES methodology in swirling jet flow simulation. As a first step towards these simulations, simulations of compressible turbulent mixing layer and back-facing step are also performed to evaluate the robust method. Initial results based on both DNS and large eddy simulations are presented in this paper. Future work will be to validate the code.Copyright

Virus propagation with randomness

Abstract Viruses are organisms that need to infect a host cell in order to reproduce. The new viruses leave the infected cell and look for other susceptible cells to infect. The mathematical models for virus propagation are very similar to population and epidemic models, and involve a relatively large number of parameters. These parameters are very difficult to establish with accuracy, while variability in the cell and virus populations and measurement errors are also to be expected. To deal with this issue, we consider the parameters to be random variables with given distributions. We use a non-intrusive variant of the polynomial chaos method to obtain statistics from the differential equations of two different virus models. The equations to be solved remain the same as in the deterministic case; thus no new computer codes need to be developed. Some examples are presented.

Stable Interface Conditions for Discontinuous Galerkin Approximations of Navier-Stokes Equations

A study of boundary and interface conditions for Discontinuous Galerkin approximations of fluid flow equations is undertaken in this paper. While the interface flux for the inviscid case is usually computed by approximate Riemann solvers, most discretizations of the Navier-Stokes equations use an average of the viscous fluxes from neighboring elements. The paper presents a methodology for constructing a set of stable boundary/interface conditions that can be thought of as “viscous” Riemann solvers and are compatible with the inviscid limit.

Nonstandard jump functions for radially symmetric shock waves

Nonstandard analysis is applied to derive generalized jump functions for radially symmetric, one-dimensional, magnetogasdynamic shock waves. It is assumed that the shock wave jumps occur on infinitesimal intervals, and the jump functions for the physical parameters occur smoothly across these intervals. Locally integrable predistributions of the Heaviside function are used to model the flow variables across a shock wave. The equations of motion expressed in nonconservative form are then applied to derive unambiguous relationships between the jump functions for the physical parameters for two families of self-similar flows. It is shown that the microstructures for these families of radially symmetric, magnetogasdynamic shock waves coincide in a nonstandard sense for a specified density jump function