Benito M. Chen-Charpentier

A nonstandard numerical scheme of predictor-corrector type for epidemic models

In this paper we construct and develop a competitive nonstandard finite difference numerical scheme of predictor-corrector type for the classical SIR epidemic model. This proposed scheme is designed with the aim of obtaining dynamical consistency between the discrete solution and the solution of the continuous model. The nonstandard finite difference scheme with Conservation Law (NSFDCL) developed here satisfies some important properties associated with the considered SIR epidemic model, such as positivity, boundedness, monotonicity, stability and conservation of frequency of the oscillations. Numerical comparisons between the NSFDCL numerical scheme developed here and Runge-Kutta type schemes show its effectiveness.

An unconditionally positivity preserving scheme for advection-diffusion reaction equations

Abstract Parabolic equations with advection, diffusion and reaction terms are used to model many physical and biological systems. In many applications the unknowns represent quantities that cannot be negative such as concentrations of chemical compounds or population sizes. Widely used schemes such as finite differences may produce negative solutions because of truncation errors and may then become unstable. We propose a new scheme that guarantees the positivity of the solutions for arbitrary step sizes. It works for reaction terms that consist of the sum of a positive function and a negative function. We develop it for one advection–diffusion reaction equation in one spatial dimension with constant velocity and diffusion and state how to generalize it. The method is applicable to both advection and diffusion dominated problems. We give some examples from different applications.

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.

Numerical simulation of biofilm growth in porous media

Biofilms are very important in controlling pollution in aquifers. The bacteria may either consume the contaminant or form biobarriers to limit its spread. In this paper we review the mathematical modeling of biofilm growth at the microscopic and macroscopic scales, together with a scale-up technique. At the pore-scale, we solve the Navier-Stokes equations for the flow, the advection-diffusion equation for the transport, together with equations for the biofilm growth. These results are scaled up using network model techniques, in order to have relations between the amount and distribution of the biomass, and macroscopic properties such as permeability and porosity. A macroscopic model is also presented. We give some results.

Combination of nonstandard schemes and Richardson's extrapolation to improve the numerical solution of population models

In this paper we combine nonstandard finite-difference (NSFD) schemes and Richardsons extrapolation method to obtain numerical solutions of two biological systems. The first biological system deals with the dynamics of phytoplankton-nutrient interaction under nutrient recycling and the second one deals with the modeling of whooping cough in the human population. Since both models requires positive solutions, the numerical solutions need to satisfy this property. In addition, it is necessary in some cases that numerical solutions reproduce correctly the dynamical behavior while in other cases it is necessary just to find the steady state. NSFD schemes can do this. In this paper Richardsons extrapolation is applied directly to the NSFD solution to increase the order of accuracy of the numerical solutions of these biological systems. Numerical results show that Richardsons extrapolation method improves accuracy.

Construction of nonstandard finite difference schemes for the SI and SIR epidemic models of fractional order

In this paper we construct nonstandard finite difference (NSFD) schemes to obtain numerical solutions of the susceptible-infected (SI) and susceptible-infected-recovered (SIR) fractional-order epidemic models. In order to deal with fractional derivatives we apply the Caputo operator and use the Grunwald-Letnikov method to approximate the fractional derivatives in the numerical simulations. According to the principles of dynamic consistency we construct NSFD schemes to numerically integrate the fractional-order epidemic models. These NSFD schemes preserve the positivity that other classical methods do not guarantee. Additionally, the NSFD schemes hold other conservation properties of the solution corresponding to the continuous epidemic model. We run some numerical comparisons with classical methods to test the behavior of the NSFD schemes using the short memory principle. We conclude that the NSFD schemes, which are explicit and computationally inexpensive, are reliable methods to obtain realistic positive numerical solutions of the SI and SIR fractional-order epidemic models.

Nonstandard discrete approximationspreserving stability properties of continuous mathematical models

Most numerical methods for differential equations introduce spurious solutions. Westudy the method presented by Mickens to obtain exact nonstandard methods for some ordinary differential equations. We show how to generalize his method to equations with no known exact analytical solution, and show that the new scheme has better stability properties than Runge-Kutta methods. We apply the method to several examples.

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.

Numerical simulation of dual-species biofilms in porous media

There are bacteria that can form strong biofilms in porous media. The biofilms can be used as biobarriers to restrict the flow of pollutants. If a second species of bacteria that can actually react with the contaminants is added to the biobarrier, the result is a much more effective way of controlling the pollutants. We propose a mathematical model for the formation of these biobarriers and numerically solve the resulting equations for the flow, transport and reactions. Qualitative comparisons with some experimental results are also given.

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.