Modeling of Subsurface Biobarrier Formation

Biofilm-forming microbes can form biobarriers to inhibit contaminant migration in groundwater and potentially biotransform organic contaminants to less harmful forms. Biofilm-forming microbes thereby provide an in situ method for treatment of contaminated groundwater. A mathematical and numerical model to describe the population distribution and growth of bacteria in porous media is presented here. The model is based on the convection-dispersion equation with nonlinear reaction terms. Accurate numerical simulations are crucial to the development of contaminant remediation strategies. We use the nonstandard numerical approach that is based on non-local treatment of nonlinear reactions and modified characteristic derivatives. This approach leads to significant qualitative improvements in the behavior of the numerical solution. Numerical results for a simple biobarrier formation model are presented to demonstrate the performance of the proposed new method. Comparisons of simulated results with experimental results obtained from the Montana State Center for Biofilm Engineering are also presented

Mathematical modeling of bioremediation of trichloroethylene in aquifers

AbstractTrichloroethylene (TCE) is a very common contaminant of groundwater. It is used as an industrial solvent and is frequently poured into the soil. There exist bacteria that can degrade TCE. In contrast with most cases of bioremediation, the bacteria that degrade TCE do not use it as a carbon source. Instead the bacteria produce an enzyme to metabolize methane. This enzyme can degrade other organics including TCE. In this paper we model in situ bioremediation of TCE in an aquifer by using two species of bacteria: one that forms biobarriers to restrict the movement of TCE and the second one to reduce TCE. The model includes flow of water, transport of TCE and the nutrients, bacterial growth and degradation of TCE. Nonstandard numerical methods are used to discretize the equations. Some results are presented

Computational uncertainty quantification for random time-discrete epidemiological models using adaptive gPC

[EN] Population dynamics models consisting of nonlinear difference equations allow us to get a better understanding of the processes involved in epidemiology. Usually, these mathematical models are studied under a deterministic approach. However, in order to take into account the uncertainties associated with the measurements of the model input parameters, a more realistic approach would be to consider these inputs as random variables. In this paper, we study the random time-discrete epidemiological models SIS, SIR, SIRS, and SEIR using a powerful unified approach based upon the so-called adaptive generalized polynomial chaos (gPC) technique. The solution to these random difference equations is a stochastic process in discrete time, which represents the number of susceptible, infected, recovered, etc individuals at each time step. We show, via numerical experiments, how adaptive gPC permits quantifying the uncertainty for the solution stochastic process of the aforementioned random time-discrete epidemiological model and obtaining accurate results at a cheap computational expense. We also highlight how adaptive gPC can be applied in practice, by means of an example using real data.This work has been supported by the Spanish Ministerio de Economia y Competitividad grant MTM2017-89664-P. Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia. The authors are grateful for the helpful and valuable reviewers' comments that have considerably improved the final form of this manuscript.Calatayud-Gregori, J.; Cortés, J.; Jornet-Sanz, M.; Villanueva Micó, RJ. (2018). Computational uncertainty quantification for random time-discrete epidemiological models using adaptive gPC. Mathematical Methods in the Applied Sciences. 41(18):9618-9627. https://doi.org/10.1002/mma.5315S96189627411

A Cellular Automata Model of Infection Control on Medical Implants

S. epidermidis infections on medically implanted devices are a common problem in modern medicine due to the abundance of the bacteria. Once inside the body, S. epidermidis gather in communities called biofilms and can become extremely hard to eradicate, causing the patient serious complications. We simulate the complex S. epidermidis-Neutrophils interactions in order to determine the optimum conditions for the immune system to be able to contain the infection and avoid implant rejection. Our cellular automata model can also be used as a tool for determining the optimal amount of antibiotics for combating biofilm formation on medical implants

Mathematical Modeling for Studying the Sustainability of Plants Subject to the Stress of Two Distinct Herbivores

Viability of plants, especially endangered species, are usually affected by multiple stressors, including insects, herbivores, environmental factors and other plant species. We present new mathematical models, based on systems of ordinary differential equations, of two distinct herbivore species feeding (two stressors) on the same plant species. The new feature is the explicit functional form modeling the simultaneous feedback interactions (synergistic or additive or antagonistic) between the three species in the ecosystem. The goal is to investigate whether the coexistence of the plant and both herbivore species is possible (a sustainable system) and under which conditions sustainability is feasible. Our theoretical analysis of the novel model without including competitions among the two herbivores reveals that the number of equilibrium states and their local stability depends on the type of interaction between the stressors: synergistic or additive or antagonistic. Our numerical results, based on value of parameters available, suggest that a sustainable system requires significant herbivore inter- or intra-species competition or both types. Additionally, our numerical findings indicate that competition and interaction of additive type promotes coexistence equilibrium states with the highest plant biomass. Furthermore, the system can exhibit periodic behavior and show the potential for multi-stability

Do the generalized polynomial chaos and Fröbenius methods retain the statistical moments of random differential equations?

The aim of this paper is to explore whether the generalized polynomial chaos (gPC) and random Fröbenius methods preserve the first three statistical moments of random differential equations. There exist exact solutions only for a few cases, so there is a need to use other techniques for validating the aforementioned methods in regards to their accuracy and convergence. Here we present a technique for indirectly study both methods. In order to highlight similarities and possible differences between both approaches, the study is performed by means of a simple but still illustrative test-example involving a random differential equation whose solution is highly oscillatory. This comparative study shows that the solutions of both methods agree very well when the gPC method is developed in terms of the optimal orthogonal polynomial basis selected according to the statistical distribution of the random input. Otherwise, we show that results provided by the gPC method deteriorate severely. A study of the convergence rates of both methods is also included.This work has been partially supported by the Spanish M.C.Y.T. grants MTM2009-08587, DPI2010-20891-C02-01 as well as the Universitat Politecnica de Valencia grants PAID06-11 (ref. 2070) and PAID00-11 (ref. 2753).Chen Charpentier, BM.; Cortés López, JC.; Romero Bauset, JV.; Roselló Ferragud, MD. (2013). Do the generalized polynomial chaos and Fröbenius methods retain the statistical moments of random differential equations?. Applied Mathematics Letters. 26(5):553-558. doi:10.1016/j.aml.2012.12.013S55355826

Combining Polynomial Chaos Expansions and the Random Variable Transformation Technique to Approximate the Density Function of Stochastic Problems, Including Some Epidemiological Models

[EN] In this paper, we deal with computational uncertainty quantification for stochastic models with one random input parameter. The goal of the paper is twofold: First, to approximate the set of probability density functions of the solution stochastic process, and second, to show the capability of our theoretical findings to deal with some important epidemiological models. The approximations are constructed in terms of a polynomial evaluated at the random input parameter, by means of generalized polynomial chaos expansions and the stochastic Galerkin projection technique. The probability density function of the aforementioned univariate polynomial is computed via the random variable transformation method, by taking into account the domains where the polynomial is strictly monotone. The algebraic/exponential convergence of the Galerkin projections gives rapid convergence of these density functions. The examples are based on fundamental epidemiological models formulated via linear and nonlinear differential and difference equations, where one of the input parameters is assumed to be a random variable.This work has been supported by the Spanish Ministerio de Economia y Competitividad grant MTM2017-89664-P. The author Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia.Calatayud-Gregori, J.; Chen-Charpentier, BM.; Cortés, J.; Jornet-Sanz, M. (2019). Combining Polynomial Chaos Expansions and the Random Variable Transformation Technique to Approximate the Density Function of Stochastic Problems, Including Some Epidemiological Models. Symmetry (Basel). 11(1):1-28. https://doi.org/10.3390/sym11010043S128111Strand, J. . (1970). Random ordinary differential equations. Journal of Differential Equations, 7(3), 538-553. doi:10.1016/0022-0396(70)90100-2Bharucha-Reid, A. T. (1964). 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Communications in Nonlinear Science and Numerical Simulation, 50, 1-15. doi:10.1016/j.cnsns.2017.02.011Chen-Charpentier, B. M., & Stanescu, D. (2010). Epidemic models with random coefficients. Mathematical and Computer Modelling, 52(7-8), 1004-1010. doi:10.1016/j.mcm.2010.01.014Lucor, D., Su, C.-H., & Karniadakis, G. E. (2004). Generalized polynomial chaos and random oscillators. International Journal for Numerical Methods in Engineering, 60(3), 571-596. doi:10.1002/nme.976Santonja, F., & Chen-Charpentier, B. (2012). Uncertainty Quantification in Simulations of Epidemics Using Polynomial Chaos. Computational and Mathematical Methods in Medicine, 2012, 1-8. doi:10.1155/2012/742086Stanescu, D., & Chen-Charpentier, B. M. (2009). Random coefficient differential equation models for bacterial growth. Mathematical and Computer Modelling, 50(5-6), 885-895. doi:10.1016/j.mcm.2009.05.017Calatayud, J., Cortés, J. C., Jornet, M., & Villanueva, R. J. (2018). 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On the convergence of adaptive gPC for non-linear random difference equations: Theoretical analysis and some practical recommendations. Journal of Nonlinear Sciences and Applications, 11(09), 1077-1084. doi:10.22436/jnsa.011.09.06Casabán, M.-C., Cortés, J.-C., Romero, J.-V., & Roselló, M.-D. (2015). Probabilistic solution of random SI-type epidemiological models using the Random Variable Transformation technique. Communications in Nonlinear Science and Numerical Simulation, 24(1-3), 86-97. doi:10.1016/j.cnsns.2014.12.016Dorini, F. A., Cecconello, M. S., & Dorini, L. B. (2016). On the logistic equation subject to uncertainties in the environmental carrying capacity and initial population density. Communications in Nonlinear Science and Numerical Simulation, 33, 160-173. doi:10.1016/j.cnsns.2015.09.009Dorini, F. A., & Cunha, M. C. C. (2008). Statistical moments of the random linear transport equation. Journal of Computational Physics, 227(19), 8541-8550. doi:10.1016/j.jcp.2008.06.002Hussein, A., & Selim, M. M. (2012). Solution of the stochastic radiative transfer equation with Rayleigh scattering using RVT technique. Applied Mathematics and Computation, 218(13), 7193-7203. doi:10.1016/j.amc.2011.12.088Hussein, A., & Selim, M. M. (2015). Solution of the stochastic generalized shallow-water wave equation using RVT technique. The European Physical Journal Plus, 130(12). doi:10.1140/epjp/i2015-15249-3Hussein, A., & Selim, M. M. (2013). A general analytical solution for the stochastic Milne problem using Karhunen–Loeve (K–L) expansion. Journal of Quantitative Spectroscopy and Radiative Transfer, 125, 84-92. doi:10.1016/j.jqsrt.2013.03.018Xu, Z., Tipireddy, R., & Lin, G. (2016). Analytical approximation and numerical studies of one-dimensional elliptic equation with random coefficients. Applied Mathematical Modelling, 40(9-10), 5542-5559. doi:10.1016/j.apm.2015.12.041Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., & Roselló, M.-D. (2017). Full solution of random autonomous first-order linear systems of difference equations. Application to construct random phase portrait for planar systems. Applied Mathematics Letters, 68, 150-156. doi:10.1016/j.aml.2016.12.015El-Tawil, M. A. (2005). The approximate solutions of some stochastic differential equations using transformations. Applied Mathematics and Computation, 164(1), 167-178. doi:10.1016/j.amc.2004.04.062Calatayud, J., Cortés, J.-C., & Jornet, M. (2018). The damped pendulum random differential equation: A comprehensive stochastic analysis via the computation of the probability density function. Physica A: Statistical Mechanics and its Applications, 512, 261-279. doi:10.1016/j.physa.2018.08.024Calatayud, J., Cortés, J. C., & Jornet, M. (2018). Uncertainty quantification for random parabolic equations with nonhomogeneous boundary conditions on a bounded domain via the approximation of the probability density function. Mathematical Methods in the Applied Sciences, 42(17), 5649-5667. doi:10.1002/mma.5333Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., & Roselló, M.-D. (2018). Solving second-order linear differential equations with random analytic coefficients about ordinary points: A full probabilistic solution by the first probability density function. Applied Mathematics and Computation, 331, 33-45. doi:10.1016/j.amc.2018.02.051Casabán, M.-C., Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., Roselló, M.-D., & Villanueva, R.-J. (2016). A comprehensive probabilistic solution of random SIS-type epidemiological models using the random variable transformation technique. Communications in Nonlinear Science and Numerical Simulation, 32, 199-210. doi:10.1016/j.cnsns.2015.08.009Kegan, B., & West, R. W. (2005). 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Transit times and mean ages for nonautonomous and autonomous compartmental systems

We develop a theory for transit times and mean ages for nonautonomous compartmental systems. Using the McKendrick-von F\"orster equation, we show that the mean ages of mass in a compartmental system satisfy a linear nonautonomous ordinary differential equation that is exponentially stable. We then define a nonautonomous version of transit time as the mean age of mass leaving the compartmental system at a particular time and show that our nonautonomous theory generalises the autonomous case. We apply these results to study a nine-dimensional nonautonomous compartmental system modeling the terrestrial carbon cycle, which is a modification of the Carnegie-Ames-Stanford approach (CASA) model, and we demonstrate that the nonautonomous versions of transit time and mean age differ significantly from the autonomous quantities when calculated for that model

A simple model of immune and muscle cell crosstalk during muscle regeneration

Muscle injury during aging predisposes skeletal muscles to increased damage due to reduced regenerative capacity. Some of the common causes of muscle injury are strains, while other causes are more complex muscle myopathies and other illnesses, and even excessive exercise can lead to muscle damage. We develop a new mathematical model based on ordinary differential equations of muscle regeneration. It includes the interactions between the immune system, healthy and damaged myonuclei as well as satellite cells. Our new mathematical model expands beyond previous ones by accounting for 21 specific parameters, including those parameters that deal with the interactions between the damaged and dead myonuclei, the immune system, and the satellite cells. An important assumption of our model is the replacement of only damaged parts of the muscle fibers and the dead myonuclei. We conduce systematic sensitivity analysis to determine which parameters have larger effects on the model and therefore are more influential for the muscle regeneration process. We propose additional validation for these parameters. We further demonstrate that these simulations are species-, muscle-, and age-dependent. In addition, the knowledge of these parameters and their interactions, may suggest targeting or selecting these interactions for treatments that accelerate the muscle regeneration process