Unconditionally positive NSFD and classical finite difference schemes for biofilm formation on medical implant using Allen-Cahn equation

• Autorzy: Tijani Yusuf O. , Appadu Appanah Rao

Identyfikatory

DOI

10.1515/dema-2022-0006

• Języki publikacji: EN

Abstrakty

EN

The study of biofilm formation is becoming increasingly important. Microbes that produce biofilms have complicated impact on medical implants. In this paper, we construct an unconditionally positive non-standard finite difference scheme for a mathematical model of biofilm formation on a medical implant. The unknowns in many applications reflect values that cannot be negative, such as chemical component concentrations or population numbers. The model employed here uses the bistable Allen-Cahn partial differential equation, which is a generalization of Fisher’s equation. We study consistency and convergence of the scheme constructed. We compare the performance of our scheme with a classical finite difference scheme using four numerical experiments. The technique used in the construction of unconditionally positive method in this study can be applied to other areas of mathematical biology and sciences. The results here elaborate the benefits of the non-standard approximations over the classical approximations in practical applications.

Słowa kluczowe

EN

biofilm; finite difference method; advection-diffusion-reaction; positivity; boundedness

PL

biofilm; metoda różnic skończonych; adwekcja-dyfuzja; ograniczoność

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2022
• Tom: Vol. 55, nr 1
• Strony: 40--60
• Opis fizyczny: Bibliogr. 31 poz., tab., wykr.

Twórcy

autor

Tijani Yusuf O.

• Department of Mathematics and Applied Mathematics, Nelson Mandela University, Gqeberha, 6031, South Africa

autor

Appadu Appanah Rao

• Department of Mathematics and Applied Mathematics, Nelson Mandela University, Gqeberha, 6031, South Africa

Bibliografia

• [1] J. Perez-Velazquez, M. Golgeli, and R. Garcia-Contreras, Mathematical modelling of bacterial quorum sensing: a review, Bull. Math. Bio. 78 (2016), 1585–1639, DOI: https://doi.org/10.1007/s11538-016-0160-6.
• [2] B. E. Rittmann and P. L. McCarty, Model of steady-state-biofilm kinetics, Biotechnol. Bioeng. 22 (1980), no. 11, 2343–2357, DOI: https://doi.org/10.1002/bit.260221110.
• [3] O. Wanner and W. Gujer, A multispecies biofilm model, Biotechnol. Bioeng. 28 (1986), no. 3, 314–328, DOI: https://doi.org/10.1002/bit.260280304.
• [4] P. Nilsson, A. Olofsson, M. Fagerlind, T. Fagerström, S. Rice, S. Kjelleberg, et al., Kinetics of the AHL regulatory system in a model biofilm system: how many bacteria constitute a “quorum”?, J. Mol. Biol. 309 (2001), no. 3, 631–640, DOI: https://doi.org/10.1006/jmbi.2001.4697.
• [5] E. Alpkvist and I. Klapper, A multidimensional multispecies continuum model for heterogeneous biofilm development, Bull. Math. Biol. 69 (2007), no. 2, 765–789, DOI: https://doi.org/10.1007/s11538-006-9168-7.
• [6] H. J. Eberl, D. F. Parker, and M. C. M. Van Loosdrecht, A new deterministic spatio-temporal continuum model for biofilm development, Theor. 3 (2001), no. 3, 161–175, http://eudml.org/doc/227116.
• [7] M. D. Moralez-Hernandez, I. E. Medina-Ramirez, F. J. Avelar-Gonzalez, and J. E. Macias-DiazAn, An efficient recursive algorithm in the computational simulation of the bounded growth of biological films, Int. J. Comput. Methods 9 (2012), no. 4, 1250050-01-15, DOI: https://doi.org/10.1142/S0219876212500508.
• [8] J. E. Macias-Diaz, S. Macias, and I. E. Medina-Ramirez, An efficient nonlinear finite-difference approach in the computational modeling of the dynamics of a nonlinear diffusion-reaction equation in microbial ecology, Comput. Biol. Chem. 47(2013), 24–30, DOI: https://doi.org/10.1016/j.compbiolchem.2013.05.003.
• [9] G. F. Sun, G. R. Liu, and M. Li, A novel explicit positivity-preserving finite-difference scheme for simulating bounded growth of biological films, Int. J. Comput. Methods 13 (2016), no. 2, 1640013, DOI: https://doi.org/10.1142/S0219876216400132.
• [10] G. F. Sun, G. R. Liu, and M. Li, An efficient explicit finite-difference scheme for simulating coupled biomass growth on nutritive substrates, Math. Probl. Eng. 17 (2014), 708497, DOI: https://doi.org/10.1155/2015/708497.
• [11] E. Balsa-Canto, A. Lopez-Nunez, and C. Vazquez, Numerical methods for a nonlinear reaction-diffusion system modeling a batch culture of biofilm, Appl. Math. Model. 41 (2017), 164–179, DOI: https://doi.org/10.1016/j.apm.2016.08.020.
• [12] M. A. Efendiev, H. J. Eberl, and S. V. Zelik, Existence and longtime behavior of solutions of a nonlinear reaction-diffusionsystem arising in the modeling of biofilms, RIMS Kokyuroko 1258 (2002), 49–71.
• [13] H. J. Eberl and L. Demaret, A finite difference scheme for a degenerated diffusion equation arising in microbial ecology, Electron. J. Differ. Equ. 15 (2007), 77–95, http://ejde.math.txstate.edu
• [14] A. Q. Cai, K. A. Landman, and B. D. Hughes, Multi-scale modeling of a wound-healing cell migration assay, J. Theor. Biol. 245 (2007), no. 3, 576–594, DOI: https://doi.org/10.1016/j.jtbi.2006.10.024.
• [15] B. M. Chen-Charpentier and D. Stanescu, Biofilm growth on medical implants with randomness, Math. Comput. Model. 54 (2011), no. 7–8, 1682–1686, DOI: https://doi.org/10.1016/j.mcm.2010.11.075.
• [16] B. M. Chen-Charpentier and H. V. Kojouharov, An unconditionally positivity preserving scheme for advection-diffusion reaction equations, Math. Comput. Model. 57 (2013), no. 9–10, 2177–2185, DOI: https://doi.org/10.1016/j.mcm.2011.05.005.
• [17] A. R. Appadu, Analysis of the unconditionally positive finite difference scheme for advection-diffusion-reaction equations with different regimes, AIP Confer. Proc. 1738 (2016), no. 1, 030005, DOI: https://doi.org/10.1063/1.4951761.
• [18] A. R. Appadu, Performance of UPFD scheme under some different regimes of advection, diffusion and reaction, Int. J. Numer. Method H. 27 (2017), no. 7, 1412–1429, DOI: https://doi.org/10.1108/HFF-01-2016-0038.
• [19] W. Qin, D. Ding, and X. Ding, Unconditionally positivity and boundedness preserving schemes for a FitzHugh-Nagumo equation, Int. J. Comput. Math. 92 (2014), no. 10, 2198–2218, DOI: https://doi.org/10.1080/00207160.2014.975696.
• [20] A. Md. Ali, H. J. Eberl, and R. Sudarsan, Numerical solution of a degenerate, diffusion-reaction based biofilm growth model on structured non-orthogonal grids, Commun. Comput. Phys. 24 (2018), no. 3, 695–741, DOI: https://doi.org/10.4208/cicp.OA-2017-0165.
• [21] M. Jornet, Modeling of Allee effect in biofilm formation via the stochastic bistable Allen-Cahn partial differential equation, Stoch. Anal. Appl. 39 (2021), no. 1, 22–32, DOI: https://doi.org/10.1080/07362994.2020.1777163.
• [22] P. F. Verhulst, Recherches mathématiques sur la loi daaccroissement de la population, Nouv. Mém. Acad. R. Sci. B.-lett. Brux. 18 (1845), 1–45.
• [23] B. Perthame, Parabolic equations in biology, Lecture Notes on Mathematical Modelling in the Life Sciences, Springer, Switzerland, 2015.
• [24] Y. O. Tijani, A. R. Appadu, and A. A. Aderogba, Some finite difference methods to model biofilm growth and decay: classical and non-standard, Computation 9 (2021), no. 11, 123, DOI: https://doi.org/10.3390/computation9110123.
• [25] R. Anguelov, J. M. S. Lubuma, and M. Shillor, Dynamically consistent nonstandard finite difference schemes for continuous dynamical systems, Discrete Contin. Dyn. Syst. 255 (2009), 34–43, DOI: https://doi.org/10.3934/proc.2009.2009.34.
• [26] T. R. Taha and M. J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations III. Numerical Korteweg-de Vries equation, J. Comput. Phys. 77 (1984), no. 2, 540–548, DOI: https://doi.org/10.1016/0021-9991(88)90184-2.
• [27] A. C. Hindmarsch, P. M. Gresho, and D. F. Griffiths, The stability of explicit Euler time-integration for certain finite difference approximations of the multi-dimensional advection-diffusion equation, Int. J. Numer. Methods Fluids 4 (1984), no. 9, 853–897, DOI: https://doi.org/10.1002/fld.1650040905.
• [28] R. E. Mickens, Application of Nonstandard Finite Difference Scheme, World Scientific, Singapore, 2000.
• [29] A. R. Appadu, B. İnan, and Y. O. Tijani, Comparative study of some numerical methods for the Burgers-Huxley equation, Symmetry 11 (2019), no. 11, 1333, DOI: https://doi.org/10.3390/sym11111333.
• [30] F. B. Hilderband, Finite-Difference Equations and Simulations, Prentice-Hall, Englewood Cliffs, NJ, USA, 1968.
• [31] B. M. Chen-Charpentier and H. V. Kojouharov, Non-standard numerical methods applied to subsurface biobarrier formation models in porous media, Bull. Math. Bio. 61 (1999), 779–798, DOI: https://doi.org/10.1006/bulm.1999.0113.

Uwagi

PL

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).