Search Advanced

Mathematical Biosciences and Engineering

2014, Volume 11, Issue 4: 679-721. doi: 10.3934/mbe.2014.11.679
Previous Article Next Article

A structured model for the spread of Mycobacterium marinum: Foundations for a numerical approximation scheme

  • 1.
    Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504-1010
  • 2.
    Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504-1010
  • 3.
    Department of Biology, University of Louisiana at Lafayette, Lafayette, LA 70504-2451
  • Received: 01 July 2013 Accepted: 29 June 2018 Published: 01 March 2014

  • MSC : Primary: 92-08; Secondary: 92B05.

  • We develop a finite difference scheme to approximate the solution of a novel size-structured mathematical model of the transmission dynamics of Mycobacterium marinum (Mm) in an aquatic environment. The model consists of a system of nonlinear hyperbolic partial differential equations coupled with three nonlinear ordinary differential equations. Existence and uniqueness results are established and convergence of the finite difference approximation to the unique bounded variation weak solution of the model is obtained. Numerical simulations demonstrating the accuracy of the method are presented. We also conducted preliminary studies on the key features of this model, such as various forms of growth rates (indicative of possible theories of development), and conditions for competitive exclusion or coexistence as determined by reproductive fitness and genetic spread in the population.

    Citation: Azmy S. Ackleh, Mark L. Delcambre, Karyn L. Sutton, Don G. Ennis. A structured model for the spread of Mycobacterium marinum: Foundations for a numerical approximation scheme[J]. Mathematical Biosciences and Engineering, 2014, 11(4): 679-721. doi: 10.3934/mbe.2014.11.679

    Related Papers:

    [1] Qihua Huang, Hao Wang . A toxin-mediated size-structured population model: Finite difference approximation and well-posedness. Mathematical Biosciences and Engineering, 2016, 13(4): 697-722. doi: 10.3934/mbe.2016015
    [2] Azmy S. Ackleh, Rainey Lyons, Nicolas Saintier . Finite difference schemes for a structured population model in the space of measures. Mathematical Biosciences and Engineering, 2020, 17(1): 747-775. doi: 10.3934/mbe.2020039
    [3] Suman Ganguli, David Gammack, Denise E. Kirschner . A Metapopulation Model Of Granuloma Formation In The Lung During Infection With Mycobacterium Tuberculosis. Mathematical Biosciences and Engineering, 2005, 2(3): 535-560. doi: 10.3934/mbe.2005.2.535
    [4] Azmy S. Ackleh, Vinodh K. Chellamuthu, Kazufumi Ito . Finite difference approximations for measure-valued solutions of a hierarchicallysize-structured population model. Mathematical Biosciences and Engineering, 2015, 12(2): 233-258. doi: 10.3934/mbe.2015.12.233
    [5] Azmy S. Ackleh, Rainey Lyons, Nicolas Saintier . High resolution finite difference schemes for a size structured coagulation-fragmentation model in the space of radon measures. Mathematical Biosciences and Engineering, 2023, 20(7): 11805-11820. doi: 10.3934/mbe.2023525
    [6] Simone De Reggi, Francesca Scarabel, Rossana Vermiglio . Approximating reproduction numbers: a general numerical method for age-structured models. Mathematical Biosciences and Engineering, 2024, 21(4): 5360-5393. doi: 10.3934/mbe.2024236
    [7] Arthur D. Lander, Qing Nie, Frederic Y. M. Wan . Spatially Distributed Morphogen Production and Morphogen Gradient Formation. Mathematical Biosciences and Engineering, 2005, 2(2): 239-262. doi: 10.3934/mbe.2005.2.239
    [8] Andrey V. Melnik, Andrei Korobeinikov . Lyapunov functions and global stability for SIR and SEIR models withage-dependent susceptibility. Mathematical Biosciences and Engineering, 2013, 10(2): 369-378. doi: 10.3934/mbe.2013.10.369
    [9] Saima Akter, Zhen Jin . A fractional order model of the COVID-19 outbreak in Bangladesh. Mathematical Biosciences and Engineering, 2023, 20(2): 2544-2565. doi: 10.3934/mbe.2023119
    [10] Inom Mirzaev, David M. Bortz . A numerical framework for computing steady states of structured population models and their stability. Mathematical Biosciences and Engineering, 2017, 14(4): 933-952. doi: 10.3934/mbe.2017049
  • We develop a finite difference scheme to approximate the solution of a novel size-structured mathematical model of the transmission dynamics of Mycobacterium marinum (Mm) in an aquatic environment. The model consists of a system of nonlinear hyperbolic partial differential equations coupled with three nonlinear ordinary differential equations. Existence and uniqueness results are established and convergence of the finite difference approximation to the unique bounded variation weak solution of the model is obtained. Numerical simulations demonstrating the accuracy of the method are presented. We also conducted preliminary studies on the key features of this model, such as various forms of growth rates (indicative of possible theories of development), and conditions for competitive exclusion or coexistence as determined by reproductive fitness and genetic spread in the population.


    [1] Mathematical and Computer Modelling, 50 (2009), 653-664.
    [2] Discrete and Continuous Dynamical Systems - Series B, 4 (2004), 1203-1222.
    [3] Journal of Biological Systems, 5 (1997), 1-16.
    [4] Mathematical Biosciences, 245 (2013), 2-11.
    [5] Applied Mathematics and Computation, 108 (2000), 103-113.
    [6] Mathematical Biosciences and Engineering, 204 (2006), 21-48.
    [7] Numerical Functional Analysis and Optimization, 18 (1997), 865-884.
    [8] Contemporary Mathematics, 513 (2010), 1-23.
    [9] Journal of Biological Systems, doi:10.1142/S0218339014500028.
    [10] Mathematical Biosciences and Engineering, 5 (2008), 601-616.
    [11] Mathematical Biosciences, 157 (1999), 169-188.
    [12] Applied Numerical Mathematics, 50 (2004), 291-327.
    [13] SIAM Journal of Numerical Analysis, 26 (1989), 1474-1486.
    [14] Mathematical Biosciences and Engineering, 1 (2004), 15-48.
    [15] International Series of Numerical Mathematics, 118 (1994), 1-19.
    [16] Current Opinion in Pulmonary Medicine, 6 (2000), 174-179.
    [17] Comparative Biochemistry and Physiology, Part C, 145 (2007), 45-54.
    [18] Comparative Biochemistry and Physiology, Part C, 149 (2009), 152-160.
    [19] Annual Review of Microbiology 57 (2003), 641-676.
    [20] Immunity, 17 (2002), 693-702.
    [21] Infection and Immunity, 69 (2001), 7310-7317.
    [22] Nature, 435 (2005), 1046.
    [23] Journal of Computational Physics, 49 (1983), 357-393.
    [24] Journal of Wildlife Diseases, 23 (1987), 391-395.
    [25] Journal of Mathematical Analysis and Applications, 181 (1994), 716-754.
    [26] Numerical Methods for Partial Differential Equations, 25 (2009), 918-930.
    [27] Zoological Science, 11 (1994), 825-839.
    [28] Journal of Fish Diseases, 32 (2009), 119-130.
    [29] Numerical Methods for Partial Differential Equations, 19 (2003), 1-21.
    [30] Microbial Pathogenesis, 40 (2006), 139-151.
    [31] Infection and Immunity, 73 (2005), 4214-4221.
    [32] International Journal for Numerical Methods in Engineering, 46 (1999), 131-150.
    [33] Ph.D. Dissertation, University of Louisiana at Lafayette, 2011.
    [34] Comparative Biochemistry and Physiology,Part C, 155 (2012), 39-48.
    [35] Mathematical Biosciences, 157 (1999), 169-188.
    [36] FEMS Microbiology Letters, 225 (2003), 177-182.
    [37] Journal of the American Medical Association, 40 (1996), 220-226.
    [38] SIAM Journal on Numerical Analysis, 45 (2007), 352-370.
    [39] Journal of Scientific Computing, 33 (2007), 279-291.
    [40] 2nd edition, Springer-Verlag, New York, 1994.
    [41] Genome Research, 18 (2008), 729-741.
    [42] Infection and Immunity, 66 (1998), 2938-2942.
    [43] Mathematical Biosciences, 225 (2010), 59-67.
    [44] Cellular Microbiology, 10 (2008), 1027-1039.
    [45] Springer, New York, 1998.
    [46] Journal of Computational and Applied Mathematics, 236 (2011), 937-949.

  • This article has been cited by:

    1. Azmy S. Ackleh, Mark L. Delcambre, Karyn L. Sutton, A second-order high-resolution finite difference scheme for a size-structured model for the spread ofMycobacterium marinum, 2015, 9, 1751-3758, 156, 10.1080/17513758.2014.962998
    2. Hayriye Gulbudak, Cameron J. Browne, Infection severity across scales in multi-strain immuno-epidemiological Dengue model structured by host antibody level, 2020, 80, 0303-6812, 1803, 10.1007/s00285-020-01480-3
    3. O. Angulo, J.C. López-Marcos, M.A. López-Marcos, Study on the efficiency in the numerical integration of size-structured population models: Error and computational cost, 2016, 291, 03770427, 391, 10.1016/j.cam.2015.03.022
    4. Azmy S. Ackleh, Baoling Ma, Tingting Tang, A high resolution finite difference method for a model of structured susceptible-infected populations coupled with the environment, 2017, 33, 0749159X, 1420, 10.1002/num.22139
  • Reader Comments
  • © 2014 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Mathematical Biosciences and Engineering
4.4

Metrics

Article views(3029) PDF downloads(495) Cited by(4)

Preview PDF
Download XML
Export Citation
Article outline
Abstract
References

Other Articles By Authors

Related pages

Tools

Copyright © AIMS Press

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog