2014, 11(4): 679-721. doi: 10.3934/mbe.2014.11.679

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, United States

2. 

Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504-1010, United States, United States

3. 

Department of Biology, University of Louisiana at Lafayette, Lafayette, LA 70504-2451, United States

Received  July 2013 Revised  December 2013 Published  March 2014

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. Mathematical Biosciences & Engineering, 2014, 11 (4) : 679-721. doi: 10.3934/mbe.2014.11.679
References:
[1]

L. M. Abia, O. Angulo, J. C. Lopez-Marcos and M. A. Lopez-Marcos, Numerical schemes for a size-structured cell population model with equal fission,, Mathematical and Computer Modelling, 50 (2009), 653. doi: 10.1016/j.mcm.2009.05.023.

[2]

L. M. Abia, O. Angulo and J. C. Lopez-Marcos, Size-structured population dynamics models and their numerical solutions,, Discrete and Continuous Dynamical Systems - Series B, 4 (2004), 1203. doi: 10.3934/dcdsb.2004.4.1203.

[3]

L. M. Abia and J. C. Lopez-Marcos, Second order schemes for age-structured population equations,, Journal of Biological Systems, 5 (1997), 1. doi: 10.1142/S0218339097000023.

[4]

A. S. Ackleh, B. Ma and J. J. Thibodeaux, A second-order high resolution finite difference scheme for a structured erythropoiesis model subject to malaria infection,, Mathematical Biosciences, 245 (2013), 2. doi: 10.1016/j.mbs.2013.03.007.

[5]

A. S. Ackleh and K. Deng, A monotone approximation for a nonlinear non autonomous size-structured population model,, Applied Mathematics and Computation, 108 (2000), 103. doi: 10.1016/S0096-3003(99)00002-8.

[6]

A. S. Ackleh, K. Deng, K. Ito and J. Thibodeaux, A structured erythropoiesis model with nonlinear cell maturation velocity and hormone decay rate,, Mathematical Biosciences and Engineering, 204 (2006), 21. doi: 10.1016/j.mbs.2006.08.004.

[7]

A. S. Ackleh and K. Ito, An implicit finite difference scheme for the nonlinear size-structured population model,, Numerical Functional Analysis and Optimization, 18 (1997), 865. doi: 10.1080/01630569708816798.

[8]

A. S. Ackleh, K. Deng and Q. Huang, Existence-uniqueness results and difference approximations for an amphibian juvenile-adult model,, Contemporary Mathematics, 513 (2010), 1. doi: 10.1090/conm/513/10072.

[9]

A. S. Ackleh, K. L. Sutton, K. N. Mutoji, A. Mallick and D. G. Ennis, A structured model for the transmission dynamics of Mycobacterium marinum between aquatic animals,, Journal of Biological Systems, ().

[10]

A. S. Ackleh and J. Thibodeaux, Parameter estimation in a structured erythropoiesis model,, Mathematical Biosciences and Engineering, 5 (2008), 601. doi: 10.3934/mbe.2008.5.601.

[11]

O. Angulo and J. C. Lopez-Marcos, Numerical schemes for size-structured population equations,, Mathematical Biosciences, 157 (1999), 169. doi: 10.1016/S0025-5564(98)10081-0.

[12]

O. Angulo and J. C. Lopez-Marcos, Numerical integration of fully nonlinear size-structured population models,, Applied Numerical Mathematics, 50 (2004), 291. doi: 10.1016/j.apnum.2004.01.007.

[13]

T. Arbogast and F. A. Milner, A finite element method for a two-sex model of population dynamics,, SIAM Journal of Numerical Analysis, 26 (1989), 1474. doi: 10.1137/0726086.

[14]

H. T. Banks, C. E. Cole, P. M. Schlosser and H. T. Tran, Modeling and optimal regulation of erythropoiesis subject to benzene intoxication,, Mathematical Biosciences and Engineering, 1 (2004), 15. doi: 10.3934/mbe.2004.1.15.

[15]

H. T. Banks, F. Kappel and C. Wang, A semigroup formulation of a nonlinear size-structured distributed rate population model,, International Series of Numerical Mathematics, 118 (1994), 1.

[16]

D. Bleed, C. Dye and M. C. Raviglione, Dynamics and control of the global tuberculosis epidemic,, Current Opinion in Pulmonary Medicine, 6 (2000), 174. doi: 10.1097/00063198-200005000-00002.

[17]

G. W. Broussard and D. G. Ennis, Mycobacterium marinum produces long-term chronic infections in medaka: A new animal model for studying human tuberculosis,, Comparative Biochemistry and Physiology, 145 (2007), 45. doi: 10.1016/j.cbpc.2006.07.012.

[18]

G. W. Broussard, M. B. Norris, R. N. Winn, J. Fournie, A. Schwindt, M. L. Kent and D. G. Ennis, Chronic mycobacterosis acts as a tumor promoter for hepatocarcinomas in Japanese medaka,, Comparative Biochemistry and Physiology, 149 (2009), 152.

[19]

C. L. Cosma, D. R. Sherman and L. Ramakrishnan, The secret lives of the pathogenic mycobacteria,, Annual Review of Microbiology 57 (2003), 57 (2003), 641.

[20]

J. M. Davis, H. Clay, J. L. Lewis, N. Ghori, P. Herbomel and L. Ramakrishnan, Real-time visualization of Mycobacterium-macrophage interactions leading of initiation of granuloma formation in zebrafish embryos,, Immunity, 17 (2002), 693. doi: 10.1016/S1074-7613(02)00475-2.

[21]

S. H. El-Etr, L. Yan and J. D. Cirillo, Fish monocytes as a model for mycobacterial host-pathogen interactions,, Infection and Immunity, 69 (2001), 7310. doi: 10.1128/IAI.69.12.7310-7317.2001.

[22]

R. E. Gozlan, S. St-Hilaire, S. W. Feist, P. Martin and M. L. Kent, Disease threat to European fish,, Nature, 435 (2005). doi: 10.1038/4351046a.

[23]

A. Harten, High resolution schemes for hyperbolic conservation laws,, Journal of Computational Physics, 49 (1983), 357. doi: 10.1016/0021-9991(83)90136-5.

[24]

R. P. Hedrick, T. McDowell and J. Groff, Mycobacteriosis in cultured striped bass from California,, Journal of Wildlife Diseases, 23 (1987), 391. doi: 10.7589/0090-3558-23.3.391.

[25]

W. Huyer, A size structured population model with dispersion,, Journal of Mathematical Analysis and Applications, 181 (1994), 716. doi: 10.1006/jmaa.1994.1054.

[26]

M. Iannelli, T. Kostova and F. A. Milner, A fourth-order method for numerical integration of age- and size-structured population models,, Numerical Methods for Partial Differential Equations, 25 (2009), 918. doi: 10.1002/num.20381.

[27]

T. Iwamatsu, Stages of normal development in the medaka oryzias latipes,, Zoological Science, 11 (1994), 825.

[28]

J. M. Jacobs, C. B. Stine, A. M. Baya and M. L. Kent, A review of mycobacteriosis in marine fish,, Journal of Fish Diseases, 32 (2009), 119. doi: 10.1111/j.1365-2761.2008.01016.x.

[29]

T. Kostova, An explicit third-order numerical method for size-structured population equations,, Numerical Methods for Partial Differential Equations, 19 (2003), 1. doi: 10.1002/num.10037.

[30]

P. K. Mehta, A. K. Pandey, S. Subbian, S. H. El-Etr, S. L. Cirillo, M. M. Samrakandi and J. D. Cirillo, Identification of Mycobacterium marinum macrophage infection mutants,, Microbial Pathogenesis, 40 (2006), 139. doi: 10.1016/j.micpath.2005.12.002.

[31]

E. Miltner, K. Daroogheh, P. K. Mehta, S. L. Cirillo, J. D. Cirillo and L. E. Bermudez, Identification of Mycobacterium avium genes that affect invasion of the intestinal epithelium,, Infection and Immunity, 73 (2005), 4214. doi: 10.1128/IAI.73.7.4214-4221.2005.

[32]

N. Moes, J. Dolbow and T. Belytschko, A finite element method for crack growth without remeshing,, International Journal for Numerical Methods in Engineering, 46 (1999), 131. doi: 10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.3.CO;2-A.

[33]

K. Nadine Mutoji, Investigation into Mechanisms of Mycobacterial Transmission Between Fish,, Ph.D. Dissertation, (2011).

[34]

K. N. Mutoji and D. G. Ennis, Expression of common fluorescent reporters may modulate virulence for Mycobacterium marinum: Dramatic attenuation results from GFP over-expression,, Comparative Biochemistry and Physiology, 155 (2012), 39. doi: 10.1016/j.cbpc.2011.05.011.

[35]

A. Oscar and J. C. Lopez-Marcos, Numerical schemes for size-structured population equations,, Mathematical Biosciences, 157 (1999), 169. doi: 10.1016/S0025-5564(98)10081-0.

[36]

M. G. Prouty, N. E. Correa, L. P. Barker, P. Jagadeeswaran and K. E. Klose, Zebrafish-Mycobacterium marinum model for mycobacterial pathogenesis,, FEMS Microbiology Letters, 225 (2003), 177.

[37]

M. C. Raviglione, D. E. Snider Jr and A. Kochi, Global epidemiology of tuberculosis: Morbidity and mortality of a worldwide epidemic,, Journal of the American Medical Association, 40 (1996), 220. doi: 10.1097/00132586-199604000-00069.

[38]

J. Shen, C. W. Shu and M. Zhang, High resolution schemes for a hierarchical size structured model,, SIAM Journal on Numerical Analysis, 45 (2007), 352. doi: 10.1137/050638126.

[39]

J. Shen, C. W. Shu and M. Zhang, A high order WENO scheme for a hierarchical size-structured population model,, Journal of Scientific Computing, 33 (2007), 279. doi: 10.1007/s10915-007-9152-x.

[40]

J. Smoller, Shock Waves and Reaction-Diffusion Equations,, 2nd edition, (1994).

[41]

T. P. Stinear, T. Seemann, P. F. Harrison, G. A. Jenkin, J. K. Davies, P. D. R. Johnson, Z. Abdellah, C. Arrowsmith, T. Chillingworth, C. Churcher, K. Clarke, A. Cronin, P. Davis, I. Goodhead, N. Holroyd, K. Jagels, A. Lord, S. Moule, K. Mungall, H. Norbertczak, M. A. Quail, E. Rabbinowitsch, D. Walker, B. White, S. Whitehead, P. L. C. Small, R. Brosch, L. Ramakrishnan, M. A. Fischbach, J. Parkhill and S. T. Cole, Insights from the complete genome sequence of Mycobacterium marinum on the evolution of Mycobacterium tuberculosis,, Genome Research, 18 (2008), 729. doi: 10.1101/gr.075069.107.

[42]

A. M. Talaat, R. Reimschuessel, S. S. Wasserman and M. Trucksis, Goldfish, Carassius auratus, a novel animal model for the study of Mycobacterium marinum pathogenesis,, Infection and Immunity, 66 (1998), 2938.

[43]

J. J. Thibodeaux, Modeling erythropoiesis subject to malaria infection,, Mathematical Biosciences, 225 (2010), 59. doi: 10.1016/j.mbs.2010.02.001.

[44]

D. M. Tobin and L. Ramakrishnan, Comparative pathogenesis of Mycobacterium marinum and Mycobacterium tuberculosis,, Cellular Microbiology, 10 (2008), 1027. doi: 10.1111/j.1462-5822.2008.01133.x.

[45]

W. Walter, Ordinary Differential Equations,, Springer, (1998). doi: 10.1007/978-1-4612-0601-9.

[46]

R. Zhang, M. Zhang and C. W. Shu, High order positivity-preserving finite volume WENO schemes for a hierarchical size-structured population model,, Journal of Computational and Applied Mathematics, 236 (2011), 937. doi: 10.1016/j.cam.2011.05.007.

show all references

References:
[1]

L. M. Abia, O. Angulo, J. C. Lopez-Marcos and M. A. Lopez-Marcos, Numerical schemes for a size-structured cell population model with equal fission,, Mathematical and Computer Modelling, 50 (2009), 653. doi: 10.1016/j.mcm.2009.05.023.

[2]

L. M. Abia, O. Angulo and J. C. Lopez-Marcos, Size-structured population dynamics models and their numerical solutions,, Discrete and Continuous Dynamical Systems - Series B, 4 (2004), 1203. doi: 10.3934/dcdsb.2004.4.1203.

[3]

L. M. Abia and J. C. Lopez-Marcos, Second order schemes for age-structured population equations,, Journal of Biological Systems, 5 (1997), 1. doi: 10.1142/S0218339097000023.

[4]

A. S. Ackleh, B. Ma and J. J. Thibodeaux, A second-order high resolution finite difference scheme for a structured erythropoiesis model subject to malaria infection,, Mathematical Biosciences, 245 (2013), 2. doi: 10.1016/j.mbs.2013.03.007.

[5]

A. S. Ackleh and K. Deng, A monotone approximation for a nonlinear non autonomous size-structured population model,, Applied Mathematics and Computation, 108 (2000), 103. doi: 10.1016/S0096-3003(99)00002-8.

[6]

A. S. Ackleh, K. Deng, K. Ito and J. Thibodeaux, A structured erythropoiesis model with nonlinear cell maturation velocity and hormone decay rate,, Mathematical Biosciences and Engineering, 204 (2006), 21. doi: 10.1016/j.mbs.2006.08.004.

[7]

A. S. Ackleh and K. Ito, An implicit finite difference scheme for the nonlinear size-structured population model,, Numerical Functional Analysis and Optimization, 18 (1997), 865. doi: 10.1080/01630569708816798.

[8]

A. S. Ackleh, K. Deng and Q. Huang, Existence-uniqueness results and difference approximations for an amphibian juvenile-adult model,, Contemporary Mathematics, 513 (2010), 1. doi: 10.1090/conm/513/10072.

[9]

A. S. Ackleh, K. L. Sutton, K. N. Mutoji, A. Mallick and D. G. Ennis, A structured model for the transmission dynamics of Mycobacterium marinum between aquatic animals,, Journal of Biological Systems, ().

[10]

A. S. Ackleh and J. Thibodeaux, Parameter estimation in a structured erythropoiesis model,, Mathematical Biosciences and Engineering, 5 (2008), 601. doi: 10.3934/mbe.2008.5.601.

[11]

O. Angulo and J. C. Lopez-Marcos, Numerical schemes for size-structured population equations,, Mathematical Biosciences, 157 (1999), 169. doi: 10.1016/S0025-5564(98)10081-0.

[12]

O. Angulo and J. C. Lopez-Marcos, Numerical integration of fully nonlinear size-structured population models,, Applied Numerical Mathematics, 50 (2004), 291. doi: 10.1016/j.apnum.2004.01.007.

[13]

T. Arbogast and F. A. Milner, A finite element method for a two-sex model of population dynamics,, SIAM Journal of Numerical Analysis, 26 (1989), 1474. doi: 10.1137/0726086.

[14]

H. T. Banks, C. E. Cole, P. M. Schlosser and H. T. Tran, Modeling and optimal regulation of erythropoiesis subject to benzene intoxication,, Mathematical Biosciences and Engineering, 1 (2004), 15. doi: 10.3934/mbe.2004.1.15.

[15]

H. T. Banks, F. Kappel and C. Wang, A semigroup formulation of a nonlinear size-structured distributed rate population model,, International Series of Numerical Mathematics, 118 (1994), 1.

[16]

D. Bleed, C. Dye and M. C. Raviglione, Dynamics and control of the global tuberculosis epidemic,, Current Opinion in Pulmonary Medicine, 6 (2000), 174. doi: 10.1097/00063198-200005000-00002.

[17]

G. W. Broussard and D. G. Ennis, Mycobacterium marinum produces long-term chronic infections in medaka: A new animal model for studying human tuberculosis,, Comparative Biochemistry and Physiology, 145 (2007), 45. doi: 10.1016/j.cbpc.2006.07.012.

[18]

G. W. Broussard, M. B. Norris, R. N. Winn, J. Fournie, A. Schwindt, M. L. Kent and D. G. Ennis, Chronic mycobacterosis acts as a tumor promoter for hepatocarcinomas in Japanese medaka,, Comparative Biochemistry and Physiology, 149 (2009), 152.

[19]

C. L. Cosma, D. R. Sherman and L. Ramakrishnan, The secret lives of the pathogenic mycobacteria,, Annual Review of Microbiology 57 (2003), 57 (2003), 641.

[20]

J. M. Davis, H. Clay, J. L. Lewis, N. Ghori, P. Herbomel and L. Ramakrishnan, Real-time visualization of Mycobacterium-macrophage interactions leading of initiation of granuloma formation in zebrafish embryos,, Immunity, 17 (2002), 693. doi: 10.1016/S1074-7613(02)00475-2.

[21]

S. H. El-Etr, L. Yan and J. D. Cirillo, Fish monocytes as a model for mycobacterial host-pathogen interactions,, Infection and Immunity, 69 (2001), 7310. doi: 10.1128/IAI.69.12.7310-7317.2001.

[22]

R. E. Gozlan, S. St-Hilaire, S. W. Feist, P. Martin and M. L. Kent, Disease threat to European fish,, Nature, 435 (2005). doi: 10.1038/4351046a.

[23]

A. Harten, High resolution schemes for hyperbolic conservation laws,, Journal of Computational Physics, 49 (1983), 357. doi: 10.1016/0021-9991(83)90136-5.

[24]

R. P. Hedrick, T. McDowell and J. Groff, Mycobacteriosis in cultured striped bass from California,, Journal of Wildlife Diseases, 23 (1987), 391. doi: 10.7589/0090-3558-23.3.391.

[25]

W. Huyer, A size structured population model with dispersion,, Journal of Mathematical Analysis and Applications, 181 (1994), 716. doi: 10.1006/jmaa.1994.1054.

[26]

M. Iannelli, T. Kostova and F. A. Milner, A fourth-order method for numerical integration of age- and size-structured population models,, Numerical Methods for Partial Differential Equations, 25 (2009), 918. doi: 10.1002/num.20381.

[27]

T. Iwamatsu, Stages of normal development in the medaka oryzias latipes,, Zoological Science, 11 (1994), 825.

[28]

J. M. Jacobs, C. B. Stine, A. M. Baya and M. L. Kent, A review of mycobacteriosis in marine fish,, Journal of Fish Diseases, 32 (2009), 119. doi: 10.1111/j.1365-2761.2008.01016.x.

[29]

T. Kostova, An explicit third-order numerical method for size-structured population equations,, Numerical Methods for Partial Differential Equations, 19 (2003), 1. doi: 10.1002/num.10037.

[30]

P. K. Mehta, A. K. Pandey, S. Subbian, S. H. El-Etr, S. L. Cirillo, M. M. Samrakandi and J. D. Cirillo, Identification of Mycobacterium marinum macrophage infection mutants,, Microbial Pathogenesis, 40 (2006), 139. doi: 10.1016/j.micpath.2005.12.002.

[31]

E. Miltner, K. Daroogheh, P. K. Mehta, S. L. Cirillo, J. D. Cirillo and L. E. Bermudez, Identification of Mycobacterium avium genes that affect invasion of the intestinal epithelium,, Infection and Immunity, 73 (2005), 4214. doi: 10.1128/IAI.73.7.4214-4221.2005.

[32]

N. Moes, J. Dolbow and T. Belytschko, A finite element method for crack growth without remeshing,, International Journal for Numerical Methods in Engineering, 46 (1999), 131. doi: 10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.3.CO;2-A.

[33]

K. Nadine Mutoji, Investigation into Mechanisms of Mycobacterial Transmission Between Fish,, Ph.D. Dissertation, (2011).

[34]

K. N. Mutoji and D. G. Ennis, Expression of common fluorescent reporters may modulate virulence for Mycobacterium marinum: Dramatic attenuation results from GFP over-expression,, Comparative Biochemistry and Physiology, 155 (2012), 39. doi: 10.1016/j.cbpc.2011.05.011.

[35]

A. Oscar and J. C. Lopez-Marcos, Numerical schemes for size-structured population equations,, Mathematical Biosciences, 157 (1999), 169. doi: 10.1016/S0025-5564(98)10081-0.

[36]

M. G. Prouty, N. E. Correa, L. P. Barker, P. Jagadeeswaran and K. E. Klose, Zebrafish-Mycobacterium marinum model for mycobacterial pathogenesis,, FEMS Microbiology Letters, 225 (2003), 177.

[37]

M. C. Raviglione, D. E. Snider Jr and A. Kochi, Global epidemiology of tuberculosis: Morbidity and mortality of a worldwide epidemic,, Journal of the American Medical Association, 40 (1996), 220. doi: 10.1097/00132586-199604000-00069.

[38]

J. Shen, C. W. Shu and M. Zhang, High resolution schemes for a hierarchical size structured model,, SIAM Journal on Numerical Analysis, 45 (2007), 352. doi: 10.1137/050638126.

[39]

J. Shen, C. W. Shu and M. Zhang, A high order WENO scheme for a hierarchical size-structured population model,, Journal of Scientific Computing, 33 (2007), 279. doi: 10.1007/s10915-007-9152-x.

[40]

J. Smoller, Shock Waves and Reaction-Diffusion Equations,, 2nd edition, (1994).

[41]

T. P. Stinear, T. Seemann, P. F. Harrison, G. A. Jenkin, J. K. Davies, P. D. R. Johnson, Z. Abdellah, C. Arrowsmith, T. Chillingworth, C. Churcher, K. Clarke, A. Cronin, P. Davis, I. Goodhead, N. Holroyd, K. Jagels, A. Lord, S. Moule, K. Mungall, H. Norbertczak, M. A. Quail, E. Rabbinowitsch, D. Walker, B. White, S. Whitehead, P. L. C. Small, R. Brosch, L. Ramakrishnan, M. A. Fischbach, J. Parkhill and S. T. Cole, Insights from the complete genome sequence of Mycobacterium marinum on the evolution of Mycobacterium tuberculosis,, Genome Research, 18 (2008), 729. doi: 10.1101/gr.075069.107.

[42]

A. M. Talaat, R. Reimschuessel, S. S. Wasserman and M. Trucksis, Goldfish, Carassius auratus, a novel animal model for the study of Mycobacterium marinum pathogenesis,, Infection and Immunity, 66 (1998), 2938.

[43]

J. J. Thibodeaux, Modeling erythropoiesis subject to malaria infection,, Mathematical Biosciences, 225 (2010), 59. doi: 10.1016/j.mbs.2010.02.001.

[44]

D. M. Tobin and L. Ramakrishnan, Comparative pathogenesis of Mycobacterium marinum and Mycobacterium tuberculosis,, Cellular Microbiology, 10 (2008), 1027. doi: 10.1111/j.1462-5822.2008.01133.x.

[45]

W. Walter, Ordinary Differential Equations,, Springer, (1998). doi: 10.1007/978-1-4612-0601-9.

[46]

R. Zhang, M. Zhang and C. W. Shu, High order positivity-preserving finite volume WENO schemes for a hierarchical size-structured population model,, Journal of Computational and Applied Mathematics, 236 (2011), 937. doi: 10.1016/j.cam.2011.05.007.

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