Geodesic
Optimal Control in Coupled Within-host and Between-host Models
E. Numfor1 ; S. Bhattacharya2 ; S. Lenhart1 ; M. Martcheva2
1 Department of Mathematics, University of Tennessee, TN 37996, USA
2 Department of Mathematics, University of Florida, FL 32611, USA
Mathematical modelling of natural phenomena, Tome 9 (2014) no. 4, pp. 171-203.

Voir la notice de l'article provenant de la source EDP Sciences

We formulate an immuno-epidemiological model of coupled “within-host” model of ODEs and “between-host” model of ODE and PDE, using the Human Immunodeficiency Virus (HIV) for illustration. Existence and uniqueness of solution to the “between-host” model is established, and an explicit expression for the basic reproduction number of the “between-host” model derived. Stability of disease-free and endemic equilibria is investigated. An optimal control problem with drug-treatment control on the within-host system is formulated and analyzed; these results are novel for optimal control of ODEs linked with such first order PDEs. Numerical simulations based on the forward-backward sweep method are obtained.
DOI : 10.1051/mmnp/20149411

E. Numfor 1 ; S. Bhattacharya 2 ; S. Lenhart 1 ; M. Martcheva 2

1 Department of Mathematics, University of Tennessee, TN 37996, USA
2 Department of Mathematics, University of Florida, FL 32611, USA 
@article{MMNP_2014_9_4_a10,
     author = {E. Numfor and S. Bhattacharya and S. Lenhart and M. Martcheva},
     title = {Optimal {Control} in {Coupled} {Within-host} and {Between-host} {Models}},
     journal = {Mathematical modelling of natural phenomena},
     pages = {171--203},
     publisher = {mathdoc},
     volume = {9},
     number = {4},
     year = {2014},
     doi = {10.1051/mmnp/20149411},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20149411/}
}
TY  - JOUR
AU  - E. Numfor
AU  - S. Bhattacharya
AU  - S. Lenhart
AU  - M. Martcheva
TI  - Optimal Control in Coupled Within-host and Between-host Models
JO  - Mathematical modelling of natural phenomena
PY  - 2014
SP  - 171
EP  - 203
VL  - 9
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20149411/
DO  - 10.1051/mmnp/20149411
LA  - en
ID  - MMNP_2014_9_4_a10
ER  - 
%0 Journal Article
%A E. Numfor
%A S. Bhattacharya
%A S. Lenhart
%A M. Martcheva
%T Optimal Control in Coupled Within-host and Between-host Models
%J Mathematical modelling of natural phenomena
%D 2014
%P 171-203
%V 9
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20149411/
%R 10.1051/mmnp/20149411
%G en
%F MMNP_2014_9_4_a10
E. Numfor; S. Bhattacharya; S. Lenhart; M. Martcheva. Optimal Control in Coupled Within-host and Between-host Models. Mathematical modelling of natural phenomena, Tome 9 (2014) no. 4, pp. 171-203. doi : 10.1051/mmnp/20149411. http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20149411/

[1] R. M. Anderson, R. M. May Phil. Trans. R. Soc. Lond. 1986 533 570

[2] S. Anita, V. Arnăutu, V. Capasso. An Introduction to Optimal Control Problems in Life Sciences. Springer Science, New York, 2011.

[3] S. Anita. Analysis and Control of Age-Dependent Population Dynamics. Kluwer Academic Publishers, Dordretcht, 2000.

[4] S. Anita J. Math. Anal. Appl. 1998 6 22

[5] R. Antia, B. Levin, R. M. May Am. Nat. 1994 457 472

[6] V. Barbu, M. Iannelli J. Optim. Theory Appl. 1999 1 14

[7] V. Barbu. Mathematical Methods in Optimization of Differential Systems. Kluwer Academic Publishers, Dordretcht, 1994.

[8] M. Brokate J. Math. Biol. 1985 75 101

[9] S. Butler, D. Kirschner, S. Lenhart. Optimal Control of Chemotherapy Affecting the Infectivity Of HIV, Advances In Mathematical Population Dynamics-Molecules, Cells And Man. Editors: O. Arino, D. Axelrod, And M. Kimmel, World Scientific Press, Singapore, (1997) 557–569.

[10] C. Castillo-Chavez, Z. Feng Math.Biosci. 1998 135 154

[11] D. Coombs, M. A. Gilchrist, J. Percus, A. S. Perelson Bull. Math. Biol. 2003 1003 1023

[12] U. Dissanayake, S. Dias, H. Polson, S. Longacre, P. Udagama-Randeniya Bio. Sci. 2008 97 105

[13] I. Ekeland J. Math. Appl. 1974 324 353

[14] L. C. Evans. Partial Differential Equations. American Mathematical Society, 2nd edition, 2010.

[15] L. C. Evans, R. F. Gariepy. Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton, 1992.

[16] Z. Feng, J. Velasco-Hernandez, B. Tapia-Santos Math. Biosci. 2013 49 55

[17] Z. Feng, J. Velasco-Hernandez, B. Tapia-Santos, M.C. Leite Nonlinear Dyn. 2012 401 411

[18] K.R. Fister, S. Lenhart, J. S. Mcnally J. of Differential Equations 1998 1 12

[19] K.R. Fister, S. Lenhart Appl. Math. Optim. 2006 1 15

[20] K.R. Fister, S. Lenhart J. Math. Anal. 2004 526 537

[21] V.V. Ganusov, C.T. Bergstrom, R. Antia Evolution 2002 213 223

[22] M. A. Gilchrist, D.A. Coombs Theor. Pop. Biol. 2006 145 153

[23] M. A. Gilchrist, A. Sasaki J. Theor. Biol. 2002 289 308

[24] A. B. Gumel, P. N. Shivakumar, B. M. Sahai Nonlinear. Anal. 2001 1773 1783

[25] M. E. Gurtin, R. C. Maccamy Arch. Rat. Mech. Anal. 1974 281 300

[26] K. Hattaf, N. Yousfi World J. Modell. Simul. 2012 27 35

[27] J. M. Heffernan Math. Popul. Stud. 2011 47 54

[28] J. M. Heffernan, R. J. Smith, L. M. Wahl J. R. Soc. Interface 2005 281 293

[29] B. Hellriegel TRENDS in Parasitology 2001 102 106

[30] Y. Hsieh, S. Sheu J. Math. Biol. 2001 69 80

[31] M. Iannelli, M. Martcheva, F. A. Milner. Gender-Structured Population Modeling: Mathematical Methods. Numerics, and Simulations, SIAM, Philadelphia, 2005.

[32] H. Joshi Optimal Control Appl. Methods 2002 199 213

[33] H. Joshi, S. Lenhart, K. Albright, K. Gipson Biosci. Eng. 2008 757 770

[34] H. Joshi, S. Lenhart, M. Y. Li, L. Wang Contemp. Math. 2006 187 207

[35] M. Kgosimore, E. M. Lungu Math. Biosci. Eng. 2006 297 312

[36] D. Kirschner AMS Notices 1996 191 202

[37] D. Kirschner, G. F. Webb Bull. Math.Biol. 1996 367 391

[38] D. Kirschner, S. Lenhart, S. Serbin J. Math. Biol. 1997 775 792

[39] S. Lenhart, J. T. Wortman. Optimal Control Applied to Biological Models. Taylor Francis, Boca Raton, FL, 2007.

[40] X. Li, J. Liu, M. Martcheva Math. Biosci. Eng. 2010 123 147

[41] M. Martcheva AIP Conf. Proc. 2011 176 183

[42] M. Martcheva, F. A. Milner Math. Population Studies 1999 111 129

[43] L. F. Murphy, S. J. Smith J. Math. Biol. 1990 77 90

[44] A. Perelson, D. E. Kirschner, R. D. Boer Math. Biosci. 1993 81 125

[45] Z. Qiu, X. Li, M. Martcheva J. Math. Anal. Appl. 2012 395 612

[46] E. Shim, Z. Feng, M. Martcheva Math. Biol. 2006 719 746

[47] R. J. Smith J. Theor Biol. 2008 227 237

[48] G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, Inc, New York, 1985.

[49] E. K. Yeargers, R. W. Shonkwiler, J. V. Herod, An Introduction to the Mathematics of Biology with Computer Algebra Models, Birkhăuser, Boston, 1996.

Cité par Sources :