-
Previous Article
Qualitative analysis of a model for co-culture of bacteria and amoebae
- MBE Home
- This Issue
-
Next Article
Longtime behavior of one-dimensional biofilm models with shear dependent detachment rates
Delay equations modeling the effects of phase-specific drugs and immunotherapy on proliferating tumor cells
1. | Zentrum Mathematik, Technische Universität München, Boltzmannstr. 3, 85748 Garching b. München, Germany, Germany, Germany |
References:
[1] |
J. Al-Omari and S. Gourley, Dynamics of a stage-structured population model incorporating a state-dependent maturation delay, Nonlinear Analysis: Real World Applications 6 (2005), 13-33.
doi: 10.1016/j.nonrwa.2004.04.002. |
[2] |
B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts and P. Walter, "Molecular Biology of the Cell," 5th edition, Taylor & Francis Ltd., New York, 2007. |
[3] |
A. Bellen and M. Zennaro, "Numerical Methods for Delay Differential Equations," Clarendon Press, New York, 2003. |
[4] |
S. P. Blythe, R. M. Nisbet and W. S. C. Gurney, The dynamics of population models with distributed maturation periods, Theoretical Population Biology, 25 (1984), 289-311.
doi: 10.1016/0040-5809(84)90011-X. |
[5] |
G. Bocharov and K. P. Hadeler, Structured population models, conservation laws, and delay equations, Journal of Differential Equations, 168 (2000), 212-237.
doi: 10.1006/jdeq.2000.3885. |
[6] |
M. Chaplain and A. Matzavinos, Mathematical modeling of spatio-temporal phenomena in tumor immunology, in "Tutorials in Mathematical Biosciences. III," Lecuture Notes in Math., 1872, Berlin, (2006), 131-183. |
[7] |
K. L. Cooke and P. van den Driessche, On zeros of some transcendental equations, Funkcialaj Ekvacioj, 29 (1986), 77-90. |
[8] |
G. M. Cooper and R. E. Hausman, "The Cell: A Molecular Approach,'' ASM Press, Washington, 1997. |
[9] |
A. d'Onofrio, A general framework for modeling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedical inferences, Physica D, 208 (2005), 220-235.
doi: 10.1016/j.physd.2005.06.032. |
[10] |
A. d'Onofrio, Tumor-immune system interaction: Modelling the tumor-stimulated proliferation of effectors and immunotherapy, Mathematical Models and Methods in Applied Science, 16 (2006), 1375-1401.
doi: 10.1142/S0218202506001571. |
[11] |
A. d'Onofrio, F. Gatti, P. Cerrai and L. Freschi, Delay-induced oscillatory dynamics of tumour-immune system interaction, Mathematical and Computer Modelling, 51 (2010), 572-591. |
[12] |
J. Dyson, R. Villella-Bressan and G. Webb, Asynchronous exponential growth in an age structured population of proliferating and quiescent cells, Deterministic and Stochastic Modeling of Biointeraction (West Lafayette, IN, 2000), Mathematical Biosciences, 177/178 (2002), 73-83.
doi: 10.1016/S0025-5564(01)00097-9. |
[13] |
J. Dyson, R. Villella-Bressan and G. Webb, A spatial model of tumor growth with cell age, cell size, and mutation of cell phenotypes, Mathematical Modelling of Natural Phenomena, 2 (2007), 69-100.
doi: 10.1051/mmnp:2007004. |
[14] |
W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21.
doi: 10.1038/287017a0. |
[15] |
Y. Kuang, "Delay Differential Equations: With Applications in Population Dynamics,'' Academic Press, New York, 2003. |
[16] |
W. Liu, T. Hillen and H. Freedman, A mathematical model for $M$-phase specific chemotherapy including the $G_0$-phase and immunoresponse, Mathematical Bioscience and Engineering, 4 (2007), 239-259.
doi: 10.3934/mbe.2007.4.239. |
[17] |
H. Lodish et al., "Molecular Cell Biology,'' 3rd Ed. Scientific American Books , New York, 1995. |
[18] |
N. MacDonald, "Biological Delay Systems: Linear Stability Theory,'' Cambridge University Press, 1989. |
[19] |
R. Nisbet and W. Gurney, The systematic formulation of population models for insects with dynamically varying instar duration, Theoretical Population Biology, 23 (1983), 114-135.
doi: 10.1016/0040-5809(83)90008-4. |
[20] |
T. Roose, S. J. Chapman and P. K. Maini, Mathematical models of avascular tumor growth, SIAM Review, 49 (2007), 179-208.
doi: 10.1137/S0036144504446291. |
[21] |
R. A. Santiago-Mozos, I. G. Khan and M. Madden, Revealing the origin and nature of drug resistance of dynamic tumour systems, International Journal of Knowledge Discovery in Bioinformatics, 1 (2010), 26-53. |
[22] |
F. R. Sharpe and A. J. Lotka, A problem in age distribution, Philosophical Magazine Series 6, 21 (1911), 435-438.
doi: 10.1080/14786440408637050. |
[23] |
H. Smith, "Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems," Mathematical Surveys and Monographs, 41, American Mathematical Society, Providence, Rhode Island, 1995. |
[24] |
H. Smith, "An Introduction to Delay Differential Equations with Applications to the Life Sciences,'' Springer, New York, 2011.
doi: 10.1007/978-1-4419-7646-8. |
[25] |
U. Veronesi and G. Quaranta, "Un Male Curabile,'' Mondadori Editore, Milano, 1986. |
[26] |
M. Villasana and A. Radunskaya, A delay differential equation model for tumor growth, Journal of Mathematical Biology, 47 (2003), 270-294.
doi: 10.1007/s00285-003-0211-0. |
[27] |
G. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics,'' Monographs and Textbooks in Pure and Applied Mathematics, 1985. |
show all references
References:
[1] |
J. Al-Omari and S. Gourley, Dynamics of a stage-structured population model incorporating a state-dependent maturation delay, Nonlinear Analysis: Real World Applications 6 (2005), 13-33.
doi: 10.1016/j.nonrwa.2004.04.002. |
[2] |
B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts and P. Walter, "Molecular Biology of the Cell," 5th edition, Taylor & Francis Ltd., New York, 2007. |
[3] |
A. Bellen and M. Zennaro, "Numerical Methods for Delay Differential Equations," Clarendon Press, New York, 2003. |
[4] |
S. P. Blythe, R. M. Nisbet and W. S. C. Gurney, The dynamics of population models with distributed maturation periods, Theoretical Population Biology, 25 (1984), 289-311.
doi: 10.1016/0040-5809(84)90011-X. |
[5] |
G. Bocharov and K. P. Hadeler, Structured population models, conservation laws, and delay equations, Journal of Differential Equations, 168 (2000), 212-237.
doi: 10.1006/jdeq.2000.3885. |
[6] |
M. Chaplain and A. Matzavinos, Mathematical modeling of spatio-temporal phenomena in tumor immunology, in "Tutorials in Mathematical Biosciences. III," Lecuture Notes in Math., 1872, Berlin, (2006), 131-183. |
[7] |
K. L. Cooke and P. van den Driessche, On zeros of some transcendental equations, Funkcialaj Ekvacioj, 29 (1986), 77-90. |
[8] |
G. M. Cooper and R. E. Hausman, "The Cell: A Molecular Approach,'' ASM Press, Washington, 1997. |
[9] |
A. d'Onofrio, A general framework for modeling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedical inferences, Physica D, 208 (2005), 220-235.
doi: 10.1016/j.physd.2005.06.032. |
[10] |
A. d'Onofrio, Tumor-immune system interaction: Modelling the tumor-stimulated proliferation of effectors and immunotherapy, Mathematical Models and Methods in Applied Science, 16 (2006), 1375-1401.
doi: 10.1142/S0218202506001571. |
[11] |
A. d'Onofrio, F. Gatti, P. Cerrai and L. Freschi, Delay-induced oscillatory dynamics of tumour-immune system interaction, Mathematical and Computer Modelling, 51 (2010), 572-591. |
[12] |
J. Dyson, R. Villella-Bressan and G. Webb, Asynchronous exponential growth in an age structured population of proliferating and quiescent cells, Deterministic and Stochastic Modeling of Biointeraction (West Lafayette, IN, 2000), Mathematical Biosciences, 177/178 (2002), 73-83.
doi: 10.1016/S0025-5564(01)00097-9. |
[13] |
J. Dyson, R. Villella-Bressan and G. Webb, A spatial model of tumor growth with cell age, cell size, and mutation of cell phenotypes, Mathematical Modelling of Natural Phenomena, 2 (2007), 69-100.
doi: 10.1051/mmnp:2007004. |
[14] |
W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21.
doi: 10.1038/287017a0. |
[15] |
Y. Kuang, "Delay Differential Equations: With Applications in Population Dynamics,'' Academic Press, New York, 2003. |
[16] |
W. Liu, T. Hillen and H. Freedman, A mathematical model for $M$-phase specific chemotherapy including the $G_0$-phase and immunoresponse, Mathematical Bioscience and Engineering, 4 (2007), 239-259.
doi: 10.3934/mbe.2007.4.239. |
[17] |
H. Lodish et al., "Molecular Cell Biology,'' 3rd Ed. Scientific American Books , New York, 1995. |
[18] |
N. MacDonald, "Biological Delay Systems: Linear Stability Theory,'' Cambridge University Press, 1989. |
[19] |
R. Nisbet and W. Gurney, The systematic formulation of population models for insects with dynamically varying instar duration, Theoretical Population Biology, 23 (1983), 114-135.
doi: 10.1016/0040-5809(83)90008-4. |
[20] |
T. Roose, S. J. Chapman and P. K. Maini, Mathematical models of avascular tumor growth, SIAM Review, 49 (2007), 179-208.
doi: 10.1137/S0036144504446291. |
[21] |
R. A. Santiago-Mozos, I. G. Khan and M. Madden, Revealing the origin and nature of drug resistance of dynamic tumour systems, International Journal of Knowledge Discovery in Bioinformatics, 1 (2010), 26-53. |
[22] |
F. R. Sharpe and A. J. Lotka, A problem in age distribution, Philosophical Magazine Series 6, 21 (1911), 435-438.
doi: 10.1080/14786440408637050. |
[23] |
H. Smith, "Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems," Mathematical Surveys and Monographs, 41, American Mathematical Society, Providence, Rhode Island, 1995. |
[24] |
H. Smith, "An Introduction to Delay Differential Equations with Applications to the Life Sciences,'' Springer, New York, 2011.
doi: 10.1007/978-1-4419-7646-8. |
[25] |
U. Veronesi and G. Quaranta, "Un Male Curabile,'' Mondadori Editore, Milano, 1986. |
[26] |
M. Villasana and A. Radunskaya, A delay differential equation model for tumor growth, Journal of Mathematical Biology, 47 (2003), 270-294.
doi: 10.1007/s00285-003-0211-0. |
[27] |
G. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics,'' Monographs and Textbooks in Pure and Applied Mathematics, 1985. |
[1] |
Wenxiang Liu, Thomas Hillen, H. I. Freedman. A mathematical model for M-phase specific chemotherapy including the $G_0$-phase and immunoresponse. Mathematical Biosciences & Engineering, 2007, 4 (2) : 239-259. doi: 10.3934/mbe.2007.4.239 |
[2] |
Gladis Torres-Espino, Claudio Vidal. Periodic solutions of a tumor-immune system interaction under a periodic immunotherapy. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4523-4547. doi: 10.3934/dcdsb.2020301 |
[3] |
Fadia Bekkal-Brikci, Giovanna Chiorino, Khalid Boushaba. G1/S transition and cell population dynamics. Networks and Heterogeneous Media, 2009, 4 (1) : 67-90. doi: 10.3934/nhm.2009.4.67 |
[4] |
Mahmoud Abouagwa, Ji Li. G-neutral stochastic differential equations with variable delay and non-Lipschitz coefficients. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1583-1606. doi: 10.3934/dcdsb.2019241 |
[5] |
Paolo Ubezio. Unraveling the complexity of cell cycle effects of anticancer drugs in cell populations. Discrete and Continuous Dynamical Systems - B, 2004, 4 (1) : 323-335. doi: 10.3934/dcdsb.2004.4.323 |
[6] |
Andrzej Swierniak, Jaroslaw Smieja. Analysis and Optimization of Drug Resistant an Phase-Specific Cancer. Mathematical Biosciences & Engineering, 2005, 2 (3) : 657-670. doi: 10.3934/mbe.2005.2.657 |
[7] |
Urszula Ledzewicz, Omeiza Olumoye, Heinz Schättler. On optimal chemotherapy with a strongly targeted agent for a model of tumor-immune system interactions with generalized logistic growth. Mathematical Biosciences & Engineering, 2013, 10 (3) : 787-802. doi: 10.3934/mbe.2013.10.787 |
[8] |
Jinhu Xu, Yicang Zhou. Bifurcation analysis of HIV-1 infection model with cell-to-cell transmission and immune response delay. Mathematical Biosciences & Engineering, 2016, 13 (2) : 343-367. doi: 10.3934/mbe.2015006 |
[9] |
Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5227-5249. doi: 10.3934/dcdsb.2020341 |
[10] |
Chandra Shekhar, Amit Kumar, Shreekant Varshney, Sherif Ibrahim Ammar. $ \bf{M/G/1} $ fault-tolerant machining system with imperfection. Journal of Industrial and Management Optimization, 2021, 17 (1) : 1-28. doi: 10.3934/jimo.2019096 |
[11] |
Ruiling Tian, Dequan Yue, Wuyi Yue. Optimal balking strategies in an M/G/1 queueing system with a removable server under N-policy. Journal of Industrial and Management Optimization, 2015, 11 (3) : 715-731. doi: 10.3934/jimo.2015.11.715 |
[12] |
Yi Peng, Jinbiao Wu. On the $ BMAP_1, BMAP_2/PH/g, c $ retrial queueing system. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3373-3391. doi: 10.3934/jimo.2020124 |
[13] |
Jinliang Wang, Lijuan Guan. Global stability for a HIV-1 infection model with cell-mediated immune response and intracellular delay. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 297-302. doi: 10.3934/dcdsb.2012.17.297 |
[14] |
Huiyan Zhu, Xingfu Zou. Dynamics of a HIV-1 Infection model with cell-mediated immune response and intracellular delay. Discrete and Continuous Dynamical Systems - B, 2009, 12 (2) : 511-524. doi: 10.3934/dcdsb.2009.12.511 |
[15] |
Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133 |
[16] |
Yong Ren, Xuejuan Jia, Lanying Hu. Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2157-2169. doi: 10.3934/dcdsb.2015.20.2157 |
[17] |
Huijie Qiao, Jiang-Lun Wu. Path independence of the additive functionals for stochastic differential equations driven by G-lévy processes. Probability, Uncertainty and Quantitative Risk, , () : -. doi: 10.3934/puqr.2022007 |
[18] |
Min Yu, Gang Huang, Yueping Dong, Yasuhiro Takeuchi. Complicated dynamics of tumor-immune system interaction model with distributed time delay. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2391-2406. doi: 10.3934/dcdsb.2020015 |
[19] |
Urszula Ledzewicz, Mozhdeh Sadat Faraji Mosalman, Heinz Schättler. Optimal controls for a mathematical model of tumor-immune interactions under targeted chemotherapy with immune boost. Discrete and Continuous Dynamical Systems - B, 2013, 18 (4) : 1031-1051. doi: 10.3934/dcdsb.2013.18.1031 |
[20] |
Sheng Zhu, Jinting Wang. Strategic behavior and optimal strategies in an M/G/1 queue with Bernoulli vacations. Journal of Industrial and Management Optimization, 2018, 14 (4) : 1297-1322. doi: 10.3934/jimo.2018008 |
2018 Impact Factor: 1.313
Tools
Metrics
Other articles
by authors
[Back to Top]