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Delay equations modeling the effects of phase-specific drugs and immunotherapy on proliferating tumor cells
Qualitative analysis of a model for co-culture of bacteria and amoebae
1. | Center for Information Technology, Bruno Kessler Foundation, via Sommarive 18, I-38123 Trento Povo, Italy |
2. | Institut de Mathématiques de Bordeaux, UMR CNRS 5251 - Case 36, Université Victor Segalen Bordeaux 2, 3ter place de la Victoire 33076 Bordeaux Cedex, France |
3. | Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China |
References:
[1] |
A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Maier, "Theory of Bifurcations of Dynamical Systems on a Plane,'' Israel Program for Scientific Translations, Jerusalem, 1971. |
[2] |
R. Bogdanov, Bifurcations of a limit cycle for a family of vector fields on the plane, Selecta. Math. Soviet., 1 (1981), 373-388. |
[3] |
R. Bogdanov, Versal deformations of a singular point on the plane in the case of zero eigenvalues, Selecta. Math. Soviet., 1 (1981), 389-421. |
[4] |
S.-N. Chow and J. K. Hale, "Methods of Bifurcation Theory,'' Springer-Verlag, New York-Heidelberg-Berlin, 1982.
doi: 10.1007/978-1-4613-8159-4. |
[5] |
P. Cosson, L. Zulianello, O. Join-Lambert, F. Faurisson, L. Gebbie, M. Benghezal, C. Van Delden, L. K. Curty and T. Khler, Pseudomonas aeruginosa virulence analyzed in a Dictyostelium discoideum host system, J. Bacteriol., 184 (2002), 3027-3033.
doi: 10.1128/JB.184.11.3027-3033.2002. |
[6] |
F. Dumortier, R. Roussarie and J. Sotomayor, Generic $3$-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension $3$, Ergodic Theory Dynam. Systems, 7 (1987), 375-413.
doi: 10.1017/S0143385700004119. |
[7] |
F. Dumortier, R. Roussarie, J. Sotomayor and H. Zoladek, "Bifurcations of Planar Vector Fields. Nilpotent Singularities and Abelian Integrals,'' Lecture Notes in Mathematics, 1480, Springer-Verlag, Berlin, 1991. |
[8] |
E. M. C. D'Agata, P. Magal, D. Olivier, S. Ruan and G. F. Webb, Modeling antibiotic resistance in hospitals: The impact of minimizing treatment duration, J. Theor. Biol., 249 (2007), 487-499.
doi: 10.1016/j.jtbi.2007.08.011. |
[9] |
E. M. C. D'Agata, M. Dupont-Rouzeyrol, P. Magal, D. Olivier and S. Ruan, The impact of different antibiotic regimens on the emergence of antimicrobial-resistant bacteria, PLoS ONE, 3 (2008), 1-9. |
[10] |
R. Froquet, N. Cherix, S. E. Burr, J. Frey, S. Vilches, J. M. Tomas and P. Cosson, Alternative host model to evaluate Aeromonas virulence, Appl. Environ. Microbiol., 73 (2007), 5657-5659.
doi: 10.1128/AEM.00908-07. |
[11] |
L. Fumanelli, M. Iannelli, H. A. Janjua and O. Jousson, Mathematical modeling of bacterial virulence and host-pathogen interactions in the Dictyostelium/Pseudomonas system, J. Theor. Biol., 270 (2011), 19-24.
doi: 10.1016/j.jtbi.2010.11.018. |
[12] |
J. K. Hale, "Asymptotic Behavior of Dissipative Systems,'' Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988. |
[13] |
J. K. Hale and P. Waltman, Persistence in infinite dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395.
doi: 10.1137/0520025. |
[14] |
E. Kipnis, T. Sawa and J. Wiener-Kronish, Targeting mechanisms of Pseudomonas aeruginosa pathogenesis, Medecine et Maladies Infectieuses, 36 (2006), 78-91.
doi: 10.1016/j.medmal.2005.10.007. |
[15] |
C. L. Kurz and J. J. Ewbank, Infection in a dish: High-throughput analyses of bacterial pathogenesis, Curr. Opin. Microbiol., 10 (2007), 10-16.
doi: 10.1016/j.mib.2006.12.001. |
[16] |
S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2001), 1445-1472.
doi: 10.1137/S0036139999361896. |
[17] |
G. F. Webb, E. M. C. D'Agata, P. Magal and S. Ruan, A model of antibiotic resistant bacterial epidemics in hospitals, Proc. Natl. Acad. Sci. USA, 102 (2005), 13343-13348.
doi: 10.1073/pnas.0504053102. |
[18] |
D. Xiao and S. Ruan, Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting, in "Differential Equations with Applications to Biology" (Halifax, NS, 1997), Fields Inst. Commun., 21, Amer. Math. Soc., Providence, RI, (1999), 493-506. |
[19] |
Z. F. Zhang, T. R. Ding, W. Z. Huang and Z. X. Dong, "Qualitative Theory of Differential Equations,'' Transl. Math. Monogr., 101, American Mathematical Society, Providence, RI, 1992. |
show all references
References:
[1] |
A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Maier, "Theory of Bifurcations of Dynamical Systems on a Plane,'' Israel Program for Scientific Translations, Jerusalem, 1971. |
[2] |
R. Bogdanov, Bifurcations of a limit cycle for a family of vector fields on the plane, Selecta. Math. Soviet., 1 (1981), 373-388. |
[3] |
R. Bogdanov, Versal deformations of a singular point on the plane in the case of zero eigenvalues, Selecta. Math. Soviet., 1 (1981), 389-421. |
[4] |
S.-N. Chow and J. K. Hale, "Methods of Bifurcation Theory,'' Springer-Verlag, New York-Heidelberg-Berlin, 1982.
doi: 10.1007/978-1-4613-8159-4. |
[5] |
P. Cosson, L. Zulianello, O. Join-Lambert, F. Faurisson, L. Gebbie, M. Benghezal, C. Van Delden, L. K. Curty and T. Khler, Pseudomonas aeruginosa virulence analyzed in a Dictyostelium discoideum host system, J. Bacteriol., 184 (2002), 3027-3033.
doi: 10.1128/JB.184.11.3027-3033.2002. |
[6] |
F. Dumortier, R. Roussarie and J. Sotomayor, Generic $3$-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension $3$, Ergodic Theory Dynam. Systems, 7 (1987), 375-413.
doi: 10.1017/S0143385700004119. |
[7] |
F. Dumortier, R. Roussarie, J. Sotomayor and H. Zoladek, "Bifurcations of Planar Vector Fields. Nilpotent Singularities and Abelian Integrals,'' Lecture Notes in Mathematics, 1480, Springer-Verlag, Berlin, 1991. |
[8] |
E. M. C. D'Agata, P. Magal, D. Olivier, S. Ruan and G. F. Webb, Modeling antibiotic resistance in hospitals: The impact of minimizing treatment duration, J. Theor. Biol., 249 (2007), 487-499.
doi: 10.1016/j.jtbi.2007.08.011. |
[9] |
E. M. C. D'Agata, M. Dupont-Rouzeyrol, P. Magal, D. Olivier and S. Ruan, The impact of different antibiotic regimens on the emergence of antimicrobial-resistant bacteria, PLoS ONE, 3 (2008), 1-9. |
[10] |
R. Froquet, N. Cherix, S. E. Burr, J. Frey, S. Vilches, J. M. Tomas and P. Cosson, Alternative host model to evaluate Aeromonas virulence, Appl. Environ. Microbiol., 73 (2007), 5657-5659.
doi: 10.1128/AEM.00908-07. |
[11] |
L. Fumanelli, M. Iannelli, H. A. Janjua and O. Jousson, Mathematical modeling of bacterial virulence and host-pathogen interactions in the Dictyostelium/Pseudomonas system, J. Theor. Biol., 270 (2011), 19-24.
doi: 10.1016/j.jtbi.2010.11.018. |
[12] |
J. K. Hale, "Asymptotic Behavior of Dissipative Systems,'' Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988. |
[13] |
J. K. Hale and P. Waltman, Persistence in infinite dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395.
doi: 10.1137/0520025. |
[14] |
E. Kipnis, T. Sawa and J. Wiener-Kronish, Targeting mechanisms of Pseudomonas aeruginosa pathogenesis, Medecine et Maladies Infectieuses, 36 (2006), 78-91.
doi: 10.1016/j.medmal.2005.10.007. |
[15] |
C. L. Kurz and J. J. Ewbank, Infection in a dish: High-throughput analyses of bacterial pathogenesis, Curr. Opin. Microbiol., 10 (2007), 10-16.
doi: 10.1016/j.mib.2006.12.001. |
[16] |
S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2001), 1445-1472.
doi: 10.1137/S0036139999361896. |
[17] |
G. F. Webb, E. M. C. D'Agata, P. Magal and S. Ruan, A model of antibiotic resistant bacterial epidemics in hospitals, Proc. Natl. Acad. Sci. USA, 102 (2005), 13343-13348.
doi: 10.1073/pnas.0504053102. |
[18] |
D. Xiao and S. Ruan, Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting, in "Differential Equations with Applications to Biology" (Halifax, NS, 1997), Fields Inst. Commun., 21, Amer. Math. Soc., Providence, RI, (1999), 493-506. |
[19] |
Z. F. Zhang, T. R. Ding, W. Z. Huang and Z. X. Dong, "Qualitative Theory of Differential Equations,'' Transl. Math. Monogr., 101, American Mathematical Society, Providence, RI, 1992. |
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