American Institute of Mathematical Sciences

Advanced
December  2011, 4(6): 1587-1597. doi: 10.3934/dcdss.2011.4.1587

Periodic solutions of a model for tumor virotherapy

Daniel Vasiliu 1, and  Jianjun Paul Tian 2,

1. 

Department of Mathematics, Christopher Newport University, Newport News VA, 23606, United States
2. 

Mathematics Department, College of William and Mary, Williamsburg, VA 23187, United States

Received  April 2009 Revised  October 2009 Published  December 2010

In this article we study periodic solutions of a mathematical model for brain tumor virotherapy by finding Hopf bifurcations with respect to a biological significant parameter, the burst size of the oncolytic virus. The model is derived from a PDE free boundary problem. Our model is an ODE system with six variables, five of them represent different cell or virus populations, and one represents tumor radius. We prove the existence of Hopf bifurcations, and periodic solutions in a certain interval of the value of the burst size. The evolution of the tumor radius is much influenced by the value of the burst size. We also provide a numerical confirmation.
Keywords: Hopf bifurcation, virotherapy, tumor model., Periodic solutions.
Mathematics Subject Classification: Primary: 92C37; Secondary: 34A3.
Citation: Daniel Vasiliu, Jianjun Paul Tian. Periodic solutions of a model for tumor virotherapy. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1587-1597. doi: 10.3934/dcdss.2011.4.1587
References:
[1]

E. Antonio Chiocca, Oncolytic viruses, Nature reviews, Cancer, 2 (2002), 938-950. doi: 10.1038/nrc948.

[2]

K. Ikeda, T. Ichikawa, H. Wakimoto, J. S. Silver, D. Finkestein, G. R. Harsh, D. N. Louis, R. T. Bartus, F. H. Hochberg and E. A. Chiocca, Oncolytic virus therapy of multiple tumors in the brain requires suppression of innate and elicited antiviral responses, Nature Med., 5 (1999), 881-888. doi: 10.1038/11320.

[3]

G. Fulci, et al, Cyclophosphamide enhances glioma virotherapy by inhibiting innate immune responses, PNAS Proceedings of the National Academy of Sciences of the United States of America, 103 (2006), 12873-12878.

[4]

H. Kambara, H. Okano, E. A. Chiocca and Y. Saeki, An oncolytic HSV-1 mutant expressing ICP34.5 under control of a nestin promoter increases survival of animals even when symptomatic from a brain tumor, Cancer Research, 65 (2005), 2832-2839. doi: 10.1158/0008-5472.CAN-04-3227.

[5]

H. Kambara, Y. Saeki and E. A Chiocca, Cyclophosphamide allows for in vivo dose reduction of a potent oncolytic virus, Cancer Research, 65 (2005), 11255-11258. doi: 10.1158/0008-5472.CAN-05-2278.

[6]

Shangbin Cui and Avner Friedman, A hyperbolic free boundary problem modeling tumor growth, Interfaces and Free Boundaries, 5 (2003), 159-181

[7]

Avner Friedman, Jianjun Paul Tian, Giulia Fulci, Antonio Chioca and Jin Wang, Glioma virotherapy: Effects of innate immune suppression and increased viral replication capacity, Cancer Research, 4 (2006), 2314-2319. doi: 10.1158/0008-5472.CAN-05-2661.

[8]

Jianjun Paul Tian, Finite-time perturbations of dynamical systems and applications to tumor therapy,, appear to Discrete and Continuous Dynamical Systems - B., (). 

[9]

Peter J. Olver, "Classical Invariant Theory," London Mathematical Society Student Texts, 1999.

[10]

Lawrence Perko, "Differential Equations and Dynamical Systems," Third Edition, Springer, 2007.

[11]

D.Roose and V. Hlavaček, A direct method for the computation of Hopf bifurcation points, SIAM Journal of Applied Mathematics, 45 (1985), 879-894. doi: 10.1137/0145053.

show all references

References:
[1]

E. Antonio Chiocca, Oncolytic viruses, Nature reviews, Cancer, 2 (2002), 938-950. doi: 10.1038/nrc948.

[2]

K. Ikeda, T. Ichikawa, H. Wakimoto, J. S. Silver, D. Finkestein, G. R. Harsh, D. N. Louis, R. T. Bartus, F. H. Hochberg and E. A. Chiocca, Oncolytic virus therapy of multiple tumors in the brain requires suppression of innate and elicited antiviral responses, Nature Med., 5 (1999), 881-888. doi: 10.1038/11320.

[3]

G. Fulci, et al, Cyclophosphamide enhances glioma virotherapy by inhibiting innate immune responses, PNAS Proceedings of the National Academy of Sciences of the United States of America, 103 (2006), 12873-12878.

[4]

H. Kambara, H. Okano, E. A. Chiocca and Y. Saeki, An oncolytic HSV-1 mutant expressing ICP34.5 under control of a nestin promoter increases survival of animals even when symptomatic from a brain tumor, Cancer Research, 65 (2005), 2832-2839. doi: 10.1158/0008-5472.CAN-04-3227.

[5]

H. Kambara, Y. Saeki and E. A Chiocca, Cyclophosphamide allows for in vivo dose reduction of a potent oncolytic virus, Cancer Research, 65 (2005), 11255-11258. doi: 10.1158/0008-5472.CAN-05-2278.

[6]

Shangbin Cui and Avner Friedman, A hyperbolic free boundary problem modeling tumor growth, Interfaces and Free Boundaries, 5 (2003), 159-181

[7]

Avner Friedman, Jianjun Paul Tian, Giulia Fulci, Antonio Chioca and Jin Wang, Glioma virotherapy: Effects of innate immune suppression and increased viral replication capacity, Cancer Research, 4 (2006), 2314-2319. doi: 10.1158/0008-5472.CAN-05-2661.

[8]

Jianjun Paul Tian, Finite-time perturbations of dynamical systems and applications to tumor therapy,, appear to Discrete and Continuous Dynamical Systems - B., (). 

[9]

Peter J. Olver, "Classical Invariant Theory," London Mathematical Society Student Texts, 1999.

[10]

Lawrence Perko, "Differential Equations and Dynamical Systems," Third Edition, Springer, 2007.

[11]

D.Roose and V. Hlavaček, A direct method for the computation of Hopf bifurcation points, SIAM Journal of Applied Mathematics, 45 (1985), 879-894. doi: 10.1137/0145053.

[1]

Dmitriy Yu. Volkov. The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori. Conference Publications, 2015, 2015 (special) : 1098-1104. doi: 10.3934/proc.2015.1098
[2]

Qigang Yuan, Jingli Ren. Periodic forcing on degenerate Hopf bifurcation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208
[3]

Bing Zeng, Pei Yu. A hierarchical parametric analysis on Hopf bifurcation of an epidemic model. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022069
[4]

Jianjun Paul Tian. The replicability of oncolytic virus: Defining conditions in tumor virotherapy. Mathematical Biosciences & Engineering, 2011, 8 (3) : 841-860. doi: 10.3934/mbe.2011.8.841
[5]

Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026
[6]

Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325
[7]

Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735
[8]

Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121
[9]

Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051
[10]

Stephen Pankavich, Nathan Neri, Deborah Shutt. Bistable dynamics and Hopf bifurcation in a refined model of early stage HIV infection. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 2867-2893. doi: 10.3934/dcdsb.2020044
[11]

Jixun Chu, Pierre Magal. Hopf bifurcation for a size-structured model with resting phase. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 4891-4921. doi: 10.3934/dcds.2013.33.4891
[12]

Jisun Lim, Seongwon Lee, Yangjin Kim. Hopf bifurcation in a model of TGF-$\beta$ in regulation of the Th 17 phenotype. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3575-3602. doi: 10.3934/dcdsb.2016111
[13]

Qingyan Shi, Junping Shi, Yongli Song. Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 467-486. doi: 10.3934/dcdsb.2018182
[14]

Hossein Mohebbi, Azim Aminataei, Cameron J. Browne, Mohammad Reza Razvan. Hopf bifurcation of an age-structured virus infection model. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 861-885. doi: 10.3934/dcdsb.2018046
[15]

Xianlong Fu, Zhihua Liu, Pierre Magal. Hopf bifurcation in an age-structured population model with two delays. Communications on Pure and Applied Analysis, 2015, 14 (2) : 657-676. doi: 10.3934/cpaa.2015.14.657
[16]

Yaodan Huang, Zhengce Zhang, Bei Hu. Bifurcation from stability to instability for a free boundary tumor model with angiogenesis. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2473-2510. doi: 10.3934/dcds.2019105
[17]

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
[18]

Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997
[19]

Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045
[20]

John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (80)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy