# American Institute of Mathematical Sciences

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

## Periodic solutions of a model for tumor virotherapy

 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.
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.
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