# American Institute of Mathematical Sciences

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 & 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, 2 (2002), 938. 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. 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. [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. 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. 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. [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. 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, (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. doi: 10.1137/0145053.

show all references

##### References:
 [1] E. Antonio Chiocca, Oncolytic viruses,, Nature reviews, 2 (2002), 938. 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. 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. [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. 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. 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. [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. 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, (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. 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] 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 [3] 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 [4] Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735 [5] Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121 [6] Jixun Chu, Pierre Magal. Hopf bifurcation for a size-structured model with resting phase. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4891-4921. doi: 10.3934/dcds.2013.33.4891 [7] Jisun Lim, Seongwon Lee, Yangjin Kim. Hopf bifurcation in a model of TGF-$\beta$ in regulation of the Th 17 phenotype. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3575-3602. doi: 10.3934/dcdsb.2016111 [8] Xianlong Fu, Zhihua Liu, Pierre Magal. Hopf bifurcation in an age-structured population model with two delays. Communications on Pure & Applied Analysis, 2015, 14 (2) : 657-676. doi: 10.3934/cpaa.2015.14.657 [9] Hossein Mohebbi, Azim Aminataei, Cameron J. Browne, Mohammad Reza Razvan. Hopf bifurcation of an age-structured virus infection model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 861-885. doi: 10.3934/dcdsb.2018046 [10] Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026 [11] Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051 [12] Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997 [13] Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045 [14] John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805 [15] Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152 [16] Kousuke Kuto. Stability and Hopf bifurcation of coexistence steady-states to an SKT model in spatially heterogeneous environment. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 489-509. doi: 10.3934/dcds.2009.24.489 [17] Xiaofeng Xu, Junjie Wei. Turing-Hopf bifurcation of a class of modified Leslie-Gower model with diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 765-783. doi: 10.3934/dcdsb.2018042 [18] Ghendrih Philippe, Hauray Maxime, Anne Nouri. Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solution. Kinetic & Related Models, 2009, 2 (4) : 707-725. doi: 10.3934/krm.2009.2.707 [19] Sebastián Ferrer, Francisco Crespo. Parametric quartic Hamiltonian model. A unified treatment of classic integrable systems. Journal of Geometric Mechanics, 2014, 6 (4) : 479-502. doi: 10.3934/jgm.2014.6.479 [20] Guowei Dai, Ruyun Ma, Haiyan Wang. Eigenvalues, bifurcation and one-sign solutions for the periodic $p$-Laplacian. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2839-2872. doi: 10.3934/cpaa.2013.12.2839

2016 Impact Factor: 0.781