# American Institute of Mathematical Sciences

October  2017, 10(5): 1079-1093. doi: 10.3934/dcdss.2017058

## Dynamical behavior of a new oncolytic virotherapy model based on gene variation

 School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

* Corresponding author: Zhiming Guo

Received  August 2016 Revised  January 2017 Published  June 2017

Fund Project: The authors are supported by NNSF of China grant 11371107 and Program for Changjiang Scholars and Innovative Research Team in University grant IRT1226.

Oncolytic virotherapy is an experimental treatment of cancer patients. This method is based on the administration of replication-competent viruses that selectively destroy tumor cells but remain healthy tissue unaffected. In order to obtain optimal dosage for complete tumor eradication, we derive and analyze a new oncolytic virotherapy model with a fixed time period $τ$ and non-local infection which is caused by the diffusion of the target cells in a continuous bounded domain, where $τ$ is assumed to be the duration that oncolytic viruses spend to destroy the target cells and to release new viruses since they enter into the target cells. This model is a delayed reaction diffusion system with nonlocal reaction term. By analyzing the global stability of tumor cell eradication equilibrium, we give different treatment strategies for cancer therapy according to the different genes mutations (oncogene and antioncogene).

Citation: Zizi Wang, Zhiming Guo, Huaqin Peng. Dynamical behavior of a new oncolytic virotherapy model based on gene variation. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1079-1093. doi: 10.3934/dcdss.2017058
##### References:
 [1] Amar, M. Ben, C. Chatelain and P. Ciarletta, Contour instabilities in early tumor growth models, Physical review letters, 106 (2011), 970-978. [2] Ž. Bajzer, T. Carr and K. Josić, Modeling of cancer virotherapy with recombinant measles viruses, Journal of Theoretical Biology, 252 (2008), 109-122.  doi: 10.1016/j.jtbi.2008.01.016. [3] Becker and M. Wayne, et al, The World of the Cell, Vol. 6. San Francisco: Benjamin Cummings, 2003. [4] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley and Sons, 2003. doi: 10.1002/0470871296. [5] D. Dingli, K. W. Peng and M. E. Harvey, Image-guided radiovirotherapy for multiple myeloma using a recombinant measles virus expressing the thyroidal sodium iodide symporter, Blood, 103 (2004), 1641-1646.  doi: 10.1182/blood-2003-07-2233. [6] L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, 1998. [7] S. Fedotov and A. Iomin, Migration and proliferation dichotomy in tumor-cell invasion, Physical Review Letters, 98 (2007), 1-5. [8] R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Research, 56 (1996), 5745-5753. [9] N. L. Komarova and D. Wodarz, ODE models for oncolytic virus dynamics, Journal of Theoretical Biology, 263 (2010), 530-543.  doi: 10.1016/j.jtbi.2010.01.009. [10] Y. Lin, H. Zhang and J. Liang, Identification and characterization of alphavirus M1 as a selective oncolytic virus targeting ZAP-defective human cancers, Proceedings of the National Academy of Sciences, 111 (2014), 4504-4512.  doi: 10.1073/pnas.1408759111. [11] R. H. Martin and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems, Transactions of the American Mathematical Society, 321 (1990), 1-44.  doi: 10.2307/2001590. [12] R. L. Martuza and A. Malick, Experimental therapy of human glioma by means of a genetically engineered virus mutant, Science, 252 (1991), 854-856.  doi: 10.1126/science.1851332. [13] S. A. Menchón and C. A. Condat, Cancer growth: Predictions of a realistic model, Physical Review E, Statistical Nonlinear & Soft Matter Physics, (78) (2008), 2pp. [14] J. D. Murray, Mathematical Biology Ⅱ Spatial Models and Biomedical Applications, Springer, 2003. [15] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [16] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Soc. , Providence, RI, 1995. [17] J. W. H. So, J. Wu and X. Zou, A reaction-diffusion model for a single species with age structure. Ⅰ Travelling wavefronts on unbounded domains, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 457 (2001), 1841-1853.  doi: 10.1098/rspa.2001.0789. [18] Y. Tao and Q. Guo, The competitive dynamics between tumor cells, a replication-competent virus and an immune response, Journal of Mathematical Biology, 51 (2005), 37-74.  doi: 10.1007/s00285-004-0310-6. [19] H. R. Thieme and X. Q. Zhao, A non-local delayed and diffusive predator-prey model, Nonlinear Analysis: Real World Applications, 2 (2001), 145-160.  doi: 10.1016/S0362-546X(00)00112-7. [20] H. R. Thieme, Convergence results and a Poincareć-Bendixson trichotomy for asymptotically autonomous differential equations, Journal of mathematical biology, 30 (1992), 755-763.  doi: 10.1007/BF00173267. [21] M. X. Wang, Y. J. Li and P. Y. Lai, Model on cell movement, growth, Differentiation and de-differentiation: Reaction-diffusion equation and wave propagation, European Physical Journal E, 36 (2013), 1-18.  doi: 10.1140/epje/i2013-13065-4. [22] S. Wang, S. Wang and X. Song, Hopf bifurcation analysis in a delayed oncolytic virus dynamics with continuous control, Nonlinear Dynamics, 67 (2012), 629-640.  doi: 10.1007/s11071-011-0015-5. [23] Z. Wang, Z. Guo and H. Peng, A mathematical model verifying potent oncolytic efficacy of M1 virus, Mathematical Biosciences, 276 (2016), 19-27.  doi: 10.1016/j.mbs.2016.03.001. [24] D. Wodarz, Gene therapy for killing p53-negative cancer cells: Use of replicating versus nonreplicating agents, Human Gene Therapy, 14 (2003), 153-159.  doi: 10.1089/104303403321070847. [25] D. Wodarz, Viruses as antitumor weapons defining conditions for tumor remission, Cancer Research, 61 (2001), 3501-3507. [26] X. Q. Zhao, Global attractivity in a class of nonmonotone reaction-diffusion equations with timedelay, Canadian Applied Mathematics Quarterly, 17 (2009), 271-281.

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##### References:
 [1] Amar, M. Ben, C. Chatelain and P. Ciarletta, Contour instabilities in early tumor growth models, Physical review letters, 106 (2011), 970-978. [2] Ž. Bajzer, T. Carr and K. Josić, Modeling of cancer virotherapy with recombinant measles viruses, Journal of Theoretical Biology, 252 (2008), 109-122.  doi: 10.1016/j.jtbi.2008.01.016. [3] Becker and M. Wayne, et al, The World of the Cell, Vol. 6. San Francisco: Benjamin Cummings, 2003. [4] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley and Sons, 2003. doi: 10.1002/0470871296. [5] D. Dingli, K. W. Peng and M. E. Harvey, Image-guided radiovirotherapy for multiple myeloma using a recombinant measles virus expressing the thyroidal sodium iodide symporter, Blood, 103 (2004), 1641-1646.  doi: 10.1182/blood-2003-07-2233. [6] L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, 1998. [7] S. Fedotov and A. Iomin, Migration and proliferation dichotomy in tumor-cell invasion, Physical Review Letters, 98 (2007), 1-5. [8] R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Research, 56 (1996), 5745-5753. [9] N. L. Komarova and D. Wodarz, ODE models for oncolytic virus dynamics, Journal of Theoretical Biology, 263 (2010), 530-543.  doi: 10.1016/j.jtbi.2010.01.009. [10] Y. Lin, H. Zhang and J. Liang, Identification and characterization of alphavirus M1 as a selective oncolytic virus targeting ZAP-defective human cancers, Proceedings of the National Academy of Sciences, 111 (2014), 4504-4512.  doi: 10.1073/pnas.1408759111. [11] R. H. Martin and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems, Transactions of the American Mathematical Society, 321 (1990), 1-44.  doi: 10.2307/2001590. [12] R. L. Martuza and A. Malick, Experimental therapy of human glioma by means of a genetically engineered virus mutant, Science, 252 (1991), 854-856.  doi: 10.1126/science.1851332. [13] S. A. Menchón and C. A. Condat, Cancer growth: Predictions of a realistic model, Physical Review E, Statistical Nonlinear & Soft Matter Physics, (78) (2008), 2pp. [14] J. D. Murray, Mathematical Biology Ⅱ Spatial Models and Biomedical Applications, Springer, 2003. [15] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [16] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Soc. , Providence, RI, 1995. [17] J. W. H. So, J. Wu and X. Zou, A reaction-diffusion model for a single species with age structure. Ⅰ Travelling wavefronts on unbounded domains, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 457 (2001), 1841-1853.  doi: 10.1098/rspa.2001.0789. [18] Y. Tao and Q. Guo, The competitive dynamics between tumor cells, a replication-competent virus and an immune response, Journal of Mathematical Biology, 51 (2005), 37-74.  doi: 10.1007/s00285-004-0310-6. [19] H. R. Thieme and X. Q. Zhao, A non-local delayed and diffusive predator-prey model, Nonlinear Analysis: Real World Applications, 2 (2001), 145-160.  doi: 10.1016/S0362-546X(00)00112-7. [20] H. R. Thieme, Convergence results and a Poincareć-Bendixson trichotomy for asymptotically autonomous differential equations, Journal of mathematical biology, 30 (1992), 755-763.  doi: 10.1007/BF00173267. [21] M. X. Wang, Y. J. Li and P. Y. Lai, Model on cell movement, growth, Differentiation and de-differentiation: Reaction-diffusion equation and wave propagation, European Physical Journal E, 36 (2013), 1-18.  doi: 10.1140/epje/i2013-13065-4. [22] S. Wang, S. Wang and X. Song, Hopf bifurcation analysis in a delayed oncolytic virus dynamics with continuous control, Nonlinear Dynamics, 67 (2012), 629-640.  doi: 10.1007/s11071-011-0015-5. [23] Z. Wang, Z. Guo and H. Peng, A mathematical model verifying potent oncolytic efficacy of M1 virus, Mathematical Biosciences, 276 (2016), 19-27.  doi: 10.1016/j.mbs.2016.03.001. [24] D. Wodarz, Gene therapy for killing p53-negative cancer cells: Use of replicating versus nonreplicating agents, Human Gene Therapy, 14 (2003), 153-159.  doi: 10.1089/104303403321070847. [25] D. Wodarz, Viruses as antitumor weapons defining conditions for tumor remission, Cancer Research, 61 (2001), 3501-3507. [26] X. Q. Zhao, Global attractivity in a class of nonmonotone reaction-diffusion equations with timedelay, Canadian Applied Mathematics Quarterly, 17 (2009), 271-281.
By Theorem 3.1, it is easy to see that the stability of tumor eradication equilibrium $E$ is independent of diffusion coefficients $d_i, (i=1, 2, 3)$ and $\tau$. Here, we set $d_1=1, d_2=1, d_3=1, \tau=0$, $d=1, \mu=1, a_1=1, b_1=1, c_1=2$, $a_2=1, b_2=1, c_2=1, B=0.2$, $\Gamma(x,y,\tau)=1$, as $x= y$, and $\Gamma(x,y,\tau)=0$, as $x\neq y$. Thus, by direct calculations, we get $\frac{d}{\mu}(a_2-\frac{a_1b_2}{b_1})=0$, $\frac{d}{\mu}(a_2-\frac{a_1c_2}{c_1})=0.5$. Then $B>\frac{d}{\mu}(a_2-\frac{a_1b_2}{b_1})$ in Theorem 3.1 holds. But the component tumor cells $u_2$ doesn't tend to 0 as $t\rightarrow\infty$
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