DOI QR코드

DOI QR Code

Hopf-bifurcation Analysis of a Delayed Model for the Treatment of Cancer using Virotherapy

  • Rajalakshmi, Maharajan (Division of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Chennai Campus) ;
  • Ghosh, Mini (Division of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Chennai Campus)
  • Received : 2020.02.23
  • Accepted : 2020.11.16
  • Published : 2022.03.31

Abstract

Virotherapy is an effective method for the treatment of cancer. The oncolytic virus specifically infects the lyse cancer cell without harming normal cells. There is a time delay between the time of interaction of the virus with the tumor cells and the time when the tumor cells become infectious and produce new virus particles. Several types of viruses are used in virotherapy and the delay varies with the type of virus. This delay can play an important role in the success of virotherapy. Our present study is to explore the impact of this delay in cancer virotherapy through a mathematical model based on delay differential equations. The partial success of virotherapy is guarenteed when one gets a stable non-trivial equilibrium with a low level of tumor cells. There exits Hopf-bifurcation by considering the delay as bifurcation parameter. We have estimated the length of delay which preserves the stability of the non-trivial equilibrium point. So when the delay is less than a threshold value, we can predict partial success of virotherapy for suitable sets of parameters. Here numerical simulations are also performed to support the analytical findings.

Keywords

References

  1. Z. Bajzer, T. Carr, K. Josic, S. J. Russell and D. Dingli, Modeling of cancer virotherapy with recombinant measles viruses, J. Theoret. Biol., 252(2008), 109-122. https://doi.org/10.1016/j.jtbi.2008.01.016
  2. S. Banerjee and R. R. Sarkar, Delay-induced model for tumorimmune interaction and control of malignant tumor growth, Biosystems, 91(1)(2008), 268-288. https://doi.org/10.1016/j.biosystems.2007.10.002
  3. N. Bellomo, K. Painter, Y. Tao and M. Winkler, Occurrence vs. absence of taxisdriven instabilities in a May-Nowak model for virus infection, SIAM J. Appl. Math., 79(5)(2019), 1990-2010. https://doi.org/10.1137/19m1250261
  4. M. Biesecker, J. Kimn, H. Lu, D. Dingli and Z. Bajzer, Optimization of Virotherapy for Cancer, Bull. Math. Biol., 72(2010), 469-489. https://doi.org/10.1007/s11538-009-9456-0
  5. C. Castillo-Chavez, Z. Feng and W. Huang, On the computation of R0 and its role on global stability, in: Mathematical Approaches for For Emerging and Reemerging Infectious Diseases, Springer-Verlag(2002), 229-250.
  6. M. A. J. Chaplain, Modelling aspects of cancer growth: insight from mathematical and numerical analysis and computational simulation, in Multiscale Problems in the Life Sciences, Vol. 1940 of Lecture Notes in Mathematics, Springer(2008), 147-200.
  7. J. J. Crivellia, J. Fldes, P. S. Kim and J. R. Wares, A mathematical model for cell cycle-specific cancer virotherapy, J. Biol. Dyn., 6(1)(2012), 104-120. https://doi.org/10.1080/17513758.2011.613486
  8. P. V. Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission Mathematical Biosciences, Mathematical Biosciences, 180(2002), 29-48. https://doi.org/10.1016/S0025-5564(02)00108-6
  9. R. Eftimie and L. Gibelli, A kinetic theory approach for modelling tumour and macrophages heterogeneity and plasticity during cancer progression, Math. Mod. Meth. App. Sci., 30(2020), 659-683. https://doi.org/10.1142/s0218202520400011
  10. A. M. Elaiw and E. D. Al Agha, Analysis of a delayed and diffusive oncolytic M1 virotherapy model with immune response, Nonlinear Anal. Real World Appl., 55(2020), 103-116.
  11. A. Eladdadi, L. D. Pillis and P. Kim, Modelling tumourimmune dynamics, disease progression and treatment, Letters in Biomathematics, 5(2018), S1-S5. https://doi.org/10.1080/23737867.2018.1483003
  12. H. Enderling and A. J. Mark, Chaplain, Mathematical Modeling of Tumor Growth and Treatment, Current Pharmaceutical Design, 20(30)(2014), 4934-4940. https://doi.org/10.2174/1381612819666131125150434
  13. S. Khajanchi and J. J. Nieto, Mathematical modeling of tumor-immune competitive system, considering the role of time delay, Appl. Math. Comput., 340(2019), 180-205. https://doi.org/10.1016/j.amc.2018.08.018
  14. S. Khajanchi and S. Banerjee, Stability and bifurcation analysis of delay induced tumor immune interaction model, Appl. Math. Comput., 248(2014), 652-671. https://doi.org/10.1016/j.amc.2014.10.009
  15. K. S. Kim, S. Kim and I. H. Jung, Dynamics of tumor virotherapy: A deterministic and stochastic model approach, Stoch. Anal. Appl., 34(3)(2016), 483-495. https://doi.org/10.1080/07362994.2016.1150187
  16. K. S. Kim, S. Kim and I. H. Jung, Hopf bifurcation analysis and optimal control of Treatment in a delayed oncolytic virus dynamics, Math. Comput. Simulation, 149(2018), 1-16. https://doi.org/10.1016/j.matcom.2018.01.003
  17. M. Rajalakshmi and M. Ghosh, Modeling treatment of cancer using virotherapy with generalized logistic growth of tumor cells, Stoch. Anal. Appl., 36(6)(2018), 1068-1086. https://doi.org/10.1080/07362994.2018.1535319
  18. F. A. Rihan, D. H. Abdelrahman, F. Al-Maskari, F. Ibrahim and M. A. Abdeen, Delay differential model for tumour-immune response with chemoimmunotherapy and optimal control, Comput. Math. Methods Med., 2014(2014), Article ID 982978.
  19. R. R. Sarkar and S. Banerjee, Cancer self remission and tumor stability-a stochastic approach, Math. Biosci., 196(2005), 65-81. https://doi.org/10.1016/j.mbs.2005.04.001
  20. Z. Wang, Z. Guo and H. Peng, A mathematical model verifying potent oncolytic efficacy of M1 virus, Math. Biosci., 276(2016), 19-27. https://doi.org/10.1016/j.mbs.2016.03.001
  21. R. Yafia, Dynamics analysis and limit cycle in a delayed model for tumor growth with quiescence, Nonlinear Anal. Model. Control, 11(1)(2006), 95-110. https://doi.org/10.15388/NA.2006.11.1.14766
  22. R. Yafia, A study of differential equation modeling malignant tumor cells in competition with immune system, Int. J. Biomath., 4(2)(2011), 185-206. https://doi.org/10.1142/S1793524511001404