Research article

Dynamics of the tumor-immune-virus interactions: Convergence conditions to tumor-only or tumor-free equilibrium points

  • Received: 17 April 2018 Accepted: 05 September 2018 Published: 17 December 2018
  • In the present paper convergence dynamics of one tumor-immune-virus model is examined with help of the localization method of compact invariant sets and the LaSalle theorem. This model was elaborated by Eftimie et al. in 2016. It is shown that this model possesses the Lagrange stability property of positive half-trajectories and ultimate upper bounds for compact invariant sets are obtained. Conditions of convergence dynamics are found. It is explored the case when any trajectory is attracted to one of tumor-only equilibrium points or tumor-free equilibrium points. Further, it is studied ultimate dynamics of one modification of Eftimie et al. model in which the immune cells injection is included. This modified system possesses the global tumor cells eradication property if the influx rate of immune cells exceeds some value which is estimated. Main results are expressed in terms simple algebraic inequalities imposed on model and treatment parameters.

    Citation: Konstantin E. Starkov, Giovana Andres Garfias. Dynamics of the tumor-immune-virus interactions: Convergence conditions to tumor-only or tumor-free equilibrium points[J]. Mathematical Biosciences and Engineering, 2019, 16(1): 421-437. doi: 10.3934/mbe.2019020

    Related Papers:

  • In the present paper convergence dynamics of one tumor-immune-virus model is examined with help of the localization method of compact invariant sets and the LaSalle theorem. This model was elaborated by Eftimie et al. in 2016. It is shown that this model possesses the Lagrange stability property of positive half-trajectories and ultimate upper bounds for compact invariant sets are obtained. Conditions of convergence dynamics are found. It is explored the case when any trajectory is attracted to one of tumor-only equilibrium points or tumor-free equilibrium points. Further, it is studied ultimate dynamics of one modification of Eftimie et al. model in which the immune cells injection is included. This modified system possesses the global tumor cells eradication property if the influx rate of immune cells exceeds some value which is estimated. Main results are expressed in terms simple algebraic inequalities imposed on model and treatment parameters.


    加载中


    [1] Ž. Bajzer, T. Carr, K. Josić, S.J Russell and D. Dingli, Modeling of cancer virotherapy with recombinant measles viruses, J. Theor. Biol. 252 (2008), 109–122.
    [2] M. Biesecker, J.H. Kimn, H. Lu, D. Dingli and Ž. Bajzer, Optimization of virotherapy for cancer, Bulletin Math. Biol., 72 (2010), 469–489.
    [3] B.W. Bridle, K.B. Stephenson, J.E. Boudreau, S. Koshy, N. Kazdhan, E. Pullenayegum, J. Brunellière, J.L. Bramson, B.D. Lichty and Y. Wan, Potentiating cancer immunotherapy using an oncolytic virus, Mol. Ther., 18 (2010), 1430–1439.
    [4] R.M. Diaz, F. Galivo, T. Kottke, P.Wongthida, J. Qiao, J. Thompson, M. Valdes, G. Barber and R.G Vile, Oncolytic immunovirotherapy for melanoma using vesicular stomatitis virus, Cancer Res., 67 (2007), 2840–2848.
    [5] R. Eftimie, J. Dushoff, B.M. Bridle, J.L. Bramson and D.J.D. Earn, Multi-stability and multiinstability phenomena in a mathematical model of tumor-immune-virus interactions, Bulletin Math. Biol., 73 (2011), 2932–2961.
    [6] R. Eftimie, C.K. Macnamara, J. Dushoff, J.L. Bramson and D.J.D. Earn, Bifurcations and chaotic dynamics in a tumour-immune-virus system, Math. Modell. Nat. Pheno., 11 (2016), 65–85.
    [7] M.W. Hirsch, Systems of differential equations that are competitive or cooperative II: Convergence almost everywhere, SIAM J. Math. Analysis, 16 (1985), 423–439.
    [8] B.Y. Hwang and D.V. Schaffer, Engineering a serum-resistant and thermostable vesicular stomatitis virus G glycoprotein for pseudotyping retroviral and lentiviral vectors, Gene Ther., 20 (2013), 807– 815.
    [9] A.P. Krishchenko, Estimation of domains with cycles, Computer Math. Appl., 34 (1997), 325–332.
    [10] A.P. Krishchenko, Localization of invariant compact sets of dynamical systems, Differ. Equations, 41 (2005), 1669–1676.
    [11] A.P. Krishchenko and K.E. Starkov, Localization of compact invariant sets of the Lorenz system, Phys. Lett. A, 353 (2006), 383–388.
    [12] A.P. Krishchenko and K.E. Starkov, On the global dynamics of a chronic myelogenous leukemia model, Commun. Nonlin. Sci. Numer. Simul., 33 (2016), 174–183.
    [13] A.P. Krishchenko and K.E. Starkov, The four-dimensional Kirschner-Panetta type cancer model: how to obtain tumor eradication? Math. Biosci. Eng., 15 (2018), 1243–1254.
    [14] S. Varghese and S.D. Rabkin, Oncolytic herpes simplex virus vectors for cancer virotherapy, Cancer Gene Ther., 9 (2002), 967–978.
    [15] M.Y. Li and S.J. Muldowney, On R.A. Smith's autonomous convergence theorem, Rocky Mountain J. Math., 25 (1995), 365–379.
    [16] R.A. Smith, Some applications of Hausdorff dimension inequalities for ordinary differential equations, Proceed. Royal Soc. Edinburgh Sec. A: Math., 104 (1986), 235–259.
    [17] K.E. Starkov, On dynamic tumor eradication conditions under combined chemical/anti-angiogenic therapies, Phys. Lett. A, 382 (2018), 387–393.
    [18] K.E. Starkov and S. Bunimovich-Mendrazitsky, Dynamical properties and tumor clearance conditions for a nine-dimensional model of bladder cancer immunotherapy, Math. Biosci. Eng., 13 (2016), 1059–1075.
    [19] K.E. Starkov and L. Jimenez Beristain, Dynamic analysis of the melanoma model: from cancer persistence to its eradication, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1750151-1– 1750151-11.
    [20] K.E. Starkov and A.P. Krishchenko, Ultimate dynamics of the Kirschner-Panetta model: Tumor eradication and related problems, Phys. Lett. A, 381 (2017), 3409–3016.
    [21] H.R. Thieme, Asymptotically autonomous differential equations in the plane, Rocky Mountain J. Math., 34 (1994), 351–380.
  • Reader Comments
  • © 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3818) PDF downloads(603) Cited by(4)

Article outline

Figures and Tables

Figures(3)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog