An Analysis of the Dynamics of a Cancerous Tumour Model with Targeted Chemotherapy


  • Department of Mathematics, Gauhati University, Guwahati – 781014, Assam
  • Department of Mathematics, Gauhati University, Guwahati – 781014, Assam
  • Department of Mathematics, Gauhati University, Guwahati – 781014, Assam
  • Department of Mathematics, Gauhati University, Guwahati – 781014, Assam



Global Stability, Hopf Bifurcation, Lotka-Volterra Type, Targeted Chemotherapy


We have analyzed a model of Lotka-Volterra type interacting between immune cell-tumour cell-normal cells, where control policy is applied in terms of targeted chemotherapy. We determined conditions for the local stability of all the equilibrium points and global stability condition for the tumour free equilibrium point, including the feasibility of the solution. Further, we have discussed the possibility of Hopf bifurcation at each equilibrium point. Numerical simulation was carried out to observe the qualitative behaviour of the system as the control parameter is varied.


Komarova NL, Wodarz D. Drug resistance in cancer: Principles of emergence and prevention. PNAS. 2005 Jul 5; 102(27):9714–9. PMid: 15980154 PMCid: PMC1172248.

Malinzi J, Sibanda P, Mambili-Mamboundou H. Analysis of virotherapy in solid tumour invasion. MB. 2015 Feb; 263:102–10. PMid: 25725123. targeted-therapies

Belostotski G. A control theory model for cancer treatment by radiotherapy. IJPAM. 2005; 25(4):447–80.

Isea R, Lonngren KE. A mathematical model of cancer under radiotherapy. IJPHR . 2015 Oct; 3(6):340–4.

Adam JA, Panetta J. A simple mathematical model and alternative paradigm for certain chemotherapeutic regimens. MCM. 1995 Apr; 22(8):49–60.

Eisen M. Mathematical models in cell biology and cancer chemotherapy. SV. 1979; 30:1–431. PMid: 447222. https://

Martin RB, Fisher ME, Minchin R.F, Teo KL. A mathematical model of cancer chemotherapy with an optimal selection of parameters. MBIJ. 1990; 99(2):205–30.

Isaeva O, Osipov V. Different strategies for cancer treatment: Mathematical modeling. CMMM. 2009; 10(4):253–72.

Dehingia K, Nath KC, Sarmah HK. Mathematical modelling of tumour-immune dynamics and cure therapy: A review of literature.JNR. 2020; 21(1):26–41.

Frascoli F, Kim PS, Hughes BD, Landman KA. A dynamical model of tumour immunotherapy. MB. 2014; 253:50–62. PMid: 24759513.

Adam JA. The dynamics growth-factor-modified immune response to cancer growth: One dimensional models. MCM. 1993; 17(3):83–106.

Owen M, Sherratt J. Modeling the macrophage invasion of tumours: Effects on growth and composition.

IMAJMAMB. 1998; 15(2):165–85.

Wilson S, Levy D. A mathematical model of the enhancement of tumour vaccine efficacy by immune therapy. BMB. 2012 Mar; 74(7):1485–500. PMid: 22438084 PMCid: PMC3822329.

Albert A, Freedman M, Perelson AS. Tumours and the immune system: The effects of a tumour growth modulator. MB. 1980 Jul; 50(1-2):25–58.

Liu P, Liu X. Dynamics of a tumour-immune model considering targeted chemotherapy. CSF. 2017 May; 98:7– 13.

Padma VV. An overview of targeted cancer therapy. BM. 2015 Nov; 5(4):1–6. PMid: 26613930 PMCid: PMC4662664.

Gerber DE. Targeted therapies: A new generation of cancer treatments. AFP. 2008 Feb; 77(3):311–9.

Seebacher NA, Stacy AE, Porter GM, Merlot AM. Clinical development of targeted and immune based anti-cancer therapies. JECCR. 2019 Apr; 38:156. PMid: 30975211 PMCid: PMC6460662.

Pento JT. Monoclonal antibodies for the treatment of cáncer. Monoclonal antibodies for the treatment of cancer. NIH. 2017 Nov; 37(11):5935–9.

de Pillis LG, Savage H, Radunskaya AE. Mathematical model of colorectal cancer with monoclonal antibody treatments. BJMCS. 2013 Dec; X(X):XX–XX.

Smith K, Garman L, Wrammert J, Zheng NY, Capra JD, Ahmed R, Wilson PC. Rapid generation of fully human monoclonal antibodies specific to a vaccinating antigen. NIHPA. 2009; 4(3):372–84. PMid: 19247287 PMCid: PMC2750034.

Anser W, Ghosh S. Monoclona antibodies: A tool in clinical Research. IJCM. 2013 Jul; 4: 9–21. IJCM.S11968

Lia Y, Wang R, Chen X, Tang D. Emerging trends and new developments in monoclonal antibodies: A scientometric analysis. HVI. 2017 Jun; 13(6):1388–97. PMid: 28301271 PMCid: PMC5489293. 017.1286433

Kirschner D. On the global dynamics of a model for tumour immunotherapy. MBE. 2009 Jul; 6(3): 573–8.

PMid: 19566127.

Gurcan F, Kartal S, Ozturk I, Bozkurt F. Stability and bifurcation analysis of a mathematical model for tumourimmune interaction with piecewise constant arguments of delay. CSF. 2014 Jun; 38(9):169–79.

Galach M. Dynamics of the tumour-immune system competition - the effect of time delay. IJAMCS. 2003; 13(3):395–406.

Dibrov BF, Zhabotinsky AM, Neyfakh YA, Orlova MP, Churikova LI. Mathematical model of cancer chemotherapy. Periodic schedules of phase-specific cytotoxic- agent administration increasing the selectivity of therapy. MB.1985 Mar; 73(1):1–31.

de Pillis LG de, Radunskaya AE. A mathematical tumour model with immune resistance and drug therapy: An optimal control approach. JTM. 2001; 3(2):79–100.

de Pillis LG, Radunskaya AE. The dynamics of optimally controlled tumour model. A case ctudy. MCM. 2003 Jun; 37(11):1221–44.

Paul R, Das A, Bhattacharya D, Sarma HK. A policy to eradicate tumour in a discrete-continuous immune celltumour cell-drug administration model with the help of stability analysis and bifurcation analysis of the model. IJAER. 2019; 14(9):2192–7.

Paul R, Das A, Sarma HK. Global stability analysis to control growth of tumour in an immune-tumour-normal cell model with drug administration in the form of chemotherapy. IJTP. 2017; 65(3–4)91–106.

Novozhilov AS, Benezovskaya FS, Koonin EV. Mathematical modelling of tumour therapy with oncolytic viruses: Regimes with complete tumour elimination within the framework of deterministic models. BD. 2006 Feb; 1:6.

Abbott LH, Michor F. Mathematical models of targeted cancer therapy. BJC. 2006; 95:1136–41. PMid: 17031409 PMCid: PMC2360553.

Yu P. Closed form conditions of bifurcation points for general differential equations. IJBC. 2005; 15(4):1467–83.






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