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Optimal Strategies in the Treatment of Cancers in the Lotka–Volterra Mathematical Model of Competition

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Abstract

The Lotka–Volterra competition model is applied to describe the interaction between the concentrations of healthy and cancerous cells in diseases associated with blood cancer. The model is supplemented with a differential equation characterizing the change in the concentration of a chemotherapeutic drug. The equation contains a scalar bounded control that specifies the rate of drug intake. We consider the problem of minimizing the weighted difference between the concentrations of cancerous and healthy cells at the end time of the treatment period. The Pontryagin maximum principle is used to establish analytically the properties of an optimal control. We describe situations in which the optimal control is a bang–bang function and situations in which the control may contain a singular arc in addition to bang–bang arcs. The results obtained are confirmed by corresponding numerical calculations.

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Funding

The work of the first two authors was supported by the Russian Foundation for Basic Research jointly with the Department of Science and Technology of the Government of India (project no. 18-51-45003 IND_a).

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Authors and Affiliations

  1. Faculty of Computational Mathematics and Cybernetics, Moscow State University, 119991, Moscow, Russia

    N. L. Grigorenko, E. N. Khailov & A. D. Klimenkova

  2. Texas Womans University, TX 76204, Denton, USA

    E. V. Grigorieva

Authors
  1. N. L. Grigorenko
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  2. E. N. Khailov
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  3. E. V. Grigorieva
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  4. A. D. Klimenkova
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Corresponding authors

Correspondence to N. L. Grigorenko, E. N. Khailov, E. V. Grigorieva or A. D. Klimenkova.

Additional information

Translated from Trudy Instituta Matematiki i Mekhaniki UrO RAN, Vol. 26, No. 1, pp. 71 - 88, 2020 https://doi.org/10.21538/0134-4889-2020-26-1-71-88.

Translated by I. Tselishcheva

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Grigorenko, N.L., Khailov, E.N., Grigorieva, E.V. et al. Optimal Strategies in the Treatment of Cancers in the Lotka–Volterra Mathematical Model of Competition. Proc. Steklov Inst. Math. 313 (Suppl 1), S100–S116 (2021). https://doi.org/10.1134/S0081543821030111

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  • DOI: https://doi.org/10.1134/S0081543821030111

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