Stability analysis of a chemotherapy model with delays

Ismail Abdulrashid; Abdallah A. M. Alsammani; Xiaoying Han

doi:10.3934/dcdsb.2019002

Article Contents

2019, Volume 24, Issue 3: 989-1005. Doi: 10.3934/dcdsb.2019002

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Stability analysis of a chemotherapy model with delays

Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA

^* Corresponding author: Xiaoying Han
^* Corresponding author: Xiaoying Han

Received: December 31, 2017

Revised: February 28, 2018

Early access: January 2019

Published: January 2019

This work was partially supported by Simons Foundation, USA (Collaboration Grants for Mathematicians No. 429717) and MINECO/FEDER, EU (Project No. MTM2015-63723-P)

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Abstract

A chemotherapy model for cancer treatment is studied, where the chemotherapy agent and cells are assumed to follow a predator-prey type relation. The time delays from the instant that the chemotherapy agent is injected to the instant that the treatment is effective are taken into account and dynamics of systems with or without delays are compared. First basic properties of solutions including existence and uniqueness, boundedness and positiveness are discussed. Then conditions on model parameters are established for different outcomes of the treatment. Numerical simulations are provided to illustrate theoretical results.

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Mathematics Subject Classification: Primary: 34D23, 34K20; Secondary: 92B05, 93D05.

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Figure 1. Chemotherapy with delays approaching the axial steady state

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Figure 2. Comparison of normal and cancer cells of chemotherapy with/without delays

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Figure 3. Chemotherapy with delays approaching a preferred steady state

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Figure 4. Comparison of normal and cancer cells of chemotherapy with/without delays

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Figure 5. Chemotherapy with delays approaching a failure steady state

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Figure 6. Comparison of normal and cancer cells of chemotherapy with/without delays

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Table 1. Description of parameters in the chemotherapy model

Parameter	Description
$ D $	Injection rate of the chemotherapy agent
$ I $	Injection concentration of the chemotherapy agent
$ {\alpha} $	Killing rate of the chemotherapy agent on cells
$ \delta $	Intraspecific competition coefficient between cancer and normal cells
$ \beta_{1} $	Intrinsic growth rate of cancer cells
$ \beta_{2} $	Intrinsic growth rate of normal cells
$ \kappa_{1} $	Environmental carrying capacity of cancer cells
$ \kappa_{2} $	Environmental carrying capacity of normal cells
$ \gamma_1 $	Effectiveness of chemotherapy agent on cancer cells
$ \gamma_2 $	Effectiveness of chemotherapy agent on normal cells

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