# American Institute of Mathematical Sciences

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Preface to the special issue "Dynamics and control in distributed systems: Dedicated to the memory of Valery S. Melnik (1952-2007)"
March  2019, 24(3): 989-1005. doi: 10.3934/dcdsb.2019002

## Stability analysis of a chemotherapy model with delays

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

* Corresponding author: Xiaoying Han

Received  January 2018 Revised  March 2018 Published  January 2019

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

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.

Citation: Ismail Abdulrashid, Abdallah A. M. Alsammani, Xiaoying Han. Stability analysis of a chemotherapy model with delays. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 989-1005. doi: 10.3934/dcdsb.2019002
##### References:

show all references

##### References:
Chemotherapy with delays approaching the axial steady state
Comparison of normal and cancer cells of chemotherapy with/without delays
Chemotherapy with delays approaching a preferred steady state
Comparison of normal and cancer cells of chemotherapy with/without delays
Chemotherapy with delays approaching a failure steady state
Comparison of normal and cancer cells of chemotherapy with/without delays
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
 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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