Global stability and tumor clearance conditions for a cancer chemotherapy system

Valle, Paul A.; Starkov, Konstantin E.; Coria, Luis N.

doi:10.1016/j.cnsns.2016.04.025

Cited by 26 publications

(23 citation statements)

References 19 publications

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“…Lemma 3.1. For (3.9), the following conclusions hold true: 2,3,4,5,6), then (3.9) has no root with zero real parts when 2,3,4,5), then (3.9) has a pair of purely imaginary roots ±iα 0 when ϱ 2 =ϱ 2l (l=1,2,…,) where…”

Section: Basic Knowledgementioning

confidence: 96%

“…(2) If (H4) is fulfilled and ε i >0(i=1, 2,3,4,5), then (3.21) has a pair of purely imaginary roots ±iβ 0 when ϱ 1 =ϱ 1h (l=1,2,…,) where…”

Section: Basic Knowledgementioning

confidence: 99%

“…Lemma 3.7. For Equation (3.5), the following results are true: 2,3,4,5,6)>0, then Equation (3.5) has no root with zero real parts. 2.…”

Section: Basic Knowledgementioning

confidence: 99%

“…If ς 6 >0 and there exists a constant ν * >0 such that ι′(ν * )<0, then Equation (3.5) has at least two pairs of purely imaginary roots. 2,3,4,5,6), then dχ 4 ðδÞ dδ > 0; ∀ δ>0 and χ 4 (0)=ς 6 >0. Then we know that Equation (3.48) has no positive real root.…”

Section: Basic Knowledgementioning

confidence: 99%

“…Various strategies have been taken to overcome this challenging problem. [2][3][4] Generally speaking, chemotherapy is a usual cancer treatment approach to slow or stop the growth of cancer via killing cancer cells. In past several decades, a large numbers of chemotherapy models have been established to investigate the cancer treatment.…”

Section: Introductionmentioning

confidence: 99%

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Bifurcation analysis for a fractional‐order chemotherapy model with two different delays

Liao

et al. 2019

Math Methods in App Sciences

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Cancer is one of the most serious diseases in the world. The investigation on cancer treatment has attracted great attention from medical workers, mathematical researchers, and scholars in various fields. To understand the intrinsic characteristics of cancer, numerous scholars have established cancer treatment models and discussed the dynamical properties. However, the main work of many scholars only focuses on the integer-order cancer treatment models, while the study on fractional-order ones is quite a few. In the present article, on the basis of previous publications, we will put up a new fractional-order chemotherapy model with two different delays. By applying the stability theory and the Hopf bifurcation of fractional-order differential equations, we obtain a series of new stability criteria of the involved model and some sufficient conditions to ensure the existence of Hopf bifurcation of the involved model. Furthermore, the impact of two different delays and the fractional order on the stability and the emergence of Hopf bifurcation of involved model is presented.To illustrate the validity of analytical predictions, we perform computer simulations with Matlab software. The theoretical results in the manuscript are innovative and play an important role in cancer treatment.

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Section: Basic Knowledgementioning

confidence: 96%

“…(2) If (H4) is fulfilled and ε i >0(i=1, 2,3,4,5), then (3.21) has a pair of purely imaginary roots ±iβ 0 when ϱ 1 =ϱ 1h (l=1,2,…,) where…”

Section: Basic Knowledgementioning

confidence: 99%

“…Lemma 3.7. For Equation (3.5), the following results are true: 2,3,4,5,6)>0, then Equation (3.5) has no root with zero real parts. 2.…”

Section: Basic Knowledgementioning

confidence: 99%

Section: Basic Knowledgementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Bifurcation analysis for a fractional‐order chemotherapy model with two different delays

Liao

et al. 2019

Math Methods in App Sciences

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A numerical approach of tumor‐immune model with B cells and monoclonal antibody drug by multi‐step differential transformation method

Dhar

Gupta

2020

Math Methods in App Sciences

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Monoclonal antibody (mAb) drugs are used to kill malignant tumors by making them able to see into the immune system. In this article, we developed a mathematical model with the help of tumor‐immune system with antibodies effect and efficacy of mAb drug. We defined the existence of solutions of the model, and by the Lyapunov method, we showed that the solutions are bounded; thereafter, we illustrated the local and global stability phenomenon, which is described by the Lyapunov method and localization of compact invariant sets (LCIS) method for disease‐free equilibrium points including the biological significance. In actual fact, this model suggested some ideas about the malignant tumors reciprocate to mAb drug dosage and its efficacy, eventually to overcome malignancy. It is likely to control the steadiness of the tumor, which is very important for any kind of treatment. The numerical calculations for the proposed model have been carried out by multi‐step differential transformation method (Ms‐DTM). The simulation of the model is depicted through Figures 1–5, which shows the effect of interaction rate of tumor cells and antibodies and dose of the drug for various values of efficacy.

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A Mathematical Model of Cancer Evolutionary Escape

Valle

Starkov

Korobeinikov

2018

Trends in Mathematics

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Global stability and tumor clearance conditions for a cancer chemotherapy system

Cited by 26 publications

References 19 publications

Bifurcation analysis for a fractional‐order chemotherapy model with two different delays

Bifurcation analysis for a fractional‐order chemotherapy model with two different delays

A numerical approach of tumor‐immune model with B cells and monoclonal antibody drug by multi‐step differential transformation method

A Mathematical Model of Cancer Evolutionary Escape

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