Bifurcation analysis in a model of cytotoxic T-lymphocyte response to viral infections

Chan, Bernard S.; Yu, Pei

doi:10.1016/j.nonrwa.2011.07.012

Cited by 26 publications

(23 citation statements)

References 14 publications

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“…7. From this The optimal control graph for the objective functional (J) using the parameter values given in Table 3 with B 1 = 5, B 2 = 30 figure, we can observe that the chemotherapeutic drug control (u 1 ) and immunotherapeutic drug control (u 2 ) minimize the objective functional (J ) given in (7). Overall the numerical analysis demonstrates that two controls u 1 (t) and u 2 (t) decrease the tumor burden minimizing total drug administered.…”

Section: Numerical Simulationsmentioning

confidence: 97%

“…Hence they are balanced by choosing weight constants B 1 = 5, B 2 = 30 in the objective functional given in (7). The numerical results for the Table 2 optimal problem are obtained by using the parameter values given in Table 3 [2,6,9,14,26,32].…”

Section: Numerical Simulationsmentioning

confidence: 99%

“…de Pillis and Radunskaya [10] presented a mathematical model, showing competition between normal cells and tumor cells considering the role of chemotherapeutic drugs. There are some other research works on tumor-immune dynamics [1,3,4,7,12,13,15,31,33,36,38,40]. Also there are some research works on the tumor growth models with optimal control strategies [8,10,11,[16][17][18][19]30,35].…”

Section: Introductionmentioning

confidence: 99%

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Analysis of the Dynamics of a Tumor–Immune System with Chemotherapy and Immunotherapy and Quadratic Optimal Control

Sharma

Samanta

2015

Differ Equ Dyn Syst

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In this paper, we consider a tumor growth model with the effect of tumor-immune interactions and chemotherapeutic as well as immunotherapeutic drugs. In our model there are four compartments, namely tumor cells, immune cells, chemotherapeutic drug concentration and immunotherapeutic drug concentration. The dynamical behaviour of our system by analyzing the existence and stability of our system at various equilibria is discussed elaborately. We set up an optimal control problem relative to the model so as to minimize the number of tumor cells and the chemotherapeutic and immunotherapeutic drugs administration. Here we use a quadratic control to quantify this goal and consider the administration of chemotherapy and immunotherapeutic drugs as controls to reduce the spread of the tumor growth. The important mathematical findings for the dynamical behaviour of the tumor-immune model are also numerically verified using MATLAB. Finally epidemiological implications of our analytical findings are addressed critically.

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Section: Numerical Simulationsmentioning

confidence: 97%

Section: Numerical Simulationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Analysis of the Dynamics of a Tumor–Immune System with Chemotherapy and Immunotherapy and Quadratic Optimal Control

Sharma

Samanta

2015

Differ Equ Dyn Syst

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“…Many mathematical models have been proposed for dynamics of HIV infection [6,16,21,27]. In 2000, Arnout et al [6] conclude that during the administration of antiretroviral drugs, priming the CTL response can lead to better immune responses from the patients.…”

Section: Introductionmentioning

confidence: 99%

“…The authors suggedt that CTLs are responsible for the killing of a considerable number of infected T helper cells, but nonetheless the majority of infected cells are killed by other cells of the immune system. Chan et al [16] study a model for primary and secondary CTL response cells. The model exhibits rich dynamics and the authors observed oscillatory patterns for a wide parameter range.…”

Section: Introductionmentioning

confidence: 99%

A latency fractional order model for HIV dynamics

Pinto

Carvalho

2017

Journal of Computational and Applied Mathematics

109

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We study a fractional order model for HIV infection where latent T helper cells are included. We compute the reproduction number of the model and study the stability of the disease free equilibrium. We observe that the reproduction number varies with the order of the fractional derivative α. In terms of epidemics, this suggests that varying α induces a change in the patients' epidemic status. Moreover, we simulate the variation of relevant parameters, such as the fraction of uninfected CD4 + T cells that become latently infected, and the CTLs proliferation rate due to infected CD4 + T cells. The model produces biologically reasonable results.

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Bifurcation analysis of HIV infection model with antibody and cytotoxic T‐lymphocyte immune responses and Beddington–DeAngelis functional response

Balasubramaniam

Tamilalagan

Mani

2014

Math Methods in App Sciences

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Communicated by M. EfendievIn this paper, a mathematical model for HIV-1 infection with antibody, cytotoxic T-lymphocyte immune responses and Beddington-DeAngelis functional response is investigated. The stability of the infection-free and infected steady states is investigated. The basic reproduction number R 0 is identified for the proposed system. If R 0 < 1, then there is an infectionfree steady state, which is locally asymptotically stable. Further, the infected steady state is locally asymptotically stable for R 0 > 1 in the absence of immune response delay. We use Nyquist criterion to estimate the length of the delay for which stability continues to hold. Also the existence of the Hopf bifurcation is investigated by using immune response delay as a bifurcation parameter. Numerical simulations are presented to justify the analytical results. Stability analysis of the infection-free steady state (E 0 )If R 0 < 1, then an infected cell produces less than one new infected cell over the course of its infectious period. Hence, the infection cannot persist. Now, we state and prove the following theorem related to R 0 < 1 for system (3).Theorem 2.1 If R 0 < 1, then the infection-free steady state E 0 is locally asymptotically stable. ProofThe infection-free steady state E 0 .x 0 , y 0 , v 0 , w 0 , z 0 / of the system (3) is given by E 0 D d , 0, 0, 0, 0 .

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Bifurcation analysis in a model of cytotoxic T-lymphocyte response to viral infections

Cited by 26 publications

References 14 publications

Analysis of the Dynamics of a Tumor–Immune System with Chemotherapy and Immunotherapy and Quadratic Optimal Control

Analysis of the Dynamics of a Tumor–Immune System with Chemotherapy and Immunotherapy and Quadratic Optimal Control

A latency fractional order model for HIV dynamics

Bifurcation analysis of HIV infection model with antibody and cytotoxic T‐lymphocyte immune responses and Beddington–DeAngelis functional response

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