American Institute of Mathematical Sciences

• Previous Article
Non-autonomous stochastic evolution equations of parabolic type with nonlocal initial conditions
• DCDS-B Home
• This Issue
• Next Article
Modeling multiple taxis: Tumor invasion with phenotypic heterogeneity, haptotaxis, and unilateral interspecies repellence

Qualitative analysis of a simple tumor-immune system with time delay of tumor action

 1 Department of Mathematics, Shaanxi University of Science and Technology, Xi'an, 710021, China 2 Department of Immunology, Xi'an Medical University, Xi'an 710021, China 3 Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, China, Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899, USA

* Corresponding author: Jianquan Li; ** Equal contributor: Dian Zhang

Received  May 2019 Revised  October 2020 Published  November 2020

Fund Project: Jianquan Li is supported by the National Natural Science Fund of China (Nos. 11971281, 12071268), Dian Zhang is supported by the Project of Xi'an Medical University (No. 2018GJFY05), Jia Li is supported by the National Natural Science Fund of China (No. 12071095), Xiaolin Lin is supported by the National Natural Science Fund of China (No. 11771259)

In this paper, we propose a simple tumor-immune system model with time delay of tumor action, where two kinds of effects of the tumor cells (i.e. stimulation and neutralization) on the effector cells are considered. The local stability of the model is obtained by analyzing the characteristic equations of the model at the corresponding equilibria, the sufficient conditions on the global stability are found by applying the Fluctuation Lemma and constructing the different convergent sequences. The obtained results show that, compared to the results for the model without time delay, the time delay of tumor action can affect the stability of tumor equilibrium of the model as the stimulation effect of the tumor cells is strong enough, while the delay is harmless for the stability of tumor equilibrium under the neutralization of tumor cells. For the appropriate neutralization of tumor cells on effector cells, the bistability of the tumor free equilibrium and the stronger tumor equilibrium can appear. In the case of stimulation of tumor cells, the sufficiently large time delay can lead to the appearance of a stable periodic solution by Hopf bifurcation, and the numerical simulation illustrates that the amplitude of the periodic orbit increases with time delay. We also discuss the dependence of the tumor equilibrium and the time delay threshold, determining the stability of the tumor equilibrium, on tumor action. The related conditions determining dynamics of the model are expressed by certain formulae with biological meanings.

Citation: Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020341
References:

show all references

References:
The stable periodic orbit of model (6) as ${\mu} K>\eta$ and $n{\sigma}<r\eta$. Here, $r = 2.5, K = 2, n = 0.8, {\sigma} = 0.5, {\mu} = 4$, $\eta = 1.5$ and ${\tau} = 0.3>0.1943 = {\tau}_0.$ Model (6) has a unique tumor equilibrium $P^*(0.3272, 2.6138)$, and a stable periodic orbit
The bifurcation diagram of model (6) with respect to the time delay ${\tau}$ when ${\mu} K> \eta$ and $n{\sigma}<r\eta$. Here, except for the time delay ${\tau}$, the values of other parameters are the same as those for Figure 1
The trajectories of $T = T(t)$ and $E = E(t)$ of model (6) corresponding to three delays ${\tau} = 1, 5, 10$ in case of $0<{\mu}<\frac{\eta}{K}$ and $n{\sigma}<r\eta$, respectively. Other parameter values are $r = 2.5, K = 2, n = 0.8, {\sigma} = 0.5, {\mu} = 0.4$ and $\eta = 1.5$
The trajectories of model (6) with two delays ${\tau} = 0.3, 2.0$ in Case C4 of Table 1, where the left column is for ${\tau} = 0.3$, the right column is for ${\tau} = 2.0$, and the initial conditions of model (6) are the same in the two columns. The first row is the orbits of model (6) in the $T$-$E$ plane, the second row is the trajectories of $T = T(t)$, and the third row is those of $E = E(t)$
The graph of function ${\tau}_0 = {\tau}_0({\mu})$ defined by (16) with ${\mu} >\frac{\eta}{K}$. Here, except for ${\mu}$ and ${\tau}$, the values of other parameters are the same as those for Figure 1, and the regions $D_1$ and $D_2$ correspond to the stable and unstable cases of the tumor equilibrium $P^*$, respectively
The existence and the stability of all feasible equilibria of model (2) without time delay. Here, US stands for unstable, GAS-1 globally asymptotically stable in the first quadrant of the $T$-$E$ plane, GAS-2 globally asymptotically stable on the set $\{(T, E): T\ge 0, E>0\}$, LAS locally asymptotically stable, SN saddle-node
 Case Conditions $P_0$ $P^*$ $P_*$ $P^*_*$ C1 ${\mu}>0, n{\sigma}0, n{\sigma}\ge r\eta$ GAS-2 — — — C7 $\begin{array}{c} {\mu}<0, \eta+ {\mu} K< 0, \\n{\sigma}>-\frac{rK}{4{\mu}}( {\mu}-\frac{\eta}{K})^2 \end{array}$ GAS-2 — — —
 Case Conditions $P_0$ $P^*$ $P_*$ $P^*_*$ C1 ${\mu}>0, n{\sigma}0, n{\sigma}\ge r\eta$ GAS-2 — — — C7 $\begin{array}{c} {\mu}<0, \eta+ {\mu} K< 0, \\n{\sigma}>-\frac{rK}{4{\mu}}( {\mu}-\frac{\eta}{K})^2 \end{array}$ GAS-2 — — —
 [1] Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074 [2] Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468 [3] Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216 [4] Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457 [5] Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164 [6] Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020342 [7] Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432 [8] Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048 [9] Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450 [10] Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020275 [11] Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103 [12] A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441 [13] Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321 [14] Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046 [15] Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045 [16] Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020273 [17] Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137 [18] Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020336 [19] Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339 [20] Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

2019 Impact Factor: 1.27

Tools

Article outline

Figures and Tables