Narrow Results

We propose a simple model of tumor-immune interactions, which involves effector cells and tumor cells. In the model, the stimulation delay of tumor antigen in the immune system is incorporated. We investigate the dynamical behavior of the model via theoretic analysis and numerical simulations. The saddle-node bifurcation can occur in both cases with and without delay. In contrast to the case without delay, stimulation delay may result in some complex dynamical behaviors and biological phenomena. In the presence of delay, conditions on absolute/conditional stability of equilibria and the existence of Hopf bifurcations are obtained. We further discuss the effect of the tumor on the switch between absolute stability and conditional stability. Numerical simulations also show the existence of homoclinic bifurcation and the dependence of the asymptotic state of the tumor progression on initial conditions for different delay values. Effects of delay on the dynamics of the model and on the region of tumor extinction are illustrated by simulations with different sets of parameter values. Finally, the corresponding biological implications are demonstrated.

One main feature of a malignant tumor is its uncontrolled growth. In this paper, we propose a simple tumor-immune model to study the progressive characteristics of malignant and benign tumors, where the anti-tumor immunity can be described by the Michaelis–Menten function or the mass action law. The model includes only two state variables for the tumor cells and the effector cells representing the immune system. Three quantities with clear biological meanings are given to determine the asymptotic states of the tumor progression. Moreover, differences in asymptotic states between the two anti-tumor immunity descriptions are drawn. Differently from existing simple models, on the one hand, the model exhibits rich dynamical behaviors including super-critical and sub-critical Bogdanov–Takens bifurcations (consisting of Hopf bifurcation, saddle–node bifurcation, and homoclinic bifurcation) and saddle–node bifurcation of nonconstant periodic solutions (leading to the appearance of two periodic orbits) as the parameters vary; on the other hand, the malignant feature, dormancy, and immune escape of the tumor are revealed with numerical simulations. Furthermore, from the perspective of qualitative analysis and numerical simulations, how the obtained results can be applied to the treatment and control of tumors is illustrated.

Dynamical analysis of system of ordinary differential equations (ODEs) is an interesting topic among researchers and several tools have been discovered since so far to deal with its several other features. There are many built-in functions designed in MATLAB that researchers can use as a tool for the system of ODEs with integer order but in the similar systems have fractional-order derivative is a difficult task. Therefore, the purpose of this work is to use the Caputo fractional-order derivative to qualitatively analyze and then numerically simulate the tumor-immune system interaction model. Moreover, the Schauder fixed point theorem is used to establish the condition for the existence of at least one solution, while the Banach fixed point theorem is utilized to guarantee the existence of a unique solution. Moreover, the stability in the trajectories of considered system achieved with the aid of Ulam–Hyers (UH) stability. In addition, the numerical computations are performed using Variational Iteration Method (VIM) and the visualized results are shown using MATLAB.

Induction of antitumor immunity by vaccination is one of the major current immunotherapy strategies. We present a mathematical model of the competition between immune cells and mammary carcinogenesis under the effect of Triplex vaccine. The model describes both humoral and cell-mediated immune responses against cancer cells. The control of the cancer cells growth occurs through the application of the pulse vaccination. Here we determine the relationship between the strength of the vaccine and the time required to eradicate cancer cells, and we present some simulations to illustrate our theoretical results, namely, the total cancer cells depletion, which is influenced by competition occurs among the immune and cancer cells.

In this paper, we consider tumor-immune interaction model systems. The numerical solutions for the tumor-immune interaction system are obtained by using the 2-point Block Backward Differentiation Formula (BBDF) methods developed by Zarina et al. in 2007. The numerical results are presented in terms of computational time and accuracy of the solutions.