Narrow Results

In this study, we develop a new dynamical model of cancer growth, which includes the interactions between tumour cells, healthy tissue cells, and activated immune system cells, clearly leading to chaotic behavior. We explain the biological relevance of our model and the ways in which it differs from the existing ones. We perform equilibria analysis, indicate the conditions where chaotic dynamics can be observed, and show rigorously the existence of chaos by calculating the Lyapunov exponents and the Lyapunov dimension of the system. Moreover, we demonstrate that Shilnikov's theorem is valid in the parameter range of interest.

We study the structure of the periodic orbits on a simple branched manifold which is a subtemplate of the branched manifold of the chaotic attractor that is obtained from a cancer model. We indicate the conditions for the knotted and linked periodic orbits by using symbolic dynamics, then we extend the results to the periodic solutions of the chaotic attractor. Furthermore, we compare the knots obtained by using same symbol sequences for the simple branched manifold and the attractor's branched manifold by using a knot invariant, specifically, we calculate the Kauffman bracket polynomial. In order to count the number of closed curves which is required to calculate the bracket polynomial we propose a new method which uses cyclic permutations.

A mathematical model describing interactions among tumor cells, healthy host cells and immune cells is extensively investigated through bifurcation analysis with all parameters fixed except one bifurcation parameter. Transcritical bifurcation and saddle-node bifurcation are studied in the vector fields restricted to the corresponding center manifolds. Hopf bifurcation is analyzed in the frequency domain. In particular, the sixth-order harmonic balance approximations to the frequency and the amplitude of the periodic solutions, and the analytical expressions for these solutions are given. Numerical simulation study demonstrates various types of bifurcations and the complex solution behaviors, such as cyclic fold bifurcation, period-doubling bifurcation, period-doubling cascade and chaotic orbits. All these results complement previous theoretical studies on the model, and contribute to a better understanding of the qualitative dynamics of the cancer model.

In this paper, we have investigated women’s malignant disease, cervical cancer, by constructing the compartmental model. An extended fractal–fractional model is used to study the disease dynamics. The points of equilibria are computed analytically and verified by numerical simulations. The key role of $R_0$ in describing the stability of the model is presented. The sensitivity analysis of $R_0$ for deciding the role of certain parameters altering the disease dynamics is carried out. The numerical simulations of the proposed numerical technique are demonstrated to test the claimed facts.

In this paper, we develop a three-dimensional fractional-order cancer model. The proposed model involves the interaction among tumor cells, healthy tissue cells and activated effector cells. The detailed analysis of the equilibrium points is studied. Also, the existence and uniqueness of the solution are investigated. The fractional derivative is considered in the Caputo sense. Numerical simulations are performed to illustrate the effectiveness of the obtained theoretical results. The outcome of the study reveals that the order of the fractional derivative has a significant effect on the dynamic process. Further, the calculated Lyapunov exponents give the existence of chaotic behavior of the proposed model. Also, it is observed from the obtained results that decrease in fractional-order $\rho$ increases the chaotic behavior of the model.

Multiscale, agent-based mathematical models of biological systems are often associated with model uncertainty and sensitivity to parameter perturbations. Here, three uncertainty and sensitivity analyses methods, that are suitable to use when working with agent-based models, are discussed. These methods are namely Consistency Analysis, Robustness Analysis and Latin Hypercube Analysis. This introductory review discusses origins, conventions, implementation and result interpretation of the aforementioned methods. Information on how to implement the discussed methods in MATLAB is included.

In an earlier paper, we developed a free boundary model to describe the growth of ductal carcinoma in situ (DCIS). Assuming that we know the coefficients of the model, we analyzed the growth tendency of DCIS. The analysis and computation of the problem show interesting results that are similar to the patterns found in DCIS. In this paper we consider an inverse problem of determining the coefficient function of the free boundary model of DCIS from information obtained by incisional biopsy. Assuming that from incisional biopsy we observe the tumor pattern and some information of its growing rate, which is corresponding to the solution of the free boundary problem and some information of its derivative at a time, we consider the inverse problem of finding the potential function of the free boundary problem. We show that mathematically we can determine the coefficient of the model from incisional biopsy information, hence we may develop mathematical methods to diagnose growth tendency of DCIS from incisional biopsy information.