Stability analysis and numerical simulations via fractional calculus for tumor dormancy models

• Published in: Communications in nonlinear science & numerical simulation Vol. 72; pp. 528 - 543
• Main Authors: Silva, Jairo G., Ribeiro, Aiara C.O., Camargo, Rubens F., Mancera, Paulo F.A., Santos, Fernando L.P.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 30.06.2019; Elsevier Science Ltd
• Subjects: Cancer; Caputo derivatives; Computer simulation; Damping; Differential calculus; Differential equations; Equilibrium; Finite difference method; Fractional calculus; Fractional stability; Immune system; Mathematical models; Numerical analysis; Stability analysis; Systems stability; Tumor dormancy; Tumors; Fractional stability; Tumor dormancy; Caputo derivatives; Immune system
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2019.01.021

•The fractional modeling produces new and challenging scenarios for tumor dormancy.•The order of the derivative can trigger qualitative changes in the solutions.•Fractional models bring scenarios with greater efficiency of the immune system.•Smaller order of the derivative can illustrate a longer time for cancer escape. Fractional calculus is a field of mathematics in considerable expansion and has been understood as a tool with a wide range of applications, including in biological systems. Cancer dormancy is a state in which cancer cells have an intrinsic rate of reduced proliferation over a period of time, after which they arise with an accelerated growth rate, usually triggering metastases. This work investigates via fractional calculus two proposed ordinary differential equation systems via fractional calculus, which address dynamics between tumor cells and the immune system. Analyses and comparison processes are performed through an analytical study about the equilibrium points of each model and numerical simulations by the Nonstandard Finite Difference (NSFD) method. The analytical and numerical results show two important behaviors associated with tumor dormancy. The first refers to qualitative changes in the stability of an equilibrium point, which strongly depend on the order of the derivative and represent scenarios where there is no cancer escape. The second behavior refers to changes in the damping of some solutions, which can represent longer periods of time to exit the state of dormancy. In this case, different orders used in the derivatives are investigated, as well as their influence on the behavior of the solutions against the main parameters of the model.