Lyapunov functions for fractional-order systems in biology: Methods and applications

• Published in: Chaos, solitons and fractals solitons and fractals, 2020-11, Vol.140, p.110224, Article 110224
• Main Authors: Boukhouima, Adnane, Hattaf, Khalid, Lotfi, El Mehdi, Mahrouf, Marouane, Torres, Delfim F.M., Yousfi, Noura
• Format: Article
• Language: English
• Subjects: Caputo derivatives; Fractional calculus; Lyapunov analysis; Mathematical biology; Nonlinear ordinary differential equations; Stability
• ISSN: 0960-0779; 1873-2887
• DOI: 10.1016/j.chaos.2020.110224

•New estimates for fractional Caputo derivatives.•General Lyapunov functions for fractional-order systems.•Global dynamics and stability of fractional epidemic systems.•Fractional population and cellular HIV models.•Real world applications in biology and medicine. We prove new estimates of the Caputo derivative of order α ∈ (0, 1] for some specific functions. The estimations are shown useful to construct Lyapunov functions for systems of fractional differential equations in biology, based on those known for ordinary differential equations, and therefore useful to determine the global stability of the equilibrium points for fractional systems. To illustrate the usefulness of our theoretical results, a fractional HIV population model and a fractional cellular model are studied. More precisely, we construct suitable Lyapunov functionals to demonstrate the global stability of the free and endemic equilibriums, for both fractional models, and we also perform some numerical simulations that confirm our choices.