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

TL;DR

This paper develops new methods for constructing Lyapunov functions for fractional-order biological systems, enabling the analysis of their global stability through theoretical estimates and practical models.

Contribution

It introduces new estimates of the Caputo derivative for specific functions, facilitating Lyapunov-based stability analysis of fractional biological systems.

Findings

Lyapunov functions successfully demonstrate stability in fractional HIV and cellular models.

Theoretical estimates aid in constructing Lyapunov functionals for fractional differential equations.

Numerical simulations confirm the effectiveness of the proposed stability analysis methods.

Abstract

We prove new estimates of the Caputo derivative of order $\alpha \in (0,1]$ 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.