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

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.

arXiv commentThis is a preprint of a paper whose final and definite form is published in 'Chaos, Solitons & Fractals', available at [http://dx.doi.org/10.1016/j.chaos.2020.110224]. Submitted 25-May-2020; Revised 25-July-2020; Accepted 19-August-2020

References in corpus (12)

• Fractional calculus and continuous-time finance
• Reaction Front in an A+B -> C Reaction-Subdiffusion Process
• Memory effects on epidemic evolution: The susceptible-infected-recovered epidemic model
• A survey on fuzzy fractional differential and optimal control nonlocal evolution equations
• A SICA compartmental model in epidemiology with application to HIV/AIDS in Cape Verde
• Optimal Control of a Tuberculosis Model with State and Control Delays
• Stability of a fractional HIV/AIDS model
• Optimal control of a fractional order epidemic model with application to human respiratory syncytial virus infection
• Stability and Optimal Control of a Delayed HIV Model
• Uniform asymptotic stability of a fractional tuberculosis model
• Dynamics of the Chain of Oscillators with Long-Range Interaction: From Synchronization to Chaos
• Analysis of a SIRI epidemic model with distributed delay and relapse

Cited by in corpus (2)

• Lyapunov Functions and Stability Analysis of Fractional-Order Systems
• A review of commonly used compartmental models in epidemiology