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

Abstract

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.

Fractional-Order Derivative Model of Rift Valley Fever in Urban Peridomestic Cycle

“…where x * is the equilibrium point of the model systems (9a)-(9f ). en, we have the following corollary as in the study by Bourkhouima et al [10]. Corollary 1.…”

“…Modelling with fractional-order derivatives has motivated many people involved in research in science including epidemiology. is is evident from the work by Silva and Torres [11], Boukhouima et al [10], Ghanbari et al [12], Alkahtani and Alzaid [13], Tuan et al [14], and Khan and Atangana [15], just to mention a few. Motivated by their work, a fractional-order derivative model of RVF infection in Caputo derivative is presented in this paper.…”

“…Fractional-order derivatives are a generalization of the integer-order derivative and integrals to an arbitrary real or complex order. ey are powerful mathematical tools which can be used to model the real-world phenomenon which involve memory, wide range of interactions, and hereditary properties, a property which is neglected by integer-order derivatives [10].…”

T-S Fuzzy Control of Uncertain Fractional-Order Systems with Time Delay

“…In the last century, fractional-order calculus has been widely used in many electrical systems, robot systems, engineering systems, and thermodynamic systems, which has attracted extensive attention [1][2][3][4][5]. e fractional-order system (FOS) has more advantages than classical integer-order system [6][7][8]. Firstly, FOSs focus on the characteristics of the whole time domain.…”

Focus point: cancer and HIV/AIDS dynamics—from optimality to modelling

“…Furthermore, the research team [13] describes the dynamics of HIV-AIDS infection by investigating the celebrated mathematical SICA model via a fractional order system. Stability and instability of equilibrium points are determined in terms of the basic reproduction number.…”