Volterra-type Lyapunov functions for fractional-order epidemic systems

Cruz Vargas-De-León

doi:10.1016/j.cnsns.2014.12.013

Volterra-type Lyapunov functions for fractional-order epidemic systems

Cruz Vargas-De-León

Research output: Contribution to journal › Article › peer-review

300 Scopus citations

Abstract

In this paper we prove an elementary lemma which estimates fractional derivatives of Volterra-type Lyapunov functions in the sense Caputo when α. ∈. (0, 1). Moreover, by using this result, we study the uniform asymptotic stability of some Caputo-type epidemic systems with a pair of fractional-order differential equations. These epidemic systems are the Susceptible-Infected-Susceptible (SIS), Susceptible-Infected-Recovered (SIR) and Susceptible-Infected-Recovered-Susceptible (SIRS) models and Ross-Macdonald model for vector-borne diseases. We show that the unique endemic equilibrium is uniformly asymptotically stable if the basic reproductive number is greater than one. We illustrate our theoretical results with numerical simulations using the Adams-Bashforth-Moulton scheme implemented in the fde12 Matlab function.

Original language	English
Pages (from-to)	75-85
Number of pages	11
Journal	Communications in Nonlinear Science and Numerical Simulation
Volume	24
Issue number	1-3
DOIs	https://doi.org/10.1016/j.cnsns.2014.12.013
State	Published - 1 Jul 2015
Externally published	Yes

Keywords

Caputo fractional derivative
Direct lyapunov method
Epidemiological models
Stability
Volterra-type lyapunov function

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10.1016/j.cnsns.2014.12.013

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abstract = "In this paper we prove an elementary lemma which estimates fractional derivatives of Volterra-type Lyapunov functions in the sense Caputo when α. ∈. (0, 1). Moreover, by using this result, we study the uniform asymptotic stability of some Caputo-type epidemic systems with a pair of fractional-order differential equations. These epidemic systems are the Susceptible-Infected-Susceptible (SIS), Susceptible-Infected-Recovered (SIR) and Susceptible-Infected-Recovered-Susceptible (SIRS) models and Ross-Macdonald model for vector-borne diseases. We show that the unique endemic equilibrium is uniformly asymptotically stable if the basic reproductive number is greater than one. We illustrate our theoretical results with numerical simulations using the Adams-Bashforth-Moulton scheme implemented in the fde12 Matlab function.",

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AB - In this paper we prove an elementary lemma which estimates fractional derivatives of Volterra-type Lyapunov functions in the sense Caputo when α. ∈. (0, 1). Moreover, by using this result, we study the uniform asymptotic stability of some Caputo-type epidemic systems with a pair of fractional-order differential equations. These epidemic systems are the Susceptible-Infected-Susceptible (SIS), Susceptible-Infected-Recovered (SIR) and Susceptible-Infected-Recovered-Susceptible (SIRS) models and Ross-Macdonald model for vector-borne diseases. We show that the unique endemic equilibrium is uniformly asymptotically stable if the basic reproductive number is greater than one. We illustrate our theoretical results with numerical simulations using the Adams-Bashforth-Moulton scheme implemented in the fde12 Matlab function.

KW - Caputo fractional derivative

KW - Direct lyapunov method

KW - Epidemiological models

KW - Stability

KW - Volterra-type lyapunov function

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DO - 10.1016/j.cnsns.2014.12.013

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VL - 24

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JO - Communications in Nonlinear Science and Numerical Simulation

JF - Communications in Nonlinear Science and Numerical Simulation

IS - 1-3

ER -

Volterra-type Lyapunov functions for fractional-order epidemic systems

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Keywords

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