TY - JOUR
T1 - Volterra-type Lyapunov functions for fractional-order epidemic systems
AU - Vargas-De-León, Cruz
N1 - Publisher Copyright:
© 2014 Elsevier B.V.
PY - 2015/7/1
Y1 - 2015/7/1
N2 - 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.
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
UR - http://www.scopus.com/inward/record.url?scp=84924986295&partnerID=8YFLogxK
U2 - 10.1016/j.cnsns.2014.12.013
DO - 10.1016/j.cnsns.2014.12.013
M3 - Artículo
SN - 1007-5704
VL - 24
SP - 75
EP - 85
JO - Communications in Nonlinear Science and Numerical Simulation
JF - Communications in Nonlinear Science and Numerical Simulation
IS - 1-3
ER -