Volterra-type Lyapunov functions for fractional-order epidemic systems

• We prove an lemma to Caputo fractional derivatives of Lyapunov functions.
• We estimate the derivatives in order 0 < α < 1 to Volterra-type functions.
• We prove the uniform asymptotic stability of some epidemic systems.
• We illustrate our theoretical results with numerical simulations.
• Our method could be useful in the stability analysis of epidemic systems.

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)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.