A fractional-order epidemic model with time-delay and nonlinear incidence rate

•We provide an epidemic SIR model with long-range temporal memory governed by delay differential equations with fractional-order.•The existence of steady states and the asymptotic stability of the steady states are discussed.•The occurrence of Hopf bifurcation is captured when the time-delay passes...

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Bibliographic Details
Published in:Chaos, solitons and fractals Vol. 126; pp. 97 - 105
Main Authors: Rihan, F.A., Al-Mdallal, Q.M., AlSakaji, H.J., Hashish, A.
Format: Journal Article
Language:English
Published: Elsevier Ltd 01.09.2019
Subjects:
Bifurcation theory
Fractional calculus
SIR
Stability
Time-delay
Time-delay
Bifurcation theory
Stability
SIR
Fractional calculus
ISSN:0960-0779, 1873-2887
Online Access:Get full text
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Summary:•We provide an epidemic SIR model with long-range temporal memory governed by delay differential equations with fractional-order.•The existence of steady states and the asymptotic stability of the steady states are discussed.•The occurrence of Hopf bifurcation is captured when the time-delay passes through a critical value.•Theoretical results are validated numerically. In this paper, we provide an epidemic SIR model with long-range temporal memory. The model is governed by delay differential equations with fractional-order. We assume that the susceptible is obeying the logistic form in which the incidence term is of saturated form with the susceptible. Several theoretical results related to the existence of steady states and the asymptotic stability of the steady states are discussed. We use a suitable Lyapunov functional to formulate a new set of sufficient conditions that guarantee the global stability of the steady states. The occurrence of Hopf bifurcation is captured when the time-delay τ passes through a critical value τ*. Theoretical results are validated numerically by solving the governing system, using the modified Adams-Bashforth-Moulton predictor-corrector scheme. Our findings show that the combination of fractional-order derivative and time-delay in the model improves the dynamics and increases complexity of the model. In some cases, the phase portrait gets stretched as the order of the derivative is reduced.
ISSN:0960-0779
1873-2887
DOI:10.1016/j.chaos.2019.05.039