Delay-Induced Hopf Bifurcation and Periodic Dynamics in an Epidemic Model with  Nonlinear Incidence

Neeraj Kumar

doi:10.71097/IJSAT.v9.i2.11618

Delay-Induced Hopf Bifurcation and Periodic Dynamics in an Epidemic Model with Nonlinear Incidence

Author(s)	Neeraj Kumar
Country	India
Abstract	Time delays are among the most natural sources of oscillation in epidemic models: incubation, latency, and the delayed response of the transmission process itself can destabilize an otherwise stable endemic state and force the disease to persist in sustained periodic waves. In this article we study a susceptible–infected–recovered (SIR) model in which the force of infection is subject to a discrete transmission delay and to a saturated nonlinear incidence rate of the form βSI\/(1+αI). The basic reproduction number is computed, and the unique endemic equilibrium is shown to be locally asymptotically stable when the delay vanishes. Linearization at the endemic state leads to the transcendental characteristic equation λ^2+pλ+q+(rλ〖+s)e〗^(-λτ)=0, whose purely imaginary roots are located explicitly and converted into a sequence of critical delays. A transversality condition is verified, and a Hopf bifurcation theorem for the model is proved. The direction and the stability of the bifurcating periodic solutions are determined by the normal form method on the center manifold. A numerical example illustrates the theory: the endemic equilibrium is stable for small delays, loses stability through a subcritical Hopf bifurcation at the first critical delay, and the bifurcating periodic orbits are unstable with decreasing period. The results surveyed and proved here are classical; detailed background may be found in the references.
Keywords	time delay; nonlinear incidence; SIR epidemic model; Hopf bifurcation; stability switch; periodic solution; normal form; center manifold.
Published In	Volume 9, Issue 2, April-June 2018
Published On	2018-04-07
DOI	https://doi.org/10.71097/IJSAT.v9.i2.11618

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Delay-Induced Hopf Bifurcation and Periodic Dynamics in an Epidemic Model with Nonlinear Incidence

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