Hopf bifurcation of a delayed worm model with two latent periods

Juan Liu,Zizhen Zhang

doi:10.1186/s13662-019-2372-1

Abstract

We investigate a delayed epidemic model for the propagation of worm in wireless sensor network with two latent periods. We derive sufficient conditions for local stability of the worm-induced equilibrium of the system and the existence of Hopf bifurcation by regarding different combination of two latent time delays as the bifurcation parameter and analyzing the distribution of roots of the associated characteristic equation. In particular, we investigate the direction and stability of the Hopf bifurcation by means of the normal form theory and center manifold theorem. To verify analytical results, we present numerical simulations. Also, the effect of some influential parameters of sensor network is properly executed so that the oscillations can be reduced and removed from the network.

Highlights

Coupled with the progress of the digital era and increasing development of various network applications, networks have become more and more popular in our daily life [1,2,3]
Some scholars at home and abroad formulated and investigated various mathematical models to study the spread of malicious codes in wireless sensor networks
We investigate the direction and stability of the Hopf bifurcation at the worm-induced equilibrium when τ2 > 0 and τ1 ∈ (0, τ10) by using the normal form theory and center manifold theorem

Summary

Introduction

Coupled with the progress of the digital era and increasing development of various network applications, networks have become more and more popular in our daily life [1,2,3]. Different types of malicious codes are available in the digital environment, and they require no any human intervention or infrastructure network for transmission Based on this consideration, Ojha et al [25] proposed the following model for the transmission of worm in wireless sensor with two latent periods:. Theorem 2 For system (3), if conditions (H1), (H31), and (H32) hold, system (3) is locally asymptotically stable when τ2 ∈ [0, τ20); system (3) undergoes a Hopf bifurcation at the worm-induced equilibrium E∗ when τ2 = τ20, and a family of periodic solutions bifurcate from the worm-induced equilibrium E∗; τ20 is defined as in Eq (15).

Direction and stability of Hopf bifurcation

Conclusions