Hopf bifurcation of an SEIRS epidemic model with delays and vertical transmission in the network

Chunlei Wang,Shouxia Chai

doi:10.1186/s13662-016-0793-7

Abstract

An SEIRS worm propagation model with two delays and vertical transmission in the network is investigated. It is proved that the positive equilibrium is locally asymptotically stable and the Hopf bifurcation can occur when the certain conditions are satisfied by regarding different combination of the two delays as bifurcation parameter. Then the properties of the Hopf bifurcation, such as direction and stability, are studied by using the normal form theory and the center manifold theorem. Finally, some numerical simulations are presented to verify the obtained results and to demonstrate the dynamics of the model.

Highlights

In recent years, many mathematical models such as SIR model [ – ], SEIR model [, ], SEIRS model [ ] and some other models [ – ] have been proposed to predict propagation of computer viruses
Delays have an important effect on dynamical models and they can cause the occurrence of the Hopf bifurcation
5 Conclusions In the present paper, we devote our attention to the stability and Hopf bifurcation of an SEIRS epidemic model which describes the transmission of worms in the network through vertical transmission with two delays based on the work in the literature [ ]

Summary

Introduction

Many mathematical models such as SIR model [ – ], SEIR model [ , ], SEIRS model [ ] and some other models [ – ] have been proposed to predict propagation of computer viruses. In [ ], Zhang and Yang studied the following SEIRS worm propagation model with time delay:. If the condition equation ( ) has (H ) holds, there exists a positive root v a pair of purely imaginary roots ±iω =. If the condition (H ) holds, there exists ω > such that equation ( ) has a pair of purely imaginary roots ±iω. We have the equation with respect to ω: If equation ( ) has one positive root ω such that equation ( ) has a pair of purely imaginary roots ±iω , we can obtain the corresponding critical value of the delay τ = ω arccos f ω. If the condition (H ) holds, there exists ω ∗ > satisfying equation ( ) and equation ( ) has a pair of purely imaginary roots ±iω ∗.

Direction and stability of the Hopf bifurcation

Conclusions