Hopf bifurcation analysis of a delayed SEIR epidemic model with infectious force in latent and infected period

Abstract

In this paper, we analyze a delayed SEIR epidemic model in which the latent and infected states are infective. The model has a globally asymptotically stable disease-free equilibrium whenever a certain epidemiological threshold, known as the basic reproduction number R 0 , is less than or equal to unity. We investigate the effect of the time delay on the stability of endemic equilibrium when R 0 > 1. We give criteria that ensure that endemic equilibrium is asymptotically stable for all time delays and a Hopf bifurcation occurs as time delay exceeds the critical value. We give formulae for the direction of Hopf bifurcations and the stability of bifurcated periodic solutions by applying the normal form theory and the center manifold reduction for functional differential equations. Numerical simulations are presented to illustrate the analytical results.

A delayed e-epidemic SLBS model for computer virus

“…Hence from the set of parameter values given in Eq. (31) we obtain that the endemic equilibrium point E * is locally asymptotically stable, which is confirmed by the Routh-Hurwitz criterion.…”

“…In addition to computer virus models with time delay, there are also some other dynamical models with time delay, which have shown that time delay causes problems such as instability and restrict the conceivable performance of the control systems. For example, the predator-prey model [27][28][29][30], the epidemic model [31][32][33][34][35], and the neural network model [36][37][38][39]. All the mentioned works about delayed dynamical models have shown that time delay can produce complicated nonlinear phenomena with the change of time.…”

“…Others shows time series of all three classes for different values of time delay. The values of other parameters are same as in(31)…”

“…The bifurcation diagram of (a) uninfected/susceptible node, (b) latent node, and (c) breaking-out node with respect to τ . The values of other parameters are as in(31) …”

“…Effect of infected rate at which uninfected computers are infected due to the influence of infected removable storage media (θ ) (a) uninfected/susceptible node, (b) latent node, and (c) breaking-out node with τ = 8. The values of other parameters are as in(31) …”