Stability and bifurcations in an epidemic model with varying immunity period

An epidemic model with distributed time delay is derived to describe the dynamics of infectious diseases with varying immunity. It is shown that solutions are always positive, and the model has at most two steady states: disease-free and endemic. It is proved that the disease-free equilibrium is locally and globally asymptotically stable. When an endemic equilibrium exists, it is possible to analytically prove its local and global stability using Lyapunov functionals. Bifurcation analysis is performed using DDE-BIFTOOL and traceDDE to investigate different dynamical regimes in the model using numerical continuation for different values of system parameters and different integral kernels.Comment: 16 pages, 5 figure

Modeling and Analysis of Bifurcation in a Delayed Worm Propagation Model

A delayed worm propagation model with birth and death rates is formulated. The stability of the positive equilibrium is studied. Through theoretical analysis, a critical value τ0 of Hopf bifurcation is derived. The worm propagation system is locally asymptotically stable when time delay is less than τ0. However, Hopf bifurcation appears when time delay τ passes the threshold τ0, which means that the worm propagation system is unstable and out of control. Consequently, time delay should be adjusted to be less than τ0 to ensure the stability of the system stable and better prediction of the scale and speed of Internet worm spreading. Finally, numerical and simulation experiments are presented to simulate the system, which fully support our analysis

Effect of General Cross-Immunity Protection and Antibody- Dependent Enhancement in Dengue Dynamics

A mathematical model to describe the dynamic of a multiserotype infectious disease at the population level is studied. Applied to dengue fever epidemiology, we analyse a mathematical model with time delay to describe the cross-immunity protection period, including a key parameter for the antibody-dependent enhancement (ADE) effect, the well-known features of dengue fever infection. Numerical experiments are performed to show the stability of the coexistence equilibrium, which is completely determined by the basic reproduction number and by the invasion reproduction number, as well as the bifurcation structures for different scenarios of dengue fever transmission in a population. The model shows a rich dynamical behavior, from fixed points and periodic oscillations up to chaotic behaviour with complex attractors.Laboratory for Industrial and Applied Mathematics (LIAM), Department of Mathematics and Statistics, York University, CA

bifurcation analysis of a delayed worm propagation model with saturated incidence

This paper is concerned with a delayed SVEIR worm propagation model with saturated incidence. The main objective is to investigate the effect of the time delay on the model. Sufficient conditions for local stability of the positive equilibrium and existence of a Hopf bifurcation are obtained by choosing the time delay as the bifurcation parameter. Particularly, explicit formulas determining direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are derived by using the normal form theory and the center manifold theorem. Numerical simulations for a set of parameter values are carried out to illustrate the analytical results

Study of an HIV-1 Model with Time Delays

We propose a mathematical model for HIV-1 infection with two time delays, one for the average latent period of cell infection and the other for the average time needed for the virus production after a virion enters a cell. The model examines a viral-therapy for controlling infections through recombining HIV-1 virus with a genetically modified virus. When only the intracellular delay is enrolled into model (1.13), the basic reproduction numbers Rq and Rd are identified and their threshold properties are discussed. When Rq \u3c 1, the infection-free equilibrium Eq is globally asymptotically stable. When Rq \u3e 1, Eq becomes unstable and there occurs the single-infection equilibrium Es. If Rq \u3e 1 and Rd \u3c 1, Es is asymptotically stable, while for Rd \u3e 1, Es loses its stability to the double-infection equilibrium. For the double-infection equilibrium Ed, we show how to determine its stability and existence of Hopf bifurcation. Some simulations are presented to demonstrate the theoretical results. Further investigation is carried over by introducing the second time lag into model (2.1). We have identified the new basic reproduction numbers Rqand Rd, and proved that for Rq \u3c 1 the infection-free equilibrium Eq is globally asymptotically stable. If Rq \u3e 1 and Rd \u3c 1, the single-infection equilibrium Es is asymptotically stable. For the double-infection equilibrium Ed, it has been found that there exist both Hopf and N double Hopf bifurcations. These theoretical predictions are verified by using some numerical examples. Evidences indicate that the viral-therapy, of recombining HIV-1 virus with a genetically modified virus may be effective in reducing the HIV-1 load, and larger delays may be able to help eradicate the virus

The Application of Saturated Incidence Rate with Delay and Awareness in Modelling the Spread of Infectious Diseases

The current state as regards the spread of the coronavirus disease 2019 indicates that the major preventive measure is minimizing individual contact with the virus. Consequently, this paper derived a mathematical model using saturated incidence rate with the disease incubating period as a delay parameter along other parameters measuring the inhibitory effects of awareness dissemination from the media, general campaign and individual interactions. The analytical evaluation of the model indicates that the stability of the disease-free and the endemic steady states of the model are determined by the basic reproduction number. In addition, the disease-free steady state is independent of the incubating period, whereas the endemic steady state is initially stable but undergoes Hopf bifurcation as the incubating period exceeds the critical value by exhibiting periodic oscillation and become unstable. The outcome of the numerical simulation of the model via MATLAB application affirmed the results obtained analytically. Keywords: Infectious Diseases, Model, Incidence rate, Basic reproduction number, Hopf bifurcatio

Modeling and Bifurcation Research of a Worm Propagation Dynamical System with Time Delay

Both vaccination and quarantine strategy are adopted to control the Internet worm propagation. By considering the interaction infection between computers and external removable devices, a worm propagation dynamical system with time delay under quarantine strategy is constructed based on anomaly intrusion detection system (IDS). By regarding the time delay caused by time window of anomaly IDS as the bifurcation parameter, local asymptotic stability at the positive equilibrium and local Hopf bifurcation are discussed. Through theoretical analysis, a threshold τ0 is derived. When time delay is less than τ0, the worm propagation is stable and easy to predict; otherwise, Hopf bifurcation occurs so that the system is out of control and the containment strategy does not work effectively. Numerical analysis and discrete-time simulation experiments are given to illustrate the correctness of theoretical analysis