EFFECT OF AWARENESS PROGRAMS IN CONTROLLING THE PREVALENCE OF AN EPIDEMIC WITH TIME DELAY

2011 ◽  
Vol 19 (02) ◽  
pp. 389-402 ◽  
Author(s):  
A. K. MISRA ◽  
ANUPAMA SHARMA ◽  
VISHAL SINGH
Keyword(s):  
Mathematical Model ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Periodic Solutions ◽  
Stability Theory ◽  
Direct Contact ◽  
Behavioral Changes ◽  
Nonlinear Mathematical Model ◽  
Awareness Programs

A nonlinear mathematical model with delay to capture the dynamics of effect of awareness programs on the prevalence of any epidemic is proposed and analyzed. It is assumed that pathogens are transmitted via direct contact between susceptibles and infectives. It is assumed further that cumulative density of awareness programs increases at a rate proportional to the number of infectives. It is considered that awareness programs are capable of inducing behavioral changes in susceptibles, which result in the isolation of aware population. The model is analyzed using stability theory of differential equations and numerical simulations. The model analysis shows that, though awareness programs cannot eradicate infection, they help in controlling the prevalence of disease. It is also found that time delay in execution of awareness programs destabilizes the system and periodic solutions may arise through Hopf-bifurcation.

scholarly journals HOPF BIFURCATION ANALYSIS FOR A MATHEMATICAL MODEL OF P53-MDM2 INTERACTION

2008 ◽  
Vol 18 (01) ◽  
pp. 275-283 ◽  
Author(s):  
MIHAELA NEAMŢU ◽  
RAUL FLORIN HORHAT ◽  
DUMITRU OPRIŞ
Keyword(s):  
Mathematical Model ◽  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Periodic Solutions ◽  
Stationary State ◽  
Bifurcation Analysis ◽  
Local Stability ◽  
Bifurcation Parameter ◽  
Simple Mathematical Model

In this paper we analyze a simple mathematical model which describes the interaction between proteins P53 and Mdm2. For the stationary state we discuss the local stability and the existence of a Hopf bifurcation. We study the direction and stability of the bifurcating periodic solutions by choosing the delay as a bifurcation parameter. Finally, we will offer some numerical simulations and present our conclusions.

scholarly journals Modelling the Influence of Awareness Programs by Media on the Drinking Dynamics

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Hai-Feng Huo ◽  
Qian Wang
Keyword(s):  
Mathematical Model ◽  
Sufficient Conditions ◽  
Alcohol Problems ◽  
Behavioral Changes ◽  
Nonlinear Mathematical Model ◽  
Effective Measure ◽  
Heavy Drinkers ◽  
Awareness Programs ◽  
The Stability ◽  
Separate Class

We develop a nonlinear mathematical model with the effect of awareness programs on the binge drinking. Due to the fact that awareness programs are capable of inducing behavioral changes in nondrinkers, we introduce a separate class by avoiding contacts with the heavy drinkers. Furthermore we assume that cumulative density of awareness programs increases at a rate proportional to the number of heavy drinkers. We establish some sufficient conditions for the stability of the alcohol free and the alcohol present equilibria and give some numerical simulations to explain our main result. Our results show that awareness programs is an effective measure in reducing alcohol problems.

scholarly journals Effect of Awareness Programs on the Epidemic Outbreaks with Time Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Lixia Zuo ◽  
Maoxing Liu
Keyword(s):  
Infectious Disease ◽  
Growth Rate ◽  
Time Delay ◽  
Numerical Simulations ◽  
Direct Contact ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Implementation Rate ◽  
Awareness Programs ◽  
Dissemination Rate

An epidemic model with time delay has been proposed and analyzed. In this model the effect of awareness programs driven by media on the prevalence of an infectious disease is studied. It is assumed that pathogens are transmitted via direct contact between the susceptible and the infective populations and further assumed that the growth rate of cumulative density of awareness programs increases at a rate proportional to the infective population. The model is analyzed by using stability theory of differential equations and numerical simulations. Both equilibria have been proved to be globally asymptotically stable. The results we obtained and numerical simulations suggest the increasing of the dissemination rate and implementation rate can reduce the proportion of the infective population.

A Mathematical Model for the Control of Infectious Diseases: Effects of TV and Radio Advertisements

2018 ◽  
Vol 28 (03) ◽  
pp. 1850037 ◽  
Author(s):  
A. K. Misra ◽  
Rajanish Kumar Rai
Keyword(s):  
Mathematical Model ◽  
Infectious Diseases ◽  
Behavioral Changes ◽  
Control Mechanisms ◽  
Cumulative Number ◽  
Nonlinear Mathematical Model ◽  
Model Formulation ◽  
Infection Period ◽  
Awareness Programs ◽  
Precautionary Measures

The broadcast of awareness programs through TV and radio advertisements (ads) makes people aware and brings behavioral changes among the individuals regarding the risk of infection and its control mechanisms. In this paper, we propose and analyze a nonlinear mathematical model for the control of infectious diseases due to impact of TV and radio advertisements. It is assumed that susceptible individuals are vulnerable to infection as well as information through TV and radio ads and they contract infection via direct contact with infected individuals. In the model formulation, it is also assumed that the growth rates in cumulative number of TV and radio ads are proportional to the number of infected individuals with decreasing function of aware individuals. Further, it is assumed that awareness among susceptible individuals induces behavioral changes and they form separate aware classes, which are fully protected from infection as they use precautionary measures for their protection during the infection period. The feasibility of equilibria and their stability properties are discussed. It is shown that the augmentation in dissemination rate of awareness among susceptible individuals due to TV and radio ads may cause stability switches through Hopf-bifurcation. The analytical findings are supported through numerical simulations.

HOPF BIFURCATION OF A DELAYED DIFFERENTIAL EQUATION

2007 ◽  
Vol 17 (04) ◽  
pp. 1367-1374 ◽  
Author(s):  
QIAN GUO ◽  
CHANGPIN LI
Keyword(s):  
Differential Equation ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Periodic Solutions ◽  
Nonlinear Differential Equation ◽  
Trivial Solution ◽  
The Stability ◽  
Analytical Expressions ◽  
Theoretical Results

In this paper, we study Hopf bifurcation of a second-order nonlinear differential equation with time delay by using the Lyapunov–Schmidt reduction. The approximate analytical expressions of the periodic solutions bifurcated from the trivial solution are given. We also discuss the stability of these periodic solutions. The numerical simulations line up with the theoretical results.

Dynamics of the impact of Twitter with time delay on the spread of infectious diseases

2018 ◽  
Vol 11 (05) ◽  
pp. 1850067 ◽  
Author(s):  
Maoxing Liu ◽  
Yuting Chang ◽  
Haiyan Wang ◽  
Benxing Li
Keyword(s):  
Infectious Disease ◽  
Mathematical Model ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Reproduction Number ◽  
Behavioral Changes ◽  
The Public ◽  
The Stability ◽  
The Impact ◽  
The Basic Reproduction Number

In this paper, a mathematical model to study the impact of Twitter in controlling infectious disease is proposed. The model includes the dynamics of “tweets” which may enhance awareness of the disease and cause behavioral changes among the public, thus reducing the transmission of the disease. Furthermore, the model is improved by introducing a time delay between the outbreak of disease and the release of Twitter messages. The basic reproduction number and the conditions for the stability of the equilibria are derived. It is shown that the system undergoes Hopf bifurcation when time delay is increased. Finally, numerical simulations are given to verify the analytical results.

Hopf Bifurcation Analysis and Stability for a Ratio-Dependent Predator–Prey Diffusive System with Time Delay

2020 ◽  
Vol 30 (03) ◽  
pp. 2050037
Author(s):  
Longyue Li ◽  
Yingying Mei ◽  
Jianzhi Cao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Periodic Solutions ◽  
Positive Equilibrium ◽  
Dimensional System ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Predator Prey ◽  
Ratio Dependent ◽  
Low Dimensional

In this paper, we are focused on a new ratio-dependent predator–prey system that introduced the diffusive and time delay effect simultaneously. By analyzing the characteristic equations and the distribution of eigenvalues, we examine the stability and boundary of positive equilibrium states, and the existence of spatially homogeneous and spatially inhomogeneous bifurcating periodic solutions, respectively. Further, we prove that when [Formula: see text], the system has Hopf bifurcation at the positive equilibrium state. By using the center manifold reduction, we simplify the system so that we can convert an infinite-dimensional system into a low-dimensional finite-dimensional system. By using the normal form theory, we obtain explicit expressions for the direction, stability and period of Hopf bifurcation periodic solutions. Finally, we have illustrated the main results in this thesis by numerical examples, our work may provide some useful measures to save time or cost and to control the ecosystem.

TWO-TO-ONE RESONANT HOPF BIFURCATIONS IN A QUADRATICALLY NONLINEAR OSCILLATOR INVOLVING TIME DELAY

2012 ◽  
Vol 22 (03) ◽  
pp. 1250060 ◽  
Author(s):  
J. C. JI ◽  
X. Y. LI ◽  
Z. LUO ◽  
N. ZHANG
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Periodic Solutions ◽  
Nonlinear Oscillator ◽  
Critical Time ◽  
Zero Solution ◽  
Delayed Feedback Control ◽  
Hopf Bifurcations ◽  
Centre Manifold ◽  
Normal Form Theory

The trivial equilibrium of a weakly nonlinear oscillator having quadratic nonlinearities under a delayed feedback control can change its stability via a single Hopf bifurcation as the time delay increases. Double Hopf bifurcation occurs when the characteristic equation has two pairs of purely imaginary solutions. An interaction of resonant Hopf–Hopf bifurcations may be possible when the two critical time delays corresponding to the two Hopf bifurcations have the same value. With the aid of normal form theory and centre manifold theorem as well as the method of multiple scales, the present paper studies the dynamics of a quadratically nonlinear oscillator involving time delay in the vicinity of the point of two-to-one resonances of Hopf–Hopf bifurcations. The ratio of the frequencies of two Hopf bifurcations is numerically found to be nearly equal to two. The two resonant Hopf bifurcations can generate two respective periodic solutions. Consequently, the centre manifold corresponding to these two solutions is determined by a set of four first-order differential equations under two-to-one internal resonances. It is shown that the amplitudes of the two bifurcating periodic solutions admit the trivial solution and two-mode solutions for the averaged equations on the centre manifolds. Correspondingly, the cumulative behavior of the original nonlinear oscillator exhibits the initial equilibrium and a quasi-periodic motion having two frequencies. Illustrative examples are given to show the unstable zero solution, stable zero solution, and stable two-mode solution of the nonlinear oscillator under the two-to-one resonant Hopf–Hopf interactions.

scholarly journals Chaos and Hopf Bifurcation Analysis of the Delayed Local Lengyel-Epstein System

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Qingsong Liu ◽  
Yiping Lin ◽  
Jingnan Cao ◽  
Jinde Cao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Local Reaction ◽  
Reaction Diffusion ◽  
Critical Value ◽  
Hopf Bifurcations ◽  
The Stability

The local reaction-diffusion Lengyel-Epstein system with delay is investigated. By choosingτas bifurcating parameter, we show that Hopf bifurcations occur when time delay crosses a critical value. Moreover, we derive the equation describing the flow on the center manifold; then we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, numerical simulations are performed to support the analytical results and the chaotic behaviors are observed.

scholarly journals Asymptotic Properties of One Mathematical Model in Food Engineering under Stochastic Perturbations

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3013
Author(s):  
Leonid Shaikhet
Keyword(s):  
Mathematical Model ◽  
Mathematical Models ◽  
Numerical Simulations ◽  
Wide Class ◽  
Asymptotic Properties ◽  
Nonlinear Mathematical Model ◽  
Simple Criterion ◽  
Stochastic Perturbations ◽  
Food Engineering ◽  
Nonlinear Mathematical Models

For the example of one nonlinear mathematical model in food engineering with several equilibria and stochastic perturbations, a simple criterion for determining a stable or unstable equilibrium is reported. The obtained analytical results are illustrated by detailed numerical simulations of solutions of the considered Ito stochastic differential equations. The proposed criterion can be used for a wide class of nonlinear mathematical models in different applications.

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