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.

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 Stability and Hopf Bifurcation Analysis of a Fractional-Order Epidemic Model with Time Delay

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Zhen Wang ◽  
Xinhe Wang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Equilibrium Point ◽  
Fractional Order ◽  
Bifurcation Analysis ◽  
Epidemic Model ◽  
Bifurcation Parameter ◽  
Theoretical Results ◽  
Disease Free

A fractional-order epidemic model with time delay is considered. Firstly, stability of the disease-free equilibrium point and endemic equilibrium point is studied. Then, by choosing the time delay as a bifurcation parameter, the existence of Hopf bifurcation is studied. Finally, numerical simulations are given to illustrate the effectiveness and feasibility of theoretical results.

Stability and Bifurcation Analysis in a Diffusive Brusselator-Type System

2016 ◽  
Vol 26 (07) ◽  
pp. 1650119 ◽  
Author(s):  
Maoxin Liao ◽  
Qi-Ru Wang
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Periodic Solutions ◽  
Bifurcation Analysis ◽  
Equilibrium Solution ◽  
Dynamic Properties ◽  
Type System ◽  
Ode System ◽  
Stability And Bifurcation Analysis ◽  
The Stability

In this paper, the dynamic properties for a Brusselator-type system with diffusion are investigated. By employing the theory of Hopf bifurcation for ordinary and partial differential equations, we mainly obtain some conditions of the stability and Hopf bifurcation for the ODE system, diffusion-driven instability of the equilibrium solution, and the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions for the PDE system. Finally, some numerical simulations are presented to verify our results.

scholarly journals Analysis of a Mathematical Model of Emerging Infectious Disease Leading to Amphibian Decline

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Muhammad Dur-e-Ahmad ◽  
Mudassar Imran ◽  
Adnan Khan
Keyword(s):  
Infectious Disease ◽  
Mathematical Model ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Bifurcation Analysis ◽  
Deterministic Model ◽  
Three Dimensional ◽  
Reproduction Ratio ◽  
Amphibian Larvae ◽  
The Stability

We formulate a three-dimensional deterministic model of amphibian larvae population to investigate the cause of extinction due to the infectious disease. The larvae population of the model is subdivided into two classes, exposed and unexposed, depending on their vulnerability to disease. Reproduction ratioℛ0has been calculated and we have shown that ifℛ0<1, the whole population will be extinct. For the case ofℛ0>1, we discussed different scenarios under which an infected population can survive or be eliminated using stability and persistence analysis. Finally, we also used Hopf bifurcation analysis to study the stability of periodic solutions.

scholarly journals A time-delay COVID-19 propagation model considering supply chain transmission and hierarchical quarantine rate

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Fangfang Yang ◽  
Zizhen Zhang
Keyword(s):  
Supply Chain ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Local Stability ◽  
Propagation Model ◽  
Bifurcation Parameter ◽  
Two Delays ◽  
Chain Transmission ◽  
Theoretical Results

AbstractIn this manuscript, we investigate a novel Susceptible–Exposed–Infected–Quarantined–Recovered (SEIQR) COVID-19 propagation model with two delays, and we also consider supply chain transmission and hierarchical quarantine rate in this model. Firstly, we analyze the existence of an equilibrium, including a virus-free equilibrium and a virus-existence equilibrium. Then local stability and the occurrence of Hopf bifurcation have been researched by thinking of time delay as the bifurcation parameter. Besides, we calculate direction and stability of the Hopf bifurcation. Finally, we carry out some numerical simulations to prove the validity of theoretical results.

A NONLINEAR DELAY MODEL DESCRIBING THE GROWTH OF TUMOR CELLS UNDER IMMUNE SURVEILLANCE AGAINST CANCER AND ITS STABILITY ANALYSIS

2012 ◽  
Vol 05 (03) ◽  
pp. 1260017 ◽  
Author(s):  
LING CHEN ◽  
WANBIAO MA
Keyword(s):  
Stability Analysis ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Tumor Cells ◽  
Local Stability ◽  
Immune Surveillance ◽  
Delay Model ◽  
Existence Of Periodic Solutions ◽  
Growth Of Tumor ◽  
Nonlinear Delay

In this paper, based on some biological meanings and a model which was proposed by Lefever and Garay (1978), a nonlinear delay model describing the growth of tumor cells under immune surveillance against cancer is given. Then, boundedness of the solutions, local stability of the equilibria and Hopf bifurcation of the model are discussed in details. The existence of periodic solutions explains the restrictive interactions between immune surveillance and the growth of the tumor cells.

scholarly journals Stability and Hopf Bifurcation Analysis on a Bazykin Model with Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Jianming Zhang ◽  
Lijun Zhang ◽  
Chaudry Masood Khalique
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Periodic Solutions ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Finite Delay ◽  
The Stability ◽  
Predator System

The dynamics of a prey-predator system with a finite delay is investigated. We show that a sequence of Hopf bifurcations occurs at the positive equilibrium as the delay increases. By using the theory of normal form and center manifold, explicit expressions for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived.

Hopf Bifurcation Analysis of a Three-Stage-Structured Prey-Predator System with Multi-Delays

2013 ◽  
Vol 756-759 ◽  
pp. 2857-2862
Author(s):  
Shun Yi Li ◽  
Wen Wu Liu
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Analysis ◽  
Local Stability ◽  
Positive Equilibrium ◽  
Characteristic Equations ◽  
Numerical Examples ◽  
Predator Model ◽  
Predator System

A three-stage-structured prey-predator model with multi-delays is considered. The characteristic equations and local stability of the equilibrium are analyzed, and the conditions for the positive equilibrium occurring Hopf bifurcation are obtained by applying the theorem of Hopf bifurcation. Finally, numerical examples and brief conclusion are given.

scholarly journals A Mathematical Model to Study the Effectiveness of Some of the Strategies Adopted in Curtailing the Spread of COVID-19

2020 ◽  
Vol 2020 ◽  
pp. 1-6 ◽  
Author(s):  
Isa Abdullahi Baba ◽  
Bashir Abdullahi Baba ◽  
Parvaneh Esmaili
Keyword(s):  
Mathematical Model ◽  
Stability Analysis ◽  
Numerical Simulations ◽  
Government Policy ◽  
Local Stability ◽  
Equilibrium Points ◽  
Compartmental Models ◽  
Effective Strategy ◽  
Local Stability Analysis ◽  
The Government

In this paper, we developed a model that suggests the use of robots in identifying COVID-19-positive patients and which studied the effectiveness of the government policy of prohibiting migration of individuals into their countries especially from those countries that were known to have COVID-19 epidemic. Two compartmental models consisting of two equations each were constructed. The models studied the use of robots for the identification of COVID-19-positive patients. The effect of migration ban strategy was also studied. Four biologically meaningful equilibrium points were found. Their local stability analysis was also carried out. Numerical simulations were carried out, and the most effective strategy to curtail the spread of the disease was shown.

scholarly journals Complex Behaviors of Epidemic Model with Nonlinear Rewiring Rate

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Ding Fang ◽  
Yongxin Zhang ◽  
Wendi Wang
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Bifurcation Analysis ◽  
Epidemic Model ◽  
Multiple Equilibria ◽  
Homoclinic Bifurcation ◽  
Propagation Model ◽  
Bifurcation Curve ◽  
Saddle Node Bifurcation ◽  
Saddle Node

An SIS propagation model with the nonlinear rewiring rate on an adaptive network is considered. It is found by bifurcation analysis that the model has the complex behaviors which include the transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation, and Bogdanov–Takens bifurcation. Especially, a bifurcation curve with “S” shape emerges due to the nonlinear rewiring rate, which leads to multiple equilibria and twice saddle-node bifurcations. Numerical simulations show that the model admits a homoclinic bifurcation and a saddle-node bifurcation of the limit cycle.

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