scholarly journals Dynamics and Stability Analysis of a Brucellosis Model with Two Discrete Delays

2018 ◽  
Vol 2018 ◽  
pp. 1-20 ◽  
Author(s):  
Paride O. Lolika ◽  
Steady Mushayabasa
Keyword(s):  
Mathematical Model ◽  
Stability Analysis ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Incubation Period ◽  
Steady States ◽  
Discrete Delays ◽  
Two Delays

We present a mathematical model for brucellosis transmission that incorporates two discrete delays and culling of infected animals displaying signs of brucellosis infection. The first delay represents the incubation period while the second account for the time needed to detect and cull infectious animals. Feasibility and stability of the model steady states have been determined analytically and numerically. Further, the occurrence of Hopf bifurcation has been established. Overall the findings from the study, both analytical and numerical, suggest that the two delays can destabilize the system and periodic solutions can arise through Hopf bifurcation.

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 Hopf Bifurcation of a Mathematical Model for Growth of Tumors with an Action of Inhibitor and Two Time Delays

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Bao Shi ◽  
Fangwei Zhang ◽  
Shihe Xu
Keyword(s):  
Mathematical Model ◽  
Hopf Bifurcation ◽  
Time Delays ◽  
Cell Loss ◽  
Bifurcation Parameter ◽  
Discrete Delays

A mathematical model for growth of tumors with two discrete delays is studied. The delays, respectively, represent the time taken for cells to undergo mitosis and the time taken for the cell to modify the rate of cell loss due to apoptosis and kill of cells by the inhibitor. We show the influence of time delays on the Hopf bifurcation when one of delays is used as a bifurcation parameter.

A model of a fishery with fish stock involving delay equations

2009 ◽  
Vol 367 (1908) ◽  
pp. 4907-4922 ◽  
Author(s):  
P. Auger ◽  
Arnaud Ducrot
Keyword(s):  
Differential Equation ◽  
Mathematical Model ◽  
Hopf Bifurcation ◽  
Delay Differential Equation ◽  
Bifurcation Analysis ◽  
Infinite Delay ◽  
Steady States ◽  
Delay Equations ◽  
Fish Stock ◽  
Delay Differential

The aim of this paper is to provide a new mathematical model for a fishery by including a stock variable for the resource. This model takes the form of an infinite delay differential equation. It is mathematically studied and a bifurcation analysis of the steady states is fulfilled. Depending on the different parameters of the problem, we show that Hopf bifurcation may occur leading to oscillating behaviours of the system. The mathematical results are finally discussed.

scholarly journals Stability analysis and Hopf bifurcation in a diffusive epidemic model with two delays

2020 ◽  
Vol 17 (4) ◽  
pp. 4127-4146
Author(s):  
Huan Dai ◽  
◽  
Yuying Liu ◽  
Junjie Wei ◽  
Keyword(s):  
Stability Analysis ◽  
Hopf Bifurcation ◽  
Epidemic Model ◽  
Two Delays

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 Global Hopf Bifurcation on Two-Delays Leslie-Gower Predator-Prey System with a Prey Refuge

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Qingsong Liu ◽  
Yiping Lin ◽  
Jingnan Cao
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Functional Differential Equations ◽  
Global Hopf Bifurcation ◽  
Bifurcation Theorem ◽  
Predator Prey ◽  
Prey System ◽  
Two Delays ◽  
Functional Differential ◽  
The Stability

A modified Leslie-Gower predator-prey system with two delays is investigated. By choosingτ1andτ2as bifurcation parameters, we show that the Hopf bifurcations occur when time delay crosses some critical values. 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. Numerical simulations are carried out to illustrate the theoretical results and chaotic behaviors are observed. Finally, using a global Hopf bifurcation theorem for functional differential equations, we show the global existence of the periodic solutions.

HOPF BIFURCATION ANALYSIS FOR A RATIO-DEPENDENT PREDATOR–PREY SYSTEM INVOLVING TWO DELAYS

2014 ◽  
Vol 55 (3) ◽  
pp. 214-231 ◽  
Author(s):  
E. KARAOGLU ◽  
H. MERDAN
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Prey System ◽  
Discrete Delays ◽  
Form Theory ◽  
Two Delays ◽  
Ratio Dependent ◽  
Centre Manifold Theorem ◽  
Analysis Stability

AbstractThe aim of this paper is to give a detailed analysis of Hopf bifurcation of a ratio-dependent predator–prey system involving two discrete delays. A delay parameter is chosen as the bifurcation parameter for the analysis. Stability of the bifurcating periodic solutions is determined by using the centre manifold theorem and the normal form theory introduced by Hassard et al. Some of the bifurcation properties including the direction, stability and period are given. Finally, our theoretical results are supported by some numerical simulations.

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 Delayed Vaccination Model for Rotavirus Infection

2020 ◽  
Vol 13 (4) ◽  
pp. 840-851
Author(s):  
Florence A. Adongo ◽  
Onyango Omondi Lawrence ◽  
Job Bonyo ◽  
G. O. Lawi ◽  
Ogada A. Elisha
Keyword(s):  
Mathematical Model ◽  
Stability Analysis ◽  
Hopf Bifurcation ◽  
Numerical Analysis ◽  
Delay Differential Equations ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Rotavirus Infection ◽  
Delay Differential ◽  
Disease Free

In this work, a mathematical model for rotavirus infection incorporating delay differential equations has been formulated. Stability analysis of the model has been performed. The result shows that the Disease Free Equilibrium is globally asymptotically stable and the Endemic Equilibrium undergoes a Hopf bifurcation. Numerical analysis has been performed to validate the analysis.

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