scholarly journals Stability and Hopf Bifurcation Analysis of a Vector-Borne Disease Model with Two Delays and Reinfection

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Yanxia Zhang ◽  
Long Li ◽  
Junjian Huang ◽  
Yanjun Liu
Keyword(s):  
Hopf Bifurcation ◽  
Disease Model ◽  
Endemic Equilibrium ◽  
Partial Protection ◽  
Center Manifold Theorem ◽  
Vector Borne Disease ◽  
Two Delays ◽  
Normal Form Method ◽  
Vector Borne ◽  
Delay Differential

In this paper, a vector-borne disease model with two delays and reinfection is established and considered. First of all, the existence of the equilibrium of the system, under different cases of two delays, is discussed through analyzing the corresponding characteristic equation of the linear system. Some conditions that the system undergoes Hopf bifurcation at the endemic equilibrium are obtained. Furthermore, by employing the normal form method and the center manifold theorem for delay differential equations, some explicit formulas used to describe the properties of bifurcating periodic solutions are derived. Finally, the numerical examples and simulations are presented to verify our theoretical conclusions. Meanwhile, the influences of the degree of partial protection for recovered people acquired by a primary infection on the endemic equilibrium and the critical values of the two delays are analyzed.

scholarly journals Stability and Hopf Bifurcation of a Vector-Borne Disease Model with Saturated Infection Rate and Reinfection

2019 ◽  
Vol 2019 ◽  
pp. 1-17 ◽  
Author(s):  
Zhixing Hu ◽  
Shanshan Yin ◽  
Hui Wang
Keyword(s):  
Hopf Bifurcation ◽  
Infection Rate ◽  
Disease Model ◽  
Endemic Equilibrium ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Cure Rate ◽  
Vector Borne Disease ◽  
Vector Borne ◽  
The Stability

This paper established a delayed vector-borne disease model with saturated infection rate and cure rate. First of all, according to the basic reproductive number R0, we determined the disease-free equilibrium E0 and the endemic equilibrium E1. Through the analysis of the characteristic equation, we consider the stability of two equilibriums. Furthermore, the effect on the stability of the endemic equilibrium E1 by delay was studied, the existence of Hopf bifurcations of this system in E1 was analyzed, and the length of delay to preserve stability was estimated. The direction and stability of the Hopf bifurcation were also been determined. Finally, we performed some numerical simulation to illustrate our main results.

scholarly journals Hopf Bifurcation and Global Periodic Solutions in a Predator-Prey System with Michaelis-Menten Type Functional Response and Two Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-16
Author(s):  
Yunxian Dai ◽  
Yiping Lin ◽  
Huitao Zhao
Keyword(s):  
Hopf Bifurcation ◽  
Functional Response ◽  
Center Manifold ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Prey System ◽  
Model Study ◽  
Two Delays ◽  
Normal Form Method

We consider a predator-prey system with Michaelis-Menten type functional response and two delays. We focus on the case with two unequal and non-zero delays present in the model, study the local stability of the equilibria and the existence of Hopf bifurcation, and then obtain explicit formulas to determine the properties of Hopf bifurcation by using the normal form method and center manifold theorem. Special attention is paid to the global continuation of local Hopf bifurcation when the delaysτ1≠τ2.

scholarly journals Global Dynamics of a Vector-Borne Disease Model with Two Transmission Routes

2020 ◽  
Vol 30 (06) ◽  
pp. 2050083
Author(s):  
Sk Shahid Nadim ◽  
Indrajit Ghosh ◽  
Joydev Chattopadhyay
Keyword(s):  
Growth Rate ◽  
Transmission Coefficient ◽  
Disease Model ◽  
Psychological Effect ◽  
Endemic Equilibrium ◽  
Global Dynamics ◽  
Direct Transmission ◽  
Vector Borne Disease ◽  
Indirect Transmission ◽  
Vector Borne

In this paper, we study the dynamics of a vector-borne disease model with two transmission paths: direct transmission through contact and indirect transmission through vector. The direct transmission is considered to be a nonmonotone incidence function to describe the psychological effect of some severe diseases among the population when the number of infected hosts is large and/or the disease possesses high case fatality rate. The system has a disease-free equilibrium which is locally asymptotically stable when the basic reproduction number ([Formula: see text]) is less than unity and may have up to four endemic equilibria. Analytical expression representing the epidemic growth rate is obtained for the system. Sensitivity of the two transmission pathways were compared with respect to the epidemic growth rate. We numerically find that the direct transmission coefficient is more sensitive than the indirect transmission coefficient with respect to [Formula: see text] and the epidemic growth rate. Local stability of endemic equilibrium is studied. Further, the global asymptotic stability of the endemic equilibrium is proved using Li and Muldowney geometric approach. The explicit condition for which the system undergoes backward bifurcation is obtained. The basic model also exhibits the hysteresis phenomenon which implies diseases will persist even when [Formula: see text] although the system undergoes a forward bifurcation and this phenomenon is rarely observed in disease models. Consequently, our analysis suggests that the diseases with multiple transmission routes exhibit bistable dynamics. However, efficient application of temporary control in bistable regions will curb the disease to lower endemicity. Additionally, numerical simulations reveal that the equilibrium level of infected hosts decreases as psychological effect increases.

scholarly journals Stability and Hopf Bifurcation Analysis in a Delayed Myc/E2F/miR-17-92 Network Involving Interlinked Positive and Negative Feedback Loops

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Guiyuan Wang ◽  
Zhuoqin Yang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Negative Feedback ◽  
Feedback Loop ◽  
Sufficient Conditions ◽  
Center Manifold Theorem ◽  
Negative Feedback Loops ◽  
Feedback Strength ◽  
Normal Form Method ◽  
Delay Differential

MiR-17-92 plays an important role in regulating the levels of the Myc/E2F protein. In this paper, we consider a coupling network between Myc/E2F/miR-17-92 delayed negative feedback loop and Myc/E2F positive feedback loop described by a two-dimensional delay differential equation. Based on linear stability analysis and bifurcation theory, sufficient conditions for stability of equilibria and oscillatory behaviors via Hopf bifurcation are derived when choosing time delay as well as negative feedback strength associated with oscillations as bifurcation parameters, respectively. Furthermore, direction and stability of Hopf bifurcation of time delay are studied by using the normal form method and center manifold theorem. Finally, several numerical simulations are performed to verify the results we obtained.

scholarly journals Dynamic Analysis and Hopf Bifurcation of a Lengyel–Epstein System with Two Delays

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Long Li ◽  
Yanxia Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Two Delays ◽  
Stability Method ◽  
The Stability ◽  
Delay Differential

In this paper, a Lengyel–Epstein model with two delays is proposed and considered. By choosing the different delay as a parameter, the stability and Hopf bifurcation of the system under different situations are investigated in detail by using the linear stability method. Furthermore, the sufficient conditions for the stability of the equilibrium and the Hopf conditions are obtained. In addition, the explicit formula determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are obtained with the normal form theory and the center manifold theorem to delay differential equations. Some numerical examples and simulation results are also conducted at the end of this paper to validate the developed theories.

scholarly journals Stability and Hopf Bifurcation Analysis of a Plant Virus Propagation Model with Two Delays

2018 ◽  
Vol 2018 ◽  
pp. 1-12
Author(s):  
Junli Liu ◽  
Tailei Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Sufficient Conditions ◽  
Endemic Equilibrium ◽  
Propagation Model ◽  
Virus Propagation ◽  
Normal Form Theory ◽  
Globally Asymptotically Stable ◽  
Center Manifold Theorem ◽  
The Stability ◽  
Delay Differential

To understand the interaction between the insects and the plants, a system of delay differential equations is proposed and studied. We prove that if R0≤1, the disease-free equilibrium is globally asymptotically stable for any length of time delays by constructing a Lyapunov functional, and the system admits a unique endemic equilibrium if R0>1. We establish the sufficient conditions for the stability of the endemic equilibrium and existence of Hopf bifurcation. Using the normal form theory and center manifold theorem, the explicit formulae which determine the stability, direction, and other properties of bifurcating periodic solutions are derived. Some numerical simulations are given to confirm our analytic results.

scholarly journals Hopf Bifurcation Analysis for a Semiratio-Dependent Predator-Prey System with Two Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Ming Zhao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Center Manifold ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Prey System ◽  
Two Delays ◽  
Normal Form Method ◽  
Nonmonotonic Functional Response

This paper is concerned with a semiratio-dependent predator-prey system with nonmonotonic functional response and two delays. It is shown that the positive equilibrium of the system is locally asymptotically stable when the time delay is small enough. Change of stability of the positive equilibrium will cause bifurcating periodic solutions as the time delay passes through a sequence of critical values. The properties of Hopf bifurcation such as direction and stability are determined by using the normal form method and center manifold theorem. Numerical simulations confirm our theoretical findings.

scholarly journals Stability and Hopf Bifurcation in a Prey-Predator System with Disease in the Prey and Two Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Juan Liu
Keyword(s):  
Hopf Bifurcation ◽  
Local Stability ◽  
Center Manifold ◽  
Positive Equilibrium ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Two Delays ◽  
Normal Form Method ◽  
The Stability ◽  
Predator System

This paper is concerned with a prey-predator system with disease in the prey and two delays. Local stability of the positive equilibrium of the system and existence of local Hopf bifurcation are investigated by choosing different combinations of the two delays as bifurcation parameters. For further investigation, the direction and the stability of the Hopf bifurcation are determined by using the normal form method and center manifold theorem. Finally, some numerical simulations are given to support the theoretical analysis.

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