scholarly journals Dynamical Analysis of Delayed Plant Disease Models with Continuous or Impulsive Cultural Control Strategies

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
Tongqian Zhang ◽  
Xinzhu Meng ◽  
Yi Song ◽  
Zhenqing Li
Keyword(s):  
Control Strategy ◽  
Plant Disease ◽  
Control Strategies ◽  
Disease Model ◽  
Infected Plant ◽  
Transcendental Equation ◽  
Cultural Control ◽  
Dynamical Analysis ◽  
Positive Equilibrium ◽  
Disease Free

Delayed plant disease mathematical models including continuous cultural control strategy and impulsive cultural control strategy are presented and investigated. Firstly, we consider continuous cultural control strategy in which continuous replanting of healthy plants is taken. The existence and local stability of disease-free equilibrium and positive equilibrium are studied by analyzing the associated characteristic transcendental equation. And then, plant disease model with impulsive replanting of healthy plants is also considered; the sufficient condition under which the infected plant-free periodic solution is globally attritive is obtained. Moreover, permanence of the system is studied. Some numerical simulations are also given to illustrate our results.

scholarly journals Stability and Sensitivity Analysis of a Plant Disease Model with Continuous Cultural Control Strategy

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Zhang Zhonghua ◽  
Suo Yaohong
Keyword(s):  
Sensitivity Analysis ◽  
Time Delay ◽  
Control Strategy ◽  
Plant Disease ◽  
Disease Model ◽  
Cultural Control ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Disease Free

In this paper, a plant disease model with continuous cultural control strategy and time delay is formulated. Then, how the time delay affects the overall disease progression and, mathematically, how the delay affects the dynamics of the model are investigated. By analyzing the transendental characteristic equation, stability conditions related to the time delay are derived for the disease-free equilibrium. Specially, whenR0=1, the Jacobi matrix of the model at the disease-free equilibrium always has a simple zero eigenvalue for allτ≥0. The center manifold reduction and the normal form theory are used to discuss the stability and the steady-state bifurcations of the model near the nonhyperbolic disease-free equilibrium. Then, the sensitivity analysis of the threshold parameterR0and the positive equilibriumE*is carried out in order to determine the relative importance of different factors responsible for disease transmission. Finally, numerical simulations are employed to support the qualitative results.

Rich Sliding Motion and Dynamics in a Filippov Plant-Disease System

2018 ◽  
Vol 28 (01) ◽  
pp. 1850012 ◽  
Author(s):  
Can Chen ◽  
Xi Chen
Keyword(s):  
Disease Transmission ◽  
Plant Disease ◽  
Sliding Mode ◽  
Control Strategies ◽  
Disease Model ◽  
Threshold Value ◽  
Plant Diseases ◽  
Positive Equilibrium ◽  
Threshold Values ◽  
Periodic Trajectories

In order to reduce the spread of plant diseases and maintain the number of infected trees below an economic threshold, we choose the number of infected trees and the number of susceptible plants as the control indexes on whether to implement control strategies. Then a Filippov plant-disease model incorporating cutting off infected branches and replanting susceptible trees is proposed. Based on the theory of Filippov system, the sliding mode dynamics and conditions for the existence of all the possible equilibria and Lotka–Volterra cycles are presented. We find that model solutions ultimately approach the positive equilibrium that lies in the region above the infected threshold value [Formula: see text], or the periodic trajectories that lie in the region below [Formula: see text], or the pseudo-attractor [Formula: see text], as we vary the susceptible and infected threshold values. It indicates that the plant-disease transmission is tolerable if the trajectories approach [Formula: see text] or the periodic trajectories lie in the region below [Formula: see text]. Hence an acceptable level of the number of infected trees can be achieved when the susceptible and infected threshold values are chosen appropriately.

scholarly journals The Dynamics and Optimal Control of a Hand-Foot-Mouth Disease Model

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Hongwu Tan ◽  
Hui Cao
Keyword(s):  
Optimal Control ◽  
Reproduction Number ◽  
Control Strategies ◽  
Disease Model ◽  
Mouth Disease ◽  
Infected People ◽  
Hand Foot Mouth Disease ◽  
The Cost ◽  
Existence Of Equilibria ◽  
Disease Free

We build and study the transmission dynamics of a hand-foot-mouth disease model with vaccination. The reproduction number is given, the existence of equilibria is obtained, and the global stability of disease-free equilibrium is proved by constructing the Lyapunov function. We also apply optimal control theory to the hand-foot-mouth disease model. The treatment and vaccination interventions are considered in the hand-foot-mouth disease model, and the optimal control strategies based on minimizing the cost of intervention and minimizing the number of the infected people are given. Numerical results show the usefulness of the optimization strategies.

scholarly journals Global asymptotic stability of a delayed plant disease model

2020 ◽  
Vol 1 (1) ◽  
pp. 27-38
Author(s):  
Yuming Chen ◽  
Chongwu Zheng
Keyword(s):  
Plant Disease ◽  
Disease Model ◽  
Plant Diseases ◽  
Threshold Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Sigma 1 ◽  
T Tau ◽  
Theoretical Results ◽  
Disease Free

In this paper, we consider the following system of delayed differentialequations,\[\left\{\begin {array}{rcl}\frac {dS(t)}{dt} & = & \sigma \phi-\beta S(t)I(t-\tau)-\eta S(t),\\\frac {dI(t)}{dt} & = & \sigma(1-\phi)+\betaS(t)I(t-\tau)-(\eta+\omega)I(t),\end {array}\right.\]which can be used to model plant diseases. Here $\phi\in (0,1]$,$\tau\ge 0$, and all other parameters are positive. The case where $\phi=1$ is well studiedand there is a threshold dynamics. Thesystem always has a disease-free equilibrium, which is globallyasymptotically stable if the basic reproduction number $R_0\triangleq\frac{\beta\sigma}{\eta(\eta+\omega)}\le 1$ and is unstable if$R_0>1$; when $R_0>1$, the system also has a unique endemic equilibrium,which is globally asymptotically stable. In this paper, we study thecase where $\phi\in (0,1)$. It turns out that the system only has anendemic equilibrium, which is globally asymptotically stable. Thelocal stability is established by the linearizationmethod while the global attractivity is obtained by the Lyapunovfunctional approach. The theoretical results are illustrated withnumerical simulations.

scholarly journals Dynamical Analysis of an SEIRS Model With Incomplete Recovery and Relapse on Networks and Its Optimal Vaccination Control

2021 ◽  
Author(s):  
Lei Zhang ◽  
Maoxing Liu ◽  
Qiang Hou ◽  
Boli Xie
Keyword(s):  
Control Strategy ◽  
Reproduction Number ◽  
Dynamical Analysis ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Seirs Epidemic Model ◽  
Theoretical Results ◽  
Disease Free ◽  
The Basic Reproduction Number

Abstract For some infectious diseases, such as herpes and tuberculosis, there is incomplete recovery and relapse. These phenomena make them difficult to control. In consequence of this status, an SEIRS epidemic model with incomplete recovery and relapse on networks is established and the global dynamics is studied. The results show that when the basic reproduction number R 0 <=1 the disease-free equilibrium is globally asymptotically stable; when R 0 > 1, the endemic equilibrium is globally asymptotically stable. In addition, in consideration of vaccination control strategy, an SVEIRS model is introduced and the optimal control is solved. At last, the theoretical results are illustrated with numerical simulations.

Dynamics of a prey predator disease model with Holling type functional response and stochastic perturbation

2020 ◽  
pp. 471-488
Author(s):  
M. N. Srinivas ◽  
G. Basava Kumar ◽  
V. Madhusudanan
Keyword(s):  
Equilibrium Point ◽  
Disease Model ◽  
Stochastic Perturbation ◽  
Positive Equilibrium ◽  
Type Ii ◽  
Asymptotically Stable ◽  
Unique Positive Solution ◽  
Stochastic Nature ◽  
Research Article ◽  
Positive Equilibrium Point

The present research article constitutes Holling type II and IV diseased prey predator ecosystem and classified into two categories namely susceptible and infected predators.We show that the system has a unique positive solution. The deterministic and stochastic nature of the dynamics of the system is investigated. We check the existence of all possible steady states with local stability. By using Routh-Hurwitz criterion we showed that the positive equilibrium point $E_{7}$ is locally asymptotically stable if $x^{*} > \sqrt{m_{1}}$ .Moreover condition of the global stability of positive equilibrium point $E_{7}$ are also entrenched with help of Lyupunov theorem. Some Numerical simulations are carried out to illustrate our analytical findings.

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