SLIDING BIFURCATION AND GLOBAL DYNAMICS OF A FILIPPOV EPIDEMIC MODEL WITH VACCINATION

2013 ◽  
Vol 23 (08) ◽  
pp. 1350144 ◽  
Author(s):  
AILI WANG ◽  
YANNI XIAO
Keyword(s):  
Epidemic Model ◽  
Sliding Mode ◽  
Vaccination Strategy ◽  
Threshold Value ◽  
Threshold Level ◽  
Vaccination Rate ◽  
Global Dynamics ◽  
Boundary Node ◽  
Threshold Policy ◽  
Sliding Bifurcation

This paper proposes a Filippov epidemic model with piecewise continuous function to represent the enhanced vaccination strategy being triggered once the proportion of the susceptible individuals exceeds a threshold level. The sliding bifurcation and global dynamics for the proposed system are investigated. It is shown that as the threshold value varies, the proposed system can exhibit variable sliding mode domains and local sliding bifurcations including boundary node (focus) bifurcation, double tangency bifurcation and other sliding mode bifurcations. Model solutions ultimately approach either one of two endemic states for two structures or the pseudo-equilibrium on the switching surface, depending on the threshold level. The findings indicate that proper combinations of threshold level and enhanced vaccination rate based on threshold policy can lead disease prevalence to a previously chosen level if eradication of disease is impossible.

Codimension-1 Sliding Bifurcations of a Filippov Pest Growth Model with Threshold Policy

2014 ◽  
Vol 24 (10) ◽  
pp. 1450122 ◽  
Author(s):  
Sanyi Tang ◽  
Guangyao Tang ◽  
Wenjie Qin
Keyword(s):  
Homoclinic Orbit ◽  
Sliding Mode ◽  
Control Strategies ◽  
Control Policy ◽  
Threshold Level ◽  
Total Density ◽  
Boundary Node ◽  
Threshold Policy ◽  
Saddle Node ◽  
Pest Outbreaks

A Filippov system is proposed to describe the stage structured nonsmooth pest growth with threshold policy control (TPC). The TPC measure is represented by the total density of both juveniles and adults being chosen as an index for decisions on when to implement chemical control strategies. The proposed Filippov system can have three pieces of sliding segments and three pseudo-equilibria, which result in rich sliding mode bifurcations and local sliding bifurcations including boundary node (boundary focus, or boundary saddle) and tangency bifurcations. As the threshold density varies the model exhibits the interesting global sliding bifurcations sequentially: touching → buckling → crossing → sliding homoclinic orbit to a pseudo-saddle → crossing → touching bifurcations. In particular, bifurcation of a homoclinic orbit to a pseudo-saddle with a figure of eight shape, to a pseudo-saddle-node or to a standard saddle-node have been observed for some parameter sets. This implies that control outcomes are sensitive to the threshold level, and hence it is crucial to choose the threshold level to initiate control strategy. One more sliding segment (or pseudo-equilibrium) is induced by the total density of a population guided switching policy, compared to only the juvenile density guided policy, implying that this control policy is more effective in terms of preventing multiple pest outbreaks or causing the density of pests to stabilize at a desired level such as an economic threshold.

Global Dynamics and Rich Sliding Motion in an Avian-Only Filippov System in Combating Avian Influenza

2020 ◽  
Vol 30 (01) ◽  
pp. 2050008
Author(s):  
Youping Yang ◽  
Lingjun Wang
Keyword(s):  
Avian Influenza ◽  
Sliding Mode ◽  
Dynamical Behavior ◽  
Threshold Level ◽  
Type Model ◽  
Positive Equilibrium ◽  
Global Dynamics ◽  
Threshold Values ◽  
Threshold Policy ◽  
Existence And Stability

Culling birds has always been an effective method to control the spread of avian influenza. Here, we introduce a Filippov avian-only model with culling of both susceptible and infected birds. The Filippov-type model is formulated by considering that no control strategy is taken if the number of infected birds is less than an infected threshold level [Formula: see text]; further, we cull infected birds once the number of infected birds exceeds [Formula: see text]; meanwhile, we cull susceptible birds if the number of susceptible birds exceeds a susceptible threshold level [Formula: see text]. The global dynamical behavior of the Filippov system, including the existence and stability of various types of equilibria, the existence of the sliding mode and its dynamics, together with bifurcation analyses with regard to local sliding bifurcations, is investigated. It is shown that model solutions ultimately converge to the positive equilibrium that lies in the region above [Formula: see text], or below [Formula: see text], or on [Formula: see text], as we vary the susceptible and infected threshold values [Formula: see text] and [Formula: see text]. Our results indicate that proper combinations of the susceptible and infected threshold values based on the threshold policy can maintain the number of infected birds either below a certain threshold level or at a previously given level.

scholarly journals Filippov Ratio-Dependent Prey-Predator Model with Threshold Policy Control

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Xianghong Zhang ◽  
Sanyi Tang
Keyword(s):  
Sliding Mode ◽  
Threshold Level ◽  
Economic Threshold ◽  
Boundary Node ◽  
Threshold Policy ◽  
Ratio Dependent ◽  
Predator Model ◽  
The Stability ◽  
Globally Stable ◽  
Sliding Mode Dynamics

The Filippov ratio-dependent prey-predator model with economic threshold is proposed and studied. In particular, the sliding mode domain, sliding mode dynamics, and the existence of four types of equilibria and tangent points are investigated firstly. Further, the stability of pseudoequilibrium is addressed by using theoretical and numerical methods, and also the local sliding bifurcations including regular/virtual equilibrium bifurcations and boundary node bifurcations are studied. Finally, some global sliding bifurcations are addressed numerically. The globally stable touching cycle indicates that the density of pest population can be successfully maintained below the economic threshold level by designing suitable threshold policy strategies.

scholarly journals Stability and Hopf Bifurcation for a Delayed SIR Epidemic Model with Logistic Growth

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Yakui Xue ◽  
Tiantian Li
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Threshold Value ◽  
Endemic Equilibrium ◽  
Logistic Growth ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Sir Epidemic Model ◽  
Globally Asymptotically Stable ◽  
Sir Epidemic

We study a delayed SIR epidemic model and get the threshold value which determines the global dynamics and outcome of the disease. First of all, for anyτ, we show that the disease-free equilibrium is globally asymptotically stable; whenR0<1, the disease will die out. Directly afterwards, we prove that the endemic equilibrium is locally asymptotically stable for anyτ=0; whenR0>1, the disease will persist. However, for anyτ≠0, the existence conditions for Hopf bifurcations at the endemic equilibrium are obtained. Besides, we compare the delayed SIR epidemic model with nonlinear incidence rate to the one with bilinear incidence rate. At last, numerical simulations are performed to illustrate and verify the conclusions.

Sliding Dynamics and Bifurcations in the Extended Nonsmooth Filippov Ecosystem

2021 ◽  
Vol 31 (08) ◽  
pp. 2150119
Author(s):  
Wenjie Qin ◽  
Xuewen Tan ◽  
Xiaotao Shi ◽  
Marco Tosato ◽  
Xinzhi Liu
Keyword(s):  
Sliding Mode ◽  
Threshold Value ◽  
Boundary Node ◽  
Refuge Strategy ◽  
Boundary Equilibrium ◽  
The Stability ◽  
Sliding Dynamics ◽  
Regular Equilibrium ◽  
Existence Of Equilibria ◽  
Sliding Mode Dynamics

We propose a nonsmooth Filippov refuge ecosystem with a piecewise saturating response function and analyze its dynamics. We first investigate some key elements to our model which include the sliding segment, the sliding mode dynamics and the existence of equilibria which are classified into regular/virtual equilibrium, pseudo-equilibrium, boundary equilibrium and tangent point. In particular, we consider how the existence of the regular equilibrium and the pseudo-equilibrium are related. Then we study the stability of the standard periodic solution (limit cycle), the sliding periodic solutions (grazing or touching cycle) and the dynamics of the pseudo equilibrium, using quantitative analysis techniques related to nonsmooth Filippov systems. Furthermore, as the threshold value is varied, the model exhibits several complex bifurcations which are classified into equilibria, sliding mode, local sliding (boundary node and focus) and global bifurcations (grazing or touching). In conclusion, we discuss the importance of the refuge strategy in a biological setting.

scholarly journals Global dynamics of a piece-wise epidemic model with switching vaccination strategy

2014 ◽  
Vol 19 (9) ◽  
pp. 2915-2940 ◽  
Author(s):  
Aili Wang ◽  
◽  
Yanni Xiao ◽  
Robert A. Cheke ◽  
Keyword(s):  
Epidemic Model ◽  
Vaccination Strategy ◽  
Global Dynamics

scholarly journals Rich dynamics of a Filippov avian-only influenza model with a nonsmooth separation line

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Youping Yang ◽  
Jingwen Wang
Keyword(s):  
Complex Dynamics ◽  
Sliding Mode ◽  
Dynamical Behavior ◽  
Threshold Level ◽  
Control Measures ◽  
Sliding Modes ◽  
Threshold Policy ◽  
Effective Measure ◽  
The Real ◽  
Separation Line

AbstractDepopulation of birds has been authenticated to be an effective measure in controlling avian influenza transmission. In this work, we establish a Filippov avian-only model incorporating a threshold policy control. We choose the index—the maximum between the infected threshold level $I_{T}$ I T and the product of the number of susceptible birds S and a ratio threshold value ξ—to decide on whether to trigger the control measures or not, which then leads to a discontinuous separation line and two pieces of sliding-mode domains. Meanwhile, one more sliding-mode domain gives birth to more complex dynamics. We investigate the global dynamical behavior of the Filippov model, including the real and/or virtual equilibria and the two sliding modes and their dynamics. The solutions will eventually stabilize at the real endemic equilibrium of the subsystem or the pseudoequilibria on the two sliding modes due to different threshold values. Therefore an effective and efficient threshold policy is essential to control the influenza by driving the number of infected birds below a certain level or at a previously given level.

Dynamics of a stochastic susceptible-infective-recovered (SIRS) epidemic model with vaccination and nonlinear incidence under regime switching and Lévy jumps

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Junna Hu ◽  
Buyu Wen ◽  
Ting Zeng ◽  
Zhidong Teng
Keyword(s):  
Epidemic Model ◽  
Regime Switching ◽  
Markov Switching ◽  
Threshold Value ◽  
Open Problems ◽  
Nonlinear Incidence ◽  
Sirs Epidemic Model ◽  
Stochastic Epidemic ◽  
Lévy Jumps ◽  
White Noises

Abstract In this paper, a stochastic susceptible-infective-recovered (SIRS) epidemic model with vaccination, nonlinear incidence and white noises under regime switching and Lévy jumps is investigated. A new threshold value is determined. Some basic assumptions with regard to nonlinear incidence, white noises, Markov switching and Lévy jumps are introduced. The threshold conditions to guarantee the extinction and permanence in the mean of the disease with probability one and the existence of unique ergodic stationary distribution for the model are established. Some new techniques to deal with the Markov switching, Lévy jumps, nonlinear incidence and vaccination for the stochastic epidemic models are proposed. Lastly, the numerical simulations not only illustrate the main results given in this paper, but also suggest some interesting open problems.

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