The Stability of an Epidemic Model With Piecewise Constant Argument by Lyapunov-Razumikhin Method

2020 ◽  
pp. 236-259
Author(s):  
Nur Cengiz ◽  
Duygu Aruğaslan-Çinçin
Keyword(s):  
Epidemic Model ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Piecewise Constant ◽  
Piecewise Constant Argument ◽  
Trivial Equilibrium ◽  
The Stability ◽  
Theoretical Results ◽  
Constant Argument

The authors propose a nonlinear epidemic model by developing it with generalized piecewise constant argument (GPCA) introduced by Akhmet. The authors investigate invariance region for the considered model. For the taken model into consideration, they obtain a useful inequality concerning relation between the values of the solutions at the deviation argument and at any time for the epidemic model. The authors reach sufficient conditions for the existence and uniqueness of the solutions. Then, based on Lyapunov-Razumikhin method developed by Akhmet and Aruğaslan for the differential equations with generalized piecewise constant argument (EPCAG), sufficient conditions for the stability of the trivial equilibrium and the positive equilibrium are investigated. Thus, the theoretical results concerning the uniform stability of the equilibriums are given.

Stability Analysis of a Nonlinear Epidemic Model With Generalized Piecewise Constant Argument

2020 ◽  
pp. 182-208
Author(s):  
Duygu Aruğaslan-Çinçin ◽  
Nur Cengiz
Keyword(s):  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Auxiliary Result ◽  
Piecewise Constant ◽  
Piecewise Constant Argument ◽  
Trivial Equilibrium ◽  
Stability Examination ◽  
The Stability ◽  
Constant Argument

The authors consider a nonlinear epidemic equation by modeling it with generalized piecewise constant argument (GPCA). The authors investigate invariance region for the considered model. Sufficient conditions guaranteeing the existence and uniqueness of the solutions of the model are given by creating integral equations. An important auxiliary result giving a relation between the values of the unknown function solutions at the deviation argument and at any time t is indicated. By using Lyapunov-Razumikhin method developed by Akhmet and Aruğaslan for the differential equations with generalized piecewise constant argument (EPCAG), the stability of the trivial equilibrium is investigated in addition to the stability examination of the positive equilibrium transformed into the trivial equilibrium. Then sufficient conditions for the uniform stability and the uniform asymptotic stability of trivial equilibrium and the positive equilibrium are given.

scholarly journals Existence and Uniqueness of Periodic Solutions for a Second-Order Nonlinear Differential Equation with Piecewise Constant Argument

2009 ◽  
Vol 2009 ◽  
pp. 1-14 ◽  
Author(s):  
Gen-qiang Wang ◽  
Sui Sun Cheng
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Periodic Solutions ◽  
Nonlinear Differential Equation ◽  
Existence And Uniqueness ◽  
Second Order ◽  
Piecewise Constant ◽  
Piecewise Constant Argument ◽  
Order Nonlinear Differential Equation ◽  
Constant Argument

Based on a continuation theorem of Mawhin, a unique periodic solution is found for a second-order nonlinear differential equation with piecewise constant argument.

Assessment of Solutions to Vibration Problems Involving Piecewise Constant Exertions

2000 ◽  
Author(s):  
L. Dai
Keyword(s):  
Nonlinear Vibration ◽  
Existence And Uniqueness ◽  
Numerical Solutions ◽  
Numerical Calculations ◽  
Piecewise Constant ◽  
Truncation Errors ◽  
Piecewise Constant Argument ◽  
Vibration Problems ◽  
Linear And Nonlinear ◽  
Constant Argument

Abstract Direct analytical and numerical solutions are constructed for linear and nonlinear vibration problems involving piecewise constant exertions. Existence and uniqueness of the solutions and the truncation errors of the numerical calculations are also analysed. With the employment of a piecewise constant argument, vibration systems with piecewise constant exertions are connected with the corresponding systems with continuous exertions.

scholarly journals Existence and Exponential Stability of Solutions for Stochastic Cellular Neural Networks with Piecewise Constant Argument

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Xiaoai Li
Keyword(s):  
Neural Networks ◽  
Exponential Stability ◽  
Sufficient Conditions ◽  
Cellular Neural Networks ◽  
Piecewise Constant ◽  
Piecewise Constant Argument ◽  
Inequality Technique ◽  
Generalized Type ◽  
Moment Exponential Stability ◽  
Constant Argument

By using the concept of differential equations with piecewise constant argument of generalized type, a model of stochastic cellular neural networks with piecewise constant argument is developed. Sufficient conditions are obtained for the existence and uniqueness of the equilibrium point for the addressed neural networks.pth moment exponential stability is investigated by means of Lyapunov functional, stochastic analysis, and inequality technique. The results in this paper improve and generalize some of the previous ones. An example with numerical simulations is given to illustrate our results.

Global Exponential Stability of Cohen-Grossberg Neural Networks with Piecewise Constant Argument of Generalized Type and Impulses

2016 ◽  
Vol 28 (1) ◽  
pp. 229-255 ◽  
Author(s):  
Qiang Xi
Keyword(s):  
Neural Networks ◽  
Exponential Stability ◽  
Global Exponential Stability ◽  
Sufficient Conditions ◽  
Uniqueness Of Solutions ◽  
Piecewise Constant ◽  
Piecewise Constant Argument ◽  
Lyapunov Function Method ◽  
Generalized Type ◽  
Constant Argument

In this letter, we consider a model of Cohen-Grossberg neural networks with piecewise constant argument of generalized type and impulses. Sufficient conditions ensuring the existence and uniqueness of solutions are obtained. Based on constructing a new differential inequality with piecewise constant argument and impulse and using the Lyapunov function method, we derive sufficient conditions ensuring the global exponential stability of equilibrium point, with approximate exponential convergence rate. An example is given to illustrate the validity and advantage of the theoretical results.

scholarly journals Fixed Point and Asymptotic Analysis of Cellular Neural Networks

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Xianghong Lai ◽  
Yutian Zhang
Keyword(s):  
Neural Networks ◽  
Fixed Point ◽  
Existence And Uniqueness ◽  
Fixed Point Theory ◽  
Sufficient Conditions ◽  
Point Theory ◽  
Cellular Neural Networks ◽  
Time Varying ◽  
Trivial Equilibrium ◽  
The Stability

We firstly employ the fixed point theory to study the stability of cellular neural networks without delays and with time-varying delays. Some novel and concise sufficient conditions are given to ensure the existence and uniqueness of solution and the asymptotic stability of trivial equilibrium at the same time. Moreover, these conditions are easily checked and do not require the differentiability of delays.

Persistence, Turing Instability and Hopf Bifurcation in a Diffusive Plankton System with Delay and Quadratic Closure

2016 ◽  
Vol 26 (03) ◽  
pp. 1650047 ◽  
Author(s):  
Jiantao Zhao ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
System With Delay ◽  
The Stability ◽  
Theoretical Results

A reaction–diffusion plankton system with delay and quadratic closure term is investigated to study the interactions between phytoplankton and zooplankton. Sufficient conditions independent of diffusion and delay are obtained for the persistence of the system. Our conclusions show that diffusion can induce Turing instability, delay can influence the stability of the positive equilibrium and induce Hopf bifurcations to occur. The computational formulas which determine the properties of bifurcating periodic solutions are given by calculating the normal form on the center manifold, and some numerical simulations are carried out for illustrating the theoretical results.

scholarly journals Stability of Impulsive Neural Networks with Time-Varying and Distributed Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Qi Luo ◽  
Xinjie Miao ◽  
Qian Wei ◽  
Zhengxin Zhou
Keyword(s):  
Neural Networks ◽  
Existence And Uniqueness ◽  
Global Exponential Stability ◽  
Fixed Point Theory ◽  
Sufficient Conditions ◽  
Point Theory ◽  
Distributed Delays ◽  
Time Varying ◽  
Trivial Equilibrium ◽  
The Stability

This work is devoted to investigating the stability of impulsive cellular neural networks with time-varying and distributed delays. We use the new method of fixed point theory to obtain some new and concise sufficient conditions to ensure the existence and uniqueness of solution and the global exponential stability of trivial equilibrium. The presented algebraic criteria are easily checked and do not require the differentiability of delays.

scholarly journals ALMOST AUTOMORPHIC SOLUTIONS FOR DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENT

2013 ◽  
Vol 90 (1) ◽  
pp. 99-112 ◽  
Author(s):  
LI-LI ZHANG ◽  
HONG-XU LI
Keyword(s):  
Differential Equations ◽  
Existence And Uniqueness ◽  
Existence And Uniqueness Theorem ◽  
Piecewise Constant ◽  
Exponential Dichotomies ◽  
Almost Automorphic ◽  
Piecewise Constant Argument ◽  
Greatest Integer Function ◽  
Image Position ◽  
Constant Argument

AbstractUsing the method of exponential dichotomies, we establish a new existence and uniqueness theorem for almost automorphic solutions of differential equations with piecewise constant argument of the form $$\begin{eqnarray*}{x}^{\prime } (t)= A(t)x(t)+ B(t)x(\lfloor t\rfloor )+ f(t), \quad t\in \mathbb{R} ,\end{eqnarray*}$$ where $\lfloor \cdot \rfloor $ denotes the greatest integer function, and $A(t), B(t): \mathbb{R} \rightarrow { \mathbb{R} }^{q\times q} $, $f(t): \mathbb{R} \rightarrow { \mathbb{R} }^{q} $ are all almost automorphic.

scholarly journals Oscillation Criteria of Certain Third-Order Differential Equation with Piecewise Constant Argument

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Haihua Liang ◽  
Gen-qiang Wang
Keyword(s):  
Differential Equation ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Third Order ◽  
Piecewise Constant ◽  
Piecewise Constant Argument ◽  
Delay Differential ◽  
Nonlinear Delay Differential Equation ◽  
Nonlinear Delay ◽  
Constant Argument

We study the oscillation and asymptotic behavior of third-order nonlinear delay differential equation with piecewise constant argument of the form(r2(t)(r1(t)x'(t))')'+p(t)x'(t)+f(t,x([t]))=0. We establish several sufficient conditions which insure that any solution of this equation oscillates or converges to zero. Some examples are given to illustrate the importance of our results.

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