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Generalization of the Kermack-McKendrick SIR Model to a Patchy Environment for a Disease with Latency

J. Li, X. Zou (2009)

Mathematical Modelling of Natural Phenomena

In this paper, with the assumptions that an infectious disease has a fixed latent period in a population and the latent individuals of the population may disperse, we reformulate an SIR model for the population living in two patches (cities, towns, or countries etc.), which is a generalization of the classic Kermack-McKendrick SIR model. The model is given by a system of delay differential equations with a fixed delay accounting for the latency and non-local terms caused by the mobility of the...

Generalized practical stability analysis of discontinuous dynamical systems

Guisheng Zhai, Anthony Michel (2004)

International Journal of Applied Mathematics and Computer Science

In practice, one is not only interested in the qualitative characterizations provided by the Lyapunov stability, but also in quantitative information concerning the system behavior, including estimates of trajectory bounds, possibly over finite time intervals. This type of information has been ascertained in the past in a systematic manner using the concept of practical stability. In the present paper, we give a new definition of generalized practical stability (abbreviated as GP-stability) and...

Generalized synchronization and control for incommensurate fractional unified chaotic system and applications in secure communication

Hongtao Liang, Zhen Wang, Zongmin Yue, Ronghui Lu (2012)

Kybernetika

A fractional differential controller for incommensurate fractional unified chaotic system is described and proved by using the Gershgorin circle theorem in this paper. Also, based on the idea of a nonlinear observer, a new method for generalized synchronization (GS) of this system is proposed. Furthermore, the GS technique is applied in secure communication (SC), and a chaotic masking system is designed. Finally, the proposed fractional differential controller, GS and chaotic masking scheme are...

Generalized synchronization in a system of several non-autonomous oscillators coupled by a medium

Rogério Martins, Gonçalo Morais (2015)

Kybernetika

An abstract theory on general synchronization of a system of several oscillators coupled by a medium is given. By generalized synchronization we mean the existence of an invariant manifold that allows a reduction in dimension. The case of a concrete system modeling the dynamics of a chemical solution on two containers connected to a third container is studied from the basics to arbitrary perturbations. Conditions under which synchronization occurs are given. Our theoretical results are complemented...

Global asymptotic stability for half-linear differential systems with coefficients of indefinite sign

Jitsuro Sugie, Masakazu Onitsuka (2008)

Archivum Mathematicum

This paper is concerned with the global asymptotic stability of the zero solution of the half-linear differential system x ' = - e ( t ) x + f ( t ) φ p * ( y ) , y ' = - g ( t ) φ p ( x ) - h ( t ) y , where p > 1 , p * > 1 ( 1 / p + 1 / p * = 1 ), and φ q ( z ) = | z | q - 2 z for q = p or q = p * . The coefficients are not assumed to be positive. This system includes the linear differential system 𝐱 ' = A ( t ) 𝐱 with A ( t ) being a 2 × 2 matrix as a special case. Our results are new even in the linear case ( p = p * = 2 ). Our results also answer the question whether the zero solution of the linear system is asymptotically stable even when Coppel’s condition does not hold...

Global attractor of a differentiable autonomous system on the plane

Nguyen Van Chau (1995)

Annales Polonici Mathematici

We study the structure of a differentiable autonomous system on the plane with non-positive divergence outside a bounded set. It is shown that under certain conditions such a system has a global attractor. The main result here can be seen as an improvement of the results of Olech and Meisters in [7,9] concerning the global asymptotic stability conjecture of Markus and Yamabe and the Jacobian Conjecture.

Global exponential stability of positive periodic solutions for an epidemic model with saturated treatment

Bingwen Liu (2016)

Annales Polonici Mathematici

This paper is concerned with an SIR model with periodic incidence rate and saturated treatment function. Under proper conditions, we employ a novel argument to establish a criterion on the global exponential stability of positive periodic solutions for this model. The result obtained improves and supplements existing ones. We also use numerical simulations to illustrate our theoretical results.

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