Dynamics of a delayed SEIQ epidemic model

Abstract

In this work we consider an epidemic model that contains four species susceptible, exposed, infected and quarantined. With this model, first we find a feasible region which is invariant and where the solutions of our model are positive. Then the persistence of the model and sufficient conditions associated with extinction of infection population are discussed. To show that the system is locally asymptotically stable, a Lyapunov functional is constructed. After that, taking the delay as the key parameter, the conditions for local stability and Hopf bifurcation are derived. Further, we estimate the properties for the direction of the Hopf bifurcation and stability of the periodic solutions. Finally, some numerical simulations are presented to support our analytical results.

A Delayed Epidemic Model for Propagation of Malicious Codes in Wireless Sensor Network

“…. Now, following the steps as in [39], we shall check the stability of the system by assuming a suitable Lyapunov function w(u)(t) as follows:…”

On a SIR Model in a Patchy Environment Under Constant and Feedback Decentralized Controls with Asymmetric Parameterizations

“…In [13], some useful numerical tools are given concerning the non-singularity of perturbed matrices which are used in this paper. Background literature on dynamic systems, including its role on epidemic modelling, is given in [14][15][16][17][18][19]. In this context, typical situations which need relevant attention when dealing with epidemic models, thinking of their usefulness in their practical implementation in health centers are:…”

“…The supervisory scheme chooses online the best appropriate controller parameterization that minimizes the loss function. These considerations could be also of potential applicability interests in the cases of quarantine evaluation on certain parts of the population [17], or occurring transfers from infectious to susceptible individuals [21]. (b) The need for a development of adequate strategies for online either commissioning data [22], or intervention strategies [23], or even the programming of useful strategies for vaccine procurement in due time towards its application to the population [24].…”

Some Formal Results on Positivity, Stability, and Endemic Steady-State Attainability Based on Linear Algebraic Tools for a Class of Epidemic Models with Eventual Incommensurate Delays

“…The frontier between both situations typically occurs when the reproduction number is unity. See, for instance, [6][7][8][9][10][11][12] and 2 Discrete Dynamics in Nature and Society some references therein. In those studies, the first orthant of the space state is an invariant subspace as a result of the nonnegativity properties of the state trajectory solutions which is also an invoked property of other biological problems related to species evolution dynamics.…”

“…It is also proved that the reproduction number is linked to the Perron root of the above auxiliary matrix, which coincides with its spectral radius in typical examples of epidemic models. The presence of delays is an important modeling tool in epidemiology [11,17] in cases when there are successive outbreaks and regrowths of the disease intensity caused by increase of the transmission vector numbers or external immigration to the environment under study in the model. Therefore, once the above general algebraic framework is set for delay-free models, the above study is extended to epidemic models under, in general, incommensurate (in the sense that they are not all integer multiple of the smaller, or base, delay) state point delays, [ ].…”