Dynamics of a non-smooth epidemic model with three thresholds

Wang, Aili; Xiao, Yanni; Smith, Robert J.

doi:10.1007/s12064-019-00297-z

Cited by 7 publications

(3 citation statements)

References 42 publications

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“…Previous studies have considered the Lambert W function in the context of epidemiological models. In Reluga (2004); Wang (2010); Pakes (2015), the Lambert W function is used to express the final sizes of the epidemiological variables, and in Xiao et al (2013); Wang et al (2020), the Lambert W function is used to study an epidemiological model with a piecewise incidence rate. The usefulness of the Lambert W function in these studies, as well as in ours, lies in the fact that it can be used to express the solutions of some nonlinear equations in closed form, that cannot be solved otherwise in terms of elementary functions.…”

Section: Introductionmentioning

confidence: 99%

Analysis of Infected Population Threshold Exceedance in an SIR Epidemiological Model

Báez-Sánchez

Bobko

2021

LiB

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Since the beginning of the COVID-19 outbreak, much attention has been given to the idea of flattening the curve of cases to reduce the harmful effects of an overloaded medical system. In this context, it is relevant to determine conditions to ensure that the health care threshold capacity will not be exceeded. If such a situation is unavoidable, it would be useful to effectively quantify the potential negative impact produced. In this paper, we consider an epidemiological SIR model and a positive threshold M. Using a parametric expression for the solution curve of the SIR model and the properties of the Lambert W function, we establish necessary and sufficient conditions on the basic reproduction number R 0 to ensure that the infected population I does not exceed M. We also introduce and numerically analyze, five different quantities to measure the impact caused by a possible threshold exceedance.

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Section: Introductionmentioning

confidence: 99%

Analysis of Infected Population Threshold Exceedance in an SIR Epidemiological Model

Báez-Sánchez

Bobko

2021

LiB

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“…Mathematically, such situations are described by Filippov systems, when the phase space is divided into two (or more) parts and the system is given by different vector fields in each of those parts. Examples include sudden changes in vaccination [8,12], hospitalization [15], transmission [13], travel patterns [6], or the combination of several effects [14]. They have been used for vector borne diseases as well [16].…”

Section: Introductionmentioning

confidence: 99%

Periodic Orbits and Global Stability for a Discontinuous SIR Model with Delayed Control

Muqbel

Vas

Röst

2020

Qual. Theory Dyn. Syst.

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We propose and analyse a mathematical model for infectious disease dynamics with a discontinuous control function, where the control is activated with some time lag after the density of the infected population reaches a threshold. The model is mathematically formulated as a delayed relay system, and the dynamics is determined by the switching between two vector fields (the so-called free and control systems) with a time delay with respect to a switching manifold. First we establish the usual threshold dynamics: when the basic reproduction number R 0 ≤ 1, then the disease will be eradicated, while for R 0 > 1 the disease persists in the population. Then, for R 0 > 1, we divide the parameter domain into three regions, and prove results about the global dynamics of the switching system for each case: we find conditions for the global convergence to the endemic equilibrium of the free system, for the global convergence to the endemic equilibrium of the control system, and for the existence of periodic solutions that oscillate between the two sides of the switching manifold. The proof of the latter result is based on the construction of a suitable return map on a subset of the infinite dimensional phase space. Our results provide insight into disease management, by exploring the effect of the interplay of the control efficacy, the triggering threshold and the delay in implementation.

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“…Mathematically, such situations are described by Filippov systems, when the phase space is divided into two (or more) parts and the system is given by different vector fields in each of those parts. Examples include sudden changes in vaccination [33,44], hospitalization [47], transmission [49], travel patterns [29], or the combination of several effects [45]. They have been used for vector borne diseases as well [52].…”

Section: Introductionmentioning

confidence: 99%

Optimal Interventions for Epidemic Outbreaks

Muqbel

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I am also grateful for the support of the Stipendium Hungaricum Foundation, the Bolyai Institute and Doctoral School in Mathematics and Computer Science. My sincere thanks also go to the Dynamical Models in Biology team and the PhD students of Bolyai Institute, who encouraged me to go a long way.I would like to thank my parents and siblings for their continued support that started from my childhood and has been continuing until this moment. Also, I thank my wife and children for their patient for my long absent.Last but not least, I would like to thank my extended family and friends for their moral support throughout the writing of this thesis and my life in general.

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Dynamics of a non-smooth epidemic model with three thresholds

Cited by 7 publications

References 42 publications

Analysis of Infected Population Threshold Exceedance in an SIR Epidemiological Model

Analysis of Infected Population Threshold Exceedance in an SIR Epidemiological Model

Periodic Orbits and Global Stability for a Discontinuous SIR Model with Delayed Control

Optimal Interventions for Epidemic Outbreaks

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