Susceptible‐infected‐recovered models with natural birth and death on complex networks. (27th December 2013)

• Record Type: Journal Article
• Authors: Huang, Qian; Min, Lequan; Chen, Xiao
• Is Part Of: Mathematical methods in the applied sciences. Volume 38:Number 1(2015:Jan. 15)
• Page Start: 37
• Page End: 50
• Publication Date: 2013-12-27
• Subjects: Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.3048
• Languages: English
• ISSNs: 0170-4214

Abstract

<abstract abstract-type="main" id="mma3048-abs-0001"> <title> <x xml:space="preserve">Abstract</x> </title> <p id="mma3048-para-0001">This paper proposes two modified susceptible‐infected‐recovered (SIRS) models on homogenous and heterogeneous networks, respectively. In the study of the homogenous network model, Lyapunov functions are used to study the globally asymptotically stable of the equilibria of the model. It is proved that if the basic reproduction number of the model is less than one, then the disease‐free equilibrium is globally asymptotically stable, otherwise, the endemic equilibrium is globally asymptotically stable. In the study of the heterogeneous network model, the existences of the disease‐free equilibrium and epidemic equilibrium of the model are discussed. A threshold value <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2qt16qgt" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:1704214:media:mma3048:mma3048-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>R</mml:mi><mml:mo stretchy="true">˜</mml:mo></mml:mover></mml:mrow><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math></alternatives></inline-formula> is given. It is proved that if the threshold value <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2qt16qjx" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:1704214:media:mma3048:mma3048-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>R</mml:mi><mml:mo stretchy="true">˜</mml:mo></mml:mover></mml:mrow><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math></alternatives></inline-formula> of the model is less than one, then the disease‐free equilibrium is globally asymptotically stable. The simulation examples on the two SIRS models are given. Copyright © 2013 John Wiley & Sons, Ltd.</p> </abstract>