<abstract xmlns="http://www.w3.org/1999/xhtml">Equilibrium analysis in autonomous evolutionary models is of central importance for developing long term treatments. This task typically includes checks on the existence and stability of some equilibria. Prior to touching on the stability, one often attempts to determine the existence where the basic reproductive number <italic>ℛ</italic>0 plays a critical role as a threshold parameter. When analyzing a nontrivial equilibrium (e.g., an endemic, boundary, or coexistence equilibrium) where <italic>ℛ</italic>0 is explicit, we usually come across a typical result: if <italic>ℛ</italic>0 >1, then a nontrivial equilibrium exists in the biological sense. However, for more sophisticated models, <italic>ℛ</italic>0 can be too complicated to be revealed in terms of the involving parameters; the task of relating the formulation of a nontrivial equilibrium to <italic>ℛ</italic>0 thus becomes intractable. This paper shows how to mitigate such a problem with the aid of functional analysis, adopting the framework of a nonlinear eigenvalue problem. An equilibrium equation is first to be transformed into a canonical equation in a lower dimension, and then the existence is confirmed under several conditions. Three models are tested showing the applicability of this approach.</abstract>

Equilibrium analysis in autonomous evolutionary models is of central importance for developing long term treatments. This task typically includes checks on the existence and stability of some equilibria. Prior to touching on the stability, one often attempts to determine the existence where the basic reproductive number ℛ0 plays a critical role as a threshold parameter. When analyzing a nontrivial equilibrium (e.g., an endemic, boundary, or coexistence equilibrium) where ℛ0 is explicit, we usually come across a typical result: if ℛ0 >1, then a nontrivial equilibrium exists in the biological sense. However, for more sophisticated models, ℛ0 can be too complicated to be revealed in terms of the involving parameters; the task of relating the formulation of a nontrivial equilibrium to ℛ0 thus becomes intractable. This paper shows how to mitigate such a problem with the aid of functional analysis, adopting the framework of a nonlinear eigenvalue problem. An equilibrium equation is first to be transformed into a canonical equation in a lower dimension, and then the existence is confirmed under several conditions. Three models are tested showing the applicability of this approach.

On the existence of a nontrivial equilibrium in relation to the basic reproductive number

International Journal of Applied Mathematics and Computer Science

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Equilibrium analysis in autonomous evolutionary models is of central importance for developing long term treatments. This task typically includes checks on the...

On the existence of a nontrivial equilibrium in relation to the basic reproductive number

Published Online: Sep 23, 2017

Page range: 623 - 636

Received: Aug 02, 2016

Accepted: Apr 18, 2017

DOI: https://doi.org/10.1515/amcs-2017-0044

Keywords
autonomous model, nontrivial equilibrium, basic reproductive number, nonlinear eigenvalue problem

© 2017 Karunia Putra Wijaya et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

On the existence of a nontrivial equilibrium in relation to the basic reproductive number

Published Online: Sep 23, 2017

Page range: 623 - 636

Received: Aug 02, 2016

Accepted: Apr 18, 2017

DOI: https://doi.org/10.1515/amcs-2017-0044

Keywordsautonomous model, nontrivial equilibrium, basic reproductive number, nonlinear eigenvalue problem

© 2017 Karunia Putra Wijaya et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
autonomous model, nontrivial equilibrium, basic reproductive number, nonlinear eigenvalue problem