[vc_empty_space][vc_empty_space]
On the existence of a nontrivial equilibrium in relation to the basic reproductive number
Wijaya K.P.a, Sutiminb, Soewono E.c, Gotz T.a
a Mathematical Institute, University of Koblenz, Koblenz, 56070, Germany
b Department of Mathematics, Diponegoro University, Semarang, 50275, Indonesia
c Department of Mathematics, Bandung Institute of Technology, Bandung, 40132, Indonesia
[vc_row][vc_column][vc_row_inner][vc_column_inner][vc_separator css=”.vc_custom_1624529070653{padding-top: 30px !important;padding-bottom: 30px !important;}”][/vc_column_inner][/vc_row_inner][vc_row_inner layout=”boxed”][vc_column_inner width=”3/4″ css=”.vc_custom_1624695412187{border-right-width: 1px !important;border-right-color: #dddddd !important;border-right-style: solid !important;border-radius: 1px !important;}”][vc_empty_space][megatron_heading title=”Abstract” size=”size-sm” text_align=”text-left”][vc_column_text]© 2017 Karunia Putra Wijaya et al., published by De Gruyter Open 2017.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.[/vc_column_text][vc_empty_space][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][vc_empty_space][megatron_heading title=”Author keywords” size=”size-sm” text_align=”text-left”][vc_column_text]Autonomous models,Basic reproductive number,Equilibrium analysis,Equilibrium equation,Existence and stability,Non-trivial equilibrium,Nonlinear eigenvalue problem,Threshold parameters[/vc_column_text][vc_empty_space][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][vc_empty_space][megatron_heading title=”Indexed keywords” size=”size-sm” text_align=”text-left”][vc_column_text]autonomous model,basic reproductive number,nonlinear eigenvalue problem,nontrivial equilibrium[/vc_column_text][vc_empty_space][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][vc_empty_space][megatron_heading title=”Funding details” size=”size-sm” text_align=”text-left”][vc_column_text]This research is supported by the Indonesia Endowment Fund for Education (LPDP) and partially supported by DAAD through the project Mathematical Models for Dengue Fever (DAAD PPP-Portugal, project ID 57128360). The authors are also grateful to anonymous reviewers for their valuable suggestions.[/vc_column_text][vc_empty_space][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][vc_empty_space][megatron_heading title=”DOI” size=”size-sm” text_align=”text-left”][vc_column_text]https://doi.org/10.1515/amcs-2017-0044[/vc_column_text][/vc_column_inner][vc_column_inner width=”1/4″][vc_column_text]Widget Plumx[/vc_column_text][/vc_column_inner][/vc_row_inner][/vc_column][/vc_row][vc_row][vc_column][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][/vc_column][/vc_row]