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

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.