Steady states in a structured epidemic model with Wentzell boundary condition

TL;DR

This paper develops a nonlinear structured population model with diffusion and dynamic boundary conditions to analyze infection spread, establishing conditions for steady states and their stability in a disease transmission context.

Contribution

It introduces a novel nonlinear structured population model with Wentzell boundary conditions and provides analytical results on existence, stability, and steady states.

Findings

Global existence of solutions established

Conditions for non-trivial steady states derived

Sufficient criteria for local stability of steady states

Abstract

We introduce a nonlinear structured population model with diffusion in the state space. Individuals are structured with respect to a continuous variable which represents a pathogen load. The class of uninfected individuals constitutes a special compartment that carries mass, hence the model is equipped with generalized Wentzell (or dynamic) boundary conditions. Our model is intended to describe the spread of infection of a vertically transmitted disease, for example Wolbachia in a mosquito population. Therefore the (infinite dimensional) nonlinearity arises in the recruitment term. First we establish global existence of solutions and the Principle of Linearised Stability for our model. Then, in our main result, we formulate simple conditions, which guarantee the existence of non-trivial steady states of the model. Our method utilizes an operator theoretic framework combined with a fixed point approach. Finally, in the last section we establish a sufficient condition for the local asymptotic stability of the positive steady state.