Asymptotic properties of a general model of immune status

TL;DR

This paper analyzes a comprehensive immune system model incorporating boosting, waning, and non-Markovian antibody dynamics, proving its long-term stability through mathematical analysis of the associated stochastic process.

Contribution

It introduces a general immune status model with non-Markovian antibody dynamics and establishes its asymptotic stability using stochastic semigroup theory.

Findings

The model captures complex immune dynamics including boosting and waning.

The associated stochastic semigroup is proven to be asymptotically stable.

The system's long-term behavior converges to a stable distribution.

Abstract

We consider a model of dynamics of the immune system. The model is based on three factors: occasional boosting and continuous waning of immunity and a general description of the period between subsequent boosting events. The antibody concentration changes according to a non-Markovian process. The density of the distribution of this concentration satisfies some partial differential equation with an integral boundary condition. We check that this system generates a stochastic semigroup and we study the long-time behaviour of this semigroup. In particular we prove a theorem on its asymptotic stability.