SIR models with vital dynamics, reinfection and randomness to investigate the spread of infectious diseases

TL;DR

This paper analyzes stochastic SIR models with vital dynamics and reinfection, examining their long-term behavior, stability, and disease outcomes through theoretical analysis and numerical simulations.

Contribution

It introduces a comprehensive analysis of SIR models incorporating randomness and vital dynamics, including stability and attractor properties, which are novel in the context of disease modeling.

Findings

Existence of attractors under various scenarios

Conditions for disease eradication or endemicity

Numerical simulations confirming theoretical results

Abstract

We investigate SIR models with vital dynamics, reinfection, and randomness at the transmission coefficient and recruitment rate. Initially, we conduct an extensive analysis of the autonomous scenario, covering aspects such as local and global well-posedness, the existence and internal structure of attractors, and the presence of gradient dynamics. Subsequently, we explore the implications of small nonautonomous random perturbations, establishing the continuity of attractors and ensuring their topological structural stability. Additionally, we study scenarios in which both the transmission coefficient and the recruitment rate exhibit time-dependent or random behavior. For each scenario, we establish the existence of attractors and delineate conditions that determine whether the disease is eradicated or reaches an endemic state. Finally, we depict numerical simulations to illustrate the theoretical results.