Oscillations in SIRS model with distributed delays

TL;DR

This paper investigates how distributed delays in the SIRS epidemic model influence oscillations, providing a comprehensive analysis of stability, oscillation characteristics, and comparisons with real disease data.

Contribution

It extends the classical SIRS model by incorporating distributed delays, analyzing their impact on oscillation stability and characteristics through both numerical and linearized methods.

Findings

Oscillations can be destabilized by the shape of delay distributions.

Delay distributions significantly affect oscillation amplitude and period.

Simulation results support linear analysis in certain parameter regimes.

Abstract

The ubiquity of oscillations in epidemics presents a long standing challenge for the formulation of epidemic models. Whether they are external and seasonally driven, or arise from the intrinsic dynamics is an open problem. It is known that fixed time delays destabilize the steady state solution of the standard SIRS model, giving rise to stable oscillations for certain parameters values. In this contribution, starting from the classical SIRS model, we make a general treatment of the recovery and loss of immunity terms. We present oscillation diagrams (amplitude and period) in terms of the parameters of the model, showing how oscillations can be destabilized by the shape of the distributions of the two characteristic (infectious and immune) times. The formulation is made in terms of delay equation which are both numerical integrated and linearized. Results from simulation are included showing where they support the linear analysis and explaining why not where they do not. Considerations and comparison with real diseases are presented along.