Dynamics of an SIRWS model with waning of immunity and varying immune boosting period

TL;DR

This paper investigates an extended SIRWS model with asymmetric waning immunity, revealing complex dynamics like bifurcations and bistability, and emphasizing the importance of immune boosting duration on disease persistence.

Contribution

It introduces an asymmetric immune waning period into the SIRWS model and analyzes its impact on disease dynamics using analytic and numerical methods.

Findings

Rich bifurcation structures including Hopf bifurcations and endemic bubbles

Asymmetric immune waning significantly affects long-term disease behavior

Immune boosting duration is a key parameter influencing epidemiological outcomes

Abstract

SIRS models capture transmission dynamics of infectious diseases for which immunity is not lifelong. Extending these models by a W compartment for individuals with waning immunity, the boosting of the immune system upon repeated exposure may be incorporated. Previous analyses assumed identical waning rates from R to W and from W to S. This implicitly assumes equal length for the period of full immunity and of waned immunity. We relax this restriction, and allow an asymmetric partitioning of the total immune period. Stability switches of the endemic equilibrium are investigated with a combination of analytic and numerical tools. Then, continuation methods are applied to track bifurcations along the equilibrium branch. We find rich dynamics: Hopf bifurcations, endemic double bubbles, and regions of bistability. Our results highlight that the length of the period in which waning immunity can be boosted is a crucial parameter significantly influencing long term epidemiological dynamics.