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

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.

Keywords:

• Waning immunity
• immune boosting
• SIRWS system
• partitioning of immunity
• Hopf bifurcation

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was completed in the National Laboratory for Health Security [RRF-2.3.1-21-2022-00006]. Additionally, the work was supported by the Ministry of Innovation and Technology of Hungary from the National Research, Development and Innovation Fund [project no. TKP2021-NVA-09]. In addition, F.B. and G.R. were supported by NKFIH [grant numbers FK 138924, KKP 129877]. M.P. was also supported by the Hungarian Scientific Research Fund [grant numbers K129322 and SNN125119]; F.B. was also supported by UNKP-21-5 and the Bolyai scholarship of the Hungarian Academy of Sciences.