Waning immunity can drive repeated waves of infections

Desmond Z. Lai; Julia R. Gog; Desmond Z. Lai; Julia R. Gog

doi:10.3934/mbe.2024088

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 2: 1979-2003. doi: 10.3934/mbe.2024088

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Waning immunity can drive repeated waves of infections

Desmond Z. Lai ^{1
,
,},
Julia R. Gog ^1,2

1.
Department of Applied Mathematics and Theoretical Physics (DAMTP), University of Cambridge, United Kingdom
2.
Joint UNIversities Pandemic and Epidemiological Research (JUNIPER) Consortium, United Kingdom

Received: 12 September 2023 Revised: 14 December 2023 Accepted: 19 December 2023 Published: 08 January 2024

In infectious disease models, it is known that mechanisms such as births, seasonality in transmission and pathogen evolution can generate oscillations in infection numbers. We show how waning immunity is also a mechanism that is sufficient on its own to enable sustained oscillations. When previously infected or vaccinated individuals lose full protective immunity, they become partially susceptible to reinfections. This partial immunity subsequently wanes over time, making individuals more susceptible to reinfections and potentially more infectious if infected. Losses of full and partial immunity lead to a surge in infections, which is the precursor of oscillations. We present a discrete-time Susceptible-Infectious-Immune-Waned-Infectious (SIRWY) model that features the waning of fully immune individuals (as a distribution of time at which individuals lose fully immunity) and the gradual loss of partial immunity (as increases in susceptibility and potential infectiousness over time). A special case of SIRWY is the discrete-time SIRS model with geometric distributions for waning and recovery. Its continuous-time analogue is the classic SIRS with exponential distributions, which does not produce sustained oscillations for any choice of parameters. We show that the discrete-time version can produce sustained oscillations and that the oscillatory regime disappears as discrete-time tends to continuous-time. A different special case of SIRWY is one with fixed times for waning and recovery. We show that this simpler model can also produce sustained oscillations. In conclusion, under certain feature and parameter choices relating to how exactly immunity wanes, fluctuations in infection numbers can be sustained without the need for any additional mechanisms.
- disease dynamics,
- bifurcation analysis,
- periodicity,
- time since infection,
- time since recovery
Citation: Desmond Z. Lai, Julia R. Gog. Waning immunity can drive repeated waves of infections[J]. Mathematical Biosciences and Engineering, 2024, 21(2): 1979-2003. doi: 10.3934/mbe.2024088

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Abstract

In infectious disease models, it is known that mechanisms such as births, seasonality in transmission and pathogen evolution can generate oscillations in infection numbers. We show how waning immunity is also a mechanism that is sufficient on its own to enable sustained oscillations. When previously infected or vaccinated individuals lose full protective immunity, they become partially susceptible to reinfections. This partial immunity subsequently wanes over time, making individuals more susceptible to reinfections and potentially more infectious if infected. Losses of full and partial immunity lead to a surge in infections, which is the precursor of oscillations. We present a discrete-time Susceptible-Infectious-Immune-Waned-Infectious (SIRWY) model that features the waning of fully immune individuals (as a distribution of time at which individuals lose fully immunity) and the gradual loss of partial immunity (as increases in susceptibility and potential infectiousness over time). A special case of SIRWY is the discrete-time SIRS model with geometric distributions for waning and recovery. Its continuous-time analogue is the classic SIRS with exponential distributions, which does not produce sustained oscillations for any choice of parameters. We show that the discrete-time version can produce sustained oscillations and that the oscillatory regime disappears as discrete-time tends to continuous-time. A different special case of SIRWY is one with fixed times for waning and recovery. We show that this simpler model can also produce sustained oscillations. In conclusion, under certain feature and parameter choices relating to how exactly immunity wanes, fluctuations in infection numbers can be sustained without the need for any additional mechanisms.

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