A discrete-time waning immunity model for the spread of an immune escape pathogen

This thesis explores the formulation of waning immunity models for the spread of an immune escape (antigenically evolving) pathogen, such as the Omicron lineage of SARS-CoV-2, and investigates the possibility of such models in generating sustained or relatively large amplitudes of damped oscillations in infections. The classic Susceptible-Infectious-Immune-Susceptible (SIRS) model is a waning immunity model but is unable to generate rapid high amplitude oscillations. The oscillatory dynamics generated by SIRS-like models are often boosted by additional mechanisms and model features such as demography, a seasonal variation in transmissibility and immune boosting. We develop and simulate discrete-time waning immunity models with only model features that are related to waning immunity and pathogen infectiousness, such as a transmissibility parameter based on time since infection and a change in susceptibility based on time since last immune event (recovery from infection or vaccination). We show that these models can generate oscillatory infection dynamics that agree qualitatively with Omicron infection dynamics observed in England and countries worldwide. In Chapter 2, we explore whether a discrete-time formulation facilitates the inclusion of model features related to waning immunity in SIRS-like models and whether such a formulation can generate sustained oscillations. We formulate a general Susceptible-Infectious-Immune-Waned-Infectious (SIRWY) waning immunity model in discrete-time and find that the discrete-time classic SIRS model is a sub-model of the general model. We show that the discrete-time formulation is a specific model feature that can generate sustained oscillations by analysing the classic SIRS model without demography in both discrete-time and continuous-time. In Chapter 3, we explore whether the oscillatory Omicron infection dynamics observed in England and countries worldwide is better explained by waning immunity (continuous antigenic drift from the pathogen point of view) or immune escape (antigenic shift). We formulate a model for each hypothesis and compare the simulation results with observed data from England. Although both models can generate results that are qualitatively similar to observed data to some extent, only the waning immunity model can provide a mechanistic explanation for the observed dynamic regime of damped oscillations in infections about the endemic equilibrium. In Chapter 4, motivated by the surges in hospitalisations associated with the surges in Omicron infections in the community, we investigate whether surges in Omicron infections in a population can be suppressed by vaccination. We formulate a waning immunity model with vaccination and simulate the model based on the Autumn 2022 COVID-19 vaccination campaign in England. We find that it is possible to reduce the amplitude of oscillations in infections without having to transition the system to the dynamic regime of disease-free equilibrium. Furthermore, a fixed period of vaccination is sufficient to increase the rate of convergence of the infection dynamics to the endemic equilibrium.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2025

Author dc:creator

• Lai, Zhiyuan Desmond

Advisor dc:contributor.advisor

• Gog, Julia

• Applied Mathematics
• Mathematical Biology
• Infectious Disease Modelling
• SARS-CoV-2
• COVID-19
• Waning Immunity
• Immune Escape
• Omicron

Licence

• https://www.repository.cam.ac.uk/bitstreams/5f308938-c195-4d2b-9083-2c119a1909fb/download
• http://purl.org/NET/rdflicense/allrightsreserved

Language dc:language

eng

Author Identifier

0000-0001-7603-3915

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/395382