Modelling the impact of vaccination-dependent contact rates on disease transmission: applications to COVID-19

Mathematical models are often used to predict the impact of infectious disease interventions. Most models, however, neglect how the intervention might influence patterns of contact within the population, and consequently change the dynamics of the epidemic. Here we introduce a mathematical model of disease transmission where contact rates change in response to a vaccination campaign. We explore two scenarios: homogeneous mixing, where contact rates increase uniformly across the population as vaccination coverage increases, and heterogeneous mixing, where the probability of contact depends on the vaccination statuses of the two individuals involved. We derive the effective reproduction number as a function of vaccine coverage and its transmission-blocking effectiveness, and show the conditions under which an increase in vaccination coverage leads to a growing number of infections. The model is then parameterised using United Kingdom COVID-19 data between 2020 and 2022 and vaccine-dependent contact rates are estimated using contact survey data. We observe that contact increases concordantly with vaccination coverage. Implementing this relationship in the model, we infer that the epidemiological impact of the rising contact rate was tempered by a mixing structure where contacts predominantly involve at least one vaccinated individual.