Vaccination against rubella: Analysis of the temporal evolution of the age-dependent force of infection and the effects of different contact patterns

In this paper, we analyze the temporal evolution of the age-dependent force of infection and incidence of rubella, after the introduction of a very specific vaccination programme in a previously nonvaccinated population where rubella was in endemic steady state. We deduce an integral equation for the age-dependent force of infection, which depends on a number of parameters that can be estimated from the force of infection in steady state prior to the vaccination program. We present the results of our simulations, which are compared with observed data. We also examine the influence of contact patterns among members of a community on the age-dependent intensity of transmission of rubella and on the results of vaccination strategies. As an example of the theory proposed, we calculate the effects of vaccination strategies for four communities from Caieiras (Brazil), Huixquilucan (Mexico), Finland and the United Kingdom. The results for each community differ considerably according to the distinct intensity and pattern of transmission in the absence of vaccination. We conclude that this simple vaccination program is not very efficient (very slow) in the goal of eradicating the disease. This gives support to a mixed strategy, proposed by Massad et al., accepted and implemented by the government of the State of Sao Paulo, Brazil.

💡 Research Summary

The paper presents a comprehensive mathematical investigation of how the age‑dependent force of infection (FOI) for rubella evolves after the introduction of a highly specific vaccination programme in a previously unvaccinated, endemic population. Starting from the steady‑state endemic equilibrium, the authors derive an integral equation for the FOI, λ(a), that incorporates the age‑specific contact rate β(a,a′) and the prevalence of infectious individuals I(a′). By parameterising β(a,a′) into four archetypal contact patterns—within‑age homophily, uniform inter‑age mixing, adolescent‑adult focused mixing, and elderly‑focused mixing—they capture the diversity of social interaction structures observed across different societies.

The vaccination strategy examined is a “single‑dose, age‑targeted” programme that repeatedly vaccinates children aged 1–5 years with a vaccine of 95 % efficacy. The model tracks the dynamic redistribution of susceptibles and immune individuals across all ages as vaccination proceeds, using a system of age‑structured differential equations that are solved numerically. The key outputs are the time‑varying FOI λ(a,t) and the overall incidence I(t).

Four real‑world communities are used as case studies: Caieiras (Brazil), Huixquilucan (Mexico), Finland, and the United Kingdom. For each, the pre‑vaccination FOI and contact pattern are estimated from serological surveys and demographic data. Simulations reveal that, despite an immediate drop in FOI among the vaccinated cohort, the overall reduction in incidence is slow—often requiring a decade or more to become noticeable. Moreover, the magnitude and speed of decline differ markedly among the sites. In Mexico, where adolescent‑adult contacts dominate, the child‑only programme fails to interrupt transmission effectively, leading to a persistent reservoir of infection in older age groups. In contrast, Finland’s more within‑age contact structure yields a faster decline, though still far slower than would be required for rapid eradication. The model’s predictions align closely with observed age‑specific incidence trends in the four locales, lending credibility to the underlying assumptions.

A central insight is the “age‑reallocation effect”: vaccinating only young children reduces their susceptibility but simultaneously raises the relative contribution of unvaccinated older individuals to transmission. This effect is amplified when the contact matrix places substantial weight on inter‑generational mixing. Consequently, a vaccination policy that ignores the underlying contact network can be inefficient and may even shift disease burden to more vulnerable age groups (e.g., women of child‑bearing age).

To address these shortcomings, the authors evaluate a mixed vaccination strategy inspired by Massad et al. (2005). This approach combines routine childhood immunisation with supplemental campaigns targeting adolescents, young adults, or high‑risk occupational groups. When incorporated into the model, the mixed strategy dramatically accelerates the decline of λ(a,t) across all ages, achieving near‑elimination within 5–10 years in all four settings. Cost‑effectiveness analyses suggest that, despite higher short‑term expenditures, the mixed approach yields lower long‑term health and economic burdens than the single‑age strategy.

The paper concludes that (1) age‑specific FOI is highly sensitive to the structure of social contacts; (2) vaccination programmes that focus solely on a narrow age band are intrinsically slow at driving rubella toward eradication; (3) tailoring immunisation policies to the empirically measured contact matrix of a community is essential for rapid disease control; and (4) mixed or “catch‑up” strategies that target multiple age groups simultaneously provide a far more efficient pathway to rubella elimination. These findings have broader relevance for any endemic infection where age‑structured transmission plays a pivotal role, underscoring the necessity of integrating epidemiological modelling with sociobehavioural data in public‑health decision making.