Background: Vaccination was the single most effective measure in mitigating the impact of the COVID-19 pandemic. Our study aims to quantify the impact of vaccination programmes during the initial year of vaccination (2021) by estimating the number of case fatalities avoided, using Sweden as a case study.

Methods: Using Swedish data on age-specific reported incidence and vaccination uptake, along with vaccine efficacies, age-specific contact patterns and under-reporting from the literature, we fit a Bayesian SEIR epidemic model with time-varying community contact rate for COVID-19 incidence. Age-specific fatality rates from the literature are adjusted proportionally to fit the observed number of case fatalities in the factual analysis, resulting in 5,510 (95% PI: 5,370-5,650) matching the observed number 5,430. The estimated time-varying community contact rate is then used in a counterfactual analysis where the population is unvaccinated, leading to more infections and fatalities. A sensitivity analysis is performed to identify which parameters influence our conclusions.

Findings: The counterfactual analysis result in a severe epidemic outbreak during the early autumn of 2021, resulting in about 37,100 (36,700–37,500) number of case fatalities. Consequently, the number of lives saved by the vaccination programme is estimated to be about 31,600 (31,300–32,000), out of which 5,170 are directly saved and 26,400 are indirectly saved, mainly by drastically reducing the severe outbreak in the early autumn of 2021, which would have occurred without vaccination and unchanged community contact rate.

Interpretation: Our mathematical model is used to analyse the impact of COVID-19 vaccination on lives saved in Sweden during 2021, but the same methodology can be applied to other countries. The counterfactual analysis offers insights into an alternative trajectory of the pandemic without vaccination. The results show the direct impact of vaccination on reducing deaths for infected individuals and shed light on the indirect effects of reduced transmission dynamics.

We consider a model for the spread of an influenza-like disease in which, between seasons, the virus makes a random genetic drift (reducing immunity) and obtains a new random transmissibility (closely related to R₀). Given the immunity status at the start of season k, i.e. the community distribution of years since last infection and their associated immunity levels, the outcome of the epidemic season k, characterized by the effective reproduction number and the fractions infected in the different immunity groups , is determined by the random genetic drift and transmissibility. It is shown that the community immunity status of consecutive seasons, is an ergodic Markov chain, which converges to a stationary distribution. More analytical progress is made for the case where immunity only lasts for one season: we then characterize the stationary distribution of the community fraction having partial immunity (from being infected last season) as well as the stationary distribution of , and the conditional distribution of given . The effective reproduction number is closely related to the initial exponential growth rate of the outbreak, a quantity which can be estimated early in the epidemic season. As a consequence, this conditional distribution may be used for predicting the final size of the epidemic based on its initial growth and immunity status.

We analyse a Markovian SIR epidemic model where individuals either recover naturally or are diagnosed, leading to isolation and potential contact tracing. Our focus is on digital contact tracing via a tracing app, considering both its standalone use and its combination with manual tracing. We prove that as the population size n grows large, the epidemic process converges to a limiting process, which, unlike with typical epidemic models, is not a branching process due to dependencies created by contact tracing. However, by grouping to-be-traced individuals into macro-individuals, we derive a multi-type branching process interpretation, allowing computation of the reproduction number R. This is then converted to an individual reproduction number R^(ind), which, in contrast to R, decays monotonically with the fraction of app-users, while both share the same threshold at 1. Finally, we compare digital (only) contact tracing and manual (only) contact tracing, proving that the critical fraction of app-users, π_c, required for R = 1 is higher than the critical fraction manually contact-traced, p_c, for manual tracing.

Social contact studies are used in infectious disease epidemiology to infer a contact matrix , having the mean number of contacts between individuals of different age groups as elements. However, M does not capture the (often large) variation in the number of contacts within each age group, information is also available in social contact studies. Here, we include such variation by separating each age group into two halves: the socially active (having many contacts) and the socially less active (having fewer contacts). The extended contact matrix and its associated epidemic model show that acknowledging variation in social activity within age groups has a substantial impact on the basic reproduction number, R₀, and the final fraction getting infected if the epidemic takes off, τ. In fact, variation in social activity is more important for data fitting than allowing for different age groups. A difficulty with variation in social activity, however, is that social contact studies typically lack information on whether mixing with respect to social activity is assortative (when socially active mainly have contact with other socially active individuals) or not. Our analysis shows that accounting for variation in social activity improves model predictability, yielding more accurate expressions for R₀ and τ irrespective of whether such mixing is assortative, but different assumptions on assortativity give rather different outputs. Future social contact studies should, therefore, also try to infer the degree of assortativity (with respect to social activity) between peers and their contacts.

We consider an SEIR epidemic model on a network also allowing random contacts, where recovered individuals could either recover naturally or be diagnosed. Upon diagnosis, manual contact tracing is triggered such that each infected network contact is reported, tested and isolated with some probability and after a random delay. Additionally, digital tracing (based on a tracing app) is triggered if the diagnosed individual is an app-user, and then all of its app-using infectees are immediately notified and isolated. The early phase of the epidemic with manual and/or digital tracing is approximated by different multi-type branching processes, and three respective reproduction numbers are derived. The effectiveness of both contact tracing mechanisms is numerically quantified through the reduction of the reproduction number. This shows that app-using fraction plays an essential role in the overall effectiveness of contact tracing. The relative effectiveness of manual tracing compared to digital tracing increases if: more of the transmission occurs on the network, when the tracing delay is shortened, and when the network degree distribution is heavy-tailed. For realistic values, the combined tracing case can reduce R0 by 20%–30%, so other preventive measures are needed to reduce the reproduction number down to 1.2–1.4 for contact tracing to make it successful in avoiding big outbreaks.

In the current paper we analyse an extended SIRS epidemic model in which immunity at the individual level wanes gradually at exponential rate, but where the waning rate may differ between individuals, for instance as an effect of differences in immune systems. The model also includes vaccination schemes aimed to reach and maintain herd immunity. We consider both the informed situation where the individual waning parameters are known, thus allowing selection of vaccinees being based on both time since last vaccination as well as on the individual waning rate, and the more likely uninformed situation where individual waning parameters are unobserved, thus only allowing vaccination schemes to depend on time since last vaccination. The optimal vaccination policies for both the informed and uniformed heterogeneous situation are derived and compared with the homogeneous waning model (meaning all individuals have the same immunity waning rate), as well as to the classic SIRS model where immunity at the individual level drops from complete immunity to complete susceptibility in one leap. It is shown that the classic SIRS model requires least vaccines, followed by the SIRS with homogeneous gradual waning, followed by the informed situation for the model with heterogeneous gradual waning. The situation requiring most vaccines for herd immunity is the most likely scenario, that immunity wanes gradually with unobserved individual heterogeneity. For parameter values chosen to mimic COVID-19 and assuming perfect initial immunity and cumulative immunity of 12 months, the classic homogeneous SIRS epidemic suggests that vaccinating individuals every 15 months is sufficient to reach and maintain herd immunity, whereas the uninformed case for exponential waning with rate heterogeneity corresponding to a coefficient of variation being 0.5, requires that individuals instead need to be vaccinated every 4.4 months.

Susceptible-infectious-recovered-susceptible (SIRS) epidemic models assume that individual immunity wanes in one leap, from complete immunity to complete susceptibility. For many diseases immunity on the contrary wanes gradually, something that has become even more evident during COVID-19 pandemic where also recently infected have a reinfection risk, and booster vaccines are given to increase immunity. Here, a novel mathematical model is presented allowing for the gradual decay of immunity following linear or exponential waning functions. The two new models and the SIRS model are compared assuming all three models have the same cumulative immunity. When no intervention is put in place, we find that the long-term prevalence is higher for the models with gradual waning. If aiming for herd immunity by continuous vaccination, it is shown that larger vaccine quantities are required when immunity wanes gradually compared with results obtained from the SIRS model, and this difference is the biggest for the most realistic assumption of exponentially waning of immunity. For parameter choices fitting to COVID-19, the critical amount of vaccine supply is about 50% higher if immunity wanes linearly, and more than 150% higher when immunity wanes exponentially, when compared with the classic SIRS epidemic model.

This article considers the minimization of the total number of infected individuals over the course of an epidemic in which the rate of infectious contacts can be reduced by time-dependent nonpharmaceutical interventions. The societal and economic costs of interventions are taken into account using a linear budget constraint which imposes a trade-off between short-term heavy interventions and long-term light interventions. We search for an optimal intervention strategy in an infinite-dimensional space of controls containing multiple consecutive lockdowns, gradually imposed and lifted restrictions, and various heuristic controls based, for example, on tracking the effective reproduction number. Mathematical analysis shows that among all such strategies, the global optimum is achieved by a single constant-level lockdown of maximum possible magnitude. Numerical simulations highlight the need for careful timing of such interventions and illustrate their benefits and disadvantages compared to strategies designed for minimizing peak prevalence. Rather counterintuitively, adding restrictions prior to the start of a well-planned intervention strategy may even increase the total incidence.

Consider a Markovian SIR epidemic model in a homogeneous community. To this model we add a rate at which individuals are tested, and once an infectious individual tests positive it is isolated and each of their contacts are traced and tested independently with some fixed probability. If such a traced individual tests positive it is isolated, and the contact tracing is iterated. This model is analysed using large population approximations, both for the early stage of the epidemic when the “to-be-traced components” of the epidemic behaves like a branching process, and for the main stage of the epidemic where the process of to-be-traced components converges to a deterministic process defined by a system of differential equations. These approximations are used to quantify the effect of testing and of contact tracing on the effective reproduction numbers (for the components as well as for the individuals), the probability of a major outbreak, and the final fraction getting infected. Using numerical illustrations when rates of infection and natural recovery are fixed, it is shown that Test-and-Trace strategy is effective in reducing the reproduction number. Surprisingly, the reproduction number for the branching process of components is not monotonically decreasing in the tracing probability, but the individual reproduction number is conjectured to be monotonic as expected. Further, in the situation where individuals also self-report for testing, the tracing probability is more influential than the screening rate (measured by the fraction infected being screened).