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). 

The real-time analysis of infectious disease surveillance data is essential in obtaining situational awareness about the current dynamics of a major public health event such as the COVID-19 pandemic. This analysis of e.g., time-series of reported cases or fatalities is complicated by reporting delays that lead to under-reporting of the complete number of events for the most recent time points. This can lead to misconceptions by the interpreter, for instance the media or the public, as was the case with the time-series of reported fatalities during the COVID-19 pandemic in Sweden. Nowcasting methods provide real-time estimates of the complete number of events using the incomplete time-series of currently reported events and information about the reporting delays from the past. In this paper we propose a novel Bayesian nowcasting approach applied to COVID-19-related fatalities in Sweden. We incorporate additional information in the form of time-series of number of reported cases and ICU admissions as leading signals. We demonstrate with a retrospective evaluation that the inclusion of ICU admissions as a leading signal improved the nowcasting performance of case fatalities for COVID-19 in Sweden compared to existing methods.

A stochastic SIR (susceptible  infective  recovered) model is considered for the spread of an epidemic on a network, described initially by an Erdős–Rényi random graph, in which susceptible individuals connected to infectious neighbors may drop or rewire such connections. A novel construction of the model is used to derive a deterministic model for epidemics started with a positive fraction initially infected and prove convergence of the scaled stochastic model to that deterministic model as the population size . For epidemics initiated by a single infective that take off, we prove that for part of the parameter space, in the limit as , the final fraction infected  is discontinuous in the infection rate  at its threshold , thus not converging to 0 as . The discontinuity is particularly striking when rewiring is necessarily to susceptible individuals in that  jumps from 0 to 1 as  passes through .

We present a stochastic epidemic model to study the effect of various preventive measures, such as uniform reduction of contacts and transmission, vaccination, isolation, screening and contact tracing, on a disease outbreak in a homogeneously mixing community. The model is based on an infectivity process, which we define through stochastic contact and infectiousness processes, so that each individual has an independent infectivity profile. In particular, we monitor variations of the reproduction number and of the distribution of generation times. We show that some interventions, i.e. uniform reduction and vaccination, affect the former while leaving the latter unchanged, whereas other interventions, i.e. isolation, screening and contact tracing, affect both quantities. We provide a theoretical analysis of the variation of these quantities, and we show that, in practice, the variation of the generation time distribution can be significant and that it can cause biases in the estimation of reproduction numbers. The framework, because of its general nature, captures the properties of many infectious diseases, but particular emphasis is on COVID-19, for which numerical results are provided.

The transmission heterogeneity of an epidemic is associated with a complex mixture of host, pathogen and environmental factors. And it may indicate superspreading events to reduce the efficiency of population-level control measures and to sustain the epidemic over a larger scale and a longer duration. Methods have been proposed to identify significant transmission heterogeneity in historic epidemics based on several data sources, such as contact history, viral genomes and spatial information, which may not be available, and more importantly ignore the temporal trend of transmission heterogeneity. Here we attempted to establish a convenient method to estimate real-time heterogeneity over an epidemic. Within the branching process framework, we introduced an instant-individualheterogenous infectiousness model to jointly characterize the variation in infectiousness both between individuals and among different times. With this model, we could simultaneously estimate the transmission heterogeneity and the reproduction number from incidence time series. We validated the model with data of both simulated and real outbreaks. Our estimates of the overall and real-time heterogeneities of the six epidemics were consistent with those presented in the literature. Additionally, our model is robust to the ubiquitous bias of under-reporting and misspecification of serial interval. By analyzing recent data from South Africa, we found evidence that the Omicron might be of more significant transmission heterogeneity than Delta. Our model based on incidence data was proved to be reliable in estimating the real-time transmission heterogeneity.