Improving pairwise approximations for network models with susceptible-infected-susceptible dynamics

Network models of disease spread play an important role in elucidating the impact of long-lasting infectious contacts on the dynamics of epidemics. Moment-closure approximation is a common method of generating low-dimensional deterministic models of epidemics on networks, which has found particular success for diseases with susceptible-infected-recovered (SIR) dynamics. However, the effect of network structure is arguably more important for sexually transmitted infections, where epidemiologically relevant contacts are comparatively rare and longstanding, and which are in general modelled via the susceptible-infected-susceptible (SIS)-paradigm. In this paper, we introduce an improvement to the standard pairwise approximation for network models with SIS-dynamics for two different network structures: the isolated open triple (three connected individuals in a line) and the -regular network. This improvement is achieved by tracking the rate of change of errors between triple values and their standard pairwise approximation. For the isolated open triple, this improved pairwise model is exact, while for -regular networks a closure is made at the level of triples to obtain a closed set of equations. This improved pairwise approximation provides an insight into the errors introduced by the standard pairwise approximation, and more closely matches both higher-order moment-closure approximations and explicit stochastic simulations with only a modest increase in dimensionality to the standard pairwise approximation

Concurrency of partnerships, consistency with data, and control of sexually transmitted infections

Sexually transmitted infections (STIs) are a globally increasing public health problem. Mathematical models, carefully matched to available epidemiological and behavioural data, have an important role to play in predicting the action of control measures. Here, we explore the effect of concurrent sexual partnerships on the control of a generic STI with susceptible-infected-susceptible dynamics. Concurrency refers to being in more than one sexual partnership at the same time, and is difficult to measure accurately. We assess the impact of concurrency through the development of three nested pair-formation models: one where infection can only be transmitted via stable sexual partnerships, one where infection can also be transmitted via casual partnerships between single individuals, and one where those individuals in stable partnerships can also acquire infection from casual partnerships. For each model, we include the action of vaccination before sexual debut to inform about the ability to control. As expected, for a fixed transmission rate, concurrency increases both the endemic prevalence of infection and critical level of vaccination required to eliminate the disease significantly. However, when the transmission rate is scaled to maintain a fixed endemic prevalence across models, concurrency has a far smaller impact upon the critical level of vaccination required. Further, when we also constrain the models to have a fixed number of new partnerships over time (both long-term and casual), then increasing concurrency can slightly decrease the critical level of vaccination. These results highlight that accurate measures and models of concurrency may not always be needed for reliable forecasts when models are closely matched to prevalence data. We find that, while increases in concurrency within a population are likely to generate public-health problems, the inclusion of concurrency may be unnecessary when constructing models to determine the efficacy of the control of STIs by vaccination

Correlation, concurrency, and clustering in network models of epidemics in human populations

The spread of an epidemic can be conceptualised as a process on a network, where vertices refer to individuals and where edges refer to epidemiologically relevant connections between individuals. Understanding the impact of network structure on epidemiological outcomes is a central task in mathematical epidemiology. Accordingly, a range of mathematical models incorporating network structure have been designed. In this thesis, we develop a range of network models in the context of epidemics in human populations. Firstly, we consider a novel moment-closure approximation for a disease with susceptible-infected- susceptible dynamics. For diseases without immunity, the possibility of reinfection can introduce correlations in infection status between indirectly connected individuals, limiting the accuracy of moment-closure approaches. By incorporating these correlations into a model, we introduce an improvement to the standard pairwise approximation for two different network structures: the isolated open triple and the k-regular network. Secondly, we assess the importance of including concurrent sexual partnerships, partnerships that overlap in time, when modelling the control of sexually transmitted infections. We do this in two distinct settings, firstly developing nested pair-formation models before developing an individual-based dynamic network model of a heterosexual population. In both instances we find that while concurrency can have a large impact on epidemiological dynamics, the inclusion of concurrency in models matched to prevalence data has only a modest impact on control measures. Thirdly, we consider the extent to which the clustering imposed by social bubbles, where two households form an exclusive social group, is an effective way of increasing social contact while minimising the resulting increase in transmission in the context of COVID-19. Using a stochastic, generation-based network model of household and bubble contacts, we find that social bubble strategies are effective at minimising transmission when compared to unclustered increases in contacts

Quantifying pupil-to-pupil SARS-CoV-2 transmission and the impact of lateral flow testing in English secondary schools

A range of measures have been implemented to control within-school SARS-CoV-2 transmission in England, including the self-isolation of close contacts and twice weekly mass testing of secondary school pupils using lateral flow device tests (LFTs). Despite reducing transmission, isolating close contacts can lead to high levels of absences, negatively impacting pupils. To quantify pupil-to-pupil SARS-CoV-2 transmission and the impact of implemented control measures, we fit a stochastic individual-based model of secondary school infection to both swab testing data and secondary school absences data from England, and then simulate outbreaks from 31st August 2020 until 23rd May 2021. We find that the pupil-to-pupil reproduction number, Rschool, has remained below 1 on average across the study period, and that twice weekly mass testing using LFTs has helped to control pupil-to-pupil transmission. We also explore the potential benefits of alternative containment strategies, finding that a strategy of repeat testing of close contacts rather than isolation, alongside mass testing, substantially reduces absences with only a marginal increase in pupil-to-pupil transmission

An analysis of school absences in England during the COVID-19 pandemic

Background: The introduction of SARS-CoV-2, the virus that causes COVID-19 infection, in the UK in early 2020, resulted in the introduction of several control policies to reduce disease spread. As part of these restrictions, schools were closed to all pupils in March (except for vulnerable and key worker children), before re-opening to certain year groups in June. Finally, all school children returned to the classroom in September. Methods: Here, we analyse data on school absences in late 2020 as a result of COVID-19 infection and how that varied through time as other measures in the community were introduced. We utilise data from the Department for Education Educational Settings database and examine how pupil and teacher absences change in both primary and secondary schools. Results: Our results show that absences as a result of COVID-19 infection rose steadily following the re-opening of schools in September. Cases in teachers declined during the November lockdown, particularly in regions previously in tier 3, the highest level of control at the time. Cases in secondary school pupils increased for the first 2 weeks of the November lockdown, before decreasing. Since the introduction of the tier system, the number of absences with confirmed infection in primary schools was observed to be (markedly) lower than that in secondary schools. In December, we observed a large rise in the number of absences per school in secondary school settings in the South East and London, but such rises were not observed in other regions or in primary school settings. We conjecture that the increased transmissibility of the new variant in these regions may have contributed to this rise in secondary school cases. Finally, we observe a positive correlation between cases in the community and cases in schools in most regions, with weak evidence suggesting that cases in schools lag behind cases in the surrounding community. Conclusions: We conclude that there is no significant evidence to suggest that schools are playing a substantial role in driving spread in the community and that careful monitoring may be required as schools re-open to determine the effect associated with open schools upon community incidence

Predictions of COVID-19 dynamics in the UK : short-term forecasting and analysis of potential exit strategies

Efforts to suppress transmission of SARS-CoV-2 in the UK have seen non-pharmaceutical interventions being invoked. The most severe measures to date include all restaurants, pubs and cafes being ordered to close on 20th March, followed by a "stay at home" order on the 23rd March and the closure of all non-essential retail outlets for an indefinite period. Government agencies are presently analysing how best to develop an exit strategy from these measures and to determine how the epidemic may progress once measures are lifted. Mathematical models are currently providing short and long term forecasts regarding the future course of the COVID-19 outbreak in the UK to support evidence-based policymaking. We present a deterministic, age-structured transmission model that uses real-time data on confirmed cases requiring hospital care and mortality to provide up-to-date predictions on epidemic spread in ten regions of the UK. The model captures a range of age-dependent heterogeneities, reduced transmission from asymptomatic infections and produces a good fit to the key epidemic features over time. We simulated a suite of scenarios to assess the impact of differing approaches to relaxing social distancing measures from 7th May 2020 on the estimated number of patients requiring inpatient and critical care treatment, and deaths. With regard to future epidemic outcomes, we investigated the impact of reducing compliance, ongoing shielding of elder age groups, reapplying stringent social distancing measures using region based triggers and the role of asymptomatic transmission. We find that significant relaxation of social distancing measures from 7th May onwards can lead to a rapid resurgence of COVID-19 disease and the health system being quickly overwhelmed by a sizeable, second epidemic wave. In all considered age-shielding based strategies, we projected serious demand on critical care resources during the course of the pandemic. The reintroduction and release of strict measures on a regional basis, based on ICU bed occupancy, results in a long epidemic tail, until the second half of 2021, but ensures that the health service is protected by reintroducing social distancing measures for all individuals in a region when required. Our work confirms the effectiveness of stringent non-pharmaceutical measures in March 2020 to suppress the epidemic. It also provides strong evidence to support the need for a cautious, measured approach to relaxation of lockdown measures, to protect the most vulnerable members of society and support the health service through subduing demand on hospital beds, in particular bed occupancy in intensive care units

The effectiveness of social bubbles as part of a Covid-19 lockdown exit strategy, a modelling study

Background: ​ During the coronavirus disease 2019 (COVID-19) lockdown, contact clustering in social bubbles may allow extending contacts beyond the household at minimal additional risk and hence has been considered as part of modified lockdown policy or a gradual lockdown exit strategy. We estimated the impact of such strategies on epidemic and mortality risk using the UK as a case study. Methods: ​ We used an individual based model for a synthetic population similar to the UK, stratified into transmission risks from the community, within the household and from other households in the same social bubble. The base case considers a situation where non-essential shops and schools are closed, the secondary household attack rate is 20% and the initial reproduction number is 0.8. We simulate social bubble strategies (where two households form an exclusive pair) for households including children, for single occupancy households, and for all households. We test the sensitivity of results to a range of alternative model assumptions and parameters. Results:  Clustering contacts outside the household into exclusive bubbles is an effective strategy of increasing contacts while limiting the associated increase in epidemic risk. In the base case, social bubbles reduced fatalities by 42% compared to an unclustered increase of contacts. We find that if all households were to form social bubbles the reproduction number would likely increase to above the epidemic threshold of R=1. Strategies allowing households with young children or single occupancy households to form social bubbles increased the reproduction number by less than 11%. The corresponding increase in mortality is proportional to the increase in the epidemic risk but is focussed in older adults irrespective of inclusion in social bubbles. Conclusions: ​ If managed appropriately, social bubbles can be an effective way of extending contacts beyond the household while limiting the increase in epidemic risk.</ns3:p

The effectiveness of social bubbles as part of a Covid-19 lockdown exit strategy, a modelling study.

Background: ​ During the coronavirus disease 2019 (COVID-19) lockdown, contact clustering in social bubbles may allow extending contacts beyond the household at minimal additional risk and hence has been considered as part of modified lockdown policy or a gradual lockdown exit strategy. We estimated the impact of such strategies on epidemic and mortality risk using the UK as a case study. Methods: ​ We used an individual based model for a synthetic population similar to the UK, stratified into transmission risks from the community, within the household and from other households in the same social bubble. The base case considers a situation where non-essential shops and schools are closed, the secondary household attack rate is 20% and the initial reproduction number is 0.8. We simulate social bubble strategies (where two households form an exclusive pair) for households including children, for single occupancy households, and for all households. We test the sensitivity of results to a range of alternative model assumptions and parameters. Results:  Clustering contacts outside the household into exclusive bubbles is an effective strategy of increasing contacts while limiting the associated increase in epidemic risk. In the base case, social bubbles reduced fatalities by 42% compared to an unclustered increase of contacts. We find that if all households were to form social bubbles the reproduction number would likely increase to above the epidemic threshold of R=1. Strategies allowing households with young children or single occupancy households to form social bubbles increased the reproduction number by less than 11%. The corresponding increase in mortality is proportional to the increase in the epidemic risk but is focussed in older adults irrespective of inclusion in social bubbles. Conclusions: ​ If managed appropriately, social bubbles can be an effective way of extending contacts beyond the household while limiting the increase in epidemic risk

Spike patterning in oxytocin neurons:Capturing physiological behaviour with Hodgkin-Huxley and integrate-and-fire models

Integrate-and-fire (IF) models can provide close matches to the discharge activity of neurons, but do they oversimplify the biophysical properties of the neurons? A single compartment Hodgkin-Huxley (HH) model of the oxytocin neuron has previously been developed, incorporating biophysical measurements of channel properties obtained in vitro. A simpler modified integrate-and-fire model has also been developed, which can match well the characteristic spike patterning of oxytocin neurons as observed in vivo. Here, we extended the HH model to incorporate synaptic input, to enable us to compare spike activity in the model with experimental data obtained in vivo. We refined the HH model parameters to closely match the data, and then matched the same experimental data with a modified IF model, using an evolutionary algorithm to optimise parameter matching. Finally we compared the properties of the modified HH model with those of the IF model to seek an explanation for differences between spike patterning in vitro and in vivo. We show that, with slight modifications, the original HH model, like the IF model, is able to closely match both the interspike interval (ISI) distributions of oxytocin neurons and the observed variability of spike firing rates in vivo and in vitro. This close match of both models to data depends on the presence of a slow activity-dependent hyperpolarisation (AHP); this is represented in both models and the parameters used in the HH model representation match well with optimal parameters of the IF model found by an evolutionary algorithm. The ability of both models to fit data closely also depends on a shorter hyperpolarising after potential (HAP); this is explicitly represented in the IF model, but in the HH model, it emerges from a combination of several components. The critical elements of this combination are identified

tsleng93/SocialBubble: Social Bubble (including density dependence)

This repository contains the underlying code and functions used in the paper "The effectiveness of social bubbles as part of a Covid-19 lockdown exit strategy, a modelling study", by Trystan Leng*, Connor White, Joe Hilton, Adam J Kucharski, Lorenzo Pellis, Helena Stage, Nicholas Davies, CMMID nCov working group, Matt Keeling & Stefan Flasche. Underlying code and functions are written in matlab, and visualisations are written in matlab or R