Epidemiological and health economic implications of symptom propagation in respiratory pathogens: A mathematical modelling investigation.

BACKGROUND: Respiratory pathogens inflict a substantial burden on public health and the economy. Although the severity of symptoms caused by these pathogens can vary from asymptomatic to fatal, the factors that determine symptom severity are not fully understood. Correlations in symptoms between infector-infectee pairs, for which evidence is accumulating, can generate large-scale clusters of severe infections that could be devastating to those most at risk, whilst also conceivably leading to chains of mild or asymptomatic infections that generate widespread immunity with minimal cost to public health. Although this effect could be harnessed to amplify the impact of interventions that reduce symptom severity, the mechanistic representation of symptom propagation within mathematical and health economic modelling of respiratory diseases is understudied. METHODS AND FINDINGS: We propose a novel framework for incorporating different levels of symptom propagation into models of infectious disease transmission via a single parameter, α. Varying α tunes the model from having no symptom propagation (α = 0, as typically assumed) to one where symptoms always propagate (α = 1). For parameters corresponding to three respiratory pathogens-seasonal influenza, pandemic influenza and SARS-CoV-2-we explored how symptom propagation impacted the relative epidemiological and health-economic performance of three interventions, conceptualised as vaccines with different actions: symptom-attenuating (labelled SA), infection-blocking (IB) and infection-blocking admitting only mild breakthrough infections (IB_MB). In the absence of interventions, with fixed underlying epidemiological parameters, stronger symptom propagation increased the proportion of cases that were severe. For SA and IB_MB, interventions were more effective at reducing prevalence (all infections and severe cases) for higher strengths of symptom propagation. For IB, symptom propagation had no impact on effectiveness, and for seasonal influenza this intervention type was more effective than SA at reducing severe infections for all strengths of symptom propagation. For pandemic influenza and SARS-CoV-2, at low intervention uptake, SA was more effective than IB for all levels of symptom propagation; for high uptake, SA only became more effective under strong symptom propagation. Health economic assessments found that, for SA-type interventions, the amount one could spend on control whilst maintaining a cost-effective intervention (termed threshold unit intervention cost) was very sensitive to the strength of symptom propagation. CONCLUSIONS: Overall, the preferred intervention type depended on the combination of the strength of symptom propagation and uptake. Given the importance of determining robust public health responses, we highlight the need to gather further data on symptom propagation, with our modelling framework acting as a template for future analysis.

Estimating the fitness cost and benefit of antimicrobial resistance from pathogen genomic data.

Increasing levels of antibiotic resistance in many bacterial pathogen populations are a major threat to public health. Resistance to an antibiotic provides a fitness benefit when the bacteria are exposed to this antibiotic, but resistance also often comes at a cost to the resistant pathogen relative to susceptible counterparts. We lack a good understanding of these benefits and costs of resistance for many bacterial pathogens and antibiotics, but estimating them could lead to better use of antibiotics in a way that reduces or prevents the spread of resistance. Here, we propose a new model for the joint epidemiology of susceptible and resistant variants, which includes explicit parameters for the cost and benefit of resistance. We show how Bayesian inference can be performed under this model using phylogenetic data from susceptible and resistant lineages and that by combining data from both we are able to disentangle and estimate the resistance cost and benefit parameters separately. We applied our inferential methodology to several simulated datasets to demonstrate good scalability and accuracy. We analysed a dataset of Neisseria gonorrhoeae genomes collected between 2000 and 2013 in the USA. We found that two unrelated lineages resistant to fluoroquinolones shared similar epidemic dynamics and resistance parameters. Fluoroquinolones were abandoned for the treatment of gonorrhoea due to increasing levels of resistance, but our results suggest that they could be used to treat a minority of around 10% of cases without causing resistance to grow again.

Optimal health and economic impact of non-pharmaceutical intervention measures prior and post vaccination in England: a mathematical modelling study.

Background. Even with good progress on vaccination, SARS-CoV-2 infections in the UK may continue to impose a high burden of disease and therefore pose substantial challenges for health policy decision makers. Stringent government-mandated physical distancing measures (lockdown) have been demonstrated to be epidemiologically effective, but can have both positive and negative economic consequences. The duration and frequency of any intervention policy could, in theory, be optimized to maximize economic benefits while achieving substantial reductions in disease. Methods. Here, we use a pre-existing SARS-CoV-2 transmission model to assess the health and economic implications of different strengths of control through time in order to identify optimal approaches to non-pharmaceutical intervention stringency in the UK, considering the role of vaccination in reducing the need for future physical distancing measures. The model is calibrated to the COVID-19 epidemic in England and we carry out retrospective analysis of the optimal timing of precautionary breaks in 2020 and the optimal relaxation policy from the January 2021 lockdown, considering the willingness to pay (WTP) for health improvement. Results. We find that the precise timing and intensity of interventions is highly dependent upon the objective of control. As intervention measures are relaxed, we predict a resurgence in cases, but the optimal intervention policy can be established dependent upon the WTP per quality adjusted life year loss avoided. Our results show that establishing an optimal level of control can result in a reduction in net monetary loss of billions of pounds, dependent upon the precise WTP value. Conclusion. It is vital, as the UK emerges from lockdown, but continues to face an on-going pandemic, to accurately establish the overall health and economic costs when making policy decisions. We demonstrate how some of these can be quantified, employing mechanistic infectious disease transmission models to establish optimal levels of control for the ongoing COVID-19 pandemic.

Epidemiological impact and cost-effectiveness analysis of COVID-19 vaccination in Kenya.

BACKGROUND: A few studies have assessed the epidemiological impact and the cost-effectiveness of COVID-19 vaccines in settings where most of the population had been exposed to SARS-CoV-2 infection. METHODS: We conducted a cost-effectiveness analysis of COVID-19 vaccine in Kenya from a societal perspective over a 1.5-year time frame. An age-structured transmission model assumed at least 80% of the population to have prior natural immunity when an immune escape variant was introduced. We examine the effect of slow (18 months) or rapid (6 months) vaccine roll-out with vaccine coverage of 30%, 50% or 70% of the adult (>18 years) population prioritising roll-out in those over 50-years (80% uptake in all scenarios). Cost data were obtained from primary analyses. We assumed vaccine procurement at US$7 per dose and vaccine delivery costs of US$3.90-US$6.11 per dose. The cost-effectiveness threshold was US$919.11. FINDINGS: Slow roll-out at 30% coverage largely targets those over 50 years and resulted in 54% fewer deaths (8132 (7914-8373)) than no vaccination and was cost saving (incremental cost-effectiveness ratio, ICER=US$-1343 (US$-1345 to US$-1341) per disability-adjusted life-year, DALY averted). Increasing coverage to 50% and 70%, further reduced deaths by 12% (810 (757-872) and 5% (282 (251-317) but was not cost-effective, using Kenya's cost-effectiveness threshold (US$919.11). Rapid roll-out with 30% coverage averted 63% more deaths and was more cost-saving (ICER=US$-1607 (US$-1609 to US$-1604) per DALY averted) compared with slow roll-out at the same coverage level, but 50% and 70% coverage scenarios were not cost-effective. INTERPRETATION: With prior exposure partially protecting much of the Kenyan population, vaccination of young adults may no longer be cost-effective.

An assessment of the vaccination of school-aged children in England against SARS-CoV-2.

BACKGROUND: Children and young persons are known to have a high number of close interactions, often within the school environment, which can facilitate rapid spread of infection; yet for SARS-CoV-2, it is the elderly and vulnerable that suffer the greatest health burden. Vaccination, initially targeting the elderly and vulnerable before later expanding to the entire adult population, has been transformative in the control of SARS-CoV-2 in England. However, early concerns over adverse events and the lower risk associated with infection in younger individuals means that the expansion of the vaccine programme to those under 18 years of age needs to be rigorously and quantitatively assessed. METHODS: Here, using a bespoke mathematical model matched to case and hospital data for England, we consider the potential impact of vaccinating 12-17 and 5-11-year-olds. This analysis is reported from an early model (generated in June 2021) that formed part of the evidence base for the decisions in England, and a later model (from November 2021) that benefits from a richer understanding of vaccine efficacy, greater knowledge of the Delta variant wave and uses data on the rate of vaccine administration. For both models, we consider the population wide impact of childhood vaccination as well as the specific impact on the age groups targeted for vaccination. RESULTS: Projections from June suggested that an expansion of the vaccine programme to those 12-17 years old could generate substantial reductions in infection, hospital admission and deaths in the entire population, depending on population behaviour following the relaxation of control measures. The benefits within the 12-17-year-old cohort were less marked, saving between 660 and 1100 (95% PI (prediction interval) 280-2300) hospital admissions and between 22 and 38 (95% PI 9-91) deaths depending on assumed population behaviour. For the more recent model, the benefits within this age group are reduced, saving on average 630 (95% PI 300-1300) hospital admissions and 11 (95% PI 5-28) deaths for 80% vaccine uptake, while the benefits to the wider population represent a reduction of 8-10% in hospital admissions and deaths. The vaccination of 5-11-year-olds is projected to have a far smaller impact, in part due to the later roll-out of vaccines to this age group. CONCLUSIONS: Vaccination of 12-170-year-olds and 5-11-year-olds is projected to generate a reduction in infection, hospital admission and deaths for both the age groups involved and the population in general. For any decision involving childhood vaccination, these benefits needs to be balanced against potential adverse events from the vaccine, the operational constraints on delivery and the potential for diverting resources from other public health campaigns.

Cost-effectiveness of sleeping sickness elimination campaigns in five settings of the Democratic Republic of Congo.

Gambiense human African trypanosomiasis (gHAT) is marked for elimination of transmission by 2030, but the disease persists in several low-income countries. We couple transmission and health outcomes models to examine the cost-effectiveness of four gHAT elimination strategies in five settings - spanning low- to high-risk - of the Democratic Republic of Congo. Alongside passive screening in fixed health facilities, the strategies include active screening at average or intensified coverage levels, alone or with vector control with a scale-back algorithm when no cases are reported for three consecutive years. In high or moderate-risk settings, costs of gHAT strategies are primarily driven by active screening and, if used, vector control. Due to the cessation of active screening and vector control, most investments (75-80%) are made by 2030 and vector control might be cost-saving while ensuring elimination of transmission. In low-risk settings, costs are driven by passive screening, and minimum-cost strategies consisting of active screening and passive screening lead to elimination of transmission by 2030 with high probability.

Cost-effectiveness modelling to optimise active screening strategy for gambiense human African trypanosomiasis in endemic areas of the Democratic Republic of Congo.

BACKGROUND: Gambiense human African trypanosomiasis (gHAT) has been brought under control recently with village-based active screening playing a major role in case reduction. In the approach to elimination, we investigate how to optimise active screening in villages in the Democratic Republic of Congo, such that the expenses of screening programmes can be efficiently allocated whilst continuing to avert morbidity and mortality. METHODS: We implement a cost-effectiveness analysis using a stochastic gHAT infection model for a range of active screening strategies and, in conjunction with a cost model, we calculate the net monetary benefit (NMB) of each strategy. We focus on the high-endemicity health zone of Kwamouth in the Democratic Republic of Congo. RESULTS: High-coverage active screening strategies, occurring approximately annually, attain the highest NMB. For realistic screening at 55% coverage, annual screening is cost-effective at very low willingness-to-pay thresholds (20.4 per disability adjusted life year (DALY) averted), only marginally higher than biennial screening (14.6 per DALY averted). We find that, for strategies stopping after 1, 2 or 3 years of zero case reporting, the expected cost-benefits are very similar. CONCLUSIONS: We highlight the current recommended strategy-annual screening with three years of zero case reporting before stopping active screening-is likely cost-effective, in addition to providing valuable information on whether transmission has been interrupted.

Developing a Framework for Public Involvement in Mathematical and Economic Modelling: Bringing New Dynamism to Vaccination Policy Recommendations.

OBJECTIVES: The Mathematical and Economic Modelling for Vaccination and Immunisation Evaluation (MEMVIE) programme aimed to explore, capture and support the potential contribution of the public to mathematical and economic modelling, in order to identify the values that underpin public involvement (PI) in modelling and co-produce a framework that identifies the nature and type of PI in modelling and supports its implementation. METHODS: We established a PI Reference Group, who worked collaboratively with the academic contributors to create a deliberative knowledge space, which valued different forms of knowledge, expertise and evidence. Together, we explored the key steps of mathematical and economic methods in 21 meetings during 2015-2020. These deliberations generated rich discussion, through which we identified potential points of public contribution and the values that underpin PI in modelling. We iteratively developed a framework to guide future practice of PI in modelling. RESULTS: We present the MEMVIE Public Involvement Framework in two forms: a short form to summarise key elements, and a long form framework to provide a detailed description of each potential type of public contribution at each stage of the modelling process. At a macro level, the public can contribute to reviewing context, reviewing relevance, assessing data and justifying model choice, troubleshooting, and interpreting and reviewing outcomes and decision making. The underpinning values that drive involvement include the public contributing to the validity of the model, potentially enhancing its relevance, utility and transparency through diverse inputs, and enhancing the credibility, consistency and continuous development through scrutiny, in addition to contextualising the model within a wider societal view. DISCUSSION AND CONCLUSION: PI in modelling is in its infancy. The MEMVIE Framework is the first attempt to identify potential points of collaborative public contribution to modelling, but it requires further evaluation and refinement that we are undertaking in a subsequent study.

Optimising age coverage of seasonal influenza vaccination in England: A mathematical and health economic evaluation.

For infectious disease prevention, policy-makers are typically required to base policy decisions in light of operational and monetary restrictions, prohibiting implementation of all candidate interventions. To inform the evidence-base underpinning policy decision making, mathematical and health economic modelling can be a valuable constituent. Applied to England, this study aims to identify the optimal target age groups when extending a seasonal influenza vaccination programme of at-risk individuals to those individuals at low risk of developing complications following infection. To perform this analysis, we utilise an age- and strain-structured transmission model that includes immunity propagation mechanisms which link prior season epidemiological outcomes to immunity at the beginning of the following season. Making use of surveillance data from the past decade in conjunction with our dynamic model, we simulate transmission dynamics of seasonal influenza in England from 2012 to 2018. We infer that modified susceptibility due to natural infection in the previous influenza season is the only immunity propagation mechanism to deliver a non-negligible impact on the transmission dynamics. Further, we discerned case ascertainment to be higher for young infants compared to adults under 65 years old, and uncovered a decrease in case ascertainment as age increased from 65 to 85 years of age. Our health economic appraisal sweeps vaccination age space to determine threshold vaccine dose prices achieving cost-effectiveness under differing paired strategies. In particular, we model offering vaccination to all those low-risk individuals younger than a given age (but no younger than two years old) and all low-risk individuals older than a given age, while maintaining vaccination of at-risk individuals of any age. All posited strategies were deemed cost-effective. In general, the addition of low-risk vaccination programmes whose coverage encompassed children and young adults (aged 20 and below) were highly cost-effective. The inclusion of elder age-groups to the low-risk programme typically lessened the cost-effectiveness. Notably, elderly-centric programmes vaccinating from 65-75 years and above had the least permitted expense per vaccine.

Vaccination or mass drug administration against schistosomiasis: a hypothetical cost-effectiveness modelling comparison.

BACKGROUND: Schistosomiasis is a neglected tropical disease, targeted by the World Health Organization for reduction in morbidity by 2020. It is caused by parasitic flukes that spread through contamination of local water sources. Traditional control focuses on mass drug administration, which kills the majority of adult worms, targeted at school-aged children. However, these drugs do not confer long-term protection and there are concerns over the emergence of drug resistance. The development of a vaccine against schistosomiasis opens the potential for control methods that could generate long-lasting population-level immunity if they are cost-effective. METHODS: Using an individual-based transmission model, matched to epidemiological data, we compared the cost-effectiveness of a range of vaccination programmes against mass drug administration, across three transmission settings. Health benefit was measured by calculating the heavy-intensity infection years averted by each intervention, while vaccine costs were assessed against robust estimates for the costs of mass drug administration obtained from data. We also calculated a critical vaccination cost, a cost beyond which vaccination might not be economically favorable, by benchmarking the cost-effectiveness of potential vaccines against the cost-effectiveness of mass drug administration, and examined the effect of different vaccine protection durations. RESULTS: We found that sufficiently low-priced vaccines can be more cost-effective than traditional drugs in high prevalence settings, and can lead to a greater reduction in morbidity over shorter time-scales. MDA or vaccination programmes that target the whole community generate the most health benefits, but are generally less cost-effective than those targeting children, due to lower prevalence of schistosomiasis in adults. CONCLUSIONS: The ultimate cost-effectiveness of vaccination will be highly dependent on multiple vaccine characteristics, such as the efficacy, cost, safety and duration of protection, as well as the subset of population targeted for vaccination. However, our results indicate that if a vaccine could be developed with reasonable characteristics and for a sufficiently low cost, then vaccination programmes can be a highly cost-effective method of controlling schistosomiasis in high-transmission areas. The population-level immunity generated by vaccination will also inevitably improve the chances of interrupting transmission of the disease, which is the long-term epidemiological goal.

Correlations between stochastic endemic infection in multiple interacting subpopulations.

Heterogeneity plays an important role in the emergence, persistence and control of infectious diseases. Metapopulation models are often used to describe spatial heterogeneity, and the transition from random- to heterogeneous-mixing is made by incorporating the interaction, or coupling, within and between subpopulations. However, such couplings are difficult to measure explicitly; instead, their action through the correlations between subpopulations is often all that can be observed. We use moment-closure methods to investigate how the coupling and resulting correlation are related, considering systems of multiple identical interacting populations on highly symmetric complex networks: the complete network, the k-regular tree network, and the star network. We show that the correlation between the prevalence of infection takes a relatively simple form and can be written in terms of the coupling, network parameters and epidemiological parameters only. These results provide insight into the effect of metapopulation network structure on endemic disease dynamics, and suggest that detailed case-reporting data alone may be sufficient to infer the strength of between population interaction and hence lead to more accurate mathematical descriptions of infectious disease behaviour.

Assessing the cost-effectiveness of HPV vaccination strategies for adolescent girls and boys in the UK.

BACKGROUND: Human papillomavirus (HPV) is the most widespread sexually transmitted infection worldwide. It causes several health consequences, in particular accounting for the majority of cervical cancer cases in women. In the United Kingdom, a vaccination campaign targeting 12-year-old girls started in 2008; this campaign has been successful, with high uptake and reduced HPV prevalence observed in vaccinated cohorts. Recently, attention has focused on vaccinating both sexes, due to HPV-related diseases in males (particularly for high-risk men who have sex with men) and an equity argument over equalising levels of protection. METHODS: We constructed an epidemiological model for HPV transmission in the UK, accounting for nine of the most common HPV strains. We complemented this with an economic model to determine the likely health outcomes (healthcare costs and quality-adjusted life years) for individuals from the epidemiological model. We then tested vaccination with the three HPV vaccines currently available, vaccinating either girls alone or both sexes. For each strategy we calculated the threshold price per vaccine dose, i.e. the maximum amount paid for the added health benefits of vaccination to be worth the cost of each vaccine dose. We calculated results at 3.5% discounting, and also 1.5%, to consider the long-term health effects of HPV infection. RESULTS: At 3.5% discounting, continuing to vaccinate girls remains highly cost-effective compared to halting vaccination, with threshold dose prices of £56-£108. Vaccination of girls and boys is less cost-effective (£25-£53). Compared to vaccinating girls only, adding boys to the programme is not cost-effective, with negative threshold prices (-£6 to -£3) due to the costs of administration. All threshold prices increase when using 1.5% discounting, and adding boys becomes cost-effective (£36-£47). These results are contingent on the UK's high vaccine uptake; for lower uptake rates, adding boys (at the same uptake rate) becomes more cost effective. CONCLUSIONS: Vaccinating girls is extremely cost-effective compared with no vaccination, vaccinating both sexes is less so. Adding boys to an already successful girls-only programme has a low cost-effectiveness, as males have high protection through herd immunity. If future health effects are weighted more heavily, threshold prices increase and vaccination becomes cost-effective.

Correlations between stochastic epidemics in two interacting populations.

It is increasingly apparent that heterogeneity in the interaction between individuals plays an important role in the dynamics, persistence, evolution and control of infectious diseases. In epidemic modelling two main forms of heterogeneity are commonly considered: spatial heterogeneity due to the segregation of populations and heterogeneity in risk at the same location. The transition from random-mixing to heterogeneous-mixing models is made by incorporating the interaction, or coupling, within and between subpopulations. However, such couplings are difficult to measure explicitly; instead, their action through the correlations between subpopulations is often all that can be observed. Here, using moment-closure methodology supported by stochastic simulation, we investigate how the coupling and resulting correlation are related. We focus on the simplest case of interactions, two identical coupled populations, and show that for a wide range of parameters the correlation between the prevalence of infection takes a relatively simple form. In particular, the correlation can be approximated by a logistic function of the between population coupling, with the free parameter determined analytically from the epidemiological parameters. These results suggest that detailed case-reporting data alone may be sufficient to infer the strength of between population interaction and hence lead to more accurate mathematical descriptions of infectious disease behaviour.

Real-time decision-making during emergency disease outbreaks.

In the event of a new infectious disease outbreak, mathematical and simulation models are commonly used to inform policy by evaluating which control strategies will minimize the impact of the epidemic. In the early stages of such outbreaks, substantial parameter uncertainty may limit the ability of models to provide accurate predictions, and policymakers do not have the luxury of waiting for data to alleviate this state of uncertainty. For policymakers, however, it is the selection of the optimal control intervention in the face of uncertainty, rather than accuracy of model predictions, that is the measure of success that counts. We simulate the process of real-time decision-making by fitting an epidemic model to observed, spatially-explicit, infection data at weekly intervals throughout two historical outbreaks of foot-and-mouth disease, UK in 2001 and Miyazaki, Japan in 2010, and compare forward simulations of the impact of switching to an alternative control intervention at the time point in question. These are compared to policy recommendations generated in hindsight using data from the entire outbreak, thereby comparing the best we could have done at the time with the best we could have done in retrospect. Our results show that the control policy that would have been chosen using all the data is also identified from an early stage in an outbreak using only the available data, despite high variability in projections of epidemic size. Critically, we find that it is an improved understanding of the locations of infected farms, rather than improved estimates of transmission parameters, that drives improved prediction of the relative performance of control interventions. However, the ability to estimate undetected infectious premises is a function of uncertainty in the transmission parameters. Here, we demonstrate the need for both real-time model fitting and generating projections to evaluate alternative control interventions throughout an outbreak. Our results highlight the use of using models at outbreak onset to inform policy and the importance of state-dependent interventions that adapt in response to additional information throughout an outbreak.

The impact of current infection levels on the cost-benefit of vaccination.

When considering a new vaccine programme or modifying an existing one, economic cost-benefit analysis, underpinned by predictive epidemiological modelling, is a key component. This analysis is intimately linked to the willingness to pay for additional QALYs (quality-adjusted life-years) gained; currently in England and Wales a health programme is economically viable if the cost per QALY gained is less than £ 20,000, and models are often used to assess if a vaccine programme is likely to fall below this threshold cost. Before a programme begins, infection levels are generally high and therefore vaccination may be expected to have substantial effects and therefore will often be economically viable. However, once a programme is established, and infection rates are lower, it might be expected that a re-evaluation of the programme (using current incidence information) will show it to be less cost-effective. This is the scenario we examine here with analytical tools and simple ODE models. Surprisingly we show that in most cases the benefits from maintaining an existing vaccination programme are at least equal to those of starting the programme initially, and in the majority of scenarios the differences between the two are minimal. In practical terms, this is an extremely helpful finding, allowing us to assert that the action of immunising individuals does not de-value the vaccination programme.

Quantifying the Value of Perfect Information in Emergency Vaccination Campaigns.

Foot-and-mouth disease outbreaks in non-endemic countries can lead to large economic costs and livestock losses but the use of vaccination has been contentious, partly due to uncertainty about emergency FMD vaccination. Value of information methods can be applied to disease outbreak problems such as FMD in order to investigate the performance improvement from resolving uncertainties. Here we calculate the expected value of resolving uncertainty about vaccine efficacy, time delay to immunity after vaccination and daily vaccination capacity for a hypothetical FMD outbreak in the UK. If it were possible to resolve all uncertainty prior to the introduction of control, we could expect savings of £55 million in outbreak cost, 221,900 livestock culled and 4.3 days of outbreak duration. All vaccination strategies were found to be preferable to a culling only strategy. However, the optimal vaccination radius was found to be highly dependent upon vaccination capacity for all management objectives. We calculate that by resolving the uncertainty surrounding vaccination capacity we would expect to return over 85% of the above savings, regardless of management objective. It may be possible to resolve uncertainty about daily vaccination capacity before an outbreak, and this would enable decision makers to select the optimal control action via careful contingency planning.

Exact and approximate moment closures for non-Markovian network epidemics.

Moment-closure techniques are commonly used to generate low-dimensional deterministic models to approximate the average dynamics of stochastic systems on networks. The quality of such closures is usually difficult to asses and furthermore the relationship between model assumptions and closure accuracy are often difficult, if not impossible, to quantify. Here we carefully examine some commonly used moment closures, in particular a new one based on the concept of maximum entropy, for approximating the spread of epidemics on networks by reconstructing the probability distributions over triplets based on those over pairs. We consider various models (SI, SIR, SEIR and Reed-Frost-type) under Markovian and non-Markovian assumption characterising the latent and infectious periods. We initially study with care two special networks, namely the open triplet and closed triangle, for which we can obtain analytical results. We then explore numerically the exactness of moment closures for a wide range of larger motifs, thus gaining understanding of the factors that introduce errors in the approximations, in particular the presence of a random duration of the infectious period and the presence of overlapping triangles in a network. We also derive a simpler and more intuitive proof than previously available concerning the known result that pair-based moment closure is exact for the Markovian SIR model on tree-like networks under pure initial conditions. We also extend such a result to all infectious models, Markovian and non-Markovian, in which susceptibles escape infection independently from each infected neighbour and for which infectives cannot regain susceptible status, provided the network is tree-like and initial conditions are pure. This works represent a valuable step in enriching intuition and deepening understanding of the assumptions behind moment closure approximations and for putting them on a more rigorous mathematical footing.

Optimal but unequitable prophylactic distribution of vaccine.

The final epidemic size (R(∞)) remains one of the fundamental outcomes of an epidemic, and measures the total number of individuals infected during a "free-fall" epidemic when no additional control action is taken. As such, it provides an idealised measure for optimising control policies before an epidemic arises. Although the generality of formulae for calculating the final epidemic size have been discussed previously, we offer an alternative probabilistic argument and then use this formula to consider the optimal deployment of vaccine in spatially segregated populations that minimises the total number of cases. We show that for a limited stockpile of vaccine, the optimal policy is often to immunise one population to the exclusion of others. However, as greater realism is included, this extreme and arguably unethical policy, is replaced by an optimal strategy where vaccine supply is more evenly spatially distributed.

Impact of regulatory perturbations to disease spread through cattle movements in Great Britain.

During the past decade the British livestock industry has suffered from several major pathogen outbreaks, and a variety of regulatory and disease control measures have been applied to the movement of livestock with the express aim of mitigating the spread of infection. The Rapid Analysis and Detection of Animal-related Risks (RADAR) project, which has been collecting data on the movement of cattle since 1998, provides a relatively comprehensive record of how these policies have influenced the movement of cattle between animal holdings, markets, and slaughterhouses in Britain. Many previous studies have focused on the properties of the network that can be derived from these movements--treating farms as nodes and movements as directed (and potentially weighted) edges in the network. However, of far greater importance is how these policy changes have influenced the potential spread of infectious diseases. Here we use a stochastic fully individual-based model of cattle in Britain to assess how the epidemic potential has varied from 2000 to 2009 as the pattern of movements has changed in response to legislation and market forces. Our simulations show that the majority of policy changes lead to significant decreases in the epidemic potential (measured in multiple ways), but that this potential then increases through time as cattle farmers modify their behaviour in response. Our results suggest that the cattle industry is likely to experience boom-bust dynamics, with the actions that farmers take during epidemic-free periods to maximise their profitability likely to increase the potential for large-scale epidemics to occur.