Multiple models for outbreak decision support in the face of uncertainty.

Policymakers must make management decisions despite incomplete knowledge and conflicting model projections. Little guidance exists for the rapid, representative, and unbiased collection of policy-relevant scientific input from independent modeling teams. Integrating approaches from decision analysis, expert judgment, and model aggregation, we convened multiple modeling teams to evaluate COVID-19 reopening strategies for a mid-sized United States county early in the pandemic. Projections from seventeen distinct models were inconsistent in magnitude but highly consistent in ranking interventions. The 6-mo-ahead aggregate projections were well in line with observed outbreaks in mid-sized US counties. The aggregate results showed that up to half the population could be infected with full workplace reopening, while workplace restrictions reduced median cumulative infections by 82%. Rankings of interventions were consistent across public health objectives, but there was a strong trade-off between public health outcomes and duration of workplace closures, and no win-win intermediate reopening strategies were identified. Between-model variation was high; the aggregate results thus provide valuable risk quantification for decision making. This approach can be applied to the evaluation of management interventions in any setting where models are used to inform decision making. This case study demonstrated the utility of our approach and was one of several multimodel efforts that laid the groundwork for the COVID-19 Scenario Modeling Hub, which has provided multiple rounds of real-time scenario projections for situational awareness and decision making to the Centers for Disease Control and Prevention since December 2020.

Evaluating targeted COVID-19 vaccination strategies with agent-based modeling.

We evaluate approaches to vaccine distribution using an agent-based model of human activity and COVID-19 transmission calibrated to detailed trends in cases, hospitalizations, deaths, seroprevalence, and vaccine breakthrough infections in Florida, USA. We compare the incremental effectiveness for four different distribution strategies at four different levels of vaccine supply, starting in late 2020 through early 2022. Our analysis indicates that the best strategy to reduce severe outcomes would be to actively target high disease-risk individuals. This was true in every scenario, although the advantage was greatest for the intermediate vaccine availability assumptions and relatively modest compared to a simple mass vaccination approach under high vaccine availability. Ring vaccination, while generally the most effective strategy for reducing infections, ultimately proved least effective at preventing deaths. We also consider using age group as a practical surrogate measure for actual disease-risk targeting; this approach also outperforms both simple mass distribution and ring vaccination. We find that quantitative effectiveness of a strategy depends on whether effectiveness is assessed after the alpha, delta, or omicron wave. However, these differences in absolute benefit for the strategies do not change the ranking of their performance at preventing severe outcomes across vaccine availability assumptions.

Design of effective outpatient sentinel surveillance for COVID-19 decision-making: a modeling study.

BACKGROUND: Decision-makers impose COVID-19 mitigations based on public health indicators such as reported cases, which are sensitive to fluctuations in supply and demand for diagnostic testing, and hospital admissions, which lag infections by up to two weeks. Imposing mitigations too early has unnecessary economic costs while imposing too late leads to uncontrolled epidemics with unnecessary cases and deaths. Sentinel surveillance of recently-symptomatic individuals in outpatient testing sites may overcome biases and lags in conventional indicators, but the minimal outpatient sentinel surveillance system needed for reliable trend estimation remains unknown. METHODS: We used a stochastic, compartmental transmission model to evaluate the performance of various surveillance indicators at reliably triggering an alarm in response to, but not before, a step increase in transmission of SARS-CoV-2. The surveillance indicators included hospital admissions, hospital occupancy, and sentinel cases with varying levels of sampling effort capturing 5, 10, 20, 50, or 100% of incident mild cases. We tested 3 levels of transmission increase, 3 population sizes, and conditions of either simultaneous transmission increase or lagged increase in the older population. We compared the indicators' performance at triggering alarm soon after, but not prior, to the transmission increase. RESULTS: Compared to surveillance based on hospital admissions, outpatient sentinel surveillance that captured at least 20% of incident mild cases could trigger an alarm 2 to 5 days earlier for a mild increase in transmission and 6 days earlier for a moderate or strong increase. Sentinel surveillance triggered fewer false alarms and averted more deaths per day spent in mitigation. When transmission increase in older populations lagged the increase in younger populations by 14 days, sentinel surveillance extended its lead time over hospital admissions by an additional 2 days. CONCLUSIONS: Sentinel surveillance of mild symptomatic cases can provide more timely and reliable information on changes in transmission to inform decision-makers in an epidemic like COVID-19.

Designing effective control of dengue with combined interventions.

Viruses transmitted by Aedes mosquitoes, such as dengue, Zika, and chikungunya, have expanding ranges and seem unabated by current vector control programs. Effective control of these pathogens likely requires integrated approaches. We evaluated dengue management options in an endemic setting that combine novel vector control and vaccination using an agent-based model for Yucatán, Mexico, fit to 37 y of data. Our intervention models are informed by targeted indoor residual spraying (TIRS) experiments; trial outcomes and World Health Organization (WHO) testing guidance for the only licensed dengue vaccine, CYD-TDV; and preliminary results for in-development vaccines. We evaluated several implementation options, including varying coverage levels; staggered introductions; and a one-time, large-scale vaccination campaign. We found that CYD-TDV and TIRS interfere: while the combination outperforms either alone, performance is lower than estimated from their separate benefits. The conventional model hypothesized for in-development vaccines, however, performs synergistically with TIRS, amplifying effectiveness well beyond their independent impacts. If the preliminary performance by either of the in-development vaccines is upheld, a one-time, large-scale campaign followed by routine vaccination alongside aggressive new vector control could enable short-term elimination, with nearly all cases avoided for a decade despite continuous dengue reintroductions. If elimination is impracticable due to resource limitations, less ambitious implementations of this combination still produce amplified, longer-lasting effectiveness over single-approach interventions.

Effect of Population Partitioning on the Probability of Silent Circulation of Poliovirus.

Polio can circulate unobserved in regions that are challenging to monitor. To assess the probability of silent circulation, simulation models can be used to understand transmission dynamics when detection is unreliable. Model assumptions, however, impact the estimated probability of silent circulation. Here, we examine the impact of having distinct populations, rather than a single well-mixed population, with a discrete-individual model including environmental surveillance. We show that partitioning a well-mixed population into networks of distinct communities may result in a higher probability of silent circulation as a result of the time it takes for the detection of a circulation event. Population structure should be considered when assessing polio control in a region with many loosely interacting communities.

The Usefulness of the Test-Positive Proportion of Severe Acute Respiratory Syndrome Coronavirus 2 as a Surveillance Tool.

Comparison of coronavirus disease 2019 (COVID-19) case numbers over time and between locations is complicated by limits to virological testing to confirm severe acute respiratory syndrome coronavirus 2 infection. The proportion of tested individuals who have tested positive (test-positive proportion, TPP) can potentially be used to inform trends in incidence. We propose a model for testing in a population experiencing an epidemic of COVID-19 and derive an expression for TPP in terms of well-defined parameters related to testing and presence of other pathogens causing COVID-19-like symptoms. In the absence of dramatic shifts of testing practices in time or between locations, the TPP is positively correlated with the incidence of infection. We show that the proportion of tested individuals who present COVID-19-like symptoms encodes information similar to the TPP but has different relationships with the testing parameters, and can thus provide additional information regarding dynamic changes in TPP and incidence. Finally, we compare data on confirmed cases and TPP from US states up to October 2020. We conjecture why states might have higher or lower TPP than average. Collection of symptom status and age/risk category of tested individuals can increase the utility of TPP in assessing the state of the pandemic in different locations and times.

The Long-Term Safety, Public Health Impact, and Cost-Effectiveness of Routine Vaccination with a Recombinant, Live-Attenuated Dengue Vaccine (Dengvaxia): A Model Comparison Study.

BACKGROUND: Large Phase III trials across Asia and Latin America have recently demonstrated the efficacy of a recombinant, live-attenuated dengue vaccine (Dengvaxia) over the first 25 mo following vaccination. Subsequent data collected in the longer-term follow-up phase, however, have raised concerns about a potential increase in hospitalization risk of subsequent dengue infections, in particular among young, dengue-naïve vaccinees. We here report predictions from eight independent modelling groups on the long-term safety, public health impact, and cost-effectiveness of routine vaccination with Dengvaxia in a range of transmission settings, as characterised by seroprevalence levels among 9-y-olds (SP9). These predictions were conducted for the World Health Organization to inform their recommendations on optimal use of this vaccine. METHODS AND FINDINGS: The models adopted, with small variations, a parsimonious vaccine mode of action that was able to reproduce quantitative features of the observed trial data. The adopted mode of action assumed that vaccination, similarly to natural infection, induces transient, heterologous protection and, further, establishes a long-lasting immunogenic memory, which determines disease severity of subsequent infections. The default vaccination policy considered was routine vaccination of 9-y-old children in a three-dose schedule at 80% coverage. The outcomes examined were the impact of vaccination on infections, symptomatic dengue, hospitalised dengue, deaths, and cost-effectiveness over a 30-y postvaccination period. Case definitions were chosen in accordance with the Phase III trials. All models predicted that in settings with moderate to high dengue endemicity (SP9 ≥ 50%), the default vaccination policy would reduce the burden of dengue disease for the population by 6%-25% (all simulations: -3%-34%) and in high-transmission settings (SP9 ≥ 70%) by 13%-25% (all simulations: 10%- 34%). These endemicity levels are representative of the participating sites in both Phase III trials. In contrast, in settings with low transmission intensity (SP9 ≤ 30%), the models predicted that vaccination could lead to a substantial increase in hospitalisation because of dengue. Modelling reduced vaccine coverage or the addition of catch-up campaigns showed that the impact of vaccination scaled approximately linearly with the number of people vaccinated. In assessing the optimal age of vaccination, we found that targeting older children could increase the net benefit of vaccination in settings with moderate transmission intensity (SP9 = 50%). Overall, vaccination was predicted to be potentially cost-effective in most endemic settings if priced competitively. The results are based on the assumption that the vaccine acts similarly to natural infection. This assumption is consistent with the available trial results but cannot be directly validated in the absence of additional data. Furthermore, uncertainties remain regarding the level of protection provided against disease versus infection and the rate at which vaccine-induced protection declines. CONCLUSIONS: Dengvaxia has the potential to reduce the burden of dengue disease in areas of moderate to high dengue endemicity. However, the potential risks of vaccination in areas with limited exposure to dengue as well as the local costs and benefits of routine vaccination are important considerations for the inclusion of Dengvaxia into existing immunisation programmes. These results were important inputs into WHO global policy for use of this licensed dengue vaccine.

Epidemic Wave Dynamics Attributable to Urban Community Structure: A Theoretical Characterization of Disease Transmission in a Large Network.

BACKGROUND: Multiple waves of transmission during infectious disease epidemics represent a major public health challenge, but the ecological and behavioral drivers of epidemic resurgence are poorly understood. In theory, community structure­aggregation into highly intraconnected and loosely interconnected social groups­within human populations may lead to punctuated outbreaks as diseases progress from one community to the next. However, this explanation has been largely overlooked in favor of temporal shifts in environmental conditions and human behavior and because of the difficulties associated with estimating large-scale contact patterns. OBJECTIVE: The aim was to characterize naturally arising patterns of human contact that are capable of producing simulated epidemics with multiple wave structures. METHODS: We used an extensive dataset of proximal physical contacts between users of a public Wi-Fi Internet system to evaluate the epidemiological implications of an empirical urban contact network. We characterized the modularity (community structure) of the network and then estimated epidemic dynamics under a percolation-based model of infectious disease spread on the network. We classified simulated epidemics as multiwave using a novel metric and we identified network structures that were critical to the network's ability to produce multiwave epidemics. RESULTS: We identified robust community structure in a large, empirical urban contact network from which multiwave epidemics may emerge naturally. This pattern was fueled by a special kind of insularity in which locally popular individuals were not the ones forging contacts with more distant social groups. CONCLUSIONS: Our results suggest that ordinary contact patterns can produce multiwave epidemics at the scale of a single urban area without the temporal shifts that are usually assumed to be responsible. Understanding the role of community structure in epidemic dynamics allows officials to anticipate epidemic resurgence without having to forecast future changes in hosts, pathogens, or the environment.

Changing COVID-19 cases and deaths detection in Florida.

Epidemic data are often difficult to interpret due to inconsistent detection and reporting. As these data are critically relied upon to inform policy and epidemic projections, understanding reporting trends is similarly important. Early reporting of the COVID-19 pandemic in particular is complicated, due to changing diagnostic and testing protocols. An internal audit by the State of Florida, USA found numerous specific examples of irregularities in COVID-19 case and death reports. Using case, hospitalization, and death data from the the first year of the COVID-19 pandemic in Florida, we present approaches that can be used to identify the timing, direction, and magnitude of some reporting changes. Specifically, by establishing a baseline of detection probabilities from the first (spring) wave, we show that transmission trends among all age groups were similar, with the exception of the second summer wave, when younger people became infected earlier than seniors, by approximately 2 weeks. We also found a substantial drop in case-fatality risk (CFR) among all age groups over the three waves during the first year of the pandemic, with the most drastic changes seen in the 0 to 39 age group. The CFR trends provide useful insights into infection detection that would not be possible by relying on the number of tests alone. During the third wave, for which we have reliable hospitalization data, the CFR was remarkably stable across all age groups. In contrast, the hospitalization-to-case ratio varied inversely with cases while the death-to-hospitalization ratio varied proportionally. Although specific trends are likely to vary between locales, the approaches we present here offer a generic way to understand the substantial changes that occurred in the relationships among the key epidemic indicators.

Agent-based modeling of the COVID-19 pandemic in Florida.

The onset of the COVID-19 pandemic drove a widespread, often uncoordinated effort by research groups to develop mathematical models of SARS-CoV-2 to study its spread and inform control efforts. The urgent demand for insight at the outset of the pandemic meant early models were typically either simple or repurposed from existing research agendas. Our group predominantly uses agent-based models (ABMs) to study fine-scale intervention scenarios. These high-resolution models are large, complex, require extensive empirical data, and are often more detailed than strictly necessary for answering qualitative questions like "Should we lockdown?" During the early stages of an extraordinary infectious disease crisis, particularly before clear empirical evidence is available, simpler models are more appropriate. As more detailed empirical evidence becomes available, however, and policy decisions become more nuanced and complex, fine-scale approaches like ours become more useful. In this manuscript, we discuss how our group navigated this transition as we modeled the pandemic. The role of modelers often included nearly real-time analysis, and the massive undertaking of adapting our tools quickly. We were often playing catch up with a firehose of evidence, while simultaneously struggling to do both academic research and real-time decision support, under conditions conducive to neither. By reflecting on our experiences of responding to the pandemic and what we learned from these challenges, we can better prepare for future demands.

Evaluating targeted COVID-19 vaccination strategies with agent-based modeling.

We evaluate approaches to vaccine distribution using an agent-based model of human activity and COVID-19 transmission calibrated to detailed trends in cases, hospitalizations, deaths, seroprevalence, and vaccine breakthrough infections in Florida, USA. We compare the incremental effectiveness for four different distribution strategies at four different levels of vaccine availability, reflecting different income settings' historical COVID-19 vaccine distribution. Our analysis indicates that the best strategy to reduce severe outcomes is to actively target high disease-risk individuals. This was true in every scenario, although the advantage was greatest for the middle-income-country availability assumptions, and relatively modest compared to a simple mass vaccination approach for rapid, high levels of vaccine availability. Ring vaccination, while generally the most effective strategy for reducing infections, ultimately proved least effective at preventing deaths. We also consider using age group as a practical, surrogate measure for actual disease-risk targeting; this approach still outperforms both simple mass distribution and ring vaccination. We also find that the magnitude of strategy effectiveness depends on when assessment occurs (e.g., after delta vs. after omicron variants). However, these differences in absolute benefit for the strategies do not change the ranking of their performance at preventing severe outcomes across vaccine availability assumptions.

Informing pandemic response in the face of uncertainty. An evaluation of the U.S. COVID-19 Scenario Modeling Hub.

Our ability to forecast epidemics more than a few weeks into the future is constrained by the complexity of disease systems, our limited ability to measure the current state of an epidemic, and uncertainties in how human action will affect transmission. Realistic longer-term projections (spanning more than a few weeks) may, however, be possible under defined scenarios that specify the future state of critical epidemic drivers, with the additional benefit that such scenarios can be used to anticipate the comparative effect of control measures. Since December 2020, the U.S. COVID-19 Scenario Modeling Hub (SMH) has convened multiple modeling teams to make 6-month ahead projections of the number of SARS-CoV-2 cases, hospitalizations and deaths. The SMH released nearly 1.8 million national and state-level projections between February 2021 and November 2022. SMH performance varied widely as a function of both scenario validity and model calibration. Scenario assumptions were periodically invalidated by the arrival of unanticipated SARS-CoV-2 variants, but SMH still provided projections on average 22 weeks before changes in assumptions (such as virus transmissibility) invalidated scenarios and their corresponding projections. During these periods, before emergence of a novel variant, a linear opinion pool ensemble of contributed models was consistently more reliable than any single model, and projection interval coverage was near target levels for the most plausible scenarios (e.g., 79% coverage for 95% projection interval). SMH projections were used operationally to guide planning and policy at different stages of the pandemic, illustrating the value of the hub approach for long-term scenario projections.

Evaluation of the US COVID-19 Scenario Modeling Hub for informing pandemic response under uncertainty.

Our ability to forecast epidemics far into the future is constrained by the many complexities of disease systems. Realistic longer-term projections may, however, be possible under well-defined scenarios that specify the future state of critical epidemic drivers. Since December 2020, the U.S. COVID-19 Scenario Modeling Hub (SMH) has convened multiple modeling teams to make months ahead projections of SARS-CoV-2 burden, totaling nearly 1.8 million national and state-level projections. Here, we find SMH performance varied widely as a function of both scenario validity and model calibration. We show scenarios remained close to reality for 22 weeks on average before the arrival of unanticipated SARS-CoV-2 variants invalidated key assumptions. An ensemble of participating models that preserved variation between models (using the linear opinion pool method) was consistently more reliable than any single model in periods of valid scenario assumptions, while projection interval coverage was near target levels. SMH projections were used to guide pandemic response, illustrating the value of collaborative hubs for longer-term scenario projections.

EpiFire: An open source C++ library and application for contact network epidemiology.

BACKGROUND: Contact network models have become increasingly common in epidemiology, but we lack a flexible programming framework for the generation and analysis of epidemiological contact networks and for the simulation of disease transmission through such networks. RESULTS: Here we present EpiFire, an applications programming interface and graphical user interface implemented in C++, which includes a fast and efficient library for generating, analyzing and manipulating networks. Network-based percolation and chain-binomial simulations of susceptible-infected-recovered disease transmission, as well as traditional non-network mass-action simulations, can be performed using EpiFire. CONCLUSIONS: EpiFire provides an open-source programming interface for the rapid development of network models with a focus in contact network epidemiology. EpiFire also provides a point-and-click interface for generating networks, conducting epidemic simulations, and creating figures. This interface is particularly useful as a pedagogical tool.

Updated projections for COVID-19 omicron wave in Florida.

In this report, we use a detailed simulation model to assess and project the COVID-19 epidemic in Florida. The model is a data-driven, stochastic, discrete-time, agent based model with an explicit representation of people and places. Using the model, we find that the omicron variant wave in Florida is likely to cause many more infections than occurred during the delta variant wave. Due to testing limitations and often mild symptoms, however, we anticipate that omicron infections will be underreported compared to delta. We project that reported cases of COVID-19 will continue to grow significantly and peak in early January 2022, and that the number of reported COVID-19 deaths due to omicron may be 1/3 of the total caused by the delta wave.

Ecology and public health burden of Keystone virus in Florida.

Keystone virus (KEYV) is an under-studied orthobunyavirus that is transmitted via both horizontal and vertical cycles involving various mosquito species and vertebrate hosts. Historical evidence indicates that KEYV causes sub-clinical infections in humans, and some case studies draw links between this virus and encephalitis. Given KEYV's potential to cause human infections, it is plausible that it causes an under-appreciated proportion of both generic viral infections and unidentified viral encephalitis cases. This review summarizes current knowledge of KEYV and its disease dynamics in order to better understand the virus' medical and economic burden on human populations.

Potential test-negative design study bias in outbreak settings: application to Ebola vaccination in Democratic Republic of Congo.

BACKGROUND: Infectious disease outbreaks present unique challenges to study designs for vaccine evaluation. Test-negative design (TND) studies have previously been used to estimate vaccine effectiveness and have been proposed for Ebola virus disease (EVD) vaccines. However, there are key differences in how cases and controls are recruited during outbreaks and pandemics of novel pathogens, whcih have implications for the reliability of effectiveness estimates using this design. METHODS: We use a modelling approach to quantify TND bias for a prophylactic vaccine under varying study and epidemiological scenarios. Our model accounts for heterogeneity in vaccine distribution and for two potential routes to testing and recruitment into the study: self-reporting and contact-tracing. We derive conventional and hybrid TND estimators for this model and suggest ways to translate public health response data into the parameters of the model. RESULTS: Using a conventional TND study, our model finds biases in vaccine effectiveness estimates. Bias arises due to differential recruitment from self-reporting and contact-tracing, and due to clustering of vaccination. We estimate the degree of bias when recruitment route is not available, and propose a study design to eliminate the bias if recruitment route is recorded. CONCLUSIONS: Hybrid TND studies can resolve the design bias with conventional TND studies applied to outbreak and pandemic response testing data, if those efforts collect individuals' routes to testing. Without route to testing, other epidemiological data will be required to estimate the magnitude of potential bias in a conventional TND study. Since these studies may need to be conducted retrospectively, public health responses should obtain these data, and generic protocols for outbreak and pandemic response studies should emphasize the need to record routes to testing.

The TIRS trial: protocol for a cluster randomized controlled trial assessing the efficacy of preventive targeted indoor residual spraying to reduce Aedes-borne viral illnesses in Merida, Mexico.

BACKGROUND: Current urban vector control strategies have failed to contain dengue epidemics and to prevent the global expansion of Aedes-borne viruses (ABVs: dengue, chikungunya, Zika). Part of the challenge in sustaining effective ABV control emerges from the paucity of evidence regarding the epidemiological impact of any Aedes control method. A strategy for which there is limited epidemiological evidence is targeted indoor residual spraying (TIRS). TIRS is a modification of classic malaria indoor residual spraying that accounts for Aedes aegypti resting behavior by applying residual insecticides on exposed lower sections of walls (< 1.5 m), under furniture, and on dark surfaces. METHODS/DESIGN: We are pursuing a two-arm, parallel, unblinded, cluster randomized controlled trial to quantify the overall efficacy of TIRS in reducing the burden of laboratory-confirmed ABV clinical disease (primary endpoint). The trial will be conducted in the city of Merida, Yucatan State, Mexico (population ~ 1million), where we will prospectively follow 4600 children aged 2-15 years at enrollment, distributed in 50 clusters of 5 × 5 city blocks each. Clusters will be randomly allocated (n = 25 per arm) using covariate-constrained randomization. A "fried egg" design will be followed, in which all blocks of the 5 × 5 cluster receive the intervention, but all sampling to evaluate the epidemiological and entomological endpoints will occur in the "yolk," the center 3 × 3 city blocks of each cluster. TIRS will be implemented as a preventive application (~ 1-2 months prior to the beginning of the ABV season). Active monitoring for symptomatic ABV illness will occur through weekly household visits and enhanced surveillance. Annual sero-surveys will be performed after each transmission season and entomological evaluations of Ae. aegypti indoor abundance and ABV infection rates monthly during the period of active surveillance. Epidemiological and entomological evaluation will continue for up to three transmission seasons. DISCUSSION: The findings from this study will provide robust epidemiological evidence of the efficacy of TIRS in reducing ABV illness and infection. If efficacious, TIRS could drive a paradigm shift in Aedes control by considering Ae. aegypti behavior to guide residual insecticide applications and changing deployment to preemptive control (rather than in response to symptomatic cases), two major enhancements to existing practice. TRIAL REGISTRATION: ClinicalTrials.gov NCT04343521 . Registered on 13 April 2020. The protocol also complies with the WHO International Clinical Trials Registry Platform (ICTRP) (Additional file 1). PRIMARY SPONSOR: National Institutes of Health, National Institute of Allergy and Infectious Diseases (NIH/NIAID).

COVID-19 reopening strategies at the county level in the face of uncertainty: Multiple Models for Outbreak Decision Support.

Policymakers make decisions about COVID-19 management in the face of considerable uncertainty. We convened multiple modeling teams to evaluate reopening strategies for a mid-sized county in the United States, in a novel process designed to fully express scientific uncertainty while reducing linguistic uncertainty and cognitive biases. For the scenarios considered, the consensus from 17 distinct models was that a second outbreak will occur within 6 months of reopening, unless schools and non-essential workplaces remain closed. Up to half the population could be infected with full workplace reopening; non-essential business closures reduced median cumulative infections by 82%. Intermediate reopening interventions identified no win-win situations; there was a trade-off between public health outcomes and duration of workplace closures. Aggregate results captured twice the uncertainty of individual models, providing a more complete expression of risk for decision-making purposes.

Evaluating the probability of silent circulation of polio in small populations using the silent circulation statistic.

As polio-endemic countries move towards elimination, infrequent first infections and incomplete surveillance make it difficult to determine when the virus has been eliminated from the population. Eichner and Dietz [American Journal of Epidemiology, 143, 8 (1996)] proposed a model to estimate the probability of silent polio circulation depending upon when the last paralytic case was detected. Using the same kind of stochastic model they did, we additionally model waning polio immunity in the context of isolated, small, and unvaccinated populations. We compare using the Eichner and Dietz assumption of an initial case at the start of the simulation to a more accurate determination that observes the first case. The former estimates a higher probability of silent circulation in small populations, but this effect diminishes with increasing model population. We also show that stopping the simulation after a specific time estimates a lower probability of silent circulation than when all replicates are run to extinction, though this has limited impact on small populations. Our extensions to the Eichner and Dietz work improve the basis for decisions concerning the probability of silent circulation. Further model realism will be needed for accurate silent circulation risk assessment.