Planning as Inference in Epidemiological Dynamics Models.

In this work we demonstrate how to automate parts of the infectious disease-control policy-making process via performing inference in existing epidemiological models. The kind of inference tasks undertaken include computing the posterior distribution over controllable, via direct policy-making choices, simulation model parameters that give rise to acceptable disease progression outcomes. Among other things, we illustrate the use of a probabilistic programming language that automates inference in existing simulators. Neither the full capabilities of this tool for automating inference nor its utility for planning is widely disseminated at the current time. Timely gains in understanding about how such simulation-based models and inference automation tools applied in support of policy-making could lead to less economically damaging policy prescriptions, particularly during the current COVID-19 pandemic.

Detecting critical slowing down in high-dimensional epidemiological systems.

Despite medical advances, the emergence and re-emergence of infectious diseases continue to pose a public health threat. Low-dimensional epidemiological models predict that epidemic transitions are preceded by the phenomenon of critical slowing down (CSD). This has raised the possibility of anticipating disease (re-)emergence using CSD-based early-warning signals (EWS), which are statistical moments estimated from time series data. For EWS to be useful at detecting future (re-)emergence, CSD needs to be a generic (model-independent) feature of epidemiological dynamics irrespective of system complexity. Currently, it is unclear whether the predictions of CSD-derived from simple, low-dimensional systems-pertain to real systems, which are high-dimensional. To assess the generality of CSD, we carried out a simulation study of a hierarchy of models, with increasing structural complexity and dimensionality, for a measles-like infectious disease. Our five models included: i) a nonseasonal homogeneous Susceptible-Exposed-Infectious-Recovered (SEIR) model, ii) a homogeneous SEIR model with seasonality in transmission, iii) an age-structured SEIR model, iv) a multiplex network-based model (Mplex) and v) an agent-based simulator (FRED). All models were parameterised to have a herd-immunity immunization threshold of around 90% coverage, and underwent a linear decrease in vaccine uptake, from 92% to 70% over 15 years. We found evidence of CSD prior to disease re-emergence in all models. We also evaluated the performance of seven EWS: the autocorrelation, coefficient of variation, index of dispersion, kurtosis, mean, skewness, variance. Performance was scored using the Area Under the ROC Curve (AUC) statistic. The best performing EWS were the mean and variance, with AUC > 0.75 one year before the estimated transition time. These two, along with the autocorrelation and index of dispersion, are promising candidate EWS for detecting disease emergence.

Forecasted Size of Measles Outbreaks Associated With Vaccination Exemptions for Schoolchildren.

Importance: Vaccine exemptions, which allow unvaccinated children to attend school, have increased by a factor of 28 since 2003 in Texas. Geographic clustering of unvaccinated children facilitates the spread of measles introductions, but the potential size of outbreaks is unclear. Objective: To forecast the range of measles outbreak sizes in each metropolitan area of Texas at 2018 and future reduced school vaccination rates. Design, Setting, and Participants: An agent-based decision analytical model using a synthetic population of Texas, derived from the 2010 US Census, was used to simulate measles transmission in the Texas population. Real schools were represented in the simulations, and the 2018 vaccination rate of each real school was applied to a simulated hypothetical equivalent. Single cases of measles were introduced, daily activities and interactions were modeled for each population member, and the number of infections over the course of 9 months was counted for 1000 simulated runs in each Texas metropolitan area. Interventions: To determine the outcomes of further decreases in vaccination coverage, additional simulations were performed with vaccination rates reduced by 1% to 10% in schools with populations that are currently undervaccinated. Main Outcomes and Measures: Expected distributions of outbreak sizes in each metropolitan area of Texas at 2018 and reduced vaccination rates. Results: At 2018 vaccination rates, the median number of cases in large metropolitan areas was typically small, ranging from 1 to 3 cases, which is consistent with outbreaks in Texas 2006 to 2017. However, the upper limit of the distribution of plausible outbreaks (the 95th percentile, associated with 1 in 20 measles introductions) exceeded 400 cases in both the Austin and Dallas metropolitan areas, similar to the largest US outbreaks since measles was eliminated in 2000. Decreases in vaccination rates in schools with undervaccinated populations in 2018 were associated with exponential increases in the potential size of outbreaks: a 5% decrease in vaccination rate was associated with a 40% to 4000% increase in potential outbreak size, depending on the metropolitan area. A mean (SD) of 64% (11%) of cases occurred in students for whom a vaccine had been refused, but a mean (SD) of 36% (11%) occurred in others (ie, bystanders). Conclusions and Relevance: This study suggests that vaccination rates in some Texas schools are currently low enough to allow large measles outbreaks. Further decreases are associated with dramatic increases in the probability of large outbreaks. Limiting vaccine exemptions could be associated with a decrease in the risk of large measles outbreaks.

Exploring scenarios of chikungunya mitigation with a data-driven agent-based model of the 2014-2016 outbreak in Colombia.

New epidemics of infectious diseases can emerge any time, as illustrated by the emergence of chikungunya virus (CHIKV) and Zika virus (ZIKV) in Latin America. During new epidemics, public health officials face difficult decisions regarding spatial targeting of interventions to optimally allocate limited resources. We used a large-scale, data-driven, agent-based simulation model (ABM) to explore CHIKV mitigation strategies, including strategies based on previous DENV outbreaks. Our model represents CHIKV transmission in a realistic population of Colombia with 45 million individuals in 10.6 million households, schools, and workplaces. Our model uses high-resolution probability maps for the occurrence of the Ae. aegypti mosquito vector to estimate mosquito density in Colombia. We found that vector control in all 521 municipalities with mosquito populations led to 402,940 fewer clinical cases of CHIKV compared to a baseline scenario without intervention. We also explored using data about previous dengue virus (DENV) epidemics to inform CHIKV mitigation strategies. Compared to the baseline scenario, 314,437 fewer cases occurred when we simulated vector control only in 301 municipalities that had previously reported DENV, illustrating the value of available data from previous outbreaks. When varying the implementation parameters for vector control, we found that faster implementation and scale-up of vector control led to the greatest proportionate reduction in cases. Using available data for epidemic simulations can strengthen decision making against new epidemic threats.

Prevention of Hepatitis C by Screening and Treatment in U.S. Prisons.

BACKGROUND: The prevalence of hepatitis C virus (HCV) in U.S. prisoners is high; however, HCV testing and treatment are rare. Infected inmates released back into society contribute to the spread of HCV in the general population. Routine hepatitis screening of inmates followed by new therapies may reduce ongoing HCV transmission. OBJECTIVE: To evaluate the health and economic effect of HCV screening and treatment in prisons on the HCV epidemic in society. DESIGN: Agent-based microsimulation model of HCV transmission and progression of HCV disease. DATA SOURCES: Published literature. TARGET POPULATION: Population in U.S. prisons and general community. TIME HORIZON: 30 years. PERSPECTIVE: Societal. INTERVENTIONS: Risk-based and universal opt-out hepatitis C screening in prisons, followed by treatment in a portion of patients. OUTCOME MEASURES: Prevention of HCV transmission and associated disease in prisons and society, costs, quality-adjusted life-years (QALYs), incremental cost-effectiveness ratio (ICER), and total prison budget. RESULTS OF BASE-CASE ANALYSIS: Implementing risk-based and opt-out screening could diagnose 41,900 to 122,700 new HCV cases in prisons in the next 30 years. Compared with no screening, these scenarios could prevent 5500 to 12,700 new HCV infections caused by released inmates, wherein about 90% of averted infections would have occurred outside of prisons. Screening could also prevent 4200 to 11,700 liver-related deaths. The ICERs of screening scenarios were $19,600 to $29,200 per QALY, and the respective first-year prison budget was $900 to $1150 million. Prisons would require an additional 12.4% of their current health care budget to implement such interventions. RESULTS OF SENSITIVITY ANALYSIS: Results were sensitive to the time horizon, and ICERs otherwise remained less than $50,000 per QALY. LIMITATION: Data on transmission network, reinfection rate, and opt-out HCV screening rate are lacking. CONCLUSION: Universal opt-out HCV screening in prisons is highly cost-effective and would reduce HCV transmission and HCV-associated diseases primarily in the outside community. Investing in U.S. prisons to manage hepatitis C is a strategic approach to address the current epidemic. PRIMARY FUNDING SOURCE: National Institutes of Health.

Is population structure sufficient to generate area-level inequalities in influenza rates? An examination using agent-based models.

BACKGROUND: In New Haven County, CT (NHC), influenza hospitalization rates have been shown to increase with census tract poverty in multiple influenza seasons. Though multiple factors have been hypothesized to cause these inequalities, including population structure, differential vaccine uptake, and differential access to healthcare, the impact of each in generating observed inequalities remains unknown. We can design interventions targeting factors with the greatest explanatory power if we quantify the proportion of observed inequalities that hypothesized factors are able to generate. Here, we ask if population structure is sufficient to generate the observed area-level inequalities in NHC. To our knowledge, this is the first use of simulation models to examine the causes of differential poverty-related influenza rates. METHODS: Using agent-based models with a census-informed, realistic representation of household size, age-structure, population density in NHC census tracts, and contact rates in workplaces, schools, households, and neighborhoods, we measured poverty-related differential influenza attack rates over the course of an epidemic with a 23 % overall clinical attack rate. We examined the role of asthma prevalence rates as well as individual contact rates and infection susceptibility in generating observed area-level influenza inequalities. RESULTS: Simulated attack rates (AR) among adults increased with census tract poverty level (F = 30.5; P < 0.001) in an epidemic caused by a virus similar to A (H1N1) pdm09. We detected a steeper, earlier influenza rate increase in high-poverty census tracts-a finding that we corroborate with a temporal analysis of NHC surveillance data during the 2009 H1N1 pandemic. The ratio of the simulated adult AR in the highest- to lowest-poverty tracts was 33 % of the ratio observed in surveillance data. Increasing individual contact rates in the neighborhood did not increase simulated area-level inequalities. When we modified individual susceptibility such that it was inversely proportional to household income, inequalities in AR between high- and low-poverty census tracts were comparable to those observed in reality. DISCUSSION: To our knowledge, this is the first study to use simulations to probe the causes of observed inequalities in influenza disease patterns. Knowledge of the causes and their relative explanatory power will allow us to design interventions that have the greatest impact on reducing inequalities. CONCLUSION: Differential exposure due to population structure in our realistic simulation model explains a third of the observed inequality. Differential susceptibility to disease due to prevailing chronic conditions, vaccine uptake, and smoking should be considered in future models in order to quantify the role of additional factors in generating influenza inequalities.

A systematic review of barriers to data sharing in public health.

BACKGROUND: In the current information age, the use of data has become essential for decision making in public health at the local, national, and global level. Despite a global commitment to the use and sharing of public health data, this can be challenging in reality. No systematic framework or global operational guidelines have been created for data sharing in public health. Barriers at different levels have limited data sharing but have only been anecdotally discussed or in the context of specific case studies. Incomplete systematic evidence on the scope and variety of these barriers has limited opportunities to maximize the value and use of public health data for science and policy. METHODS: We conducted a systematic literature review of potential barriers to public health data sharing. Documents that described barriers to sharing of routinely collected public health data were eligible for inclusion and reviewed independently by a team of experts. We grouped identified barriers in a taxonomy for a focused international dialogue on solutions. RESULTS: Twenty potential barriers were identified and classified in six categories: technical, motivational, economic, political, legal and ethical. The first three categories are deeply rooted in well-known challenges of health information systems for which structural solutions have yet to be found; the last three have solutions that lie in an international dialogue aimed at generating consensus on policies and instruments for data sharing. CONCLUSIONS: The simultaneous effect of multiple interacting barriers ranging from technical to intangible issues has greatly complicated advances in public health data sharing. A systematic framework of barriers to data sharing in public health will be essential to accelerate the use of valuable information for the global good.

A large-scale immuno-epidemiological simulation of influenza A epidemics.

BACKGROUND: Agent based models (ABM) are useful to explore population-level scenarios of disease spread and containment, but typically characterize infected individuals using simplified models of infection and symptoms dynamics. Adding more realistic models of individual infections and symptoms may help to create more realistic population level epidemic dynamics. METHODS: Using an equation-based, host-level mathematical model of influenza A virus infection, we develop a function that expresses the dependence of infectivity and symptoms of an infected individual on initial viral load, age, and viral strain phenotype. We incorporate this response function in a population-scale agent-based model of influenza A epidemic to create a hybrid multiscale modeling framework that reflects both population dynamics and individualized host response to infection. RESULTS: At the host level, we estimate parameter ranges using experimental data of H1N1 viral titers and symptoms measured in humans. By linearization of symptoms responses of the host-level model we obtain a map of the parameters of the model that characterizes clinical phenotypes of influenza infection and immune response variability over the population. At the population-level model, we analyze the effect of individualizing viral response in agent-based model by simulating epidemics across Allegheny County, Pennsylvania under both age-specific and age-independent severity assumptions. CONCLUSIONS: We present a framework for multi-scale simulations of influenza epidemics that enables the study of population-level effects of individual differences in infections and symptoms, with minimal additional computational cost compared to the existing population-level simulations.

FRED (a Framework for Reconstructing Epidemic Dynamics): an open-source software system for modeling infectious diseases and control strategies using census-based populations.

BACKGROUND: Mathematical and computational models provide valuable tools that help public health planners to evaluate competing health interventions, especially for novel circumstances that cannot be examined through observational or controlled studies, such as pandemic influenza. The spread of diseases like influenza depends on the mixing patterns within the population, and these mixing patterns depend in part on local factors including the spatial distribution and age structure of the population, the distribution of size and composition of households, employment status and commuting patterns of adults, and the size and age structure of schools. Finally, public health planners must take into account the health behavior patterns of the population, patterns that often vary according to socioeconomic factors such as race, household income, and education levels. RESULTS: FRED (a Framework for Reconstructing Epidemic Dynamics) is a freely available open-source agent-based modeling system based closely on models used in previously published studies of pandemic influenza. This version of FRED uses open-access census-based synthetic populations that capture the demographic and geographic heterogeneities of the population, including realistic household, school, and workplace social networks. FRED epidemic models are currently available for every state and county in the United States, and for selected international locations. CONCLUSIONS: State and county public health planners can use FRED to explore the effects of possible influenza epidemics in specific geographic regions of interest and to help evaluate the effect of interventions such as vaccination programs and school closure policies. FRED is available under a free open source license in order to contribute to the development of better modeling tools and to encourage open discussion of modeling tools being used to evaluate public health policies. We also welcome participation by other researchers in the further development of FRED.

Dynamic simulation of crime perpetration and reporting to examine community intervention strategies.

OBJECTIVE: To develop a conceptual computational agent-based model (ABM) to explore community-wide versus spatially focused crime reporting interventions to reduce community crime perpetrated by youth. METHOD: Agents within the model represent individual residents and interact on a two-dimensional grid representing an abstract nonempirically grounded community setting. Juvenile agents are assigned initial random probabilities of perpetrating a crime and adults are assigned random probabilities of witnessing and reporting crimes. The agents' behavioral probabilities modify depending on the individual's experience with criminal behavior and punishment, and exposure to community crime interventions. Cost-effectiveness analyses assessed the impact of activating different percentages of adults to increase reporting and reduce community crime activity. Community-wide interventions were compared with spatially focused interventions, in which activated adults were focused in areas of highest crime prevalence. RESULTS: The ABM suggests that both community-wide and spatially focused interventions can be effective in reducing overall offenses, but their relative effectiveness may depend on the intensity and cost of the interventions. Although spatially focused intervention yielded localized reductions in crimes, such interventions were shown to move crime to nearby communities. Community-wide interventions can achieve larger reductions in overall community crime offenses than spatially focused interventions, as long as sufficient resources are available. CONCLUSION: The ABM demonstrates that community-wide and spatially focused crime strategies produce unique intervention dynamics influencing juvenile crime behaviors through the decisions and actions of community adults. It shows how such models might be used to investigate community-supported crime intervention programs by integrating community input and expertise and provides a simulated setting for assessing dimensions of cost comparison and intervention effect sustainability. ABM illustrates how intervention models might be used to investigate community-supported crime intervention programs.

Policies to reduce influenza in the workplace: impact assessments using an agent-based model.

OBJECTIVES: We examined the impact of access to paid sick days (PSDs) and stay-at-home behavior on the influenza attack rate in workplaces. METHODS: We used an agent-based model of Allegheny County, Pennsylvania, with PSD data from the US Bureau of Labor Statistics, standard influenza epidemic parameters, and the probability of staying home when ill. We compared the influenza attack rate among employees resulting from workplace transmission, focusing on the effects of presenteeism (going to work when ill). RESULTS: In a simulated influenza epidemic (R0 = 1.4), the attack rate among employees owing to workplace transmission was 11.54%. A large proportion (72.00%) of this attack rate resulted from exposure to employees engaging in presenteeism. Universal PSDs reduced workplace infections by 5.86%. Providing 1 or 2 "flu days"-allowing employees with influenza to stay home-reduced workplace infections by 25.33% and 39.22%, respectively. CONCLUSIONS: PSDs reduce influenza transmission owing to presenteeism and, hence, the burden of influenza illness in workplaces.

School closure as an influenza mitigation strategy: how variations in legal authority and plan criteria can alter the impact.

BACKGROUND: States' pandemic influenza plans and school closure statutes are intended to guide state and local officials, but most faced a great deal of uncertainty during the 2009 influenza H1N1 epidemic. Questions remained about whether, when, and for how long to close schools and about which agencies and officials had legal authority over school closures. METHODS: This study began with analysis of states' school-closure statutes and pandemic influenza plans to identify the variations among them. An agent-based model of one state was used to represent as constants a population's demographics, commuting patterns, work and school attendance, and community mixing patterns while repeated simulations explored the effects of variations in school closure authority, duration, closure thresholds, and reopening criteria. RESULTS: The results show no basis on which to justify statewide rather than school-specific or community-specific authority for school closures. Nor do these simulations offer evidence to require school closures promptly at the earliest stage of an epidemic. More important are criteria based on monitoring of local case incidence and on authority to sustain closure periods sufficiently to achieve epidemic mitigation. CONCLUSIONS: This agent-based simulation suggests several ways to improve statutes and influenza plans. First, school closure should remain available to state and local authorities as an influenza mitigation strategy. Second, influenza plans need not necessarily specify the threshold for school closures but should clearly define provisions for early and ongoing local monitoring. Finally, school closure authority may be exercised at the statewide or local level, so long as decisions are informed by monitoring incidence in local communities and schools.

Preparedness for pandemics: does variation among states affect the nation as a whole?

OBJECTIVE: Since states' public health systems differ as to pandemic preparedness, this study explored whether such heterogeneity among states could affect the nation's overall influenza rate. DESIGN: The Centers for Disease Control and Prevention produced a uniform set of scores on a 100-point scale from its 2008 national evaluation of state preparedness to distribute materiel from the Strategic National Stockpile (SNS). This study used these SNS scores to represent each state's relative preparedness to distribute influenza vaccine in a timely manner and assumed that "optimal" vaccine distribution would reach at least 35% of the state's population within 4 weeks. The scores were used to determine the timing of vaccine distribution for each state: each 10-point decrement of score below 90 added an additional delay increment to the distribution time. SETTING AND PARTICIPANTS: A large-scale agent-based computational model simulated an influenza pandemic in the US population. In this synthetic population each individual or agent had an assigned household, age, workplace or school destination, daily commute, and domestic intercity air travel patterns. MAIN OUTCOME MEASURES: Simulations compared influenza case rates both nationally and at the state level under 3 scenarios: no vaccine distribution (baseline), optimal vaccine distribution in all states, and vaccine distribution time modified according to state-specific SNS score. RESULTS: Between optimal and SNS-modified scenarios, attack rates rose not only in low-scoring states but also in high-scoring states, demonstrating an interstate spread of infections. Influenza rates were sensitive to variation of the SNS-modified scenario (delay increments of 1 day versus 5 days), but the interstate effect remained. CONCLUSIONS: The effectiveness of a response activity such as vaccine distribution could benefit from national standards and preparedness funding allocated in part to minimize interstate disparities.

A game dynamic model for vaccine skeptics and vaccine believers: measles as an example.

Widespread avoidance of Measles-Mumps-Rubella vaccination (MMR), with a consequent increase in the incidence of major measles outbreaks, demonstrates that the effectiveness of vaccination programs can be thwarted by the public misperceptions of vaccine risk. By coupling game theory and epidemic models, we examine vaccination choice among populations stratified into two behavioral groups: vaccine skeptics and vaccine believers. The two behavioral groups are assumed to be heterogeneous with respect to their perceptions of vaccine and infection risks. We demonstrate that the pursuit of self-interest among vaccine skeptics often leads to vaccination levels that are suboptimal for a population, even if complete coverage is achieved among vaccine believers. The demand for measles vaccine across populations driven by individual self-interest was found to be more sensitive to the proportion of vaccine skeptics than to the extent to which vaccine skeptics misperceive the risk of vaccine. Furthermore, as the number of vaccine skeptics increases, the probability of infection among vaccine skeptics increases initially, but it decreases once the vaccine skeptics begin receiving the vaccination, if both behavioral groups are vaccinated according to individual self-interest. Our results show that the discrepancy between the coverages of measles vaccine that are driven by self-interest and those driven by population interest becomes larger when the cost of vaccination increases. This research illustrates the importance of public education on vaccine safety and infection risk in order to maintain vaccination levels that are sufficient to maintain herd immunity.

The role of subway travel in an influenza epidemic: a New York City simulation.

The interactions of people using public transportation in large metropolitan areas may help spread an influenza epidemic. An agent-based model computer simulation of New York City's (NYC's) five boroughs was developed that incorporated subway ridership into a Susceptible-Exposed-Infected-Recovered disease model framework. The model contains a total of 7,847,465 virtual people. Each person resides in one of the five boroughs of NYC and has a set of socio-demographic characteristics and daily behaviors that include age, sex, employment status, income, occupation, and household location and membership. The model simulates the interactions of subway riders with their workplaces, schools, households, and community activities. It was calibrated using historical data from the 1957-1958 influenza pandemics and from NYC travel surveys. The surveys were necessary to enable inclusion of subway riders into the model. The model results estimate that if influenza did occur in NYC with the characteristics of the 1957-1958 pandemic, 4% of transmissions would occur on the subway. This suggests that interventions targeted at subway riders would be relatively ineffective in containing the epidemic. A number of hypothetical examples demonstrate this feature. This information could prove useful to public health officials planning responses to epidemics.

The benefits to all of ensuring equal and timely access to influenza vaccines in poor communities.

When influenza vaccines are in short supply, allocating vaccines equitably among different jurisdictions can be challenging. But justice is not the only reason to ensure that poorer counties have the same access to influenza vaccines as do wealthier ones. Using a detailed computer simulation model of the Washington, D.C., metropolitan region, we found that limiting or delaying vaccination of residents of poorer counties could raise the total number of influenza infections and the number of new infections per day at the peak of an epidemic throughout the region-even in the wealthier counties that had received more timely and abundant vaccine access. Among other underlying reasons, poorer counties tend to have high-density populations and more children and other higher-risk people per household, resulting in more interactions and both increased transmission of influenza and greater risk for worse influenza outcomes. Thus, policy makers across the country, in poor and wealthy areas alike, have an incentive to ensure that poorer residents have equal access to vaccines.

Would school closure for the 2009 H1N1 influenza epidemic have been worth the cost?: a computational simulation of Pennsylvania.

BACKGROUND: During the 2009 H1N1 influenza epidemic, policy makers debated over whether, when, and how long to close schools. While closing schools could have reduced influenza transmission thereby preventing cases, deaths, and health care costs, it may also have incurred substantial costs from increased childcare needs and lost productivity by teachers and other school employees. METHODS: A combination of agent-based and Monte Carlo economic simulation modeling was used to determine the cost-benefit of closing schools (vs. not closing schools) for different durations (range: 1 to 8 weeks) and symptomatic case incidence triggers (range: 1 to 30) for the state of Pennsylvania during the 2009 H1N1 epidemic. Different scenarios varied the basic reproductive rate (R(0)) from 1.2, 1.6, to 2.0 and used case-hospitalization and case-fatality rates from the 2009 epidemic. Additional analyses determined the cost per influenza case averted of implementing school closure. RESULTS: For all scenarios explored, closing schools resulted in substantially higher net costs than not closing schools. For R(0) = 1.2, 1.6, and 2.0 epidemics, closing schools for 8 weeks would have resulted in median net costs of $21.0 billion (95% Range: $8.0 - $45.3 billion). The median cost per influenza case averted would have been $14,185 ($5,423 - $30,565) for R(0) = 1.2, $25,253 ($9,501 - $53,461) for R(0) = 1.6, and $23,483 ($8,870 - $50,926) for R(0) = 2.0. CONCLUSIONS: Our study suggests that closing schools during the 2009 H1N1 epidemic could have resulted in substantial costs to society as the potential costs of lost productivity and childcare could have far outweighed the cost savings in preventing influenza cases.