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PyMC is a probabilistic programming library for Python that provides tools for constructing and fitting Bayesian models. It offers an intuitive, readable syntax that is close to the natural syntax statisticians use to describe models. PyMC leverages the symbolic computation library PyTensor, allowing it to be compiled into a variety of computational backends, such as C, JAX, and Numba, which in turn offer access to different computational architectures including CPU, GPU, and TPU. Being a general modeling framework, PyMC supports a variety of models including generalized hierarchical linear regression and classification, time series, ordinary differential equations (ODEs), and non-parametric models such as Gaussian processes (GPs). We demonstrate PyMC's versatility and ease of use with examples spanning a range of common statistical models. Additionally, we discuss the positive role of PyMC in the development of the open-source ecosystem for probabilistic programming., Competing Interests: The authors declare that they have no competing interests. Colin Carroll, Ravin Kumar and Junpeng Lao are employed by Google Inc., Christopher J. Fonnesbeck is employed by Baseball Operations Research and Development, Maxim Kochurov, Ricardo Vieira and Thomas Wiecki are employed by PyMC Labs and Michael Osthege is employed by Forschungszentrum Jülich GmbH., (© 2023 Abril-Pla et al.)

The number of all possible epidemics of a given infectious disease that could occur on a given landscape is large for systems of real-world complexity. Furthermore, there is no guarantee that the control actions that are optimal, on average, over all possible epidemics are also best for each possible epidemic. Reinforcement learning (RL) and Monte Carlo control have been used to develop machine-readable context-dependent solutions for complex problems with many possible realizations ranging from video-games to the game of Go. RL could be a valuable tool to generate context-dependent policies for outbreak response, though translating the resulting policies into simple rules that can be read and interpreted by human decision-makers remains a challenge. Here we illustrate the application of RL to the development of context-dependent outbreak response policies to minimize outbreaks of foot-and-mouth disease. We show that control based on the resulting context-dependent policies, which adapt interventions to the specific outbreak, result in smaller outbreaks than static policies. We further illustrate two approaches for translating the complex machine-readable policies into simple heuristics that can be evaluated by human decision-makers. This article is part of the theme issue 'Modelling infectious disease outbreaks in humans, animals and plants: epidemic forecasting and control'. This theme issue is linked with the earlier issue 'Modelling infectious disease outbreaks in humans, animals and plants: approaches and important themes'.

Determining how best to manage an infectious disease outbreak may be hindered by both epidemiological uncertainty (i.e. about epidemiological processes) and operational uncertainty (i.e. about the effectiveness of candidate interventions). However, these two uncertainties are rarely addressed concurrently in epidemic studies. We present an approach to simultaneously address both sources of uncertainty, to elucidate which source most impedes decision-making. In the case of the 2014 West African Ebola outbreak, epidemiological uncertainty is represented by a large ensemble of published models. Operational uncertainty about three classes of interventions is assessed for a wide range of potential intervention effectiveness. We ranked each intervention by caseload reduction in each model, initially assuming an unlimited budget as a counterfactual. We then assessed the influence of three candidate cost functions relating intervention effectiveness and cost for different budget levels. The improvement in management outcomes to be gained by resolving uncertainty is generally high in this study; appropriate information gain could reduce expected caseload by more than 50%. The ranking of interventions is jointly determined by the underlying epidemiological process, the effectiveness of the interventions and the size of the budget. An epidemiologically effective intervention might not be optimal if its costs outweigh its epidemiological benefit. Under higher-budget conditions, resolution of epidemiological uncertainty is most valuable. When budgets are tight, however, operational and epidemiological uncertainty are equally important. Overall, our study demonstrates that significant reductions in caseload could result from a careful examination of both epidemiological and operational uncertainties within the same modelling structure. This approach can be applied to decision-making for the management of other diseases for which multiple models and multiple interventions are available.

Rotavirus is the leading global cause of diarrheal mortality for unvaccinated children under 5 years of age. The outer capsid of rotavirus virions consists of VP7 and VP4 proteins, which determine viral G and P types, respectively, and are primary targets of neutralizing antibodies. Successful vaccination depends upon generating broadly protective immune responses following exposure to rotaviruses presenting a limited number of G- and P-type antigens. Vaccine introduction resulted in decreased rotavirus disease burden but also coincided with the emergence of uncommon G and P genotypes, including G12. To gain insight into the recent predominance of G12P[8] rotaviruses in the United States, we evaluated 142 complete rotavirus genome sequences and metadata from 151 clinical specimens collected in Nashville, TN, from 2011 to 2013 through the New Vaccine Surveillance Network. Circulating G12P[8] strains were found to share many segments with other locally circulating strains but to have distinct constellations. Phylogenetic analyses of G12 sequences and their geographic sources provided evidence for multiple separate introductions of G12 segments into Nashville, TN. Antigenic epitopes of VP7 proteins of G12P[8] strains circulating in Nashville, TN, differ markedly from those of vaccine strains. Fully vaccinated children were found to be infected with G12P[8] strains more frequently than with other rotavirus genotypes. Multiple introductions and significant antigenic mismatch may in part explain the recent predominance of G12P[8] strains in the United States and emphasize the need for continued monitoring of rotavirus vaccine efficacy against emerging rotavirus genotypes. IMPORTANCE Rotavirus is an important cause of childhood diarrheal disease worldwide. Two immunodominant proteins of rotavirus, VP7 and VP4, determine G and P genotypes, respectively. Recently, G12P[8] rotaviruses have become increasingly predominant. By analyzing rotavirus genome sequences from stool specimens obtained in Nashville, TN, from 2011 to 2013 and globally circulating rotaviruses, we found evidence of multiple introductions of G12 genes into the area. Based on sequence polymorphisms, VP7 proteins of these viruses are predicted to present themselves to the immune system very differently than those of vaccine strains. Many of the sick children with G12P[8] rotavirus in their diarrheal stools also were fully vaccinated. Our findings emphasize the need for continued monitoring of circulating rotaviruses and the effectiveness of the vaccines against strains with emerging G and P genotypes., (Copyright © 2018 American Society for Microbiology.)

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., Competing Interests: The authors have declared that no competing interests exist.

Resurgent outbreaks of vaccine-preventable diseases that have previously been controlled or eliminated have been observed in many settings. Reactive vaccination campaigns may successfully control outbreaks but must necessarily be implemented in the face of considerable uncertainty. Real-time surveillance may provide critical information about at-risk population and optimal vaccination targets, but may itself be limited by the specificity of disease confirmation. We propose an integrated modelling approach that synthesizes historical demographic and vaccination data with real-time outbreak surveillance via a dynamic transmission model and an age-specific disease confirmation model. We apply this framework to data from the 1996-1997 measles outbreak in São Paulo, Brazil. To simulate the information available to decision-makers, we truncated the surveillance data to what would have been available at 1 or 2 months prior to the realized interventions. We use the model, fitted to real-time observations, to evaluate the likelihood that candidate age-targeted interventions could control the outbreak. Using only data available prior to the interventions, we estimate that a significant excess of susceptible adults would prevent child-targeted campaigns from controlling the outbreak and that failing to account for age-specific confirmation rates would underestimate the importance of adult-targeted vaccination., (© 2018 The Author(s).)

Early resolution of uncertainty during an epidemic outbreak can lead to rapid and efficient decision making, provided that the uncertainty affects prioritization of actions. The wide range in caseload projections for the 2014 Ebola outbreak caused great concern and debate about the utility of models. By coding and running 37 published Ebola models with five candidate interventions, we found that, despite this large variation in caseload projection, the ranking of management options was relatively consistent. Reducing funeral transmission and reducing community transmission were generally ranked as the two best options. Value of information (VoI) analyses show that caseloads could be reduced by 11% by resolving all model-specific uncertainties, with information about model structure accounting for 82% of this reduction and uncertainty about caseload only accounting for 12%. Our study shows that the uncertainty that is of most interest epidemiologically may not be the same as the uncertainty that is most relevant for management. If the goal is to improve management outcomes, then the focus of study should be to identify and resolve those uncertainties that most hinder the choice of an optimal intervention. Our study further shows that simplifying multiple alternative models into a smaller number of relevant groups (here, with shared structure) could streamline the decision-making process and may allow for a better integration of epidemiological modeling and decision making for policy., Competing Interests: The authors declare no conflict of interest.

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.

Objectives: To determine the overall use of extracorporeal membranous oxygenation for influenza-associated illness and describe risk factors associated with mortality in these patients., Design: Retrospective multicenter cohort analysis., Setting: The international Extracorporeal Life Support Organization database was queried for patients with influenza-associated illness on extracorporeal membranous oxygenation from 1992 to 2014., Patients: In total, 1,654 patients with influenza-associated illness on extracorporeal membranous oxygenation., Interventions: None., Measurements and Main Results: Demographic and clinical data collected included age, type of support, duration of support, type of microbial codetection, complications, and survival status at discharge. The primary outcome of interest was survival to hospital discharge. From 1992 to 2014, 1,688 (3%) of the 61,336 extracorporeal membranous oxygenation runs were due to influenza-associated illness reflecting 1,654 unique patients: 30 (2%) were neonates, 521 (31%) were pediatric patients, and 1,103 (67%) were adults. Extracorporeal membranous oxygenation use for influenza-associated illness increased from 1992 to 2014, with a marked increase in use after the 2009 H1N1 pandemic. Survival to hospital discharge of patients with influenza-associated illness on extracorporeal membranous oxygenation was 63% and was not affected by bacterial codetection. However, when patients with Staphylococcus aureus codetection were compared with those with another bacterial codetection, their survival to hospital discharge was significantly lower (52% vs 67%; p < 0.01). In a logistic regression model, the effect of S. aureus on in-hospital mortality varied by age group, with younger patients with S. aureus having increased in-hospital mortality., Conclusions: Extracorporeal membranous oxygenation use for individuals with influenza increased over time, particularly after the 2009 H1N1 pandemic, most notably among older adults. Survival to hospital discharge for patients with influenza on extracorporeal membranous oxygenation was slightly higher than survival to hospital discharge for respiratory illness due to any cause. Bacterial codetection was common among patients with influenza on extracorporeal membranous oxygenation and was associated with increased days on extracorporeal membranous oxygenation but not increased mortality. Only S. aureus codetection in children was associated with increased in-hospital mortality.

Crown-of-thorns starfish (CoTS; Acanthaster spp.) are an outbreaking pest among many Indo-Pacific coral reefs that cause substantial ecological and economic damage. Despite ongoing CoTS research, there remain critical gaps in observing CoTS populations and accurately estimating their numbers, greatly limiting understanding of the causes and sources of CoTS outbreaks. Here we address two of these gaps by (1) estimating the detectability of adult CoTS on typical underwater visual count (UVC) surveys using covariates and (2) inter-calibrating multiple data sources to estimate CoTS densities within the Cairns sector of the Great Barrier Reef (GBR). We find that, on average, CoTS detectability is high at 0.82 [0.77, 0.87] (median highest posterior density (HPD) and [95% uncertainty intervals]), with CoTS disc width having the greatest influence on detection. Integrating this information with coincident surveys from alternative sampling programs, we estimate CoTS densities in the Cairns sector of the GBR averaged 44 [41, 48] adults per hectare in 2014., Competing Interests: Steven Moon is an employee of Association of Marine Park Tourism Operators, Cairns, Australia.

Formal decision-analytic methods can be used to frame disease control problems, the first step of which is to define a clear and specific objective. We demonstrate the imperative of framing clearly-defined management objectives in finding optimal control actions for control of disease outbreaks. We illustrate an analysis that can be applied rapidly at the start of an outbreak when there are multiple stakeholders involved with potentially multiple objectives, and when there are also multiple disease models upon which to compare control actions. The output of our analysis frames subsequent discourse between policy-makers, modellers and other stakeholders, by highlighting areas of discord among different management objectives and also among different models used in the analysis. We illustrate this approach in the context of a hypothetical foot-and-mouth disease (FMD) outbreak in Cumbria, UK using outputs from five rigorously-studied simulation models of FMD spread. We present both relative rankings and relative performance of controls within each model and across a range of objectives. Results illustrate how control actions change across both the base metric used to measure management success and across the statistic used to rank control actions according to said metric. This work represents a first step towards reconciling the extensive modelling work on disease control problems with frameworks for structured decision making., (Copyright © 2015 The Authors. Published by Elsevier B.V. All rights reserved.)

Optimal intervention for disease outbreaks is often impeded by severe scientific uncertainty. Adaptive management (AM), long-used in natural resource management, is a structured decision-making approach to solving dynamic problems that accounts for the value of resolving uncertainty via real-time evaluation of alternative models. We propose an AM approach to design and evaluate intervention strategies in epidemiology, using real-time surveillance to resolve model uncertainty as management proceeds, with foot-and-mouth disease (FMD) culling and measles vaccination as case studies. We use simulations of alternative intervention strategies under competing models to quantify the effect of model uncertainty on decision making, in terms of the value of information, and quantify the benefit of adaptive versus static intervention strategies. Culling decisions during the 2001 UK FMD outbreak were contentious due to uncertainty about the spatial scale of transmission. The expected benefit of resolving this uncertainty prior to a new outbreak on a UK-like landscape would be £45-£60 million relative to the strategy that minimizes livestock losses averaged over alternate transmission models. AM during the outbreak would be expected to recover up to £20.1 million of this expected benefit. AM would also recommend a more conservative initial approach (culling of infected premises and dangerous contact farms) than would a fixed strategy (which would additionally require culling of contiguous premises). For optimal targeting of measles vaccination, based on an outbreak in Malawi in 2010, AM allows better distribution of resources across the affected region; its utility depends on uncertainty about both the at-risk population and logistical capacity. When daily vaccination rates are highly constrained, the optimal initial strategy is to conduct a small, quick campaign; a reduction in expected burden of approximately 10,000 cases could result if campaign targets can be updated on the basis of the true susceptible population. Formal incorporation of a policy to update future management actions in response to information gained in the course of an outbreak can change the optimal initial response and result in significant cost savings. AM provides a framework for using multiple models to facilitate public-health decision making and an objective basis for updating management actions in response to improved scientific understanding., Competing Interests: The authors have declared that no competing interests exist.

Background: The United States has the highest rate of lumbar spine surgery in the world, with rates increasing over 200% since 1990. Medicare spends over $1 billion annually on lumbar spine surgery. Despite surgical advances, up to 40% of patients report chronic pain and disability following surgery. Our work has demonstrated that fear of movement is a risk factor for increased pain and disability and decreased physical function in patients following lumbar spine surgery for degenerative conditions. Cognitive-behavioral therapy and self-management treatments have the potential to address psychosocial risk factors and improve outcomes after spine surgery, but are unavailable or insufficiently adapted for postoperative care. Our research team developed a cognitive-behavioral based self-management approach to postoperative rehabilitation (Changing Behavior through Physical Therapy (CBPT)). Pilot testing of the CBPT program demonstrated greater improvement in pain, disability, physical and mental health, and physical performance compared to education. The current study compares which of two treatments provided by telephone - a CBPT Program or an Education Program about postoperative recovery - are more effective for improving patient-centered outcomes in adults following lumbar spine surgery for degenerative conditions., Methods/design: A multi-center, comparative effectiveness trial will be conducted. Two hundred and sixty patients undergoing lumbar spine surgery for degenerative conditions will be recruited from two medical centers and community surgical practices. Participants will be randomly assigned to CBPT or Education at 6 weeks following surgery. Treatments consist of six weekly telephone sessions with a trained physical therapist. The primary outcome will be disability and secondary outcomes include pain, general health, and physical activity. Outcomes will be assessed preoperatively and at 6 weeks, 6 months and 12 months after surgery by an assessor masked to group allocation., Discussion: Effective rehabilitation treatments that can guide clinicians in their recommendations, and patients in their actions will have the potential to effect change in current clinical practice., Trial Registration: NCT02184143.

Environmental regulations can only be effective if they are adhered to, but the motivations for regulatory compliance are not always clear. We assessed vessel operator compliance with a December 2008 regulation aimed at reducing collisions with the endangered North Atlantic right whale that requires vessels 65 feet or greater in length to travel at speeds of 10 knots or less at prescribed times and locations along the U.S. eastern seaboard. Extensive outreach efforts were undertaken to notify affected entities both before and after the regulation went into effect. Vessel speeds of 201,862 trips made between November 2008 and August 2013 by 8,009 individual vessels were quantified remotely, constituting a nearly complete census of transits made by the regulated population. Of these, 437 vessels (or their parent companies), some of whom had been observed exceeding the speed limit, were contacted through one of four non-punitive information programs. A fraction (n = 26 vessels/companies) received citations and fines. Despite the efforts to inform mariners, initial compliance was low (<5% of the trips were completely <10 knots) but improved in the latter part of the study. Each notification/enforcement program improved compliance to some degree and some may have influenced compliance across the entire regulated community. Citations/fines appeared to have the greatest influence on improving compliance in notified vessels/companies, followed in order of effectiveness by enforcement-office information letters, monthly summaries of vessel operations, and direct at-sea radio contact. Trips by cargo vessels exhibited the greatest change in behavior followed by tanker and passenger vessels. These results have application to other regulatory systems, especially where remote monitoring is feasible, and any setting where regulatory compliance is sought.

The explosion of the Deepwater Horizon drilling platform created the largest marine oil spill in U.S. history. As part of the Natural Resource Damage Assessment process, we applied an innovative modeling approach to obtain upper estimates for occupancy and for number of manatees in areas potentially affected by the oil spill. Our data consisted of aerial survey counts in waters of the Florida Panhandle, Alabama and Mississippi. Our method, which uses a Bayesian approach, allows for the propagation of uncertainty associated with estimates from empirical data and from the published literature. We illustrate that it is possible to derive estimates of occupancy rate and upper estimates of the number of manatees present at the time of sampling, even when no manatees were observed in our sampled plots during surveys. We estimated that fewer than 2.4% of potentially affected manatee habitat in our Florida study area may have been occupied by manatees. The upper estimate for the number of manatees present in potentially impacted areas (within our study area) was estimated with our model to be 74 (95%CI 46 to 107). This upper estimate for the number of manatees was conditioned on the upper 95%CI value of the occupancy rate. In other words, based on our estimates, it is highly probable that there were 107 or fewer manatees in our study area during the time of our surveys. Because our analyses apply to habitats considered likely manatee habitats, our inference is restricted to these sites and to the time frame of our surveys. Given that manatees may be hard to see during aerial surveys, it was important to account for imperfect detection. The approach that we described can be useful for determining the best allocation of resources for monitoring and conservation.

Though the control of blood phenylalanine (Phe) levels is essential for minimizing impairment in individuals with phenylketonuria (PKU), the empirical basis for the selection of specific blood Phe levels as targets has not been evaluated. We evaluated the current evidence that particular Phe levels are optimal for minimizing or avoiding cognitive impairment in individuals with PKU. This work uses meta-estimates of blood Phe-IQ correlation to predict the probability of low IQ for a range of Phe levels. We believe this metric is easily interpretable by clinicians, and hence useful in making recommendations for Phe intake. The median baseline association of Phe with IQ was estimated to be negative, both in the context of historical (median = -0.026, 95 % BCI = [-0.040, -0.013]) and concurrent (-0.007, [-0.014, 0.000]) measurement of Phe relative to IQ. The estimated additive fixed effect of critical period Phe measurement was also nominally negative for historical measurement (-0.010, [-0.022, 0.003]) and positive for concurrent measurement (0.007, [-0.018, 0.035]). Probabilities corresponding to historical measures of blood Phe demonstrated an increasing chance of low IQ with increasing Phe, with a stronger association seen between blood Phe measured during the critical period than later. In contrast, concurrently-measured Phe was more weakly correlated with the probability of low IQ, though the correlation is still positive, irrespective of whether Phe was measured during the critical or non-critical period. This meta-analysis illustrates the utility of a Bayesian hierarchical approach for not only combining information from a set of candidate studies, but also for combining different types of data to estimate parameters of interest.

Purpose: Progestogen has been investigated as a preventive intervention among women with increased preterm birth risk. Our objective was to systematically review the effectiveness of intramuscular (IM), vaginal, and oral progestogens for preterm birth and neonatal death prevention., Methods: We included articles published from January 1966 to January 2013 and found 27 randomized trials with data for Bayesian meta-analysis., Results: Across all studies, only vaginal and oral routes were effective at reducing preterm births (IM risk ratio [RR] 0.95, 95 % Bayesian credible interval [BCI]: 0.88-1.03; vaginal RR 0.87, 95 % BCI: 0.80-0.94; oral RR 0.64, 95 % BCI: 0.49-0.85). However, when analyses were limited to only single births all routes were effective at reducing preterm birth (IM RR 0.77, 95 % BCI: 0.69-0.87; vaginal RR 0.80, 95 % BCI: 0.69-0.91; oral RR 0.66, 95 % BCI: 0.47-0.84). Only IM progestogen was effective at reducing neonatal deaths (IM RR 0.78, 95 % BCI: 0.56-0.99; vaginal RR 0.75, 95 % BCI: 0.45-1.09; oral RR 0.72, 95 % BCI: 0.09-1.74). Vaginal progestogen was effective in reducing neonatal deaths when limited to singletons births., Conclusions: All progestogen routes reduce preterm births but not neonatal deaths. Future studies are needed that directly compare progestogen delivery routes.

This user guide describes a Python package, PyMC, that allows users to efficiently code a probabilistic model and draw samples from its posterior distribution using Markov chain Monte Carlo techniques.

The recent development of statistical models such as dynamic site occupancy models provides the opportunity to address fairly complex management and conservation problems with relatively simple models. However, surprisingly few empirical studies have simultaneously modeled habitat suitability and occupancy status of organisms over large landscapes for management purposes. Joint modeling of these components is particularly important in the context of management of wild populations, as it provides a more coherent framework to investigate the population dynamics of organisms in space and time for the application of management decision tools. We applied such an approach to the study of water hole use by African elephants in Hwange National Park, Zimbabwe. Here we show how such methodology may be implemented and derive estimates of annual transition probabilities among three dry-season states for water holes: (1) unsuitable state (dry water holes with no elephants); (2) suitable state (water hole with water) with low abundance of elephants; and (3) suitable state with high abundance of elephants. We found that annual rainfall and the number of neighboring water holes influenced the transition probabilities among these three states. Because of an increase in elephant densities in the park during the study period, we also found that transition probabilities from low abundance to high abundance states increased over time. The application of the joint habitat-occupancy models provides a coherent framework to examine how habitat suitability and factors that affect habitat suitability influence the distribution and abundance of organisms. We discuss how these simple models can further be used to apply structured decision-making tools in order to derive decisions that are optimal relative to specified management objectives. The modeling framework presented in this paper should be applicable to a wide range of existing data sets and should help to address important ecological, conservation, and management problems that deal with occupancy, relative abundance, and habitat suitability.

Many organisms are patchily distributed, with some patches occupied at high density, others at lower densities, and others not occupied. Estimation of overall abundance can be difficult and is inefficient via intensive approaches such as capture-mark-recapture (CMR) or distance sampling. We propose a two-phase sampling scheme and model in a Bayesian framework to estimate abundance for patchily distributed populations. In the first phase, occupancy is estimated by binomial detection samples taken on all selected sites, where selection may be of all sites available, or a random sample of sites. Detection can be by visual surveys, detection of sign, physical captures, or other approach. At the second phase, if a detection threshold is achieved, CMR or other intensive sampling is conducted via standard procedures (grids or webs) to estimate abundance. Detection and CMR data are then used in a joint likelihood to model probability of detection in the occupancy sample via an abundance-detection model. CMR modeling is used to estimate abundance for the abundance-detection relationship, which in turn is used to predict abundance at the remaining sites, where only detection data are collected. We present a full Bayesian modeling treatment of this problem, in which posterior inference on abundance and other parameters (detection, capture probability) is obtained under a variety of assumptions about spatial and individual sources of heterogeneity. We apply the approach to abundance estimation for two species of voles (Microtus spp.) in Montana, USA. We also use a simulation study to evaluate the frequentist properties of our procedure given known patterns in abundance and detection among sites as well as design criteria. For most population characteristics and designs considered, bias and mean-square error (MSE) were low, and coverage of true parameter values by Bayesian credibility intervals was near nominal. Our two-phase, adaptive approach allows efficient estimation of abundance of rare and patchily distributed species and is particularly appropriate when sampling in all patches is impossible, but a global estimate of abundance is required.