Matching decision support modeling frameworks to disease emergence stages and associated management objectives

Wildlife disease management decisions often require rapid responses to situations that are fraught with uncertainty. By recognizing that management is implemented to achieve specific objectives, resource managers and science partners can identify an analysis technique and develop a monitoring plan to evaluate management effectiveness. For emerging infectious diseases, objectives may take several distinct forms, dependent on the perceived stage of disease emergence (i.e., pre‐epidemic, early outbreak, mid‐epidemic, and endemic), the expected rate of spread, and the anticipated effect of the disease on host populations. Identifying modeling techniques and metrics that are linked to management objectives will require early and consistent communication between managers and science partners. We link modeling approaches that can be used to forecast and evaluate the performance of intervention strategies with a range of disease management objectives. Our aim is to help scientists recognize alternative modeling approaches which may better align with different forms of disease management objectives, and to help managers evaluate the relevance of proposed modeling approaches to their specified objectives for disease management. Recognizing that disease management objectives can take different forms, and thus require different modeling approaches, can help wildlife disease response teams (i.e., natural resource managers, scientists, and stakeholders working collaboratively) better prepare and respond to disease threats.


| INTRODUCTION
Emerging infectious diseases are an increasingly common challenge faced by wildlife managers (Decker et al., 2006).Identifying an effective response strategy requires identification and evaluation of the performance of different management interventions.Effectiveness is measured as the performance of an intervention on a fundamental objective, which represents the management goals of a decision-maker.Goals for mitigating disease risk in host species may include maximizing species distributions, abundances, or demographic parameters such as reproduction or survival (Converse et al., 2013;Gregory et al., 2013).Identifying specific, fundamental management objectives a priori, and expressing those objectives in terms of quantifiable metrics, serves several important purposes in a decision-making process.First, it clarifies the goals of a disease management program and specifies measurable outcomes useful for assessing whether objectives can be sufficiently met.Second, it determines the analysis approach for forecasting the effectiveness of a range of possible management actions.Finally, it can improve the design of research and surveillance programs.
Wildlife disease management may be reactive when a disease threat goes undetected (Bernard & Grant, 2019;Bloom et al., 2017;Grider et al., 2022;Schwab et al., 2018).Thus, specifying objectives may not occur prior to developing a surveillance plan or conducting research, as these activities are perceived to be of principal importance to developing predictive models of disease dynamics.However, when surveillance data and research results are not explicitly linked to well-reasoned objectives, they may fail to inform estimates of quantities of fundamental interest for managers and may not translate into improved management decisions (Thompson et al., 1998;Yoccoz et al., 2001).This disconnect can impede the identification and implementation of effective management strategies, particularly during the early stages of an epizootic when management is most useful (Langwig et al., 2015).
Given the complexity of wildlife disease systems, quantitative models are necessary for exploring data, elucidating mechanisms, and projecting population and disease status into the future (Tredennick et al., 2021).For instance, determining which factors are most influential to a disease process can be helpful for determining where and when management may be most effective, while model projections can be useful for evaluating relative performance among potential management strategies.To harness the full utility for disease decisions, models must be selected to use state variables tied to fundamental management objectives.Many of the models we describe here can be used for both inference and prediction, and in both applications, careful consideration of objectives is necessary for determining model implementation.
Our work with resource managers across the United States, supporting decisions for managing the threat of disease introduction (Box 1) has provided insights into objectives, data, and model needs (e.g., Bernard & Grant, 2019, 2021;Grant et al., 2017).Here we describe common wildlife disease management objectives and identify appropriate analytical approaches for each objective.Our aim is to provide an overview of objective-informed approaches for developing models to assess the current state of the host-pathogen system, determine which management strategies are most likely to achieve stated objectives, and provide guidance (via model predictions) to decision makers to support management decisions.This is expected to improve communication between researchers and managers about the different ways in which disease management objectives may be specified, and the associated types of modeling approaches which may be useful for supporting management decisions.

| OBJECTIVES AND THEIR MEASURABLE ATTRIBUTES
The identification of clear objectives is a recognized challenge in conservation biology (Game et al., 2013;Riley & Gregory, 2012;Tear et al., 2005).By specifying what management seeks to achieve, mitigation options can be evaluated to determine whether objectives are expected to be met provide guidance regarding how to evaluate success, and provide direction on what model(s) may be most useful.Structured decision-making (e.g., Conroy & Peterson, 2013) provides a useful approach for identifying fundamentally important objectives.Objectives can be identified through an agency's mission statement, laws or regulations, stakeholder concerns, or guided by understanding ecological thresholds (Martin et al., 2009).Once objectives are specified, appropriate models to inform management decisions may be identified.

| FREQUENTLY IDENTIFIED DISEASE MANAGEMENT OBJECTIVES
Common goals for disease management may be specific to the stage of the epidemic (Langwig et al., 2015), and include minimize invasion risk (pre-epidemic), minimize spatial spread of disease (early outbreak), minimize prevalence in affected populations (mid-epidemic), and reduction of harm (endemic).For example, early in an epidemic managers may focus on preventing pathogen introduction to a focal area or population (i.e., reserved hunting areas, populations of conservation concern; Figure 1a-'Minimize invasion risk').Once a disease arrives in a particular geographic area, the focus may shift to minimizing spread of the pathogen to other populations (Figure 1b-'Minimize spatial spread'; e.g., Bernard & Grant, 2019;Hefley et al., 2017), and minimizing infection prevalence within the focal area (Figure 1c-'Minimize local prevalence'; e.g., Cross et al., 2007).Finally, if the pathogen becomes established and is widespread through a focal area, controlling the impacts of outbreaks by maximizing survival of infected hosts may be the desired objective (Figure 1d-'Minimize effect on hosts'; e.g., Hohenlohe et al., 2019).
Once the objective(s) have been identified for a particular decision problem, models that reflect the important parameters can be selected (Table 1).Models are useful for elucidating uncertainty, quantifying assumptions, estimating parameters from field data, understanding the effects of assumptions on results (i.e., via sensitivity analyses), and creating transparency for the decision making process (Fuller et al., 2020;Miller & Pepin, 2019).Models are essential for estimating effects of mitigation strategies and evaluating trade-offs among different strategies (Gregory et al., 2013;Williams & Johnson, 2015).Models used for forecasting can be parameterized with surveillance or survey data (if available), previous research in a closelyrelated system, or via a process of expert elicitation (e.g., DiRenzo & Grant, 2019;O'Hagan et al., 2006;Russell et al., 2015).The common disease management objectives seek to minimize: invasion, pathogen spread, number of hosts infected, and effects of disease.These are formalized as the state variables in a variety of model types (Table 1).

| Overview of modeling approaches
Models estimate 'states' (or 'compartments') that represent the condition of a population, habitat patch, or individual hosts.These states can represent the presence of the pathogen at a site (occupied/unoccupied) or in an individual (infected/uninfected).Host dynamics may be considered nested within patch dynamics, where the presence of the pathogen on a host is conditional on the presence of the pathogen within a population or habitat patch (McClintock et al., 2010).In general, transitions

BOX 1 CASE STUDY: DEMONSTRATING HOW TO DEVELOP OBJECTIVES FOR DISEASE MANAGEMENT
The emerging fungal pathogen Batrachochytrium salamandrivorans (Bsal) causes chytridiomycosis in amphibians (Martel et al., 2013).The pathogen is native to Asia (Martel et al., 2014) and has spread to Europe where it has led to significant declines in many fire salamander populations (Spitzen-van der Sluijs et al., 2013).Though the pathogen has not yet been detected in North America (Waddle et al., 2020), lab trials suggest numerous North American amphibian species are susceptible to Bsal infection (Carter et al., 2020;Martel et al., 2014), making Bsal a major concern to North American land managers.
We are working with natural resource managers in National Wildlife Refuges in the northeastern U.S. (USFWS Region 5) using a structured decision-making workshop approach to develop a rapid prototype of their Bsal-amphibian management decision problems, including describing challenges and trade-offs and identifying the objectives (Grant et al., 2017).The decision these protected lands managers are facing is what actions can be taken to minimize the impacts of Bsal on amphibian populations within a refuge.During the process of identifying management objectives, we identified four disease-focused objectives shared across multiple management entities: 1. Maximize Bsal-free space across the refuge 2. Minimize annual prevalence (numbers of infected sites) of Bsal across the refuge 3. Maximize abundance of native amphibian species populations on the refuge 4. Maximize occupancy of native amphibian species within the refuge While these objectives are worded differently than the four expressed in Figure 1 and Table 1, they are congruent with the temporal and spatial scales of the general objectives we identify.These objectives were identified during a structured decision-making workshop; similar objectives were expressed when managers across the United States were asked about their concerns and goals for Bsal management (Bernard & Grant, 2019).
We note that these objectives may be considered to be hierarchically structured, creating a decision framework that can be adapted as the system state changes.As an example, ahead of an invasion a manager may want to maintain Bsalfree space, therefore putting greater emphasis on actions that may best achieve this objective.However, if actions to restrict the colonization of Bsal fail and the pathogen becomes established, objectives related to control (e.g., minimize prevalence, minimize impacts on host abundance) may be determined to have a greater likelihood of achievement given the available actions.
among states are modeled in a similar fashion across scales (i.e., a transition from unoccupied to occupied, or susceptible to infected, is analogous to a colonization [patch level] or infection [host level] process).The probabilities of transitioning from one state to another can have complex functional forms, with the probability of pathogen colonization of a patch dependent on patch size or distance to the nearest occupied patch (e.g., Gerber et al., 2018), or the probability of host infection dependent on the size of the infected population in the patch (e.g., DiRenzo et al., 2019).At the host level, probabilities of survival, movement, or reproduction may be dependent on the infection state or infection intensity.Selecting the 'best' approach is identified ideally through a structured decision-making process (Box 1), where managers and scientists jointly identify modeling approaches that relate to management objectives (e.g., Conroy & Peterson, 2013).In the following sections, we suggest modeling approaches and metrics that are relevant for each objective (and disease invasion state), but note that each modeling framework may apply in multiple contexts.

| Minimize invasion risk to a focal site (pre-epidemic)
Preventing establishment is important for slowing pathogen spread, and minimizing the invasion risk is a common objective in the pre-epidemic stages (Figure 1a).This objective focuses on disease prevention and may be most easily achieved for populations that are isolated, or can be isolated, from pathogen sources.For far-ranging wildlife species this objective may be costly to achieve.
Occupancy modeling can be a useful tool for determining factors related to pathogen spread (MacKenzie et al., 2002(MacKenzie et al., , 2006)).By including covariates, such as temperature or site connectivity, inferences can be made regarding the expected pathogen's behavior in the future.By identifying factors associated with pathogen colonization and local extinction, the potential distribution of the pathogen can be projected to identify areas at high risk of invasion.Additionally, by determining covariates that lead to pathogen colonization, management strategies can be developed that target these factors.Finally, these models can be used to estimate the effectiveness of disease mitigation strategies by comparing occupancy rates at sites under different types of management.
Models of site-occupancy rely on detection/ non-detection data for the pathogen and host at sites over time.Probabilities of detecting a pathogen when it is present can be less than one due to imperfect lab tests, field sampling errors, environmental conditions, host abundance, or pathogen loads.Therefore, the analysis of these types of data benefit from the incorporation of detection probability in a model (e.g., Miller et al., 2012;Mosher et al., 2019).With multiple visits to a site, a 'multiseason' occupancy model (MacKenzie et al., 2003) can estimate the probability a site is occupied, the rate at which sites transition from occupied to unoccupied ('local extinction'), and the rate at which sites transition from unoccupied to occupied ('colonization').These models can be extended to more than two states (e.g., pathogen not present, pathogen present and no clinical disease, and pathogen present and clinical disease) in a multi-state framework (Elmore et al., 2014;Nichols et al., 2007), or to a two-species model for hosts and pathogens (Mosher et al., 2018; Richmond F I G U R E 1 A representation of the system state of infection (gray = host is not infected, red = host is infected) across local populations within a broader geographic extent during stages of an invading epidemic.Four common objectives are specified and are state-dependent.States are identified as: (a) pre-epidemic is when no hosts are infected; (b) early outbreak, occurs when a low number of hosts are infected after the pathogen first invades a population or region; (c) mid-epidemic occurs when the pathogen spreads between hosts within populations in a region; and (d) endemic, or enzootic, infection occurs when the pathogen transmits among hosts and populations and is endemic to a population.et al., 2010); the basic sampling design and parameters remain the same.By explicitly linking the objective of reducing the risk of pathogen invasion to models that estimate environmental-or individual-level factors, managers can identify potential strategies to reduce introduction and establishment risk and data needs from surveillance programs.

| Minimize the spatial spread of the pathogen
This objective is an extension of the previous objective (e.g., preventing pathogens from invading new sites) from a focal site to a larger geographic region (Figure 1b).This objective is often adopted once a disease has been established in a core geographic area, and may also be appropriate for wildlife populations that live in social groups or discrete patches.The explicit incorporation of space (i.e., distance among populations and landscape connectivity) into models allows for the identification of factors that promote disease spread, and investigation of how management may slow the spread of disease.These models can be used to investigate how the prevention of movement between populations or the spatial distribution of treatments may slow the spread of disease into new geographic areas (Asano et al., 2008;Grider et al., 2022;Sanchez & Hudgens, 2020, numerous others).Model approaches, described in more detail below, include (1) occupancy models accounting for spatial structure of populations, (2) metapopulation models, and (3) ecological diffusion models.
T A B L E 1 Questions about data and model needs for disease management objectives can be linked to parameters of interest and suggested data requirements ('What to monitor?') and associated measurable attribute of the objective ('Metric of success'), possible modeling approaches ('How to estimate and forecast?'),and modeled state variables ('What to estimate from models?').The basic multi-season site-occupancy models introduced above assume random mixing among hosts or sites and do not account for the spatial structure of populations.However, spatial structure is important for pathogens that are spread by movements of host animals, and can be incorporated into occupancy models with additional parameters or functional relationships with the local colonization or extinction rates (e.g., Converse et al., 2017;Gerber et al., 2018).Traditional metapopulation models generally incorporate spatial dynamics implicitly.However, these models are often less data-driven and more theoretical than the occupancy models we describe here.The basic metapopulation models are formulated as a set of differential equations and describe the colonization and extinction of populations connected through immigration and emigration of (infected and uninfected) hosts (Hanski, 1994).The equations that govern the relative rates of colonization and extinction among patches can take a variety of functional forms; for example, colonization can be modeled as a function of distance (Hanski & Thomas, 1994) or connectivity (Sutherland et al., 2014), allowing for spatially-explicit dynamics (Hanski, 1994).

State-dependent objectives
While multi-population (e.g., metapopulation) occupancy models may be useful for host species living in discrete populations, diffusion models may be more appropriate for populations distributed continuously across the landscape (Oh et al., 2023;Hefley et al., 2017).For the analyses of landscapes in a continuous-time framework (vs. a patch occupancy framework in discrete time), ecological diffusion models have also been used to estimate the spread of pathogens at large spatial scales (e.g., Hefley et al., 2017;Wu, 2008).These models can use detection/non-detection data from surveillance programs to estimate the effects of covariates on pathogen growth and spread, and have been used to forecast disease occurrence dynamics, and to develop targeted surveillance programs (e.g., by identifying the leading edge of pathogen invasion; Hanley et al., 2022).

| Minimize number of infected hosts
A common objective once a pathogen has been introduced into a population is to minimize prevalence (Figure 1c).Models that track the disease status of individuals can be particularly useful once disease has become endemic.Occupancy models may be used where the individual is a "site."Alternatively, compartmental models can track the disease status of individuals through time.These models can be used to investigate disease transmission dynamics (Russell et al., 2019), or forecast the effects of mitigation strategies on host populations (Grider et al., 2022).Understanding the mechanisms that lead to reductions in disease prevalence can help identify potential mitigation strategies for an endemic disease.
Multi-scale occupancy models that take into account the hierarchical structure of pathogen presence can be used to obtain estimates of the probability of host infection (Mosher et al., 2019;Nichols et al., 2008).This framework can accommodate multiple years of data in a dynamic model of disease in which infection of individuals (i.e., colonization) or clearing of an infection (i.e., extinction) may occur between years (Green et al., 2019).However, treating infection as multiple occupancy states may not be sufficient for inference in more complex disease systems.
Compartmental disease models, such as SIR (susceptible, infected, and resistant/recovered), SEI (susceptible, exposed, and infected), and other forms of compartmental models accommodate more detail in describing disease dynamics.Canessa et al. (2019), for example, develop a management-focused compartmental SEI model for Batrachochytrium salamandrivorans (Bsal), an emerging fungal pathogen of amphibians, which reflected current knowledge of Bsal transmission dynamics (Stegen et al., 2017).Finally, compartmental models can be made spatially explicit to account for the spatial distribution of hosts.This formulation captures more realistic host-pathogen dynamics, especially for animals that live in groups (e.g., Haydon et al., 2006), and similar to the spatially-explicit occupancy models described above, allows for the evaluation of landscape-level management actions (such as vaccine 'firewalls' or movement barriers to prevent disease spread; Berg et al., 2018, Bakker et al., 2019).

| Minimize the effect of the pathogen on hosts
Once a pathogen is established, managers may want to identify actions to minimize negative impacts on a population (Figure 1d).Estimates of individual survival rates or abundance of hosts in populations where disease is endemic can be useful metrics for evaluating the effect of a disease, or the result of a mitigation action.Models which include these parameters can be particularly useful when the objective for disease management has shifted from prevention to mitigation.Evaluating the effects of disease on demographic processes (e.g., reproduction, adult and juvenile mortality, and movement) may help identify effective management strategies.For example, Rocke et al. (2017) used robust-design mark recapture models (Kendall et al., 1995) to evaluate the effects of vaccination on survival of adult and juvenile prairie dogs in colonies with and without plague.Such models can also be used to determine what environmental factors or individual host traits contribute to the survival of infect hosts, which can also improve our ability to evaluate disease management options.
Mark-recapture models are useful for estimating population demographic rates.These models also can accommodate imperfect detection when fit to field data.Multi-state versions of these models can incorporate discrete host states (such as susceptible, infected, exposed, and recovered), and estimate state-transition probabilities (e.g., of surviving while infected, surviving and remaining uninfected, and surviving and clearing infection).Model extensions accommodate cases where clinical signs of host infection or molecular detection (e.g., PCR) results are ambiguous (Conn & Cooch, 2009;Kendall et al., 2012).Additional model extensions may accommodate individuals that are not captured and therefore not observed.Similar to other types of models, transitions between states and survival probabilities can be modeled as a function of covariates (e.g., Muths et al., 2020).The markrecapture model framework has been expanded to incorporate space and estimate density directly (Efford et al., 2008;Royle et al., 2010).
Other types of models that can estimate survival and abundance include time-to-event, space-to-event, or Nmixture models.Time-to-event models be used to estimate survival rates given known fates such as from a laboratory study (Russell et al., 2019) or animals that have been closely monitored through, for example, GPS collars (Margenau et al., 2023).Time-to-event models and space-to-event models have also been used to estimate abundance (Loonam et al., 2020;Moeller et al., 2018;Moeller & Lukacs, 2022) from cameras and unmarked animals.Finally, N-mixture models can be used with unmarked animal data from repeated surveys (Borschers & Efford, 2008;Efford, 2004;Royle et al., 2013).These models are most useful for quantifying effects of management strategies and evaluating population-level impacts of disease.Survival models allow managers to determine whether characteristics of individuals (such as life stage, body size, or sex) are influencing an individual's ability to survive the effects of the pathogen.
These models can be used to forecast the effect of a management action, without fully capturing the mechanisms behind population declines (but see DiRenzo et al., 2019).More detailed matrix population models are useful for forecasting the effect of a disease on demographic rates (e.g., Thogmartin et al., 2013), and can be useful for evaluating possible management decisions (e.g., Wasserberg et al., 2009).

| CONCLUSIONS
Developing a management strategy in the face of an emerging disease is challenging; available data are usually sparse, and uncertainty regarding the disease processes can be high (Bernard & Grant, 2019;Langwig et al., 2015).Expressing ideas regarding how a system works with natural resource managers in a quantitative fashion is a useful approach when faced with decisions involving uncertainty and which require a rapid response.Despite a need to estimate and forecast disease status and species responses, the use of quantitative models in wildlife disease management applications remains challenging (Huyvaert et al., 2018).Quantitative models are useful for evaluating interventions to meet specific disease management goals, and can assess whether objectives are likely to be met.The specific management objectives should drive the type of modeling framework selected (Merkle et al., 2019;Mosher et al., 2020).
Importantly, there are assumptions and caveats associated with each model we have described, but meeting these should not be a barrier to their use in decision making.Additionally, we make a distinction between models that are used for estimating parameters, and those used for forecasting future population status and the expected effect of management actions on the objectives.This underscores the importance of identifying the specific management objectives and matching both the estimation and the forecast model to the state variables of management interest (Box 1; Martin et al., 2009).
Objectives may be differentially important at a given stage of disease emergence and establishment, and available interventions may be expected to have differential success in meeting management goals.By quantitatively describing the system, identifying data or data collection efforts to inform and improve model performance and inference, and explicitly evaluating the ability to achieve disease management objectives, collaborative response teams (i.e., natural resource managers, researchers, and stakeholders) can more effectively work together to confront disease threats.