Strategic planning of prevention and surveillance for emerging diseases and invasive species

Significance

As the global burden of emerging diseases and invasive species grows, efficient use of limited resources becomes increasingly critical. By the time the first case is detected, the disease or species may have been spreading unnoticed for an extended period. We developed a resource allocation model to minimize this unnoticed spread. The model allocates a given budget between prevention and surveillance across many geographical sites. It applies to areas where disease or species are likely to emerge but have not yet been found. We show that a stable allocation strategy is optimal in the long run. A case study on wildlife disease shows that the optimal strategy can achieve significantly earlier detection and cost savings.

Abstract

Emerging infectious diseases and biological invasions pose increasing threats to public and ecosystems health. Proactive measures—such as prevention and surveillance taken before initial detection of the pathogen or species—are essential to ensure minimal spread prior to first detection. We developed an optimization model to determine where, when, and how much effort should be allocated to prevention versus surveillance. The model accounts for imperfect detection, system dynamics, spatial heterogeneity in risk and costs and is scalable to large landscapes. We found that the most cost-effective strategy is to maintain the prevention and surveillance efforts at stable equilibrium for the majority of the time, with deviations occurring only initially to steer the system toward the equilibrium. The equilibrium effort is jointly determined by the introduction risk, management costs, and total budget. Application of this model to chronic wasting disease in New York State suggests that the optimal strategy could reduce the cumulative disease cases before initial detection by an average of 22% compared to current practice. The optimal surveillance strategy could detect the disease on average over 8 mo earlier than the current strategy.

Emerging infectious diseases pose a growing threat to public and ecosystem health worldwide (1). Climate change causes alterations in pathogens, vectors, the behavior of animal reservoirs and people, and interactions among these components, leading to more incidents of novel and endemic diseases (2, 3). Land-use change and expanding human population is also contributing to a greater number of emergent diseases as spatial overlap increases between wildlife and humans and agricultural animals (4). Similarly, invasive species are causing increasing ecological and economic damage globally (5).

Currently, the management of emerging infectious disease is often reactive, with resources allocated only after the disease or species is detected. By the time detection occurs, the pathogen may have spread extensively. There is growing recognition of benefits of proactive management (6, 7). For example, prevention and surveillance activities conducted before the disease is detected (the predetection phase) can reduce the risk of disease introduction and limit the spatial footprint and intensity of the pathogen before the time of first detection, increasing the success rate of postdetection control. This requires a strategic balance between prevention and surveillance efforts, which has previously been determined without formal theory to guide allocation of effort across these activities (8).

While prevention can lower risks associated with disease introduction and spread, it is unlikely to eliminate risks entirely (9, 10). Surveillance is necessary to ensure new introductions are detected early, during which control measures are more effective (11). However, surveillance is often resource-intensive due to low prevalence in the early stages of a disease invasion, requiring extensive sampling to achieve rapid detection, and is only a precursor to actual disease management (12). Prior to disease discovery, a holistic approach to disease control requires implementing both prevention and surveillance efforts. Effectiveness of such an approach may be enhanced by optimally allocating resources across these activities, allowing surveillance and prevention efforts to work in tandem. Thus, a key question is how to allocate limited resources between prevention and surveillance, as well as how to distribute them optimally across space and time. For example, practitioners face the dilemma of whether to concentrate efforts in a small region or distribute them more broadly. Focusing surveillance in a confined area enhances detection power but increases the risk of missing outbreaks elsewhere, whereas distributing resources across multiple sites provides broader coverage but may delay detection at any given location. Similarly, questions arise as to changing resource allocations over time and determining appropriate funding levels necessary to achieve management objectives.

To answer these important questions in a principled way, we develop a partially observable Markov decision process (POMDP) model. Although disease should ideally be managed based on complete information about its current status in a population (13), such perfect information is rarely attainable, as new cases are rarely detected immediately (14). POMDP uses the probabilistic estimate (belief) of the true disease status to guide decision making. It is a sequential approach that updates decisions based on new information, accounting for imperfect detection, operational costs, and system dynamics (15–18). While we focus on diseases, the model also applies to invasive species. For brevity, we refer to “disease” throughout.

Previous studies have explored the optimization of surveillance efforts (19, 20) and the trade-off between prevention and surveillance for managing invasive species up to three sites (21–25). However, practical disease management often spans multiple sites, and each site can fall into one of multiple prevalence levels. This leads to a POMDP with large state and action spaces, often exceeding the computational limits of existing approaches. Building on the concept of piecewise deterministic control (26)—where the belief evolves deterministically between consecutive detections and jumps upon detection—we develop a POMDP model to optimize prevention and surveillance across a large number of sites (Materials and Methods). The model determines how much effort should ideally be allocated to each activity at each site and at each time. It reformulates the POMDP as a deterministic optimal control problem, offering scalability to large instances. Importantly, it can yield explicit expression for the optimal strategy in certain cases, which is rarely available for POMDP, but see refs. 16, 27, and 28 for valuable attempts on small-size problems.

We now describe the decision problem and its objective (Materials and Methods). An organization responsible for disease management (hereafter, organization) oversees multiple jurisdictions (hereafter, sites) with host species susceptible to an emerging disease. We consider situations where the disease has not yet been detected in hosts at any site but may already be present. Each site may represent a large geographic unit, such as a county, and collectively, they form a large landscape, such as a U.S. state. Each site is identified as either disease-free or infected with a certain prevalence, and the disease status at each site can change over time due to, for example, the spread of the disease over long or short range host movement (Fig. 1A). The disease state at each site is modeled as a Markov chain. The rate of progression to higher prevalence depends on the prevention effort at the site as well as the disease status of other sites (Fig. 1B). The presence and prevalence of the disease at each site are not directly observable and must be inferred through surveillance. If the disease is present at a site, the detection probability (i.e., the probability of finding a positive case) depends on the underlying prevalence as well as the surveillance effort put forth by the organization. The organization receives a fixed budget for each period (e.g., annual) and must divide that budget between surveillance and prevention activities at each site.

Fig. 1.

(A) Illustration of the resource allocation problem. An infectious disease or invasive species can be introduced to and spread among populations across multiple spatial units (sites). Introduction rates, operation costs, and historical surveillance data are used to determine the optimal allocation of a fixed budget between prevention and surveillance activities across all sites; the allocation can change in response to new data. (B) Each site can be disease-free or infected with certain prevalence, which is not directly observable and must be inferred through surveillance. (C) The objective of the model is to minimize the expected cumulative disease cases up to the initial detection. This is equivalent to minimizing the area under the disease growth curve between the times of disease introduction and initial detection. Prevention delays introduction while surveillance accelerates detection.

When the first case is detected, the disease has likely been spreading unnoticed for some time. Our goal is to minimize this unnoticed spread. Specifically, the objective is to minimize the expected cumulative number of cases across all sites up to the time of initial detection (Fig. 1C). The first positive case often triggers a paradigm shift in management, leading to more funding, stricter regulations, and higher public awareness. We therefore assume that decisions before the initial detection can be decoupled from those made afterward. Note that all test results remain negative prior to the first detection, this allows us to recast the POMDP as deterministic optimal control.

We illustrate this model with a case study on an infectious disease in a free-ranging wildlife population. Wildlife face invasions of novel infectious diseases (29), a trend expected to worsen as suitable habitat dwindles (30). Natural resource agencies worldwide operate with limited budgets to ensure the sustainability of wild populations (31) and must allocate limited conservation funds between disease prevention and surveillance over large landscapes (32). In the United States, state wildlife agencies often receive allocations from a combination of license sales, state government, and federal distributions through the Pittman-Robertson Federal Aid in Wildlife Restoration Act of 1937 (33) and Federal Aid in Sport Fish Restoration Act of 1950 (Dingell-Johnson Sport Fish Restoration Act) (34), but these funds may be insufficient as agencies are forced to manage a dynamic landscape of disease outbreaks (35).

Chronic wasting disease (CWD) is thought to be one of the most serious threats facing wildlife populations (36, 37). It is a fatal condition spreading among cervid (family Cervidae) species across North America and Fennoscandia (38–40). As of 2025, CWD has not been confirmed in wild deer in New York State (NYS) (41), but is known to affect wild deer in neighboring states (35). We partner with the New York State Department of Environmental Conservation (NYSDEC) to plan disease management operations for CWD in wild white-tailed deer (Odocoileus virginianus). NYSDEC aims to prevent the introduction of CWD into wild deer population and ensure rapid detection if the disease is introduced. The existing CWD management program in NYS (42) prioritizes surveillance based on introduction risks and deer population density but does not explicitly account for variations in sampling costs between counties (SI Appendix, Table S1). Greater distances from the NYS Veterinary Diagnostic Laboratory in Ithaca or the Wildlife Health Unit in Delmar can increase the costs of transporting samples for testing. Given a total annual budget and the spatial variation of costs, we apply our model to optimize prevention and surveillance actions across all 62 counties in NYS in the long term, in order to minimize the total number of positive cases that go unnoticed before the initial detection.

Results

The Equilibrium.

The optimal strategy is to maintain efforts near a joint steady state for most of the planning horizon. This steady state is known as a turnpike equilibrium (hereafter equilibrium) in optimal control theory (43). Intuitively, this property can be understood by imagining a road trip: after the initial acceleration and merging onto the highway (the equilibrium), the driver maintains a steady speed for most of the journey, which is the most efficient way to travel. Similarly, in managing emerging disease, time-varying efforts are limited to the beginning of the planning horizon (Fig. 2A), where a series of adjustments are made to guide the system to the equilibrium. Once the system reaches this equilibrium, efforts are maintained at constant levels—just as a driver maintains a steady cruising speed—until the initial detection occurs.

Fig. 2.

(A) If all test results remain negative, the optimal allocation scheme will stabilize into an equilibrium (turnpike). An initial off-equilibrium phase appears only when the prior belief differs from the equilibrium belief. (B) Belief of the disease will stabilize into the equilibrium belief, if all tests remain negative. (C) Optimal allocation of prevention and surveillance in the equilibrium. Prevention is optimal only when its effectiveness exceeds a threshold, or when the introduction rate falls below a threshold (see SI Appendix, section 3 for explicit formula and parameter values).

The equilibrium arises from the balance between two opposing effects. Under the continuous risk of introduction, the probability that a site is disease-free declines over time (disease-spread effect). On the other hand, consecutive negative tests provide accumulating evidence that increases the disease-free probability (negative-test effect). Together, these effects can produce a stable belief about disease status (belief is the posterior probability of system state given all available information). In this equilibrium, new negative tests keep offsetting the risk of new introductions, stabilizing the belief at an equilibrium level. Note that equilibrium may not be reached if detection occurs during the transition phase, or if sampling effort is insufficient to counterbalance the introduction risk—such as when the budget is too low or the introduction rate is too high.

The trajectory of convergence toward the equilibrium depends on the position of the prior belief relative to the equilibrium belief. A prior belief represents the knowledge about the disease at the beginning of the planning horizon, which can be obtained from experts or estimated from historical surveillance data. If the prior matches the equilibrium belief, it is optimal to apply stable control efforts to keep the system at the equilibrium level. Otherwise, adjustments are needed to steer the system toward the equilibrium. It takes longer to converge to the equilibrium when the prior belief is further from the equilibrium belief, or when the budget is low. Note that if the prior is greater than the equilibrium belief, the disease-free belief can decline over time, even as all test results remain negative (Fig. 2B).

Balancing Prevention and Surveillance.

Prevention reduces the risk of introduction, while surveillance catches any cases that slip through. To gain insights into the optimal balance between them, we considered a stylized model for a single site and derived an explicit formula for the equilibrium (see Eq. 17 in SI Appendix, section 3). The formula suggests that surveillance is essential, while prevention should be used only when it is effective enough or when the budget is sufficiently high. We illustrate this formula in Fig. 2C. When the effectiveness of prevention falls below a certain threshold, it is optimal to focus exclusively on surveillance. Otherwise, surveillance should be combined with prevention. Indeed, if prevention is not effective enough, it is better to invest in monitoring to ensure early detection. The formula also suggests that it is never optimal to forgo surveillance unless prevention can eliminate the introduction risk—which is rare. Without surveillance, the organization would be blind to new introductions. The optimal allocation strategy is also influenced by the budget. As the budget increases, a larger proportion of the budget should be allocated to prevention. With a low budget, we should focus on the essential activity, surveillance. But if the budget exceeds a certain threshold, it is better to implement prevention along with surveillance. When the introduction rate is low, the prevalence is likely low and surveillance becomes looking for small signal with high cost, which is less cost-effective. In such cases, it is better to spend more on prevention.

To Concentrate or Distribute Surveillance Effort.

To answer the question whether to concentrate all efforts in a small area or to distribute them across a larger area, we found that the optimal surveillance strategy is jointly determined by several factors. The first factor is the spatial heterogeneity of the introduction rate. It is optimal to distribute efforts if the introduction rates are relatively uniform across sites, and to concentrate if certain sites have significantly higher introduction rates (Fig. 3A). The second factor is whether the system has reached the equilibrium. A phased-expansion strategy should be used during the transition to the equilibrium (Fig. 3B). The strategy begins by concentrating efforts in a single site and, if no detection occurs, gradually expanding the scope to include two, three, or more sites, until the system approaches the equilibrium (SI Appendix, Fig. S1 shows a three-site case). The order in which sites are included in the expansion is jointly determined by the prior beliefs, costs, and equilibrium beliefs across all sites. The initial site(s) in which to concentrate efforts might not necessarily be the sites believed to have the highest disease risk. The third factor is budget. Smaller budgets favor a concentrated strategy. However, when the budget exceeds a critical threshold (SI Appendix, section 4), a distributive strategy is optimal. Finally, the optimal strategy depends on the cost structure. The phased-expansion strategy is optimal when the surveillance cost is linear or convex with respect to the effort (Fig. 3B). If the surveillance cost is concave, however, a rotational strategy can become optimal, where all effort is concentrated on one site at a time, and the focal site alternates between two sites (Fig. 3C) or rotates through multiple sites. Concave cost function arises when sampling exhibits economies of scale.

Fig. 3.

(A) The equilibrium surveillance strategy for two sites (A and B) is jointly determined by the budget and the difference in disease introduction rates (under a linear cost function). When the difference is below a threshold (explicit formula given in Eq. 23 in SI Appendix section 4), it is optimal to distribute sampling effort across both sites; otherwise, it is optimal to focus all effort on the site with the greater introduction rate. The region where it is optimal to sample both sites expands as the budget increases. (B) Under linear surveillance costs, the optimal surveillance strategy begins by focusing all effort on one site (site B) and, if no detection occurs within a certain time window, expands the efforts to both sites (phased-expansion strategy). The initial site of focus (site B) may receive less effort in the equilibrium. (C) Under concave surveillance costs, the optimal strategy may initially concentrate all effort on one site, then shift focus to another site, and alternate among sites (rotational strategy).

CWD in Cervids in New York State.

Based on the annual introduction rates estimated using refs. 42 and 44, and the per-deer testing costs at the county level provided by NYSDEC (45), we computed the optimal allocation of an annual budget of $500,000 (illustrative) between prevention and surveillance efforts across all 62 counties of NYS over a 10-y planning horizon (2025 to 2034).

Since 2013, a statewide surveillance system has been in place that allocates efforts across counties based on the risk of disease introduction and indices of deer population density (46). Other states where CWD has been detected consistently see a higher prevalence of disease in older age class male deer (e.g., bucks); young-of-year (e.g., fawns) are unlikely to test positive because of their shortened time for exposures and long disease incubation period. Therefore, efforts are focused on collecting older animals, particularly bucks from taxidermists, since they are more valuable samples for disease detection (47). Part of these efforts include a payment incentive (48).

Prevention actions include measures that reduce the risk of pathogen introduction, such as enforcing regulations on the movement of deer and carcasses and implementing public education campaigns (49). Another preventive strategy is to limit the likelihood that an outbreak takes hold after introduction. We examine culling as a proactive prevention measure to reduce deer population density (50), thereby lowering the basic reproduction number ($R_0$) below one and reducing the likelihood that CWD becomes established in the population (see SI Appendix, section 5 for details).

The optimal strategy considers the total budget, spatial variation in costs, and is adaptive to surveillance results. As shown in Fig. 4A, the optimal allocation strategy in NYS concentrates prevention efforts in counties with the highest surveillance costs in the first year of the planning horizon (2025). In the subsequent year, prevention resources are distributed across additional counties. Similarly, the optimal strategy concentrates surveillance efforts in a few counties initially but distributes surveillance efforts across more counties in 2026 (phased-expansion strategy). Should CWD remain undetected, it will take approximately 2 y of dynamic effort before reaching the equilibrium. From 2027 onward, the allocation scheme will be fixed, where surveillance is generally prioritized at state margins and prevention is prioritized in the interior. Simulation studies suggest that the optimal strategy would reduce the cumulative cases before initial detection by an average of 22% compared with the existing practice (from 0.132 to 0.103; Fig. 4B).

Fig. 4.

(A) The joint optimal strategy for prevention and surveillance of CWD in wild white-tailed deer (O. virginianus) across 62 counties in New York State, US from 2025 onward. The allocation plan terminates upon the first detection of CWD. If all disease test results remain negative, the equilibrium will be approached in 2027. In the equilibrium, surveillance is prioritized in border counties, where direct CWD introduction risk from neighboring infected states is high, while prevention is prioritized in interior counties, where introduction risk is lower. The figure excludes five counties constituting New York City. (B) Comparison of the average cumulative cases of CWD at first detection between the current disease management strategy and the optimal strategy (annual budget $500,000). (C) Comparison of the total delay time in detection of CWD under the current surveillance strategy and optimal surveillance strategy under different annual budgets. Detecting new introductions of CWD within 4 y of disease arrival requires spending at least $320,000 per year.

If we assume all efforts are focused on surveillance (the current practice), there will be a delay in initial detection of CWD for the current surveillance strategy in NYS relative to the optimized strategy (Fig. 4C). Given a fixed annual budget of $500,000, the optimal surveillance strategy can detect the disease on average 8.4 mo earlier than the current strategy. On the other hand, the optimal strategy could achieve the same performance as current strategy with 24% less cost per year ($380,000 annually).

Budget planning.

The optimization model also allows us to estimate the minimum budget required to meet a given performance target. For example, Fig. 4C illustrates how the expected detection delay under the optimal policy varies with the annual budget. The results suggest that NYSDEC would need at least $320,000 per year to detect new introductions within an average of 4 y, and over $1 million annually to reduce the average detection time to 2 y.

Discussion

Proactive management of diseases and invasive species presents critical challenges because of incomplete information, high operation costs, and funding constraints. We introduce a model that optimizes both action (prevention) and information (surveillance) across large areas, by balancing the cost and value of information and action. The model determines the optimal management strategy to minimize disease burden, accounting for the fact that it may be impossible to precisely infer disease status (12, 51). The model is forward-looking, as it accounts for how current actions influence the future state of the system.

Although the organization has the flexibility to reallocate funds over time, our model suggests that frequent reallocation may be unnecessary in the long run. If there is no detection after some time, it is optimal to maintain a stable distribution plan, which also simplifies logistics. This, however, does not imply that one should abandon dynamic reallocation, because the first detection may occur during the transition phase to equilibrium.

The funds allocated to each site are the result of joint optimization, considering not only the local introduction risk and operation costs but also the risks and costs at other sites and the total budget. Dynamic resource allocation is needed when the prior belief is not aligned with the equilibrium belief. Because the prior belief is shaped by past actions and observations, suboptimal decisions in the past can require adjustment to align the belief with the equilibrium. A common strategy for achieving this alignment is to initially concentrate efforts in a small area and then gradually expand to larger areas. A time-invariant surveillance strategy has been proposed by Epanchin-Niell et al. (19), based on the assumption that the actual state of the system has stabilized. In contrast, the turnpike equilibrium in our model can arise even when the system continues to change (e.g., diseases keep spreading). What stabilizes is the belief of the system, which can settle into an equilibrium when the tests remain negative.

According to the model, surveillance should always be maintained, but allocating some resources to prevention may be optimal when prevention is sufficiently effective or when the budget is relatively large. Effort should be distributed across multiple sites only when the budget is sufficiently high. Otherwise, it is better to target a small number of sites with highest risks. In the case of managing CWD in NYS, the optimal strategy prioritizes prevention at sites with low introduction risk while concentrating surveillance on sites with high introduction risk, exhibiting the complementary effect from a joint optimization. Sites with high surveillance costs are more suitable for prevention, while those with high prevention costs are more suitable for surveillance.

The model provides valuable guidance for budget planning. Since optimization identifies the best possible performance within a given budget, it can determine the minimum budget required to achieve a specific management objective (Fig. 4C). Further, the model can evaluate how advancements in surveillance technology, preventive measures, or additional funding can improve overall performance.

The model has several limitations that present opportunities for future research. First, it requires estimates of introduction rates and prevention effectiveness as inputs, which can be challenging to specify. Although various methods are available for estimation (42, 52–54), substantial uncertainty often remains around these key parameters. One way to address the uncertainty is robust optimization, which seeks policies that perform well under worst-case realizations when parameters vary within specified bounds (55). An alternative approach is adaptive learning, in which the organization continuously updates its beliefs about both the system state and parameters. This formulation leads to Bayes-adaptive POMDP (56). While this formulation can be computationally demanding, low-dimensional approximations—such as the use of sufficient statistics—may help preserve tractability (26). Notably, a key feature of our approach—the deterministic evolution of belief between detections—remains applicable in this more general setting. When no detection occurs, the agent may gradually revise the introduction rate downward. This raises an open question about whether a turnpike equilibrium could emerge.

In practice, the model parameters could vary over time. If such variation is predictable, such as seasonal changes in introduction risk, it can be incorporated directly into the model as time-varying parameters. If the variation is unpredictable, such as an abrupt increase in introduction risk due to an outbreak in a neighboring region, the model can be applied in a rollout policy (57).

Second, our model assumes that surveillance samples are unbiased and representative of true prevalence within each planning unit. In practice, for diseases such as CWD, samples are often obtained from hunters and taxidermists, which may introduce some sampling bias. Incorporating such biases into the model could further improve inference accuracy. Moreover, disease prevalence may vary within a planning unit. Extending the framework to capture this within-unit heterogeneity could further enhance the precision.

The third limitation is that the decision process is assumed to end upon detection, regardless of where it occurs. A valuable extension would be to explicitly incorporate the location of detection by, for example, assigning a location-specific reward to the detection. How to estimate such rewards remains an open question. We focus on the predetection phase, treating the postdetection phase as a separate problem. In practice, some agencies must manage both phases concurrently. For example, several U.S. states have detected CWD in certain counties but not in others, so state agencies must balance resources between monitoring prevalence at known sites and detecting introductions in new areas. Further research is needed to integrate pre- and postdetection sites into a unified allocation framework.

A promising extension is to combine the decision model with simulation. We modeled the disease spread process as a Markov chain, but simulation models such as agent-based models can capture more details of the disease spread and incorporate additional sources of information.

We have presented a method that leverages decision theory to guide disease management under imperfect information. The method provides a principled means of allocating finite resources in dynamic disease systems that maximizes the time of introduction and minimizes the time to detection, which are fundamental goals of human and animal disease programs. It could be utilized to enhance zoonotic disease surveillance in low-income countries with limited resources, capacity, and funding (58). The model and most insights also extend to the management of invasive species.

Data Archival.

Data appear in SI Appendix, Tables S1–S3.

Materials and Methods

We present a detailed description of the model. Consider multiple sites indexed by i with host species susceptible to an emerging disease, which has not yet been detected in any of the sites. The organization makes decisions periodically, as new information becomes available, over a finite planning horizon, $t=1,\dots ,T$, or an infinite horizon ($T=\infty$). The underlying true status of each site is either disease-free or infected. For the infected site, the prevalence is further discretized into N levels. The disease dynamics are modeled with a Markov chain, where the state transition probabilities at a specific site may depend on the infection states of all sites. This dependence captures the spatial structure of disease dynamics. For example, the disease can be introduced to a disease-free site via long-range anthropogenic introduction or short-range spread of a host from a neighboring infected site. The disease-free state and prevalence are not directly observable but can be inferred through surveillance of the host species. Let $u_{\mathit{it}}$ denote the surveillance effort (e.g., testing sample size) allocated to site i at time t, and let $v_{\mathit{it}}$ denote the corresponding prevention effort, which can reduce the rates of disease introduction and spread (thereby increasing time to introduction and keeping prevalence low when the disease is first detected). Both efforts can be adjusted based on new information from surveillance. The evolution of disease at a given site is modeled as a birth–death process, with the local transition rates influenced by the infection states of other sites and the local prevention effort. A pure birth process is appropriate for diseases that are difficult to eradicate once established (e.g., CWD), in which case the prevalence keeps increasing. The cost of deploying surveillance effort u at site i is denoted by $s_i(u)$, and the cost of deploying prevention effort v is $c_i(v)$. Most of our results are obtained under the linear cost functions $s_i(u)=c_{\mathit{ui}}u$ and $c_i(v)=c_{\mathit{vi}}v$ but we also explored the nonlinear cost in Fig. 3C. Our model applies to general cost functions. The total spending over prevention and surveillance in each decision period is capped by a budget B. Unspent funds do not carry over to future periods.

If a single host tests positive, the site is confirmed to be infected, so a positive test establishes disease presence. However, negative tests cannot rule out infection, because surveillance may fail to detect any disease cases at an infected site. In this sense, surveillance is imperfect as it is prone to false negative. The probability of detecting the disease in an infected site depends on the underlying prevalence and the surveillance effort allocated to that site. Given prevalence p at site i, the probability of detecting the disease under surveillance effort u is denoted by ${\rho }_i(p,u)$, which can be a general function. In the case study, we assumed that the number of detections in a given period follows a Poisson distribution with rate ${\lambda }_{i}p{u}_{\mathit{it}}$, where ${\lambda }_i$ represents the capture rate of host species. Thus, the probability of detecting at least one case in a period is ${\rho }_i(p,u)=1-exp(-{\lambda }_{i}pu)$. One can also add a constant to the detection probability to account for opportunistic detections (e.g., collection and testing of an animal with clinical signs), which can occur even when the formal surveillance effort is zero.

If site i is infected with prevalence p, we assume a penalty $d_i(p)$ is incurred per period. The penalty function $d_i(·)$ is often increasing as higher prevalence leads to more cumulative disease burden or damage. The penalty function can also reflect the potential for pathogen or invasive species to spread, assigning higher penalties to sites where spread is more likely. For infinite horizon problems, we can also assign a reward $r>0$ to the disease-free state by setting the penalty $d_i(0)=-r$, thereby incentivizing prevention efforts that prolong the disease-free period. For finite planning horizons, we can introduce a terminal penalty if the disease remains undetected at the end of the planning horizon. The objective is to minimize the expected total penalty from the disease before the initial detection. When the penalty is a linear function of the prevalence and the terminal penalty is zero, the objective is equivalent to minimizing the expected cumulative number of unnoticed infections that amass before the initial detection, or equivalently, to minimizing the expected area under the disease growth curve from the initial disease introduction time to the time of detecting the first disease case. A reward can also be assigned to initial detection, depending on the location. For example, higher rewards for detecting the disease at sites with higher spread potential.

We model the decision problem as a POMDP across multiple sites, and each site can occupy one of several states representing different prevalence, leading to a high-dimensional state and action space. The size of the action space (i.e., all possible allocation schemes) grows exponentially with the number of sites. For example, in NYS, if the sample size in each of the 62 counties can range from 0 to 200, the number of possible allocations would reach $5.84\times {10}^{50}$. Traditional solution methods based on backward induction suffer from the “curse of dimensionality” and cannot scale to large problems. While recent approximate solution methods have improved computational efficiency via forward induction (59, 60), they remain insufficient for handling the size of state and action spaces encountered in real-world disease management. Deep reinforcement learning (DRL) is an emerging framework for solving complex control problems including POMDPs (61–63). However, it can be difficult to understand how DRL policies depend on model parameters. Another challenge is that DRL typically requires extensive interactions with the environment (sample inefficiency).

A key feature of our solution method lies in exploiting the inherent structure of the decision problem before the initial detection. Notably, all disease test results are negative prior to detection. Given these constant observations, the belief of the system evolves deterministically as a function of actions. This structure allows us to use a forward induction approach to predetermine belief trajectories under different strategies (26), so that the POMDP can be reformulated as a deterministic control problem and solved efficiently. This solution method yields exact optimal solutions that can scale to many sites. It also allows for explicit expressions of the optimal policy in equilibrium, offering managerial insights. The resulting optimal strategy takes the form of an open-loop control policy: the actions are selected based on the duration for which all sampling results remain negative. Note that the strategy is adaptive to new observations, because negative test results also contribute to our knowledge about infection status. If all tests in the upcoming period are negative, the belief of the system will be updated by the consecutive negative tests, and the actions for the following period will be selected based on that updated belief.

We now present the theoretical framework for reformulating POMDP as a deterministic optimal control. It extends the approach of Wang et al. (26) into multiple sites. Let $S_{\mathit{it}}$ denote the infection state at site $i=1,\cdots ,I$ at time t, with state space $\mathcal{S}=\{0,1,\dots ,N\}$. State 0 represents disease-free and state $i>0$ represents infection with prevalence rate $p_i>0$. The prevalence rates are ordered as $p_1<p_2<\cdots <p_N$. Let ${\mathit{S}}_t=(S_{1t},\dots ,S_{\mathit{It}})$ be the vector containing the infection states across all sites. The state space of ${\mathit{S}}_t$ is denoted by ${\mathcal{S}}^I$. At period t, the organization chooses the surveillance efforts ${\mathit{u}}_t=(u_{1t},\dots ,u_{\mathit{It}})$ in conjunction with prevention efforts ${\mathit{v}}_t=(v_{1t},\dots ,v_{\mathit{It}})$ for all sites. Let $\mathbb{P}(\mathit{v})=\{p_{j{j}^{\prime }}(\mathit{v}),j,j^{\prime }\in {\mathcal{S}}^I\}$ denote the transition matrix of the infection state given the prevention efforts $\mathit{v}$. Prevention efforts can reduce the rate of new introductions and the transition rate to higher prevalence states.

Define the belief, ${\mathit{b}}_t$, which is a probability distribution over $\mathit{S}$, given all information available at time t. It is a row vector with ${(N+1)}^I$ components, with each component corresponding to a specific configuration of the infection state across all sites. The belief takes on value in a probability simplex. Let $f_i(\mathit{u})$ denote the no-detection probability at composite site $i\in {\mathcal{S}}^I$ over one period under sampling effort $\mathit{u}$. The column vector $\mathit{F}(u)$ contains the no-detection probabilities at different composite states.

The sequence of events is as follows: At the beginning of period t, the organization receives test results from the previous period and updates its belief to $\mathit{b}$. Based on this belief, the organization decides the prevention and surveillance efforts $({\mathit{v}}_t,{\mathit{u}}_t)$. Next, disease introduction and spread occur (i.e., state transitions under the prevention scheme). This leads to an updated preobservation belief $\mathit{b}\mathbb{P}({\mathit{v}}_t)$. Then, surveillance results are collected based on the new state at period $t+1$. If the results include a positive case, then the decision process is terminated. If all results are negative, the posterior belief will be updated to

$$\begin{array}{c}\hfill {\mathit{b}}^{-}({\mathit{u}}_t,{\mathit{v}}_t):=\frac{\mathit{b}\mathbb{P}({\mathit{v}}_t) \text{diag}\left(\mathit{F}({\mathit{u}}_t)\right)}{\mathit{b}\mathbb{P}({\mathit{v}}_t)\mathit{F}({\mathit{u}}_t)},\end{array}$$ [1]

by Bayes’ rule, and the organization will choose the next action $({\mathit{v}}_{t+1},{\mathit{u}}_{t+1})$ based on the updated belief. In the above, $\text{diag}\left(\mathit{F}({\mathit{u}}_t)\right)$ is a diagonal matrix with entries of $\mathit{F}({\mathit{u}}_t)$ along its main diagonal. The denominator $\mathit{b}\mathbb{P}({\mathit{v}}_t)\mathit{F}({\mathit{u}}_t)$ is a normalization factor. This formulation assumes that test results reflect the posttransition state. If instead test results reflect the pretransition state, the order of the matrices $\mathbb{P}$ and $\mathit{F}$ should be reversed. In the continuous-time limit (SI Appendix sections 2–4), the distinction between the two orderings becomes negligible, as decisions and observations occur continuously.

Let $V_t(\mathit{b})$ denote the minimum expected penalty (i.e., value function) from period t onward given the beliefs across all sites. We omit the time argument in the belief for simplicity. The value function $V_t$ satisfies the following Bellman equation

$$\begin{array}{cc}\hfill V_t(\mathit{b})=& {\displaystyle \underset{\mathit{u},\mathit{v}\in A}{min}}\{\mathit{b}{\mathbf{c}}^{\top }+V_{t+1}\left({\mathit{b}}^{-}(\mathit{u},\mathit{v})\right)·\mathit{b}\mathbb{P}(\mathit{v})\mathit{F}(\mathit{u})\}\hfill \\ \hfill V_T(\mathit{b})=& \mathit{b}{\mathbf{d}}^{\top },\hfill \end{array}$$ [2]

where ${\mathit{b}}^{-}$, as defined in Eq. 1, is the posterior belief given no detection under action $(u_i,v_i)$. Here, $\mathbf{c}$ is the vector of per-period penalty ($d_i(·)$) associated with each prevalence state, and $\mathbf{d}$ is the terminal penalty for each state at the end of horizon. The feasible action set $\mathcal{A}$ is the set of all possible actions that satisfy the budget constraints. Given the beliefs ${\mathit{b}}_1,\cdots ,{\mathit{b}}_I$ at time t, the agency chooses the action that minimizes the right-hand side of the Bellman equation. We denote the optimal policy by ${\sigma }_t^u,{\sigma }_t^v$. ${\sigma }_t^u$ maps the belief to the surveillance effort while ${\sigma }_t^v$ maps the belief to the prevention effort.

Deterministic Control Formulation.

We now introduce a deterministic optimal control problem:

$$\begin{array}{c}\hfill {\displaystyle \underset{\mathit{u}(t),\mathit{v}(t)\in \mathrm{\Omega }}{min}}\left\{{\displaystyle \sum _{t=1}^{T-1}}{\mathit{\pi }}_t{\mathbf{c}}^{\top }+{\mathit{\pi }}_T{\mathbf{d}}^{\top }\right\}\end{array}$$ [3]

subject to the state dynamics

$$\begin{array}{cc}\hfill {\mathit{\pi }}_{t+1}=& {\mathit{\pi }}_t\mathbb{P}({\mathit{v}}_t)\text{diag}\left(\mathbf{F}({\mathit{u}}_t)\right), t=0,\dots ,T-1,\hfill \\ \hfill {\mathit{\pi }}_0=& {\mathit{b}}_0,\hfill \end{array}$$ [4]

and budget constraint

$$\begin{array}{c}\hfill {\displaystyle \sum _{i=1}^I}{s}_i(u_{\mathit{it}})+{\displaystyle \sum _{i=1}^I}{c}_i(v_{\mathit{it}})<B,\end{array}$$ [5]

for all $t=1,\dots ,T$. Next, we will show that, given the prior belief ${\mathit{b}}_0$ and consecutive negative test results (i.e., no detection) up to period t, the course of optimal actions in the POMDP is identical to the optimal open-loop policy in the above deterministic control problem.

Equivalency between the Formulations.

Given the prior ${\mathit{b}}_0$ and the optimal policy $({\sigma }_t^u,{\sigma }_t^v)$ for the stochastic control problem Eq. 2, let $\{{\mathit{b}}_t^{\ast },t=1,\dots ,T\}$ be the optimal sample path of the belief state conditional on no detection up to time t. Let ${\sigma }_t^u({\mathit{b}}_t^{\ast })$ and ${\sigma }_t^v({\mathit{b}}_t^{\ast })$ denote the corresponding optimal survey and protection actions, respectively, at time t. Define $\{{\mathit{\pi }}_t^{\ast },t=1,\dots ,T\}$ as the optimal state trajectory of the deterministic control problem Eqs. 3–5, given the prior ${\mathit{b}}_0$. Let $\{u_t^{\ast },t=1,\dots ,T\}$ and $\{v_t^{\ast },t=1,\dots ,T\}$ be the corresponding open-loop control trajectories. The set Ω is the set of actions that satisfy the constraints in each period.

Theorem 1.

If all tests up to time t are negative, the closed-loop optimal actions match the open-loop actions, namely, $u_t^{\ast }={\sigma }_t^u({\mathit{b}}_t^{\ast })$ and $v_t^{\ast }={\sigma }_t^v({\mathit{b}}_t^{\ast })$.

The proof of this theorem is in SI Appendix, section 1. Reformulating the problem as an optimal control framework enables us to leverage Pontryagin’s maximum principle to derive managerial insights into the optimal strategy (64).

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

All study data are included in the article and/or SI Appendix.