The early detection of disease epidemics reduces the chance of successful introductions into new locales, minimizes the number of infections, and reduces the financial impact. We develop a framework to determine the optimal sampling strategy for disease detection in zoonotic host-vector epidemiological systems when a disease goes from below detectable levels to an epidemic. We find that if the time of disease introduction is known then the optimal sampling strategy can switch abruptly between sampling only from the vector population to sampling only from the host population. We also construct time-independent optimal sampling strategies when conducting periodic sampling that can involve sampling both the host and the vector populations simultaneously. Both time-dependent and -independent solutions can be useful for sampling design, depending on whether the time of introduction of the disease is known or not. We illustrate the approach with West Nile virus, a globally-spreading zoonotic arbovirus. Though our analytical results are based on a linearization of the dynamical systems, the sampling rules appear robust over a wide range of parameter space when compared to nonlinear simulation models. Our results suggest some simple rules that can be used by practitioners when developing surveillance programs. These rules require knowledge of transition rates between epidemiological compartments, which population was initially infected, and of the cost per sample for serological tests.
The effectiveness of disease control measures often depend on when outbreaks are first discovered. Early detection can significantly reduce the costs associated with disease eradication, human illnesses, and devastation of livestock or crops. For example, the 2001 epidemic of foot and mouth in Great Britain was reported only 2 weeks after the epidemic began [1] yet had an estimated financial impact of $11. 9–$18. 4 billion dollars [2]. A hypothetical foot and mouth epidemic in California not detected for 2 weeks could have a financial impact of over $15 billion dollars, and an epidemic not detected for 3 weeks could have an impact of up to $69 billion dollars [3]. Although many studies have examined alternative control strategies and the impact of detection time on control [1], [2], [4], the complementary question of how to achieve early detection has been relatively neglected by theory. Greater attention to the design of disease surveillance methods may facilitate earlier detection and reduce the economic impacts of disease epidemics. Passive surveillance methods are the voluntary reporting of cases by primary care providers and citizens to public health officials [5]. Recent work on passive surveillance methods for human infectious diseases has progressed rapidly and includes developing methods to optimize the placement [6], [7] and performance [8] of surveillance sites. Integrating these physical surveillance systems with internet search data has led to improvements in the performance of traditional physical reporting systems [7], [9]. Active surveillance methods of zoonotic diseases are the periodic sampling by health authorities [5]. For vector-borne diseases active surveillance may include the use of sentinel animals and the longitudinal sampling of vector populations [10]. Active surveillance may often perform better for targeted objectives than passive methods [5], and recent work has begun to link active zoonotic surveillance data to epidemiological models. For example, Gerardo-Giorda et al. [11] combined surveillance data and epidemiological models to identify counties that were most important for surveillance efforts of rabies in New York State. It is likely that analytical approaches will prove useful in making active zoonotic surveillance methods more cost effective, an important consideration for surveillance organizations with limited resources [12]. Past analytical work on active disease detection examined how sampling for infected individuals in a susceptible population affects the time at which an epidemic is detected [13], [14] and the subsequent incidence of a disease at the time of discovery [15]. These studies have examined the dynamics of diseases that are directly transmitted and thus lack a disease vector. As a result, we still have little knowledge to guide early detection theory for zoonotic diseases (e. g. , Lyme disease, malaria, Rift Valley fever virus, West Nile virus, dengue fever), where sampling could occur in vector populations or host populations. Here, we studied the optimal sampling design for early disease detection using formulations of a disease with one host population and one vector population. We combined models of host-vector dynamics with a periodic sampling procedure in which sample size is constrained by economic limitations. We used a susceptible-infected (SI) model to examine how to allocate sampling effort between the vector and host populations, and we used a susceptible-infected-recovered (SIR) model to look at allocating sampling effort between the vector population, infected hosts, and recovered hosts. The CDC guidelines for evaluating public health surveillance of human based diseases [16] are standards that have been used in many assessments of zoonotic surveillance systems, although differences may exist between human and zoonotic surveillance goals [17]. A recent survey on the assessment of surveillance systems found that a number of different metrics have been used to determine zoonotic surveillance performance; two of the most frequently mentioned criteria are the sensitivity of surveillance (the ability to detect outbreaks or infection rates) and the time to outbreak detection from initial exposure [17]. Here we assume the goal of surveillance is to detect the outbreak as early as possible to minimize financial damages or spillover human infections, a common goal for zoonoses [12]. Our results provide some basic rules of thumb for practitioners designing active surveillance protocols for vector-borne diseases.
We made assumptions common to other SI models of vector-borne diseases: vectors and hosts can be in either a susceptible or an infected state at time, the disease epidemic (the dynamics of interest) occur on a relatively short time-scale and thus infected individuals cannot recover nor do individuals give birth or die over the course of the epidemic, and infection spreads only through interspecific interactions [18], [19]. Subscripts are used to denote population-level parameters: e. g. , and denote the number of infected hosts () and vectors () at time, respectively (Table 1). These assumptions give the following system of equations for the dynamics of infection of the SI model: (1a) (1b) where and correspond to the disease transmission rates from vectors to hosts and hosts to vectors, respectively. and correspond to the total host and vector population sizes, respectively. Throughout this work we assume that population sizes are constant over the course of the epidemic and that individuals are in the population only if they can potentially contract the disease. This implies that individuals that are epidemiologically isolated are not a part of the population. Note that the dynamics of susceptible host () and vector () populations are completely determined by system (1) because and. Because we are interested in detecting a disease as early as possible, we focus on the dynamics of the system immediately after disease introduction. Therefore we linearized system (1) about the disease-free state and obtained: (2a) (2b) We focus our subsequent analyses on the specific scenario of an epidemic started by an infected host with the initial conditions though analyses of alternative initial conditions are presented in Text S3 and S4. Because of the symmetric nature of system (2), this analysis yields similar results. With the assumptions listed above, the solution to system (2) is: (3a) (3b) where. For the SIR model we assume that recovered hosts obtain immunity over the timescale of the epidemic. As in the SI model the disease cannot spread through direct contacts within host and vector populations, transmission is frequency-dependent, and individuals are not born and do not die over the course of the epidemic. The full model for a single host population and single vector population is given by (4a) (4b) (4c) where designates recovered individuals and is the recovery rate of infected individuals. Note that the dynamics of susceptible host () and vector () classes are completely determined by system (4) because and. The corresponding linearized model evaluated at is (5a) (5b) (5c) For an epidemic begun by an infected host, the solution of (5) is: (6a) (6b) (6c) where. Solutions for the alternative initial conditions for system (5) are given in Text S4. This analysis is slightly more complicated than that presented in the main text, but the core ideas remain the same. Consider sampling at time from a population with potentially infected hosts and vectors. Let denote the set of events such that the disease is detected from a sample of size. If the total population abundance is much greater than the sample size then, the probability of detecting the disease, can be modeled as a binomial random variable. When the sample size is comparable to the population abundance, the hypergeometric distribution is a suitable sampling model. We do not consider the hypergeometric model here as the binomial distribution provides a reasonable approximation for realistic sample sizes. The proportion of infecteds at time is given by. In a sample of size, is the complement of not detecting any infected individuals, (7) When the disease is rare () equation (7) is well approximated asIf there are two sampling strata (e. g. , a host and vector, although the approach works as well for two host species or two strata of hosts in a single species), we need the probability of detecting the disease in either of those strata, . For two strata this quantity is given by (8) (9) A more general form when there are sampled strata is given by (10) as shown in Text S1. We also consider the probability of detecting the disease for the first time in the sample when sampling occurs regularly at discrete time intervals. A model of detecting the disease in the current sampling period, but not before, is a geometric distribution with time-dependent detection probabilities: (11) Here the number of infecteds in stratum in the sampling period is given by. Sampling strata are defined by both the animal population being sampled and the type of test that is run. For example, immunological tests on bird populations for West Nile virus can test whether individuals are currently infected or have been previously infected by the type of antibody present in the sample. Antibody-specific tests therefore distinguish between infected birds and recovered birds. The first term on the right-hand side of expression (11) represents the probability of detecting the disease in time period. The remaining terms (given in capital Pi notation,) represent the probability of not detecting the disease in sampling period, where runs from to. The product of these terms gives the probability of not detecting the disease in any of the to sampling periods. We minimized the time until detection of the epidemic using the geometric probability distribution defined in equation (11) and by using the expected time to detection, given by (12) This expected value is an infinite series that converges to an unknown quantity, therefore we numerically approximated. We have so far considered the possibility of sampling and testing infected and recovered individuals in populations. However, a common practice in zoonotic surveillance is to combine samples from multiple individuals in the stratum of interest in order to save money (e. g. , [20]). Though this pooled sampling does not identify which individual tested positive for the virus, the goal of surveillance is often to identify the presence of the virus instead of a specific infected individual. Pooling sizes must be constrained to prevent the possibility of a positive individual sample being diluted below detectable levels, often determined using experimental dilutions in the laboratory (e. g. , [21]). To incorporate pooled sampling into our sampling model, we rescaled the probability of a positive detection in a single sample by the number of individuals in a pool, : the probability of detection in a single pooled sample of strata at time can then be approximated using a linearized binomial expansion: (13) Approximation (13) works well when. Our simulations indicate that a 10% approximation error occurs when, suggesting the approximation is robust for the purpose of early detection. Pooled sampling modifies (10) and (11) to (14a) (14b) Our goal is to determine the resource allocation that will allow us to detect a disease as early as possible. We therefore introduce economic constraints on this sampling process in the next section. Agencies are often faced with monitoring endemic and emerging diseases with finite resources. This necessitates allocating those resources in the most efficient way possible. We applied a cost function to describe these constraints: we let be the budget for a set of samples taken periodically and be the cost of sampling individuals from stratum, . If we assume that we spend our entire budget then. For example, a linear cost function for a vector stratum and a host stratum can be written as (15) where and are the overhead costs (operating costs) associated with sampling vectors and hosts respectively, while and are the corresponding costs per sample. We used the Karush-Kuhn-Tucker (KKT) conditions [22] to find the sampling strategy that maximizes the probability of detection (given by equation (14a) ) or minimizes the time to disease detection (given by equation (14b) ). The KKT approach allows the minimization of a function subject to inequality constraints, e. g. , constraining the sample sizes to be nonnegative. Further details on this method, as well as some general results for cases with linear objective functions, are provided in Text S2 and S3.
Active surveillance is an important tool for decision makers; treating the process analytically can provide some important insights on how to conduct cost-efficient surveillance. Very little past work in mathematical epidemiology has focused on early detection despite these potential benefits. One of the important products of this analysis has been to explicitly define the kinds of data that will be needed to design basic surveillance studies. Specific knowledge about the costs associated with sampling different populations and information about disease transmission rates will be necessary when making very specific predictions, but, as we have shown, applying the procedure with only basic knowledge of these quantities can make predictions that may be robust. This is fortunate for monitoring agencies as ecological and epidemiological parameters can be difficult and costly to obtain. Our analyses of West Nile virus illustrates robustness to parameters that are often unknown over a variety of models and assumptions. Although we focused on basic SI and SIR models, this framework can be easily extended to include more specific models when they are available. It is likely that West Nile virus models that incorporate more biological realism (e. g. , [32]) will be necessary to provide more targeted advice concerning surveillance practices for specific management agencies, whose monitoring capabilities may differ from what was assumed in this work. However, because the timescales we are examining are relatively short, our models may provide robust predictions when sampling is conducted over limited spatial scales. Therefore, even the simplified models examined here may be useful for designing sampling strategies when more detailed ecological and epidemiological information is not available. Our results suggest that the optimal sampling design will often focus all sampling effort on a particular species or compartment. This result is due to the linear nature of the cost functions and the approximately linear nature of the dynamical systems as functions of our control variable, the sample sizes, . These on-off or “bang-bang” types of solutions arise in other epidemiological problems when determining how to treat or remove individuals in infected populations to stop an epidemic [33]–[35]. More recent work on the control of epidemics suggests that when considering multiple control strategies the optimal solution is not simply an additive combination of the independent control solutions [36]. Similar results may hold for surveillance methods when combining different types of surveillance strategies, for example active and passive sampling strategies. In cases where linearity and large population approximations for the dynamics do not hold, our analysis suggests that the optimal sampling design can be a mixture of sampling strata but this occurs over a very limited parameter space for West Nile virus (Figure 4). Nonlinear cost functions may also arise when the cost per sample changes when performing a large number of samples due to reductions in the associated personnel costs or in the laboratory fees incurred in performing a large number of tests. Changing the dynamical model by incorporating more detailed ecological and epidemiological considerations may also reduce the robustness of our linearization approximation. For example, introducing spatial structured populations [37] or heterogeneous contact rates are known to lead to additional nonlinearities in incidence functions [38], [39]. There are several additional considerations that may improve upon our efforts. Many disease models include exposed compartments (e. g. , malaria [40]) in the host and/or vector population that can delay the onset of infectiousness once bitten. This may lead to additional possibilities in the switch time analysis that we did not consider. For example, if a host population is initially infected but has a long exposed period then there may be a quick switch to sampling the vector population followed by switches at longer time scales back to the host population. Additional important developments include treating the initial conditions and transmission process as random variables. This will likely lead to a distribution of optimal strategies rather than a single, fixed strategy [41]. Recognizing uncertainty in the initial conditions may be especially important when the source of infection is unclear given the potential sensitivity of the sampling process to the initial infections. We also did not consider the possibility of testing for multiple pathogens in this analysis. For example in Florida, mosquito control agencies regularly screen for malaria, West Nile virus, and dengue fever among others [12]. Applying a mixed sampling strategy may allow managers to hedge their bets because the optimal strategy for West Nile virus may not necessarily translate to the early detection of other pathogens. Finally, our assumption that diagnostic tests for pathogen or antibodies provide perfect indicators of an individual' s state may be violated by several factors. First, immunological dynamics can lead to low viral or antibody levels even when individuals have been infected, which may lead to low test reliability [29]. Extending the approach to coupled immunological-epidemiological models may account for this source of uncertainty. Second, and perhaps more importantly, imperfect diagnostic test reliability can arise due to stochastic factors that cannot be accounted for in conventional lab techniques. These effects can be incorporated into a sampling model by multiplying the economic efficiency by a random variable representing the test sensitivity and specificity [42]. Despite the recognized impact of emerging zoonoses on human health [43] we are aware of no work that attempts to integrate the active surveillance systems explored here with disease surveillance in humans. In diseases where humans are spillover hosts, such as West Nile virus, low human incidence is expected. Passive surveillance is often more economically efficient when dealing with rare events [44] but this reporting process differs from the assumptions made in this work. In passive surveillence the reporting effort will often vary through time due to seasonal and institutional effects. Incorporating these factors into a predictive framework will require the statistical analyses of these patterns [11]. When including the surveillance of humans for West Nile virus we expect that reductions in the time to detection will occur when the recovery rate () is high or the human population size is low relative the the vector population, as this is when hosts are most efficient to sample for detecting the disease (Figure 4), though the particular effects will depend on the amount of sampling effort and the transmission rate to humans from the vector. In general we expect that accounting for passive human surveillance of zoonoses may change the optimal active surveillance strategy for wildlife populations as it may not be necessary to sample hosts that have strong interactions with humans or species that significantly lag behind the epidemiological response of humans. Another important case that we did not consider here are zoonotic diseases such as avian influenza, which spread much more easily within one zoologic species than across-species. For these diseases, the goal of surveillance is to detect a subtype of the disease more virulent in humans, indicated by sustained human to human transmission. This sampling needs to be tailored to detect clusters of human cases linked to a single avian-to-human transmission that deviate from what is to be expected from low-level human-to-human and bird-to-human transmission [45]. This kind of surveillance will require more detailed contact tracing that is not accounted for in our framework, though the basic structure we have described here could still be applied. More complex statistical analyses will also be needed to determine whether levels of infecteds and recovereds are significantly higher than background levels in order to determine if an outbreak is occurring. Analyses such as those determining epidemic thresholds from public health data (e. g. [45]–[47]) will be useful starting points for integrating thresholds into detecting epidemics of endemic zoonotic diseases.
Outbreaks of zoonoses can have large costs to society through public health and agricultural impacts. Because many zoonoses co-occur in multiple animal populations simultaneously, detection of zoonotic outbreaks can be especially difficult. We evaluated how to design sampling strategies for the early detection of disease outbreaks of vector-borne diseases. We built a framework to integrate epidemiological dynamical models with a sampling process that accounts for budgetary constraints, such as those faced by many management agencies. We illustrate our approach using West Nile virus, a globally-spreading zoonotic arbovirus that has significantly affected North American bird populations. Our results suggest that simple formulas can often make robust predictions about the proper sampling procedure, though we also illustrate how computational methods can be used to extend our framework to more realistic modeling scenarios when these simple predictions break down.
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Optimal Sampling Strategies for Detecting Zoonotic Disease Epidemics