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Conceived and designed the experiments: SR TM YW. Performed the experiments: SR. Analyzed the data: SR TM YW. Contributed reagents/materials/analysis tools: SR YW. Wrote the paper: SR TM YW.

The authors have declared that no competing interests exist.

Developing control policies for zoonotic diseases is challenging, both because of the complex spread dynamics exhibited by these diseases, and because of the need for implementing complex multi-species surveillance and control efforts using limited resources. Mathematical models, and in particular network models, of disease spread are promising as tools for control-policy design, because they can provide comprehensive quantitative representations of disease transmission.

A layered dynamical network model for the transmission and control of zoonotic diseases is introduced as a tool for analyzing disease spread and designing cost-effective surveillance and control. The model development is achieved using brucellosis transmission among wildlife, cattle herds, and human sub-populations in an agricultural system as a case study. Precisely, a model that tracks infection counts in interacting animal herds of multiple species (e.g., cattle herds and groups of wildlife for brucellosis) and in human subpopulations is introduced. The model is then abstracted to a form that permits comprehensive targeted design of multiple control capabilities as well as model identification from data. Next, techniques are developed for such quantitative design of control policies (that are directed to both the animal and human populations), and for model identification from snapshot and time-course data, by drawing on recent results in the network control community.

The modeling approach is shown to provide quantitative insight into comprehensive control policies for zoonotic diseases, and in turn to permit policy design for mitigation of these diseases. For the brucellosis-transmission example in particular, numerous insights are obtained regarding the optimal distribution of resources among available control capabilities (e.g., vaccination, surveillance and culling, pasteurization of milk) and points in the spread network (e.g., transhumance vs. sedentary herds). In addition, a preliminary identification of the network model for brucellosis is achieved using historical data, and the robustness of the obtained model is demonstrated. As a whole, our results indicate that network modeling can aid in designing control policies for zoonotic diseases.

Zoonotic diseases (ones that infect both animals and humans) exact a significant economic and human cost, especially in developing economies. Developing effective policies for mitigating zoonotic infections is often challenging, both because of the complexity of their spread and because very limited resources must be allocated among a range of control options. It is increasingly becoming clear that mathematical modeling, and in particular network modeling, of disease spread can aid in analyzing and mitigating these spreads. Here, we develop a network model for the spread and control of a zoonotic infection, focusing particularly on a case study of brucellosis transmission and control among wildlife, cattle herds, and human subpopulations in an agricultural community. After motivating and formulating the model, we introduce tools for 1) parameterization of the model from time-course and snapshot data, 2) simulation and analysis of the model, and 3) optimal design of control policies using the model. The study shows that the network model can inform design of heterogeneous control policies that mitigate zoonotic disease spread with limited resources.

Zoonoses–infectious diseases that can be transmitted to humans from other animals–incur significant cost though their impact on both agricultural production and human communities. Zoonoses have particular prevalence and impact in the developing world, where low-cost yet effective strategies for their control and eventual eradication are badly needed (e.g.,

Recently, several studies have demonstrated that mathematical modeling can aid practitioners in developing control strategies, by allowing

Network modeling of infection spread has been a particular area of burgeoning interest over the last few years, see the articles

Models for zoonotic diseases (including for brucellosis) classically have been simple compartmental models rather than network-structured models, with each compartment capturing homogeneous transmission within an entire species, or perhaps for an age group of that species (e.g.,

The purpose of this study is to give a comprehensive treatment of the modeling, surveillance, and cost-effective control of zoonoses, by bringing to bear and enhancing a network-control-theory approach to virus-spread control. As a specific case study, we explore modeling and design of surveillance and control for brucellosis in a prototypical agricultural setting in a resource-constrained area such as sub-Saharan Africa. To this end, a network model for brucellosis transmission among animal herds and to human populations is developed, that captures the mechanism of transmission of this zoonotic bacterium, allows repesentation of realistic surveillance and control mechanisms and their costs, and measures the performance of the control scheme with regard to the disease's financial and societal impact. Once the model has been formulated in a general way, we discuss approaches for inferring important network-model parameters from limited experimental data, which include both simple heuristic approaches and new network-estimation tools from the control sciences. Using the parametrized models, design of surveillance and control capabilities is pursued, with the aim of suggesting good targeted control/surveillance strategies as well as improvements to existing strategies. A particular focus of the design is to obtain simple insights into high-performance strategies that do not rely on model details, so that the developed strategies are in some measure robust to inaccuracies and limitations in the models and available data.

This study advances the existing efforts on network modeling of infection spread, including our own earlier work, in several ways. First, it makes explicit the grapical modeling of spread in multiple species/breeds using a layered network structure. Second, it carefully models a family of surveillance and control capabilities for zoonoses, and brings to bear a

We find most illustrative to introduce the proposed layered-network modeling framework for zoonotic diseases using a case study of brucellosis transmission, both to allow careful illustration of how disease-specific characteristics can be captured using the modeling framework, and to permit development of quantitative control policies for this particular neglected zoonotic disease. To begin, Let us briefly overview the methods for modeling brucellosis spread and designing control strategies. In many communities with high brucellosis prevalence (e.g. in West Africa), transmission dynamics and control capabilities/costs vary significantly from herd to herd, because of variabilities in agricultural system, herd and pasture sizes, accessibility, and financial resources

The section is organized as follows. After a brief overview of the brucellosis case study, we present the nominal network model for brucellosis transmission among multiple animal herds and human subpopulations that was developed. Subsequently, we present the modeling of control efforts and costs. Finally, methods for control design and model identification are discussed briefly.

Brucellosis is a zoonotic bacterium with several species that cause illness in livestock (including cattle, small ruminants, pigs, camels, and bison, see e.g.

The most common strain of

The developed model tracks brucellosis prevalence (numbers of infectives) in individual herds for multiple animal species and prevalence in human subpopulations (divided by susceptibility). Specifically, brucellosis infection is modeled in

Here, we first develop a predictive mathematical model for the dynamics of the

Next, the model captures transmission from other herds of the same species. Specifically, consider transmission from a herd

The rate of infection from herd

The model also captures changes in infection counts in a herd due to 1) natural death or (rarely) remission, and 2) incorporation of infected animals from outside the modeled system (e.g., through purchase of the animals by a pastoralist). The rate of decrease in infectives due to death/remission is well-modeled as proportional to the number of infectives in the herd. Specifically, in herd

In the case where the linear model is in force, the following family of differential equations for nominal brucellosis transmission in the livestock/wildlife population is obtained, by summing the transmission rates to each herd and subtracting the natural death rate:

A second core aspect of the nominal model is the representation of brucellosis transmission from livestock to the human population prior to control. As with the animal model, infection of the human population is captured through representation of the infection

The nominal model for brucellosis transmission described above can be viewed as a

A spread graph for brucellosis transmission is illustrated, for an example with several cattle herds, wildlife that serve as a reservoir for the bacterium, and two human subpopulations.

In that the spread graph specifies the contact network (i.e., the transmission rates) among modeled populations, we note that specifying the spread graph accurately for examples of interest is of critical importance. Let us stress that, in general, three different approaches may be used to specify the spread graph: 1) the control-strategy planner may use intimate knowledge of an agricultural system to identify herd contacts and postulate transmission rates; 2) a formal system-identification methodology may be used to infer the spread graph; or 3) for large-scale networks, existing characterizations of typical network topologies (for instance, “small-world topologies” wherein all subunits are within a few steps of each other in the graph despite relatively spare connectivity) can be used to generate plausible spread graphs. In the illustrative example presented in the results, we will motivate and describe the specification of the spread graph that we have used for this example.

Next, the nominal multi-group model for brucellosis transmission is enhanced to explicitly represent surveillance and control capabilities 1) within the animal population and 2) for transmission from animals to humans.

In the animal population, control of spread is achieved through several means, including 1) vaccination, 2) surveillance for and culling of diseased animals, 3) limitation of herd commingling (through restriction of trade or movement of livestock), and 4) improvement in sanitation at farms (e.g.,

Vaccines for brucellosis have been developed for cattle and small ruminants. Application of a vaccine to a herd serves to make the members of the herd less susceptible to the disease. Vaccination capabilities are abstractly incorporated into our model as follows: vaccination of a particular herd

Surveillance for brucellosis is non-trivial, since the signs of the disease are non-specific. Very broadly, to test for brucellosis with high fidelity requires either identification of the organism cultured from fluid or tissue samples, or identification of infected animals through serological tests; these tests typically trade off specificity and sensitivity, and may be costly. Typically, surveillance is used for control by depopulating livestock that are identified as infected. Thus, a surveillance and culling program is abstractly modeled as one that removes infectives from herds at some rate. Specifically, the rate at which infectives are removed from herd

Two further methods for mitigating brucellosis spread among livestock are 1) limitation of commingling among herds, and 2) improved sanitation. Like vaccination and surveillance/culling controls, these further control methods can be naturally modeled as altering the nominal model and so the nominal interaction graph in various ways. Specifically, reducing commingling among herds will serve to reduce the interaction weights

Brucellosis is a severe and in many cases incapacitating disease in humans, whose prevention is paramount. No vaccine for brucellosis exists for humans. Instead, transmission from livestock to humans is primarily limited in three ways: 1) pasteurization of milk products and proper preparation of meat products; 2) brucellosis surveillance and control programs; and 3) reduction of transmission to those handling livestock products through improved sanitation and training in safe handling of livestock products. These control methods fundamentally serve to reduce the transmission rates to one or more human subpopulations from some (or possibly all) of the animal herds. That is, the controls serve to scale the nominal rates of transmission

Broadly, designing effective control strategies requires achieving a proper tradeoff between the costs resulting from disease prevalence and the costs of control. Altenatively, control design can be viewed as an effort to minimize disease prevalence or associated costs, while using limited control resources. Thus, to properly design control strategies, models are needed for the cost of infection as well as the costs associated with using the various control actions. As a framework for cost modeling, each infected individual in each herd and human subpopulation at each particular time is viewed as incurring a cost, and these costs are summed to obtain the full infection cost. Precisely, the infection cost at a particular time

In addition to the costs incurred by brucellosis spread, the cost of control is modeled. For each possible control action (e.g., surveillance using serological tests followed by culling), the cost is assumed to have three parts: 1) a global overhead cost for the infrastructure needed to implement the control strategy, 2) a per-herd (or per- human subpopulation) cost representing the additional cost of providing controls to the herd (which may depend on the size and type of herd), and 3) a cost per infected individual in the herd. For instance, for a testing and culling strategy, there is a fixed cost for developing the laboratory infrastructure for testing, a cost associated with testing each herd, and a cost to the farmer for each culled animal. Finally, the total cost of control is calculated as the sum of the costs for each implemented control action. The total cost of control during a period of time is denoted by

A control strategy's overall performance is captured by the sum of the infection and control costs over a period of time,

The dynamics of the spread and control model, and associated cost model, are illustrated in the following examples.

The layered network model for zoonotic disease transmission that we have introduced above, using brucellosis as a case study, is promising as a tool for systematic design of mulitfaceted control capabilities. Precisely, the formulation permits application of some new methodologies for

Before describing the methodologies for control design and parameterization, let us stress that the modeling framework permits simple simulation and analysis of the spread dynamics, for a specified control policy. In particular, we note that both the linear and nonlinear differential-equation models can be solved numerically using standard derivative-approximation methods, and can be implemented readily using e.g. the Matlab software. The linearized model also readily permits analysis of the dynamics, including closed-form computation of steady-state and transient dynamics, and computation of features of the dynamics such as the basic reproductive number. This analysis is based directly on the classical analysis of linear systems or linear differential equations; we kindly refer readers who are not familiar with the classical methodology to see

Control-systems engineers have recently engaged in a major effort to design surveillance and control capabilities in

Previous work has already applied network controller design methods to spread-control problems, albeit for simpler models than the one considered here

Initially, classical optimization machinery is applied to the design problem. That is, we consider the problem of minimizing the full cost

Numerical solution of the Lagrangian can be achieved via numerous standard recursive solvers, such as are available in the Matlab software suite (or can easily be developed by hand). Under certain broad conditions on the network topology and cost definition (which guarantee convexity of the optimization problem), these methods can be shown to obtain a globally optimal design.

Often, it is much more instructive to determine structural characteristics of high-performance or optimal designs, rather than to obtain a numerical design. This is especially true of the zoonotic disease control problems considered here, for which socio-political and geographic concerns may make implementation of a precise policy difficult and model parameterization difficult, so that an optimal design may only serve as a guideline rather than an implementable routine. Additionally, by understanding characteristics of high-performance designs, we can make explicit the role played by the network's graph topology in achieving control. For reasons such as these ones, a primary focus of our ongoing work has been to been characterize the system/graph structure of high-performance designs. This structural characterization of high-performance design can be achieved as follows: from the Lagrangian, relationships between the optimal design parameters and the state matrix of the linearized dynamics can be obtained: that is, the dynamics imposes a structure on the optimal resource allocation. Noting that the dynamics of the infection spread are specified using the network graph, we can thus immediately connect the optimal values of the design parameters with the structure of the network graph (e.g., the degrees of the vertices, etc). Further, we can in turn relate the dynamics upon application of the optimal controller to the nominal graph's structure. In this way, we obtain very simple graphical insights into high-performance control designs, that are robust to implementation and modeling limitations. We will present the outcomes of this design methodology in the

In the interest of conciseness and readability, we have not included many details of the mathematical techniques used and especially their justifications: we kindly ask the reader to see

As we will describe in detail in the

We have considered the model-identification task for the brucellosis spread application, from two viewpoints: first, from the viewpoint of identifying a model for a single herd using which a network model can be constructed; and, second, from the viewpoint of directly identifying the full network dynamics or important statistics thereof. In particular, a single-herd model has been parameterized using time-snapshot herd-size and seroprevalence data from several countries in West Africa and the Jackon bison herd, as well as using temporal data from the Jackson Bison herd

We find it most illustrative to present the results of the modeling and control design methodology introduced here, in the context of a specific example of brucellosis spread. Specifically, model dynamics and control design are illustrated in an example that is representative at a small scale of largely agricultural communities with both transhumance and sedentary herding (such as in Ethiopia or the Sudan). Specifically, a non-intensive agricultural system with one breed of cattle, comprising

The nominal spread graph, which illustrates the contact network prior to control, is shown in

Brucellosis among herds of cattle is simulated in a non-intensive agricultural system, in the nominal case that controls are not used. In this small example, a network of

The nominal model for brucellosis spread is again simulated for the 20-herd example, without control, but the disease is initiated in a small and sedentary herd (herd 20). The disease again becomes prevalent, but the spread is much slower. Interestingly, brucellosis becomes more prevalent in the large nomadic herds than the small or sedentary ones even though the infection was initiated in a small sedentary herd. This model characteristic matches with field measurements for brucellosis prevalence in pastoralist communities, e.g. in the Tigray region of Ethiopia

Several strategies for vaccination, namely ones that are targeted to tranhumance herds vs. ones that distribute resources between tranhumance and sedentary herds, are compared in the 20-herd example. In particular, herds are modeled as being vaccinated at a certain frequency, with the cost needed for each vaccination of a herd assumed identical in this illustrative example. (The relative cost of vaccinating sedentary and transhumance herds may sometimes be rather varied in practice: in some settings, transhumance herds may be incredibly difficult to reach for vaccination, while in other cases medical personnel may be able to take advantage of their mobility by placing vaccination capabilities at a location frequented by these herds. We take the simplest assumption of identical cost for ease in illustration, but the design can be achieved for other cost structures also.) Three strategies with identical total resource cost are compared: one that vaccinates only the transhumance herds, one that vaccinates sedentary and transhumance herds, and one that vaccinates only the sedentary herds. For the three vaccination strategies, the basic reproductive ratio

A uniform vaccination policy is compared with one that targets only the transhumance herds and one that only targets the sedentary herds. At low resource levels, the vaccination policy targeted to the transhumance herds outperforms the uniform one, while the uniform policy becomes more effective at higher resource levels. Both outperform sedentary-herd vaccination.

It is hypothesized that improved surveillance techniques that permit fast and cheap decentralized surveillance/culling could significantly improve brucellosis control. The model permits evaluation of the benefits of faster surveillance/culling. Here, a uniform surveillance/culling policy at all herds is considered for the

Improvement of brucellosis surveillance procedures so as to permit fast/cheap distributed surveillance and culling is an important policy goal. The model permits computation of infection costs as a function of the surveillance and culling rate, and hence indicates the cost benefit of improving surveillance/culling techniques.

Here, we consider a homogeneous surveillance/culling procedure for all herds, and calculate the integrated infection size (the number of infected animals integrated over the duration of the infection) for the 20-herd network, for a random initial condition. In this example, a culling rate of at least 0.43 (43% of the infected population per annum) is needed to make

The modeling methodology that we have introduced permits systematic optimization of control resources, according to the methods outlined above. For illustration, we have found the optimal allocation of vaccination resources to minimize the basic reproductive number in the illustrative example, as a function of the level of available resources. The optimal basic reproductive number, and the fraction of the resources that are allocated to the transhumance herds, are plotted in

In the upper plot, The basic reproductive number when the optimal vaccination policy is used is shown, for the twenty-herd example. Here, the three light, solid lines indicate the performance of the only-transhumance, only-sedentary, and uniform vaccination policies as developed in

The rate of infection in the human pastoralist subpopulation in a brucellosis outbreak with an initial low level of infection in the cattle population is shown. The upper figure shows the case where no control is used, and the lower shows the case where an optimal animal-vaccination policy is applied. It is worth noting that the nonlinear model for spread was used to simulate the infection rate, since the linear model quickly becomes inaccurate in the no-control case.

The basic reproductive number for the optimal surveillance/control policy is shown as a function of the resource level, for the twenty-herd example.

The subdivision of resources between animal-level control policies and animal-to-human control policies is examined, in the

The techniques described above for identification of the bovine brucellosis model from snapshot and time-course described were applied. The results of the identification are displayed in

Using a heuristic method, a nonlinear SIR model for brucellosis transmission within a herd has been developed, using snapshot herd-size and seroprevalence data from several West-African countries as well as from the Jackson Bison Herd (JBH). The ability of the model to predict seroprevalence vs. herd size is shown

An SIR model for brucellosis transmission is identified, based on time-course data from the Jackson bison herd upon initiation of a vaccination program. This simple model is not as accurate as the multi-state model described in

We have found it convenient to illustrate the results of our methodology using an example. Let us stress, however, that the methodology has much broader application: the developed network modeling framework can be used to quantitatively capture transmission of a zoonotic disease within a particular agricultural system or community, and in turn can be used for control and surveillance policy design for such transmission. In addition to permitting concrete designs for examples, the tractability of the model also allows us to obtain broad insights into control and surveillance design for brucellosis (and other zoonotic diseases), and conceptual insight into model parameterization from historical data. Let us discuss some broader insights obtained using the modeling methodology, drawing on the example developed in the

The network control methods allow a comprehensive study of the brucellosis controller design problem, and more generally of zooonotic-disease control. Below is a list of several aspects of the control design task that can be addressed using these methods, as well as a few of the key insights obtained through the design. These insights are envisioned as informing policy decision-making.

Let us present some insights on the model identification methodology, using the results displayed in

The snapshot data permits characterization of the ratio between the per-individual infection rate and the remission rate (through recovery or death and replacement), for a single herd. As seen in

The time-course data further permits inference of the absolute per-individual infection rate, so that (together with the snapshot data-based analysis) a full single-herd model can be identified.

In parameterizing the single-herd model, we did not have any quantitative data on the rates of infection in each herd from outside the herd: thus, the parameterization has been done assuming an identical rate of interaction/infection from outside the herd. This crude assumption clearly yields error in the parameterization result. For the data from West African herds, we have some qualitative insight into herds that have more significant interactions with other ones (e.g., share common feed lots) and hence are more susceptible to infection from outside the herd. In

Given the very limited and highly variable data available for model parameterization, the robustness of the model to parameter variations is of importance. As a first step in this direction, we have studied the ability of the model to predict single-herd brucellosis prevalence, when there is up to

The uncertainty inherent to the data used for model parameterization highlights the importance of considering stochastics in modeling disease spread. Both intrinsic variability in transmission and uncertainties/variability in model parameters may significantly impact the model dynamics and hence modulate control design (see e.g.

Given the very limited and uncertain data available for model parameterization, the robustness of the model to parameter variations is of importance. As a first step in this direction, we have studied the ability of the model to predict single-herd brucellosis prevalence, when there is up to

A network modeling methodology for capturing the spread of zoonotic agents at a herd/subpopulation granularity has been introduced, and used to compare and design control strategies for stopping the spread of zoonoses. The introduced methodology has been developed in detail in the context of a case study, namely modeling and control of brucellosis spread in animal and human populations. Parameter identification of the model from historical data has been pursued.

This modeling and controller design effort should be viewed as a foundational step toward obtaining comprehensive policies for controlling zoonoses from mathematical models: the policies suggested by our methodology must be tested in experimental herds, and the social and political ramifications of control policies and spread mitigation must be considered carefully in defining costs for a zoonosis of interest. It is also important to stress that a wide range of experimental and conceptual methodologies from outside the mathematical-modeling domain must be brought to bear to address policy design for zoonotic diseases, and that mathematical modeling efforts (and policy design more generally) are complementary to core advances in epidemiological methods. Nevertheless, we believe that some promising outcomes have been obtained through this foundational study.

Network modeling can give a clear pictorial representation of the spread of a zoonotic disease among subgroups of one or more species. The layered network model that we have developed also permits simulation and simple quantitative analysis of the spread dynamics. Also, a variety of control measures can naturally be captured in the layered network modeling framework.

The network modeling approach that we have put forth here also permits systematic and quantitative comprehensive design of resources for control in the animal and human population. Specifically, for the linearized network model studied here, recently-developed tools from the network control theory literature can be applied to design heterogeneous control capabilities to minimize a resource and spread cost. We stress that much work still needs to be done to verify that the abstract controls suggested by the design can be implemented in reality, and can achieve performance similar to that predicted by the model.

Network modeling approaches provide sufficient detail to permit comparison and design of heterogeneous control strategies, for instance ones that use surveillance and culling vs. ones that use vaccination, as well as ones that cater to the topology of interactions among the herds. Both qualitative and quantitative predictions and designs can be obtained.

Although optimal controller design requires a precise model of spread dynamics and of costs, our design methodology can provide useful insights even when such precise models are unavailable. First, the methodology provides simple insights into good designs–for instance, that more resources should be placed in large and tightly-connected herds–that do not require a precise model to implement. Second, the designs obtained through the methodology show some degree of robustness to variations in model parameters.

We have given some preliminary results on parameterizing the network model from data, and have also identified some challenges in fully addressing the parameterization problem. Given the very limited data available on the spread of various zoonotic diseases, model parameterization remains a significant challenge in using methods such as the ones proposed here. Nevertheless, our preliminary efforts on parameterization show promise, and also the model displays a degree of robustness to inaccuracies in parameterization.

Clearly, when transmission among small heterogeneous groups is considered (as in our model), stochastics in transmission very significantly impact the spread dynamics. In particular, both intrinsic variabilities in the interactions among individuals that cause spread, and also environmental uncertainties that modify transmission patterns, can critically impact the spread dynamics. Not surprisingly, prevalences of zoonotic diseases such as brucellosis show a large herd-to-herd variability, that can only be explained by considering stochastics in transmission. We have chosen to exclude consideration of stochastics in transmission in this first design effort, 1) because of the great difficulty in parameterizing such stochastic models with limited data, and 2) because our key focus here is on gaining very simple insights into policy design for which even the deterministic multi-group model may be sufficient. We consider the development of stochastic models for transmission to be an outstanding research task of critical importance.

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