1
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Hassan A, Tapp ZA, Tran DK, Rychtář J, Taylor D. Mathematical model of rabies vaccination in the United States. Theor Popul Biol 2024; 157:47-54. [PMID: 38521097 DOI: 10.1016/j.tpb.2024.03.004] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/09/2023] [Revised: 03/15/2024] [Accepted: 03/20/2024] [Indexed: 03/25/2024]
Abstract
Rabies is one of the oldest viral diseases and it has been present on every continent except Antarctica. Within the U.S. human rabies cases are quite rare. In the eastern USA, raccoons are the main reservoir hosts and pet vaccination serves as an important barrier against human rabies exposure. In this paper, we develop a compartmental model for rabies transmission amongst raccoons and domestic pets. We find the disease-free equilibria, reproduction numbers for the raccoons and domestic pets. We also determine the vaccination coverage/rates, both for raccoons and pets, needed to achieve the elimination of rabies.
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Affiliation(s)
- Annalise Hassan
- Department of Biology, Virginia Commonwealth University, Richmond, VA 23284, USA.
| | - Zoe A Tapp
- Department of Biology, Virginia Commonwealth University, Richmond, VA 23284, USA.
| | - Dan K Tran
- Department of Computer Science, Virginia Commonwealth University, Richmond, VA 23284, USA.
| | - Jan Rychtář
- Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA 23284, USA.
| | - Dewey Taylor
- Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA 23284, USA.
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2
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Igoe M, Casagrandi R, Gatto M, Hoover CM, Mari L, Ngonghala CN, Remais JV, Sanchirico JN, Sokolow SH, Lenhart S, de Leo G. Reframing Optimal Control Problems for Infectious Disease Management in Low-Income Countries. Bull Math Biol 2023; 85:31. [PMID: 36907932 PMCID: PMC10008208 DOI: 10.1007/s11538-023-01137-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/27/2022] [Accepted: 02/20/2023] [Indexed: 03/14/2023]
Abstract
Optimal control theory can be a useful tool to identify the best strategies for the management of infectious diseases. In most of the applications to disease control with ordinary differential equations, the objective functional to be optimized is formulated in monetary terms as the sum of intervention costs and the cost associated with the burden of disease. We present alternate formulations that express epidemiological outcomes via health metrics and reframe the problem to include features such as budget constraints and epidemiological targets. These alternate formulations are illustrated with a compartmental cholera model. The alternate formulations permit us to better explore the sensitivity of the optimal control solutions to changes in available budget or the desired epidemiological target. We also discuss some limitations of comprehensive cost assessment in epidemiology.
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Affiliation(s)
- Morganne Igoe
- Department of Mathematics, University of Tennessee, Knoxville, TN, USA.
| | - Renato Casagrandi
- Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Milano, Italy
| | - Marino Gatto
- Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Milano, Italy
| | - Christopher M Hoover
- Division of Environmental Health Sciences, University of California, Berkeley, Berkeley, CA, USA
| | - Lorenzo Mari
- Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Milano, Italy
| | | | - Justin V Remais
- Division of Environmental Health Sciences, University of California, Berkeley, Berkeley, CA, USA
| | - James N Sanchirico
- Environmental Science and Policy, University of California, Davis, Davis, CA, USA
| | - Susanne H Sokolow
- Stanford Program for Diseases Ecology, Health and the Environment, Stanford University, Pacific Grove, CA, USA
| | - Suzanne Lenhart
- Department of Mathematics, University of Tennessee, Knoxville, TN, USA
| | - Giulio de Leo
- Department of Earth System Science and Department of Oceans, Hopkins Marine Station, Stanford Doerr School of Sustainability, Stanford University, Pacific Grove, CA, USA
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3
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Campo VN, Palacios JL, Nagahashi H, Oh H, Rychtář J, Taylor D. A game-theoretic model of rabies in domestic dogs with multiple voluntary preventive measures. J Math Biol 2022; 85:57. [PMID: 36264390 PMCID: PMC9583067 DOI: 10.1007/s00285-022-01826-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/12/2022] [Revised: 08/14/2022] [Accepted: 10/02/2022] [Indexed: 11/26/2022]
Abstract
Game theory is now routinely applied to quantitatively model the decision making of individuals presented with various voluntary actions that can prevent a given disease. Most models consider only a single preventive strategy and the case where multiple preventative actions are available is severely understudied. In our paper, we consider a very simple SI compartmental model of rabies in the domestic dog population. We study two choices of the dog owners: to vaccinate their dogs or to restrict the movements of unvaccinated dogs. We analyze the relatively rich patterns of Nash equilibria (NE). We show that there is always at least one NE at which the owners utilize only one form of prevention. However, there can be up to three different NEs at the same time: two NEs at which the owners use exclusively only the vaccination or movement restriction, and the third NE when the owners use both forms of prevention simultaneously. However, we also show that, unlike the first two types of NEs, the third kind of NE is not convergent stable.
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Affiliation(s)
- Vince N. Campo
- School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85281 USA
| | - John Lawrence Palacios
- Division of Mathematics and Computer Science, University of Guam, Mangilao, GU 96913 USA
- Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA 23284 USA
| | - Hideo Nagahashi
- Division of Mathematics and Computer Science, University of Guam, Mangilao, GU 96913 USA
| | - Hyunju Oh
- Division of Mathematics and Computer Science, University of Guam, Mangilao, GU 96913 USA
| | - Jan Rychtář
- Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA 23284 USA
| | - Dewey Taylor
- Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA 23284 USA
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4
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On the optimal control of SIR model with Erlang-distributed infectious period: isolation strategies. J Math Biol 2021; 83:36. [PMID: 34550465 PMCID: PMC8456197 DOI: 10.1007/s00285-021-01668-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/18/2021] [Revised: 07/22/2021] [Accepted: 09/08/2021] [Indexed: 11/15/2022]
Abstract
Mathematical models are formal and simplified representations of the knowledge related to a phenomenon. In classical epidemic models, a major simplification consists in assuming that the infectious period is exponentially distributed, then implying that the chance of recovery is independent on the time since infection. Here, we first attempt to investigate the consequences of relaxing this assumption on the performances of time-variant disease control strategies by using optimal control theory. In the framework of a basic susceptible–infected–removed (SIR) model, an Erlang distribution of the infectious period is considered and optimal isolation strategies are searched for. The objective functional to be minimized takes into account the cost of the isolation efforts per time unit and the sanitary costs due to the incidence of the epidemic outbreak. Applying the Pontryagin’s minimum principle, we prove that the optimal control problem admits only bang–bang solutions with at most two switches. In particular, the optimal strategy could be postponing the starting intervention time with respect to the beginning of the outbreak. Finally, by means of numerical simulations, we show how the shape of the optimal solutions is affected by the different distributions of the infectious period, by the relative weight of the two cost components, and by the initial conditions.
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5
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Mysterud A, Viljugrein H, Rolandsen CM, Belsare AV. Harvest strategies for the elimination of low prevalence wildlife diseases. ROYAL SOCIETY OPEN SCIENCE 2021; 8:210124. [PMID: 33959374 PMCID: PMC8074915 DOI: 10.1098/rsos.210124] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 01/25/2021] [Accepted: 02/03/2021] [Indexed: 06/12/2023]
Abstract
The intensive harvesting of hosts is often the only practicable strategy for controlling emerging wildlife diseases. Several harvesting approaches have been explored theoretically with the objective of lowering transmission rates, decreasing the transmission period or specifically targeting spatial disease clusters or high-risk demographic groups. Here, we present a novel model-based approach to evaluate alternative harvest regimes, in terms of demographic composition and rates, intended to increase the probability to remove all infected individuals in the population during the early phase of an outbreak. We tested the utility of the method for the elimination of chronic wasting disease based on empirical data for reindeer (Rangifer tarandus) in Norway, in populations with (Nordfjella) and without (Hardangervidda) knowledge about exact disease prevalence and population abundance. Low and medium harvest intensities were unsuccessful in eliminating the disease, even at low prevalence. High-intensity harvesting had a high likelihood of eliminating the disease, but probability was strongly influenced by the disease prevalence. We suggest that the uncertainty about disease prevalence can be mitigated by using an adaptive management approach: forecast from models after each harvest season with updated data, derive prevalence estimates and forecast further harvesting. We identified the problems arising from disease surveillance with large fluctuations in harvesting pressure and hence sample sizes. The elimination method may be suitable for pathogens that cause long-lasting infections and with slow epidemic growth, but the method should only be attempted if there is a low risk of reinfection, either by a new disease introduction event (e.g. dispersing hosts) or due to environmental reservoirs. Our simulations highlighted the short time window when such a strategy is likely to be successful before approaching near complete eradication of the population.
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Affiliation(s)
- Atle Mysterud
- Centre for Ecological and Evolutionary Synthesis (CEES), Department of Biosciences, University of Oslo, PO Box 1066, Blindern, 0316 Oslo, Norway
| | - Hildegunn Viljugrein
- Centre for Ecological and Evolutionary Synthesis (CEES), Department of Biosciences, University of Oslo, PO Box 1066, Blindern, 0316 Oslo, Norway
- Norwegian Veterinary Institute, PO Box 750 Sentrum, 0106 Oslo, Norway
| | - Christer M. Rolandsen
- Norwegian Institute for Nature Research (NINA), PO Box 5685 Torgarden, 7485 Trondheim, Norway
| | - Aniruddha V. Belsare
- Boone and Crockett Quantitative Wildlife Center, Department of Fisheries and Wildlife, Michigan State University, East Lansing, MI, USA
- Department of Biology, Emory College of Arts and Sciences, Emory University, Atlanta, GA, USA
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6
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Bozzuto C, Schmidt BR, Canessa S. Active responses to outbreaks of infectious wildlife diseases: objectives, strategies and constraints determine feasibility and success. Proc Biol Sci 2020; 287:20202475. [PMID: 33234080 DOI: 10.1098/rspb.2020.2475] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/21/2022] Open
Abstract
Emerging wildlife diseases are taking a heavy toll on animal and plant species worldwide. Mitigation, particularly in the initial epidemic phase, is hindered by uncertainty about the epidemiology and management of emerging diseases, but also by vague or poorly defined objectives. Here, we use a quantitative analysis to assess how the decision context of mitigation objectives, available strategies and practical constraints influences the decision of whether and how to respond to epidemics in wildlife. To illustrate our approach, we parametrized the model for European fire salamanders affected by Batrachochytrium salamandrivorans, and explored different combinations of conservation, containment and budgetary objectives. We found that in approximately half of those scenarios, host removal strategies perform equal to or worse than no management at all during a local outbreak, particularly where removal cannot exclusively target infected individuals. Moreover, the window for intervention shrinks rapidly if an outbreak is detected late or if a response is delayed. Clearly defining the decision context is, therefore, vital to plan meaningful responses to novel outbreaks. Explicitly stating objectives, strategies and constraints, if possible before an outbreak occurs, avoids wasting precious resources and creating false expectations about what can and cannot be achieved during the epidemic phase.
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Affiliation(s)
- Claudio Bozzuto
- Wildlife Analysis GmbH, Oetlisbergstrasse 38, 8053 Zurich, Switzerland
| | - Benedikt R Schmidt
- Institut für Evolutionsbiologie und Umweltwissenschaften, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland.,info fauna karch, UniMail, Bâtiment G, Bellevaux 51, 2000 Neuchâtel, Switzerland
| | - Stefano Canessa
- Wildlife Health Ghent, Department of Pathology, Bacteriology and Avian Diseases, Faculty of Veterinary Medicine, Ghent University, Salisburylaan 133, 9820 Merelbeke, Belgium
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7
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Mysterud A, Benestad SL, Rolandsen CM, Våge J. Policy implications of an expanded chronic wasting disease universe. J Appl Ecol 2020. [DOI: 10.1111/1365-2664.13783] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Atle Mysterud
- Centre for Ecological and Evolutionary Synthesis (CEES) Department of Biosciences University of Oslo Oslo Norway
| | - Sylvie L. Benestad
- OIE Reference Laboratory for CWD Norwegian Veterinary Institute Oslo Norway
| | | | - Jørn Våge
- OIE Reference Laboratory for CWD Norwegian Veterinary Institute Oslo Norway
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8
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Bolzoni L, Bonacini E, Della Marca R, Groppi M. Optimal control of epidemic size and duration with limited resources. Math Biosci 2019; 315:108232. [PMID: 31330135 DOI: 10.1016/j.mbs.2019.108232] [Citation(s) in RCA: 22] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/20/2018] [Revised: 07/18/2019] [Accepted: 07/18/2019] [Indexed: 11/18/2022]
Abstract
The total number of infections (epidemic size) and the time needed for the infection to go extinct (epidemic duration) represent two of the main indicators for the severity of infectious disease epidemics in human and livestock. However, few attempts have been made to address the problem of minimizing at the same time the epidemic size and duration from a theoretical point of view by using optimal control theory. Here, we investigate the multi-objective optimal control problem aiming to minimize, through either vaccination or isolation, a suitable combination of epidemic size and duration when both maximum control effort and total amount of resources available during the entire epidemic period are limited. Application of Pontryagin's Maximum Principle to a Susceptible-Infected-Removed epidemic model, shows that, when the resources are not sufficient to maintain the maximum control effort for the entire duration of the epidemic, the optimal vaccination control admits only bang-bang solutions with one or two switches, while the optimal isolation control admits only bang-bang solutions with one switch. We also find that, especially when the maximum control effort is low, there may exist a trade-off between the minimization of the two objectives. Consideration of this conflict among objectives can be crucial in successfully tackling real-world problems, where different stakeholders with potentially different objectives are involved. Finally, the particular case of the minimum time optimal control problem with limited resources is discussed.
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Affiliation(s)
- Luca Bolzoni
- Risk Analysis and Genomic Epidemiology Unit, Istituto Zooprofilattico Sperimentale della Lombardia e dell'Emilia Romagna, Via dei Mercati 13, Parma 43126, Italy.
| | - Elena Bonacini
- Department of Mathematical, Physical and Computer Sciences, University of Parma, Parco Area delle Scienze 53/A, Parma 43124, Italy
| | - Rossella Della Marca
- Department of Mathematical, Physical and Computer Sciences, University of Parma, Parco Area delle Scienze 53/A, Parma 43124, Italy
| | - Maria Groppi
- Department of Mathematical, Physical and Computer Sciences, University of Parma, Parco Area delle Scienze 53/A, Parma 43124, Italy
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9
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Optimal time-profiles of public health intervention to shape voluntary vaccination for childhood diseases. J Math Biol 2018; 78:1089-1113. [PMID: 30390103 DOI: 10.1007/s00285-018-1303-1] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/18/2017] [Revised: 09/10/2018] [Indexed: 10/27/2022]
Abstract
In order to seek the optimal time-profiles of public health systems (PHS) Intervention to favor vaccine propensity, we apply optimal control (OC) to a SIR model with voluntary vaccination and PHS intervention. We focus on short-term horizons, and on both continuous control strategies resulting from the forward-backward sweep deterministic algorithm, and piecewise-constant strategies (which are closer to the PHS way of working) investigated by the simulated annealing (SA) stochastic algorithm. For childhood diseases, where disease costs are much larger than vaccination costs, the OC solution sets at its maximum for most of the policy horizon, meaning that the PHS cannot further improve perceptions about the net benefit of immunization. Thus, the subsequent dynamics of vaccine uptake stems entirely from the declining perceived risk of infection (due to declining prevalence) which is communicated by direct contacts among parents, and unavoidably yields a future decline in vaccine uptake. We find that for relatively low communication costs, the piecewise control is close to the continuous control. For large communication costs the SA algorithm converges towards a non-monotone OC that can have oscillations.
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10
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Lolika PO, Modnak C, Mushayabasa S. On the dynamics of brucellosis infection in bison population with vertical transmission and culling. Math Biosci 2018; 305:42-54. [PMID: 30138637 DOI: 10.1016/j.mbs.2018.08.009] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/15/2017] [Revised: 08/10/2018] [Accepted: 08/18/2018] [Indexed: 11/17/2022]
Abstract
We introduce a new mathematical modeling framework that seek to improve our quantitative understanding of the influence of chronic brucellosis and culling control on brucellosis dynamics in periodic and non-periodic environments. We conduct both epidemic and endemic analysis, with a focus on the threshold dynamics characterized by the basic reproduction numbers. In addition, we also perform an optimal control study to explore optimal culling strategy in periodic and non-periodic environment.
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Affiliation(s)
- Paride O Lolika
- Department of Mathematics, University of Zimbabwe, P.O. Box MP 167, Harare, Zimbabwe; Department of Mathematics, University of Juba, P.O. Box 82 Juba, Central Equatoria, South Sudan
| | - Chairat Modnak
- Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
| | - Steady Mushayabasa
- Department of Mathematics, University of Zimbabwe, P.O. Box MP 167, Harare, Zimbabwe.
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Bolzoni L, Bonacini E, Soresina C, Groppi M. Time-optimal control strategies in SIR epidemic models. Math Biosci 2017; 292:86-96. [PMID: 28801246 PMCID: PMC7094293 DOI: 10.1016/j.mbs.2017.07.011] [Citation(s) in RCA: 45] [Impact Index Per Article: 6.4] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/06/2016] [Revised: 04/11/2017] [Accepted: 07/14/2017] [Indexed: 11/25/2022]
Abstract
We investigate the time-optimal control problem in SIR (Susceptible-Infected-Recovered) epidemic models, focusing on different control policies: vaccination, isolation, culling, and reduction of transmission. Applying the Pontryagin's Minimum Principle (PMP) to the unconstrained control problems (i.e. without costs of control or resource limitations), we prove that, for all the policies investigated, only bang-bang controls with at most one switch are admitted. When a switch occurs, the optimal strategy is to delay the control action some amount of time and then apply the control at the maximum rate for the remainder of the outbreak. This result is in contrast with previous findings on the unconstrained problems of minimizing the total infectious burden over an outbreak, where the optimal strategy is to use the maximal control for the entire epidemic. Then, the critical consequence of our results is that, in a wide range of epidemiological circumstances, it may be impossible to minimize the total infectious burden while minimizing the epidemic duration, and vice versa. Moreover, numerical simulations highlighted additional unexpected results, showing that the optimal control can be delayed also when the control reproduction number is lower than one and that the switching time from no control to maximum control can even occur after the peak of infection has been reached. Our results are especially important for livestock diseases where the minimization of outbreaks duration is a priority due to sanitary restrictions imposed to farms during ongoing epidemics, such as animal movements and export bans.
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Affiliation(s)
- Luca Bolzoni
- Risk Analysis Unit, Istituto Zooprofilattico Sperimentale della Lombardia e dell'Emilia Romagna, Via dei Mercati 13, Parma 43126, Italy.
| | - Elena Bonacini
- Department of Mathematics and Computer Science, University of Parma, Parco Area delle Scienze 53/A, Parma 43124, Italy
| | - Cinzia Soresina
- Department of Mathematics "F. Enriques", University of Milano, Via C. Saldini 50, Milano 20133, Italy
| | - Maria Groppi
- Department of Mathematics and Computer Science, University of Parma, Parco Area delle Scienze 53/A, Parma 43124, Italy
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Elmore SA, Chipman RB, Slate D, Huyvaert KP, VerCauteren KC, Gilbert AT. Management and modeling approaches for controlling raccoon rabies: The road to elimination. PLoS Negl Trop Dis 2017; 11:e0005249. [PMID: 28301480 PMCID: PMC5354248 DOI: 10.1371/journal.pntd.0005249] [Citation(s) in RCA: 39] [Impact Index Per Article: 5.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/26/2022] Open
Abstract
Rabies is an ancient viral disease that significantly impacts human and animal health throughout the world. In the developing parts of the world, dog bites represent the highest risk of rabies infection to people, livestock, and other animals. However, in North America, where several rabies virus variants currently circulate in wildlife, human contact with the raccoon rabies variant leads to the highest per capita population administration of post-exposure prophylaxis (PEP) annually. Previous rabies variant elimination in raccoons (Canada), foxes (Europe), and dogs and coyotes (United States) demonstrates that elimination of the raccoon variant from the eastern US is feasible, given an understanding of rabies control costs and benefits and the availability of proper tools. Also critical is a cooperatively produced strategic plan that emphasizes collaborative rabies management among agencies and organizations at the landscape scale. Common management strategies, alone or as part of an integrated approach, include the following: oral rabies vaccination (ORV), trap-vaccinate-release (TVR), and local population reduction. As a complement, mathematical and statistical modeling approaches can guide intervention planning, such as through contact networks, circuit theory, individual-based modeling, and others, which can be used to better understand and predict rabies dynamics through simulated interactions among the host, virus, environment, and control strategy. Strategies derived from this ecological lens can then be optimized to produce a management plan that balances the ecological needs and program financial resources. This paper discusses the management and modeling strategies that are currently used, or have been used in the past, and provides a platform of options for consideration while developing raccoon rabies virus elimination strategies in the US.
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Affiliation(s)
- Stacey A. Elmore
- United States Department of Agriculture, National Wildlife Research Center, Fort Collins, Colorado, United States of America
| | - Richard B. Chipman
- United States Department of Agriculture, Wildlife Services, National Rabies Management Program, Concord, New Hampshire, United States of America
| | - Dennis Slate
- United States Department of Agriculture, Wildlife Services, National Rabies Management Program, Concord, New Hampshire, United States of America
| | - Kathryn P. Huyvaert
- Department of Fish, Wildlife, and Conservation Biology, Colorado State University, Fort Collins, Colorado, United States of America
| | - Kurt C. VerCauteren
- United States Department of Agriculture, National Wildlife Research Center, Fort Collins, Colorado, United States of America
| | - Amy T. Gilbert
- United States Department of Agriculture, National Wildlife Research Center, Fort Collins, Colorado, United States of America
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13
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Hoverman JT, Searle CL. Behavioural influences on disease risk: implications for conservation and management. Anim Behav 2016. [DOI: 10.1016/j.anbehav.2016.05.013] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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