# American Institute of Mathematical Sciences

2014, 11(6): 1275-1294. doi: 10.3934/mbe.2014.11.1275

## Modeling colony collapse disorder in honeybees as a contagion

 1 Departments of Mathematics and Curriculum & Instruction, University of Texas at Arlington, Box 19408, Arlington, TX 76019-0408, United States 2 Department of Mathematics, University of Texas at Arlington, Box 19408, Arlington, TX 76019-0408, United States

Received  March 2014 Revised  July 2014 Published  September 2014

Honeybee pollination accounts annually for over $14 billion in United States agriculture alone. Within the past decade there has been a mysterious mass die-off of honeybees, an estimated 10 million beehives and sometimes as much as 90% of an apiary. There is still no consensus on what causes this phenomenon, called Colony Collapse Disorder, or CCD. Several mathematical models have studied CCD by only focusing on infection dynamics. We created a model to account for both healthy hive dynamics and hive extinction due to CCD, modeling CCD via a transmissible infection brought to the hive by foragers. The system of three ordinary differential equations accounts for multiple hive population behaviors including Allee effects and colony collapse. Numerical analysis leads to critical hive sizes for multiple scenarios and highlights the role of accelerated forager recruitment in emptying hives during colony collapse. Citation: Christopher M. Kribs-Zaleta, Christopher Mitchell. Modeling colony collapse disorder in honeybees as a contagion. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1275-1294. doi: 10.3934/mbe.2014.11.1275 ##### References:  [1] G. Amdam, The hive bee to forager transition in honeybee colonies: The double repressor hypothesis, Journal of Theoretical Biology, 223 (2003), 451-464. doi: 10.1016/S0022-5193(03)00121-8. [2] F. S. Bodenheimer, Studies in animal populations II. Seasonal population-trends in the honey-bee, Quat. Rev. Zool., 12 (1937), 406-425. doi: 10.1086/394540. [3] CCD Steering Committee, Colony Collapse Disorder Progress Report, United States Department of Agriculture, June 2010. Retrieved 2013-08-28 from http://www.ars.usda.gov/is/br/ccd/ccdprogressreport2010.pdf. [4] D. Cramp, A Practical Manual of Beekeeping, How To Books, London, 2008. [5] B. Dahle, The role of Varroa destructor for honeybee colony losses in Norway, Journal of Apicultural Research, 49 (2010), 124-125. doi: 10.3896/IBRA.1.49.1.26. [6] G. DeGrandi-Hoffman, S. A. Roth, G. L. Loper and E. H. Erickson, Jr., Beepop: A honeybee population dynamics simulation model, Ecological Modelling, 45 (1989), 133-150. [7] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio$R_0$in models for infectious diseases in heterogeneous population, Journal of Mathematical Biology, 28 (1990), 365-382. doi: 10.1007/BF00178324. [8] L. Dornberger, C. Mitchell, B. Hull, W. Ventura, H. Shopp, C. Kribs-Zaleta, H. Kojouharov and J. Grover, Death of the Bees: A Mathematical Model of Colony Collapse Disorder,, Technical Report 2012-12, (): 2012. [9] H. J. Eberl, M. R. Frederick and P. G. Kevan, Importance of brood maintenance terms in simple models of the honeybee-Varroa destructor-acute bee paralysis virus complex, Electronic Journal of Differential Equations (EJDE) [electronic only] Conf. 19 (2010), 85-98. Available from: http://eudml.org/doc/232749. [10] N. Gallai, J. M. Salles, J. Settele and B. E. Vaissiere, Economic valuation of the vulnerability of world agriculture confronted with pollinator decline, Ecological Economics, 68 (2009), 810-821. doi: 10.1016/j.ecolecon.2008.06.014. [11] M. Higes, R. Martin-Hernandez, C. Botias, E. G. Bailon, A. V. Gonzalez-Porto, How natural infection by Nosema ceranae causes honeybee colony collapse, Environmental Microbiology, 10 (2008), 2659-2669. doi: 10.1111/j.1462-2920.2008.01687.x. [12] B. K. Hopkins, C. Herr and W. S. Sheppard, Sequential generations of honey bee (Apis mellifera) queens produced using cryopreserved semen, Reproduction, Fertility, and Development, 24 (2012), 1079-1083. [13] D. S. Khoury, Colony Collapse Disorder: Quantitative Models of Honey Bee Population Dynamics, Unpublished Thesis, School of Mathematics and Statistics, University of Sydney, Sydney, 2009. [14] D. S. Khoury, A. B. Barron and M. R. Myerscough, Modelling food and population dynamics in honey bee colonies, PLoS ONE, 8 (2013), e59084. doi: 10.1371/journal.pone.0059084. [15] D. S. Khoury, M. R. Myerscough and A. B. Barron, A quantitative model of honey bee colony population dynamics, PLoS ONE, 6 (2011), e18491. doi: 10.1371/journal.pone.0018491. [16] B. P. Oldroyd, What's killing American honeybees?, PLoS Biology, 5 (2007), e168. doi: 10.1371/journal.pbio.0050168. [17] H. Özbek, Arılar ve İnsektisitler, Bees and insecticides, Uludag Bee Journal, 10 (2010), 85-95. [18] J. S. Pettis, E. M. Lichtenberg, M. Andree, J. Stitzinger and R. Rose, Crop pollination exposes honeybees to pesticides which alters their susceptibility to the gut pathogen, Nosema Ceranae, PLoS ONE, 8 (2013), e70182. doi: 10.1371/journal.pone.0070182. [19] S. Russell, A. B. Barron and D. Harris, Dynamic modelling of honey bee (Apis mellifera) colony growth and failure, Ecological Modelling, 265 (2013), 158-169. doi: 10.1016/j.ecolmodel.2013.06.005. [20] T. Schmickl and K. Crailsheim, HoPoMo: A model of honeybee intracolonial population dynamics and resource management, Ecological Modelling, 204 (2007), 219-245. doi: 10.1016/j.ecolmodel.2007.01.001. [21] D. J. T. Sumpter and S. J. Martin, The dynamics of virus epidemics in Varroa-infested honey bee colonies, Journal of Animal Ecology, 73 (2004), 51-63. [22] S. Suryanarayanan and D. Kleinman, Be(e)coming experts: The controversy over insecticides in the honeybee colony collapse disorder, Social Studies Of Science (Sage Publications, Ltd.), 43 (2013), 215-240. [23] J. Traynor, Evaluating pollen production of plants, American Bee Journal, 141 (2001), 287-288. [24] U. S. Department of Agriculture (USDA) CCD Steering Committee, Colony Collapse Disorder Progress Report, USDA, 2010. Available from: http://www.ars.usda.gov/is/br/ccd/ccdprogressreport2010.pdf. [25] P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [26] D. VanEngelsdorp, J. Hayes Jr., R. M. Underwood and J. S. Pettis, A survey of honeybee colony losses in the United States, fall 2008 to spring 2009, Journal of Apicultural Research, 49 (2010), 7-14. [27] C. W. Whitfield, A-M. Cziko and G. E. Robinson, Gene expression profiles in the brain predict behavior in individual honey bees, Science, 302 (2003), 296-299. doi: 10.1126/science.1086807. show all references ##### References:  [1] G. Amdam, The hive bee to forager transition in honeybee colonies: The double repressor hypothesis, Journal of Theoretical Biology, 223 (2003), 451-464. doi: 10.1016/S0022-5193(03)00121-8. [2] F. S. Bodenheimer, Studies in animal populations II. Seasonal population-trends in the honey-bee, Quat. Rev. Zool., 12 (1937), 406-425. doi: 10.1086/394540. [3] CCD Steering Committee, Colony Collapse Disorder Progress Report, United States Department of Agriculture, June 2010. Retrieved 2013-08-28 from http://www.ars.usda.gov/is/br/ccd/ccdprogressreport2010.pdf. [4] D. Cramp, A Practical Manual of Beekeeping, How To Books, London, 2008. [5] B. Dahle, The role of Varroa destructor for honeybee colony losses in Norway, Journal of Apicultural Research, 49 (2010), 124-125. doi: 10.3896/IBRA.1.49.1.26. [6] G. DeGrandi-Hoffman, S. A. Roth, G. L. Loper and E. H. Erickson, Jr., Beepop: A honeybee population dynamics simulation model, Ecological Modelling, 45 (1989), 133-150. [7] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio$R_0$in models for infectious diseases in heterogeneous population, Journal of Mathematical Biology, 28 (1990), 365-382. doi: 10.1007/BF00178324. [8] L. Dornberger, C. Mitchell, B. Hull, W. Ventura, H. Shopp, C. Kribs-Zaleta, H. Kojouharov and J. Grover, Death of the Bees: A Mathematical Model of Colony Collapse Disorder,, Technical Report 2012-12, (): 2012. [9] H. J. Eberl, M. R. Frederick and P. G. Kevan, Importance of brood maintenance terms in simple models of the honeybee-Varroa destructor-acute bee paralysis virus complex, Electronic Journal of Differential Equations (EJDE) [electronic only] Conf. 19 (2010), 85-98. Available from: http://eudml.org/doc/232749. [10] N. Gallai, J. M. Salles, J. Settele and B. E. Vaissiere, Economic valuation of the vulnerability of world agriculture confronted with pollinator decline, Ecological Economics, 68 (2009), 810-821. doi: 10.1016/j.ecolecon.2008.06.014. [11] M. Higes, R. Martin-Hernandez, C. Botias, E. G. Bailon, A. V. Gonzalez-Porto, How natural infection by Nosema ceranae causes honeybee colony collapse, Environmental Microbiology, 10 (2008), 2659-2669. doi: 10.1111/j.1462-2920.2008.01687.x. [12] B. K. Hopkins, C. Herr and W. S. Sheppard, Sequential generations of honey bee (Apis mellifera) queens produced using cryopreserved semen, Reproduction, Fertility, and Development, 24 (2012), 1079-1083. [13] D. S. Khoury, Colony Collapse Disorder: Quantitative Models of Honey Bee Population Dynamics, Unpublished Thesis, School of Mathematics and Statistics, University of Sydney, Sydney, 2009. [14] D. S. Khoury, A. B. Barron and M. R. Myerscough, Modelling food and population dynamics in honey bee colonies, PLoS ONE, 8 (2013), e59084. doi: 10.1371/journal.pone.0059084. [15] D. S. Khoury, M. R. Myerscough and A. B. Barron, A quantitative model of honey bee colony population dynamics, PLoS ONE, 6 (2011), e18491. doi: 10.1371/journal.pone.0018491. [16] B. P. Oldroyd, What's killing American honeybees?, PLoS Biology, 5 (2007), e168. doi: 10.1371/journal.pbio.0050168. [17] H. Özbek, Arılar ve İnsektisitler, Bees and insecticides, Uludag Bee Journal, 10 (2010), 85-95. [18] J. S. Pettis, E. M. Lichtenberg, M. Andree, J. Stitzinger and R. Rose, Crop pollination exposes honeybees to pesticides which alters their susceptibility to the gut pathogen, Nosema Ceranae, PLoS ONE, 8 (2013), e70182. doi: 10.1371/journal.pone.0070182. [19] S. Russell, A. B. Barron and D. Harris, Dynamic modelling of honey bee (Apis mellifera) colony growth and failure, Ecological Modelling, 265 (2013), 158-169. doi: 10.1016/j.ecolmodel.2013.06.005. [20] T. Schmickl and K. Crailsheim, HoPoMo: A model of honeybee intracolonial population dynamics and resource management, Ecological Modelling, 204 (2007), 219-245. doi: 10.1016/j.ecolmodel.2007.01.001. [21] D. J. T. Sumpter and S. J. Martin, The dynamics of virus epidemics in Varroa-infested honey bee colonies, Journal of Animal Ecology, 73 (2004), 51-63. [22] S. Suryanarayanan and D. Kleinman, Be(e)coming experts: The controversy over insecticides in the honeybee colony collapse disorder, Social Studies Of Science (Sage Publications, Ltd.), 43 (2013), 215-240. [23] J. Traynor, Evaluating pollen production of plants, American Bee Journal, 141 (2001), 287-288. [24] U. S. Department of Agriculture (USDA) CCD Steering Committee, Colony Collapse Disorder Progress Report, USDA, 2010. Available from: http://www.ars.usda.gov/is/br/ccd/ccdprogressreport2010.pdf. [25] P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [26] D. VanEngelsdorp, J. Hayes Jr., R. M. Underwood and J. S. Pettis, A survey of honeybee colony losses in the United States, fall 2008 to spring 2009, Journal of Apicultural Research, 49 (2010), 7-14. [27] C. W. Whitfield, A-M. Cziko and G. E. Robinson, Gene expression profiles in the brain predict behavior in individual honey bees, Science, 302 (2003), 296-299. doi: 10.1126/science.1086807.  [1] J. Leonel Rocha, Abdel-Kaddous Taha, Danièle Fournier-Prunaret. 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