# 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. Explosion birth and extinction: Double big bang bifurcations and Allee effect in Tsoularis-Wallace's growth models. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3131-3163. doi: 10.3934/dcdsb.2015.20.3131 [2] Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129 [3] Maria Conceição A. Leite, Yunjiao Wang. Multistability, oscillations and bifurcations in feedback loops. Mathematical Biosciences & Engineering, 2010, 7 (1) : 83-97. doi: 10.3934/mbe.2010.7.83 [4] Nika Lazaryan, Hassan Sedaghat. Extinction and the Allee effect in an age structured Ricker population model with inter-stage interaction. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 731-747. doi: 10.3934/dcdsb.2018040 [5] Elena Braverman, Alexandra Rodkina. Stochastic difference equations with the Allee effect. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 5929-5949. doi: 10.3934/dcds.2016060 [6] Jim M. Cushing. The evolutionary dynamics of a population model with a strong Allee effect. Mathematical Biosciences & Engineering, 2015, 12 (4) : 643-660. doi: 10.3934/mbe.2015.12.643 [7] Dianmo Li, Zhen Zhang, Zufei Ma, Baoyu Xie, Rui Wang. Allee effect and a catastrophe model of population dynamics. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 629-634. doi: 10.3934/dcdsb.2004.4.629 [8] Yi Yang, Robert J. Sacker. Periodic unimodal Allee maps, the semigroup property and the$\lambda\$-Ricker map with Allee effect. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 589-606. doi: 10.3934/dcdsb.2014.19.589 [9] Chuang Xu. Strong Allee effect in a stochastic logistic model with mate limitation and stochastic immigration. Discrete and Continuous Dynamical Systems - B, 2016, 21 (7) : 2321-2336. doi: 10.3934/dcdsb.2016049 [10] Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051 [11] Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256 [12] Xiaoyuan Chang, Junping Shi. Bistable and oscillatory dynamics of Nicholson's blowflies equation with Allee effect. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021242 [13] Na Min, Mingxin Wang. Dynamics of a diffusive prey-predator system with strong Allee effect growth rate and a protection zone for the prey. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1721-1737. doi: 10.3934/dcdsb.2018073 [14] Wenjie Ni, Mingxin Wang. Dynamical properties of a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3409-3420. doi: 10.3934/dcdsb.2017172 [15] Moitri Sen, Malay Banerjee, Yasuhiro Takeuchi. Influence of Allee effect in prey populations on the dynamics of two-prey-one-predator model. Mathematical Biosciences & Engineering, 2018, 15 (4) : 883-904. doi: 10.3934/mbe.2018040 [16] Qizhen Xiao, Binxiang Dai. Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1065-1081. doi: 10.3934/mbe.2015.12.1065 [17] Yuying Liu, Yuxiao Guo, Junjie Wei. Dynamics in a diffusive predator-prey system with stage structure and strong allee effect. Communications on Pure and Applied Analysis, 2020, 19 (2) : 883-910. doi: 10.3934/cpaa.2020040 [18] Yujing Gao, Bingtuan Li. Dynamics of a ratio-dependent predator-prey system with a strong Allee effect. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2283-2313. doi: 10.3934/dcdsb.2013.18.2283 [19] Nalin Fonseka, Jerome Goddard II, Ratnasingham Shivaji, Byungjae Son. A diffusive weak Allee effect model with U-shaped emigration and matrix hostility. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5509-5517. doi: 10.3934/dcdsb.2020356 [20] Chuangxia Huang, Xiaojin Guo, Jinde Cao, Ardak Kashkynbayev. Bistable dynamics on a tick population equation incorporating Allee effect and two different time-varying delays. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022122

2018 Impact Factor: 1.313