# American Institute of Mathematical Sciences

2015, 12(1): 209-231. doi: 10.3934/mbe.2015.12.209

## Aggregation and environmental transmission in chronic wasting disease

 1 Department of Mathematics, Christopher Newport University, 1 Avenue of the Arts, Newport News, VA 23606, United States 2 Department of Mathematics, The University of Texas-Pan American, 1201 W. University Drive, Edinburg, TX 78539, United States 3 Department of Mathematics, Univeristy of Ottawa, 585 King Edward, Ottawa, ON K1N 6N5, Canada

Received  March 2013 Revised  October 2014 Published  December 2014

Disease transmission depends on the interplay between the infectious agent and the behavior of the host. Some diseases, such as Chronic Wasting Disease, can be transmitted directly between hosts as well as indirectly via the environment. The social behavior of hosts affects both of these pathways, and a successful intervention requires knowledge of the relative influence of the different etiological and behavioral aspects of the disease. We develop a strategic differential equation model for Chronic Wasting Disease and include direct and indirect transmission as well as host aggregation into our model. We calculate the basic reproduction number and perform a sensitivity analysis based on Latin hypercube sampling from published parameter values. We find conditions for the existence of an endemic equilibrium, and show that, under a certain mild assumption on parameters, the model does not exhibit a backward bifurcation or bistability. Hence, the basic reproduction number constitutes the disease elimination threshold. We find that the prevalence of the disease decreases with host aggregation and increases with the lifespan of the infectious agent in the environment.
Citation: Olga Vasilyeva, Tamer Oraby, Frithjof Lutscher. Aggregation and environmental transmission in chronic wasting disease. Mathematical Biosciences & Engineering, 2015, 12 (1) : 209-231. doi: 10.3934/mbe.2015.12.209
##### References:
 [1] A. Aguzzi, M. Heikenwalder and M. Polymenidou, Insights into prion strains and neurotoxicity, Nature Reviews Molecular Cell Biology, 8 (2007), 552-561. doi: 10.1038/nrm2204. [2] E. Almberg, P. Cross, C. Johnson, D. Heisey and B. Richards, Modeling routes of Chronic Wasting Disease transmission: Environmental prion persistence promotes deer population decline and extinction, PLoS One, 6 (2011), e19896. doi: 10.1371/journal.pone.0019896. [3] M. Begon, M. Bennett, R. G. Bowers, N. P. French, S. M. Hazel and J. Turner, A clarification of transmission terms in host-microparasite models: Numbers, densities and areas, Epidemiology and Infection, 129 (2002), 147-153. doi: 10.1017/S0950268802007148. [4] R. Breban, Role of environmental persistence in pathogen transmission: A mathematical modeling approach, J. Math. Biol., 66 (2013), 535-546. doi: 10.1007/s00285-012-0520-2. [5] C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications, Mathematical Biosciences and Engineering, 1 (2004), 361-404. doi: 10.3934/mbe.2004.1.361. [6] J. Collinge and A. R. Clarke, A general model of prion strains and their pathogenicity, Science, 318 (2007), 930-936. doi: 10.1126/science.1138718. [7] M. M. Conner, M. R. Ebinger, J. A. Blanchong and P. C. Cross, Infectios disease in cervids of North America, Ann. N.Y. Acad. Sci., 1134 (2008), 146-172. doi: 10.1196/annals.1439.005. [8] M. M. Conner, M. W. Miller, M. R. Ebinger and K. P. Burnham, A meta-BACI approach for evaluating management intervention on chronic wasting disease in mule deer, Ecological Applications, 17 (2007), 140-153. doi: 10.1890/1051-0761(2007)017[0140:AMAFEM]2.0.CO;2. [9] O. Diekmann and J. A. M. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, John Wiley & Sons, 2000. [10] K. Dietz, Overall population patterns in the transmission cycle of infectious disease agents, Population Biology of Infectious Diseases, ed. R.M. Anderson and R.M. May, Dahlem Konferenzen, Springer-Verlag, 25 (1982), 87-102. doi: 10.1007/978-3-642-68635-1_6. [11] G. L. Dusek, R. J. MacKie, J. D. Herriges and B. B. Compton, Population ecology of white-tailed deer along the Lower Yellowstone River, Wildlife Monographs, 104 (1989), 3-68. [12] H. R. Fryer and A. R. McLean, There is no safe dose of prions, PLoS ONE, 6 (2011), e23664. doi: 10.1371/journal.pone.0023664. [13] J. E. Gross and M. W. Miller, Chronic wasting disease in mule deer: Disease dynamics and control, The Journal of Wildlife Management, 65 (2001), 205-215. doi: 10.2307/3802899. [14] T. Habib, E. Merrill, M. J. Pybus and D. Coltman, Modelling landscape effects on density-contact rate relationships in eastern Alberta: Inplications for chronic wasting disease, Ecological Modelling, 222 (2011), 2722-2732. doi: 10.1016/j.ecolmodel.2011.05.007. [15] J. O. Lloyd-Smith, P. C. Cross, C. J. Briggs, M. Daugherty, W. M. Getz, J. Latto, M. S. Sanchez, A. B. Smith and A. Swei, Should we expect population thresholds for wildlife disease?, Trends in Ecology & Evolution, 20 (2005), 511-519. doi: 10.1016/j.tree.2005.07.004. [16] C. K. Mathiason, S. A. Hays, J. Powers, J. Hayes-Klug and J. Langenberg, et al, Infectious prions in pre-clinical deer and transmission of chronic wasting disease solely by environmental exposure, PLoS ONE, 4 (2009), e5916. doi: 10.1371/journal.pone.0005916. [17] R. M. May, Host-parasitoid systems in patchy environments: A phenomenological model, Journal of Animal Ecology, 47 (1978), 833-844. doi: 10.2307/3674. [18] M. W. Miller, N. T. Hobbs and S. J. Tavener, Dynamics of prion disease transmission in mule deer, Ecological Applications, 16 (2006), 2208-2214. doi: {10.1890/1051-0761(2006)016[2208:DOPDTI]2.0.CO;2}. [19] M. W. Miller, E. S. Williams, N. T. Hobbs and L. L. Wolfe, Environmental sources of prion transmission in mule deer, Emerg. Infect. Dis., 10 (2004), 1003-1006. doi: 10.3201/eid1006.040010. [20] P. R. Moorcroft and M. A. Lewis, Mechanistic Home Range Analysis, Princeton University Press, 2006. [21] E. M. Schauber and A. Woolf, Chronic wasting disease in deer and elk: A critique of current models and their application, Wildlife Society Bulletin, 31 (2003), 610-616. [22] H. R. Thieme, Mathematics in Population Biology, Princeton Series in Theoretical and Computational Biology, Princeton University Press, 2003. [23] G. Wasserberg, E. E. Osnas, R. E. Rolley and M. D. Samuel, Host culling as an adaptive management tool for chronic wasting disease in white-tailed deer: A modelling study, Journal of Applied Ecology, 46 (2009), 457-466. doi: 10.1111/j.1365-2664.2008.01576.x. [24] E. S. Williams and M. W. Miller, Chronic wasting disease in deer and elk in North America, Revue Scientifique et technique de l'Office International des Epizzoties, 21 (2002), 305-316.

show all references

##### References:
 [1] A. Aguzzi, M. Heikenwalder and M. Polymenidou, Insights into prion strains and neurotoxicity, Nature Reviews Molecular Cell Biology, 8 (2007), 552-561. doi: 10.1038/nrm2204. [2] E. Almberg, P. Cross, C. Johnson, D. Heisey and B. Richards, Modeling routes of Chronic Wasting Disease transmission: Environmental prion persistence promotes deer population decline and extinction, PLoS One, 6 (2011), e19896. doi: 10.1371/journal.pone.0019896. [3] M. Begon, M. Bennett, R. G. Bowers, N. P. French, S. M. Hazel and J. Turner, A clarification of transmission terms in host-microparasite models: Numbers, densities and areas, Epidemiology and Infection, 129 (2002), 147-153. doi: 10.1017/S0950268802007148. [4] R. Breban, Role of environmental persistence in pathogen transmission: A mathematical modeling approach, J. Math. Biol., 66 (2013), 535-546. doi: 10.1007/s00285-012-0520-2. [5] C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications, Mathematical Biosciences and Engineering, 1 (2004), 361-404. doi: 10.3934/mbe.2004.1.361. [6] J. Collinge and A. R. Clarke, A general model of prion strains and their pathogenicity, Science, 318 (2007), 930-936. doi: 10.1126/science.1138718. [7] M. M. Conner, M. R. Ebinger, J. A. Blanchong and P. C. Cross, Infectios disease in cervids of North America, Ann. N.Y. Acad. Sci., 1134 (2008), 146-172. doi: 10.1196/annals.1439.005. [8] M. M. Conner, M. W. Miller, M. R. Ebinger and K. P. Burnham, A meta-BACI approach for evaluating management intervention on chronic wasting disease in mule deer, Ecological Applications, 17 (2007), 140-153. doi: 10.1890/1051-0761(2007)017[0140:AMAFEM]2.0.CO;2. [9] O. Diekmann and J. A. M. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, John Wiley & Sons, 2000. [10] K. Dietz, Overall population patterns in the transmission cycle of infectious disease agents, Population Biology of Infectious Diseases, ed. R.M. Anderson and R.M. May, Dahlem Konferenzen, Springer-Verlag, 25 (1982), 87-102. doi: 10.1007/978-3-642-68635-1_6. [11] G. L. Dusek, R. J. MacKie, J. D. Herriges and B. B. Compton, Population ecology of white-tailed deer along the Lower Yellowstone River, Wildlife Monographs, 104 (1989), 3-68. [12] H. R. Fryer and A. R. McLean, There is no safe dose of prions, PLoS ONE, 6 (2011), e23664. doi: 10.1371/journal.pone.0023664. [13] J. E. Gross and M. W. Miller, Chronic wasting disease in mule deer: Disease dynamics and control, The Journal of Wildlife Management, 65 (2001), 205-215. doi: 10.2307/3802899. [14] T. Habib, E. Merrill, M. J. Pybus and D. Coltman, Modelling landscape effects on density-contact rate relationships in eastern Alberta: Inplications for chronic wasting disease, Ecological Modelling, 222 (2011), 2722-2732. doi: 10.1016/j.ecolmodel.2011.05.007. [15] J. O. Lloyd-Smith, P. C. Cross, C. J. Briggs, M. Daugherty, W. M. Getz, J. Latto, M. S. Sanchez, A. B. Smith and A. Swei, Should we expect population thresholds for wildlife disease?, Trends in Ecology & Evolution, 20 (2005), 511-519. doi: 10.1016/j.tree.2005.07.004. [16] C. K. Mathiason, S. A. Hays, J. Powers, J. Hayes-Klug and J. Langenberg, et al, Infectious prions in pre-clinical deer and transmission of chronic wasting disease solely by environmental exposure, PLoS ONE, 4 (2009), e5916. doi: 10.1371/journal.pone.0005916. [17] R. M. May, Host-parasitoid systems in patchy environments: A phenomenological model, Journal of Animal Ecology, 47 (1978), 833-844. doi: 10.2307/3674. [18] M. W. Miller, N. T. Hobbs and S. J. Tavener, Dynamics of prion disease transmission in mule deer, Ecological Applications, 16 (2006), 2208-2214. doi: {10.1890/1051-0761(2006)016[2208:DOPDTI]2.0.CO;2}. [19] M. W. Miller, E. S. Williams, N. T. Hobbs and L. L. Wolfe, Environmental sources of prion transmission in mule deer, Emerg. Infect. Dis., 10 (2004), 1003-1006. doi: 10.3201/eid1006.040010. [20] P. R. Moorcroft and M. A. Lewis, Mechanistic Home Range Analysis, Princeton University Press, 2006. [21] E. M. Schauber and A. Woolf, Chronic wasting disease in deer and elk: A critique of current models and their application, Wildlife Society Bulletin, 31 (2003), 610-616. [22] H. R. Thieme, Mathematics in Population Biology, Princeton Series in Theoretical and Computational Biology, Princeton University Press, 2003. [23] G. Wasserberg, E. E. Osnas, R. E. Rolley and M. D. Samuel, Host culling as an adaptive management tool for chronic wasting disease in white-tailed deer: A modelling study, Journal of Applied Ecology, 46 (2009), 457-466. doi: 10.1111/j.1365-2664.2008.01576.x. [24] E. S. Williams and M. W. Miller, Chronic wasting disease in deer and elk in North America, Revue Scientifique et technique de l'Office International des Epizzoties, 21 (2002), 305-316.
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