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2011, 8(2): 463-473. doi: 10.3934/mbe.2011.8.463

Preliminary analysis of an agent-based model for a tick-borne disease

Holly Gaff 1,

1. 

Department of Biological Sciences, Old Dominion University, Norfolk, VA 23529, United States

Received  March 2010 Revised  September 2010 Published  April 2011

Ticks have a unique life history including a distinct set of life stages and a single blood meal per life stage. This makes tick-host interactions more complex from a mathematical perspective. In addition, any model of these interactions must involve a significant degree of stochasticity on the individual tick level. In an attempt to quantify these relationships, I have developed an individual-based model of the interactions between ticks and their hosts as well as the transmission of tick-borne disease between the two populations. The results from this model are compared with those from previously published differential equation based population models. The findings show that the agent-based model produces significantly lower prevalence of disease in both the ticks and their hosts than what is predicted by a similar differential equation model.
Keywords: Agent-based model, tick-borne disease..
Mathematics Subject Classification: Primary: 92B08; Secondary: 90B1.
Citation: Holly Gaff. Preliminary analysis of an agent-based model for a tick-borne disease. Mathematical Biosciences & Engineering, 2011, 8 (2) : 463-473. doi: 10.3934/mbe.2011.8.463
References:
[1]

Centers for Disease Control and Prevention, Summary of Notifiable Diseases - United States, 2006, MMWR, 55 (2008), 1-94.

[2]

D. E. Sonenshine and T. N. Mather, "Ecological Dynamics of Tick-Borne Zoonoses," Oxford University Press, 1994.

[3]

H. Gaff and L. J. Gross, Analysis of a tick-borne disease model with varying population sizes in various habitats, Bulletin of Mathematical Biology, 69 (2007), 265-288. doi: 10.1007/s11538-006-9125-5.

[4]

H. Gaff and E. Schaefer, Metapopulation models in tick-borne disease transmission modelling, In "Modelling parasitic Disease Transmission: Biology to Control," eds. Michael, E. & Spear, R. Landes Bioscience, Eurekah: Austin, TX, USA, 2008.

[5]

H. Gaff, L. Gross and E. Schaefer, Results from a mathematical model for human monocytic ehrlichiosis, Proceedings of the 5th Conference on Rickettsiae and Rickettsial diseases, Supplement to Clinical Microbiology and Infection, 15 (2008), 1-2.

[6]

D. G. Haile and G. A. Mount, Computer simulation of population dynamics of the lone star tick, Amblyomma americanum (Acari: Ixodidae), Journal of Medical Entomology, 24 (1987), 356-369.

[7]

G. A. Mount and D. G. Haile, Computer simulation of population dynamics of the American dog tick (Acari: Ixodidae), Journal of Medical Entomology, 26 (1989), 60-76.

[8]

G. A. Mount, D. G. Haile, R. B. Davey and L. M. Cooksey, Computer simulation of boophilus cattle tick (Acari: Ixodidae) population dynamics, Journal of Medical Entomology, 28 (1991), 223-240.

[9]

G. A. Mount, D. G. Haile, D. R. Barnard and E. Daniels, New version of LSTSIM for computer simulation of Amblyomma americanum (Acari: Ixodidae) population dynamics, Journal of Medical Entomology, 30 (1993), 843-857.

[10]

G. A. Mount, D. G. Haile and E. Daniels, Simulation of blacklegged tick (Acari: Ixodidae) population dynamics and transmission of Borrelia burgdorferi, Journal of Medical Entomology, 34 (1997), 461-484.

[11]

G. A. Mount, D. G. Haile and E. Daniels, Simulation of management strategies for the blacklegged tick (Acari: Ixodidae) and the Lyme disease spirochete, Borrelia burgdorferi, Journal of Medical Entomology, 90 (1997), 672-683.

[12]

S. Sandberg, T. E. Awerbuch and A. Spielman, A comprehensive multiple matrix model representing the life cycle of the tick that transmits the age of Lyme disease, Journal of Theoretical Biology, 157 (1992), 203-220. doi: 10.1016/S0022-5193(05)80621-6.

[13]

T. E. Awerbuch and S. Sandberg, Trends and oscillations in tick population dynamics, Journal of Theoretical Biology, 175 (1995), 511-516. doi: 10.1006/jtbi.1995.0158.

[14]

S. Randolph, Epidemiological uses of a population model for the tick Rhipicephalus appendiculatus, Tropical Medicine and International Health, 4 (1999), A34-A42. doi: 10.1046/j.1365-3156.1999.00449.x.

[15]

J. E. Bunnell, S. D. Price, A. Das, T. M. Shields and G. E. Glass, Geographic Information Systems and Spatial Analysis of Adult Ixodes scapularis (Acari: Ixodidae) in the Middle Atlantic Region of the U.S.A., Journal of Medical Entomology, 40 (2003), 570-576. doi: 10.1603/0022-2585-40.4.570.

[16]

A. Das, S. R. Lele, G. E. Glass, T. Shields and J. Petz, Modelling a discrete spatial response using generalized linear mixed models: Application to Lyme disease vectors, International Journal of Geographical Information Science, 16 (2002), 151-166. doi: 10.1080/13658810110099134.

[17]

G. E. Glass, B. S. Schwartz, J. M. Morgan, D. T. Johnson, P. M. Noy and E. Israel, Environmental risk factors for Lyme disease identified with geographic information systems, American Journal of Public Health, 85 (1995), 944-948. doi: 10.2105/AJPH.85.7.944.

[18]

W. E. Fitzgibbon, M. E. Parrott and G. F. Webb, A diffusive epidemic model for a host-vector system, In "Differential Equations and Applications to Biology and Industry," M. Martelli, K. Cooke, E. Cumberbatch, B. Tang, and H. Thieme, (Eds.), World Scientific Press, Singapore, 1996.

[19]

J. Radcliffe and L. Rass, The spatial spread and final size of models for the deterministic host-vector epidemic, Mathematical Biosciences, 70 (1984), 123-146. doi: 10.1016/0025-5564(84)90094-4.

[20]

J. Radcliffe and L. Rass, The rate of spread of infection in models for the deterministic host-vector epidemic, Mathematical Biosciences, 74 (1985), 257-273. doi: 10.1016/0025-5564(85)90059-8.

[21]

A. R. Giardina, K. A. Schmidt, E. M. Schauber and R. S. Ostfeld, Modeling the role of songbirds and rodents in the ecology of Lyme disease, Canadian Journal of Zoology, 78 (2000), 2184-2197. doi: 10.1139/cjz-78-12-2184.

[22]

K. LoGiudice, R. S. Ostfeld, K. A. Schmidt and F. Keesing, The ecology of infectious disease: Effects of host diversity and community composition on Lyme disease risk, Proceedings of the National Academies of Science, USA, 100 (2003), 567-571.

[23]

M. Ghosh and A. Pugliese, Seasonal population dynamics of ticks, and its influence on infection transmission: A semi- discrete approach, Bulletin of Mathematical Biology, 66 (2004), 1659-1684. doi: 10.1016/j.bulm.2004.03.007.

[24]

W. Ding, Optimal Control on Hybrid ODE Systems with Application to a Tick Disease Model, Mathematical Biosciences and Engineering, 4 (2007), 633-659.

[25]

D. L. DeAngelis and L. J. Gross, "Individual-based Models and Approaches in Ecology: Populations, Communities and Ecosystems," Taylor and Francis, 1992.

[26]

D. L. DeAngelis, L. J. Gross, W. F. Wolff, D. M. Fleming, M. P. Nott and E. J. Comiskey, Individual-based models on the landscape: Applications to the Everglades, in "Landscape Ecology: A Top-Down Approach," J. Sanderson and L. D. Harris (eds.), Lewis Publishers, Boca Raton, FL, 2000.

[27]

S. Eubank, H. Guclu, V. S. A. Kumar, M. V. Marathe, A. Srinivasan, Z. Toroczkai and N. Wang, Modelling disease outbreaks in realistic urban social networks, Nature, 429 (2004), 180-184. doi: 10.1038/nature02541.

[28]

A. G. Barbour, "Lyme Disease: The Cause, the Cure, the Controversy," John Hopkins University Press, Baltimore, Maryland, 1996.

[29]

F. Des Vignes, M. L. Levin and D. Fish, Comparative vector competence of Dermacentor variabilis and Ixodes scapularis (Acari: Ixodidae) for the agent of human granulocytic ehrlichiosis, Journal of Medical Entomology, 36 (1999), 182-185.

[30]

Dania Richter, Andrew Spielman, Nicholas Komar and Franz-Rainer Matuschka, Competence of American robins as reservoir hosts for Lyme disease spirochetes, Emerging Infectious Diseases, 6 (2000), 133-138. doi: 10.3201/eid0602.000205.

[31]

V. Grimm, U. Berger, F. Bastiansen, S. Eliassen, V. Ginot, J. Giske, J. Goss-Custard, T. Grand, S. K. Heinz, G. Huse, A. Huth, J. U. Jepsen, C. Jørgensen, W. M. Mooij, B. Müller, G. Peer, C. Piou, S. F. Railsback, A. M. Robbins, M. M. Robbins, E. Rossmanith, N. Rüger, E. Strand, S. Souissi, R. A. Stillman, R. Vabø, U. Visser and D. L. DeAngelis, A standard protocol for describing individual-based and agent-based models, Ecological Modelling 198 (2006), 115-26. doi: 10.1016/j.ecolmodel.2006.04.023.

[32]

A. L. Bauer, C. A. A. Beauchemin and A. S. Perelson, Agent-based modeling of host-pathogen systems: The successes and the challenges, Information Sciences, 179 (2009), 1379-1389. doi: 10.1016/j.ins.2008.11.012.

[33]

V. Grimm and S. F. Railsbeck, "Individual-based Modeling and Ecology," Princeton University Press, 2005.

[34]

J. M. Lockhart, W. R. Davidson, J. E. Dawson and D. E. Stallknecht, Temporal association of Amblyomma americanum with the presence of Ehrlichia chaffeensis reactive antibodies in white-tailed deer, Journal of Wildlife Diseases, 31 (1995), 119-124.

[35]

J. E. Dawson, J. E. Childs, K. L. Biggie, C. Moore, D. Stallknecht, J. Shaddock, J. Bouseman, E. Hofmeister and J. G. Olson, White-tailed deer as a potential reservoir of Ehrlichia spp., Journal of Wildlife Diseases, 30 (1994), 162-168.

[36]

B. E. Anderson, K. G. Sims, J. G. Olson, J. E. Childs, J. F. Piesman, C. M. Happ, G. O. Maupin and B. J. B. Johnson, Amblyomma americanum: A potential vector of human ehrlichiosis, American Journal of Tropical Medicine and Hygiene, 49 (1993), 239-244.

show all references

References:
[1]

Centers for Disease Control and Prevention, Summary of Notifiable Diseases - United States, 2006, MMWR, 55 (2008), 1-94.

[2]

D. E. Sonenshine and T. N. Mather, "Ecological Dynamics of Tick-Borne Zoonoses," Oxford University Press, 1994.

[3]

H. Gaff and L. J. Gross, Analysis of a tick-borne disease model with varying population sizes in various habitats, Bulletin of Mathematical Biology, 69 (2007), 265-288. doi: 10.1007/s11538-006-9125-5.

[4]

H. Gaff and E. Schaefer, Metapopulation models in tick-borne disease transmission modelling, In "Modelling parasitic Disease Transmission: Biology to Control," eds. Michael, E. & Spear, R. Landes Bioscience, Eurekah: Austin, TX, USA, 2008.

[5]

H. Gaff, L. Gross and E. Schaefer, Results from a mathematical model for human monocytic ehrlichiosis, Proceedings of the 5th Conference on Rickettsiae and Rickettsial diseases, Supplement to Clinical Microbiology and Infection, 15 (2008), 1-2.

[6]

D. G. Haile and G. A. Mount, Computer simulation of population dynamics of the lone star tick, Amblyomma americanum (Acari: Ixodidae), Journal of Medical Entomology, 24 (1987), 356-369.

[7]

G. A. Mount and D. G. Haile, Computer simulation of population dynamics of the American dog tick (Acari: Ixodidae), Journal of Medical Entomology, 26 (1989), 60-76.

[8]

G. A. Mount, D. G. Haile, R. B. Davey and L. M. Cooksey, Computer simulation of boophilus cattle tick (Acari: Ixodidae) population dynamics, Journal of Medical Entomology, 28 (1991), 223-240.

[9]

G. A. Mount, D. G. Haile, D. R. Barnard and E. Daniels, New version of LSTSIM for computer simulation of Amblyomma americanum (Acari: Ixodidae) population dynamics, Journal of Medical Entomology, 30 (1993), 843-857.

[10]

G. A. Mount, D. G. Haile and E. Daniels, Simulation of blacklegged tick (Acari: Ixodidae) population dynamics and transmission of Borrelia burgdorferi, Journal of Medical Entomology, 34 (1997), 461-484.

[11]

G. A. Mount, D. G. Haile and E. Daniels, Simulation of management strategies for the blacklegged tick (Acari: Ixodidae) and the Lyme disease spirochete, Borrelia burgdorferi, Journal of Medical Entomology, 90 (1997), 672-683.

[12]

S. Sandberg, T. E. Awerbuch and A. Spielman, A comprehensive multiple matrix model representing the life cycle of the tick that transmits the age of Lyme disease, Journal of Theoretical Biology, 157 (1992), 203-220. doi: 10.1016/S0022-5193(05)80621-6.

[13]

T. E. Awerbuch and S. Sandberg, Trends and oscillations in tick population dynamics, Journal of Theoretical Biology, 175 (1995), 511-516. doi: 10.1006/jtbi.1995.0158.

[14]

S. Randolph, Epidemiological uses of a population model for the tick Rhipicephalus appendiculatus, Tropical Medicine and International Health, 4 (1999), A34-A42. doi: 10.1046/j.1365-3156.1999.00449.x.

[15]

J. E. Bunnell, S. D. Price, A. Das, T. M. Shields and G. E. Glass, Geographic Information Systems and Spatial Analysis of Adult Ixodes scapularis (Acari: Ixodidae) in the Middle Atlantic Region of the U.S.A., Journal of Medical Entomology, 40 (2003), 570-576. doi: 10.1603/0022-2585-40.4.570.

[16]

A. Das, S. R. Lele, G. E. Glass, T. Shields and J. Petz, Modelling a discrete spatial response using generalized linear mixed models: Application to Lyme disease vectors, International Journal of Geographical Information Science, 16 (2002), 151-166. doi: 10.1080/13658810110099134.

[17]

G. E. Glass, B. S. Schwartz, J. M. Morgan, D. T. Johnson, P. M. Noy and E. Israel, Environmental risk factors for Lyme disease identified with geographic information systems, American Journal of Public Health, 85 (1995), 944-948. doi: 10.2105/AJPH.85.7.944.

[18]

W. E. Fitzgibbon, M. E. Parrott and G. F. Webb, A diffusive epidemic model for a host-vector system, In "Differential Equations and Applications to Biology and Industry," M. Martelli, K. Cooke, E. Cumberbatch, B. Tang, and H. Thieme, (Eds.), World Scientific Press, Singapore, 1996.

[19]

J. Radcliffe and L. Rass, The spatial spread and final size of models for the deterministic host-vector epidemic, Mathematical Biosciences, 70 (1984), 123-146. doi: 10.1016/0025-5564(84)90094-4.

[20]

J. Radcliffe and L. Rass, The rate of spread of infection in models for the deterministic host-vector epidemic, Mathematical Biosciences, 74 (1985), 257-273. doi: 10.1016/0025-5564(85)90059-8.

[21]

A. R. Giardina, K. A. Schmidt, E. M. Schauber and R. S. Ostfeld, Modeling the role of songbirds and rodents in the ecology of Lyme disease, Canadian Journal of Zoology, 78 (2000), 2184-2197. doi: 10.1139/cjz-78-12-2184.

[22]

K. LoGiudice, R. S. Ostfeld, K. A. Schmidt and F. Keesing, The ecology of infectious disease: Effects of host diversity and community composition on Lyme disease risk, Proceedings of the National Academies of Science, USA, 100 (2003), 567-571.

[23]

M. Ghosh and A. Pugliese, Seasonal population dynamics of ticks, and its influence on infection transmission: A semi- discrete approach, Bulletin of Mathematical Biology, 66 (2004), 1659-1684. doi: 10.1016/j.bulm.2004.03.007.

[24]

W. Ding, Optimal Control on Hybrid ODE Systems with Application to a Tick Disease Model, Mathematical Biosciences and Engineering, 4 (2007), 633-659.

[25]

D. L. DeAngelis and L. J. Gross, "Individual-based Models and Approaches in Ecology: Populations, Communities and Ecosystems," Taylor and Francis, 1992.

[26]

D. L. DeAngelis, L. J. Gross, W. F. Wolff, D. M. Fleming, M. P. Nott and E. J. Comiskey, Individual-based models on the landscape: Applications to the Everglades, in "Landscape Ecology: A Top-Down Approach," J. Sanderson and L. D. Harris (eds.), Lewis Publishers, Boca Raton, FL, 2000.

[27]

S. Eubank, H. Guclu, V. S. A. Kumar, M. V. Marathe, A. Srinivasan, Z. Toroczkai and N. Wang, Modelling disease outbreaks in realistic urban social networks, Nature, 429 (2004), 180-184. doi: 10.1038/nature02541.

[28]

A. G. Barbour, "Lyme Disease: The Cause, the Cure, the Controversy," John Hopkins University Press, Baltimore, Maryland, 1996.

[29]

F. Des Vignes, M. L. Levin and D. Fish, Comparative vector competence of Dermacentor variabilis and Ixodes scapularis (Acari: Ixodidae) for the agent of human granulocytic ehrlichiosis, Journal of Medical Entomology, 36 (1999), 182-185.

[30]

Dania Richter, Andrew Spielman, Nicholas Komar and Franz-Rainer Matuschka, Competence of American robins as reservoir hosts for Lyme disease spirochetes, Emerging Infectious Diseases, 6 (2000), 133-138. doi: 10.3201/eid0602.000205.

[31]

V. Grimm, U. Berger, F. Bastiansen, S. Eliassen, V. Ginot, J. Giske, J. Goss-Custard, T. Grand, S. K. Heinz, G. Huse, A. Huth, J. U. Jepsen, C. Jørgensen, W. M. Mooij, B. Müller, G. Peer, C. Piou, S. F. Railsback, A. M. Robbins, M. M. Robbins, E. Rossmanith, N. Rüger, E. Strand, S. Souissi, R. A. Stillman, R. Vabø, U. Visser and D. L. DeAngelis, A standard protocol for describing individual-based and agent-based models, Ecological Modelling 198 (2006), 115-26. doi: 10.1016/j.ecolmodel.2006.04.023.

[32]

A. L. Bauer, C. A. A. Beauchemin and A. S. Perelson, Agent-based modeling of host-pathogen systems: The successes and the challenges, Information Sciences, 179 (2009), 1379-1389. doi: 10.1016/j.ins.2008.11.012.

[33]

V. Grimm and S. F. Railsbeck, "Individual-based Modeling and Ecology," Princeton University Press, 2005.

[34]

J. M. Lockhart, W. R. Davidson, J. E. Dawson and D. E. Stallknecht, Temporal association of Amblyomma americanum with the presence of Ehrlichia chaffeensis reactive antibodies in white-tailed deer, Journal of Wildlife Diseases, 31 (1995), 119-124.

[35]

J. E. Dawson, J. E. Childs, K. L. Biggie, C. Moore, D. Stallknecht, J. Shaddock, J. Bouseman, E. Hofmeister and J. G. Olson, White-tailed deer as a potential reservoir of Ehrlichia spp., Journal of Wildlife Diseases, 30 (1994), 162-168.

[36]

B. E. Anderson, K. G. Sims, J. G. Olson, J. E. Childs, J. F. Piesman, C. M. Happ, G. O. Maupin and B. J. B. Johnson, Amblyomma americanum: A potential vector of human ehrlichiosis, American Journal of Tropical Medicine and Hygiene, 49 (1993), 239-244.

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