2014, 19(10): 3147-3167. doi: 10.3934/dcdsb.2014.19.3147

Range expansion of Ixodes scapularis ticks and of Borrelia burgdorferi by migratory birds

1. 

Mprime Centre for Disease Modelling, York Institute of Health Research, Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3

2. 

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

Received  April 2014 Revised  May 2014 Published  October 2014

Recent studies have suggested that the risk of exposure to Lyme disease is emerging in Canada because of the expanding range of I. scapularis ticks. The wide geographic breeding range of I. scapularis-carrying migratory birds is consistent with the widespread geographical occurrence of I. scapularis in Canada. However, how important migratory birds from the United States are for the establishment and the stable endemic transmission cycle of Lyme disease in Canada remains an issue of theoretical challenge and practical significance. In this paper, we design and analyze a periodic model of differential equations with a forcing term modeling the annual bird migration to address the aforementioned issue. Our results show that ticks can establish in any migratory bird stopovers and breeding sites. Moreover, bird-transported ticks may increase the probability of B. burgdorferi establishment in a tick-endemic habitat.
Citation: Jane M. Heffernan, Yijun Lou, Jianhong Wu. Range expansion of Ixodes scapularis ticks and of Borrelia burgdorferi by migratory birds. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3147-3167. doi: 10.3934/dcdsb.2014.19.3147
References:
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S. Ai, Global stability of equilibria in a tick-borne disease model,, Math. Biosci. Eng., 4 (2007), 567. doi: 10.3934/mbe.2007.4.567.

[2]

J. L. Aron and R. M. May, The population dynamics of malaria,, in Population Dynamics and Infectious Disease (Ed. R. M. Anderson), (1982), 139. doi: 10.1007/978-1-4899-2901-3_5.

[3]

T. Awerbuch-Friedlander, R. Levins and M. Predescu, The role of seasonality in the dynamics of deer tick populations,, Bull. Math. Biol., 67 (2005), 467. doi: 10.1016/j.bulm.2004.08.003.

[4]

T. E. Awerbuchb and S. Sandberg, Trends and oscillations in tick population dynamics,, J. Theor. Biol., 175 (1995), 511. doi: 10.1006/jtbi.1995.0158.

[5]

N. Bacaër, Approximation of the basic reproduction number $R_0$ for vector-borne diseases with a periodic vector population,, Bull. Math. Biol., 69 (2007), 1067. doi: 10.1007/s11538-006-9166-9.

[6]

N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality. The case of cutaneous leishmaniasis in Chichaoua, Morocco,, J. Math. Biol., 53 (2006), 421. doi: 10.1007/s00285-006-0015-0.

[7]

N. Bacaër and R. Ouifki, Growth rate and basic reproduction number for population models with a simple periodic factor,, Math. Biosci., 210 (2007), 647. doi: 10.1016/j.mbs.2007.07.005.

[8]

L. Bourouiba, J. Wu, S. Newman, J. Takekawa, T. Natdorj, N. Batbayar, C. M. Bishop, L. A. Hawkes, P. J. Butler and M. Wikelski M, Spatial dynamics of bar-headed geese migration in the context of H5N1,, J. R. Soc. Interface, 52 (2010), 1627. doi: 10.1098/rsif.2010.0126.

[9]

R. J. Brinkerhoff, C. M. Folsom-O'Keefe, K. Tsao and M. A. Diuk-Wasser, Do birds affect Lyme disease risk? Range expansion of the vector-borne pathogen Borrelia burgdorferi,, Frontiers in Ecology and the Environment, 9 (2011), 103. doi: 10.1890/090062.

[10]

J. L. Brunner, K. LoGiudice and R. Ostfeld, Estimating reservior competence of Borrelia burgdorferi hosts: prevalence and infectivity, sensitivity and specificity,, J. Med. Entomol., 45 (2008), 139.

[11]

Canadian Migration Monitoring Network, CMMV 10-year report,, , ().

[12]

T. Caraco, G. Gardner, W. Maniatty, E. Deelman and B. K. Szymanski, Lyme disease: self regulation and pathogen invasion,, J. Theor. Biol., 193 (1998), 561. doi: 10.1006/jtbi.1998.0722.

[13]

T. Caraco, S. Glavanakov, G. Chen, J. E Flaherty, T. K Ohsumi and B. K. Szymanski, Stage-structured infection transmission and a spatial epidemic: A model for Lyme disease,, Am. Nat., 160 (2002), 348. doi: 10.1086/341518.

[14]

Center for Disease Control and Prevention (CDC), Lyme disease-United States, 2003-2005,, MMWR Morb. Mortal. Wkly. Rep., 56 (2007), 573.

[15]

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 populations,, J. Math. Biol., 28 (1990), 365. doi: 10.1007/BF00178324.

[16]

Z. Feng and V. Vealsco-Hernández, Competitive exclusion in a vector-host model for the dengue fever,, J. Math. Biol., 35 (1997), 523. doi: 10.1007/s002850050064.

[17]

H. D. Gaff and L. J. Gross, Modelling tick-borne disease: A metapopulation model,, Bull. Math. Biol., 69 (2007), 265. doi: 10.1007/s11538-006-9125-5.

[18]

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

[19]

S. A. Gourley, R. Liu and J. Wu, Spatiotemporal distributions of migratory birds: Patchy models with delay,, SIAM J. Appl. Dyn. Syst., 9 (2010), 589. doi: 10.1137/090767261.

[20]

J. M. Heffernan, R. J Smith and L. M Wahl, Perspectives on the basic reproductive ratio,, J. R. Soc. Interface, 2 (2005), 281. doi: 10.1098/rsif.2005.0042.

[21]

M. W. Hirsch, H. L. Smith and X.-Q. Zhao, Chain transitivity, attractivity, and strong repellors for semidynamical systems,, J. Dynam. Differential Equations, 13 (2001), 107. doi: 10.1023/A:1009044515567.

[22]

G. R. Hosack, P. A. Rossignol and P. van den Driessche, The control of vector-borne disease epidemics,, J. Theor. Biol., 255 (2008), 16. doi: 10.1016/j.jtbi.2008.07.033.

[23]

H. Inaba, The Malthusian parameter and $R_0$ for heterogeneous populations in periodic environments,, Math. Biosci. Eng., 9 (2012), 313. doi: 10.3934/mbe.2012.9.313.

[24]

J. G. Kingsolver, Mosquito host choice and the epidemiology of malaria,, Am. Nat., 130 (1987), 811. doi: 10.1086/284749.

[25]

K. Kurtenbach, K. Hanincová, J. I. Tsao, G. Margos, D. Fish and H. Nicholas, Fundamental processes in the evolutionary ecology of Lyme borreliosis,, Nat. Rev. Microbiol., 4 (2006), 660. doi: 10.1038/nrmicro1475.

[26]

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,, Proc. Natl. Acad. Sci. USA, 100 (2003), 567. doi: 10.1073/pnas.0233733100.

[27]

Y. Lou and X.-Q. Zhao, The periodic Ross-Macdonald model with diffusion and advection,, Appl. Anal., 89 (2010), 1067. doi: 10.1080/00036810903437804.

[28]

N. H. Ogden, M. Bigras-Poulin, K. Hanincová, A. Maarouf A, C. J. O'Callaghan, K. Kurtenbach, Projected effects of climate change on tick phenology and fitness of pathogens transmitted by the North American tick Ixodes scapularis,, J. Theor. Biol., 254 (2008), 621. doi: 10.1016/j.jtbi.2008.06.020.

[29]

N. H. Ogden, M. Bigras-Poulin, C. J. O'Callaghan, I. K. Barker, L. R. Lindsay, A. Maarouf, K. E. Smoyer-Tomic, D. Waltner-Toews and D. Charron, A dynamical population model to investigate effects of climate on geographic range and seasonality of the tick Ixodes scapularis,, International Journal of Parasitology, 35 (2005), 375. doi: 10.1016/j.ijpara.2004.12.013.

[30]

N. H. Ogden, L. R. Lindsay, K. Hanincova, I. K. Barker, M. Bigras-Poulin, D. F. Charron, A. Heagy, C. M. Francis, C. J. O'Callaghan, I. Schwartz and R. A. Thompson, Role of Migratory Birds in Introduction and Range Expansion of Ixodes scapularis Ticks and of Borrelia burgdorferi and Anaplasma phagocytophilum in Canada,, Appl. Environ. Microbiol., 74 (2008), 1780. doi: 10.1128/AEM.01982-07.

[31]

N. H. Ogden, L. R. Lindsay, M. Morshed, P. N. Sockett and H. Artsob, The emergence of Lyme disease in Canada,, CMAJ, 180 (2009), 1221. doi: 10.1503/cmaj.080148.

[32]

R. S. Ostfeld, The ecology of Lyme-disease risk,, Am. Sci., 85 (1997), 338.

[33]

S. E. Randolph, Epidemiological uses of a population model for the tick Rhipicephalus appendiculatus,, Trop. Med. Int. Health., 4 (1999).

[34]

S. E. Randolph and D. J. Rogers, A generic population model for the African tick Rhipicephalus appendiculatus,, Parasitology, 115 (1997), 265.

[35]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,, Math. Surveys Monogr., (1995).

[36]

H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations,, J. Math. Biol., 30 (1992), 755. doi: 10.1007/BF00173267.

[37]

C. Thompson, A. Spielman and P. J. Krause, Coinfecting deer-dssociated zoonoses: Lyme disease, babesiosis, and ehrlichiosis,, Clin. Infect. Dis., 33 (2001), 676.

[38]

J. van Buskirk and R. S. Ostfeld, Controlling Lyme disease by modifying the density and species composition of tick hosts,, Ecol. Appl., 5 (1995), 1133. doi: 10.2307/2269360.

[39]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 180 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6.

[40]

W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments,, J. Dynam. Differ. Equations, 20 (2008), 699. doi: 10.1007/s10884-008-9111-8.

[41]

C. L. Wesley and L. J. S. Allen, The basic reproduction number in epidemic models with periodic demographics,, J. Biol. Dynam., 3 (2009), 116. doi: 10.1080/17513750802304893.

[42]

M. L. Wilson, A. M. Ducey, T. S. Litwin, T. A. Gavin and A. Spielman, Microgeographic distribution of immature Ixodes dammini correlated with that of deer,, Med. Vet. Entomol., 4 (1990), 151. doi: 10.1111/j.1365-2915.1990.tb00273.x.

[43]

X. Wu, V. R. Duvvuri, Y. Lou, N. H. Ogden, Y. Pelcat and J. Wu, Developing a temperature-driven map of the basic reproductive number of the emerging tick vector of Lyme disease Ixodes Scapularis in Canada,, J. Theo. Biol., 319 (2013), 50. doi: 10.1016/j.jtbi.2012.11.014.

[44]

X.-Q. Zhao, Global attractivity in monotone and subhomogeneous almost periodic systems,, J. Differ. Equations, 187 (2003), 494. doi: 10.1016/S0022-0396(02)00054-2.

[45]

X.-Q. Zhao, Dynamical Systems in Population Biology,, Springer-Verlag, (2003). doi: 10.1007/978-0-387-21761-1.

show all references

References:
[1]

S. Ai, Global stability of equilibria in a tick-borne disease model,, Math. Biosci. Eng., 4 (2007), 567. doi: 10.3934/mbe.2007.4.567.

[2]

J. L. Aron and R. M. May, The population dynamics of malaria,, in Population Dynamics and Infectious Disease (Ed. R. M. Anderson), (1982), 139. doi: 10.1007/978-1-4899-2901-3_5.

[3]

T. Awerbuch-Friedlander, R. Levins and M. Predescu, The role of seasonality in the dynamics of deer tick populations,, Bull. Math. Biol., 67 (2005), 467. doi: 10.1016/j.bulm.2004.08.003.

[4]

T. E. Awerbuchb and S. Sandberg, Trends and oscillations in tick population dynamics,, J. Theor. Biol., 175 (1995), 511. doi: 10.1006/jtbi.1995.0158.

[5]

N. Bacaër, Approximation of the basic reproduction number $R_0$ for vector-borne diseases with a periodic vector population,, Bull. Math. Biol., 69 (2007), 1067. doi: 10.1007/s11538-006-9166-9.

[6]

N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality. The case of cutaneous leishmaniasis in Chichaoua, Morocco,, J. Math. Biol., 53 (2006), 421. doi: 10.1007/s00285-006-0015-0.

[7]

N. Bacaër and R. Ouifki, Growth rate and basic reproduction number for population models with a simple periodic factor,, Math. Biosci., 210 (2007), 647. doi: 10.1016/j.mbs.2007.07.005.

[8]

L. Bourouiba, J. Wu, S. Newman, J. Takekawa, T. Natdorj, N. Batbayar, C. M. Bishop, L. A. Hawkes, P. J. Butler and M. Wikelski M, Spatial dynamics of bar-headed geese migration in the context of H5N1,, J. R. Soc. Interface, 52 (2010), 1627. doi: 10.1098/rsif.2010.0126.

[9]

R. J. Brinkerhoff, C. M. Folsom-O'Keefe, K. Tsao and M. A. Diuk-Wasser, Do birds affect Lyme disease risk? Range expansion of the vector-borne pathogen Borrelia burgdorferi,, Frontiers in Ecology and the Environment, 9 (2011), 103. doi: 10.1890/090062.

[10]

J. L. Brunner, K. LoGiudice and R. Ostfeld, Estimating reservior competence of Borrelia burgdorferi hosts: prevalence and infectivity, sensitivity and specificity,, J. Med. Entomol., 45 (2008), 139.

[11]

Canadian Migration Monitoring Network, CMMV 10-year report,, , ().

[12]

T. Caraco, G. Gardner, W. Maniatty, E. Deelman and B. K. Szymanski, Lyme disease: self regulation and pathogen invasion,, J. Theor. Biol., 193 (1998), 561. doi: 10.1006/jtbi.1998.0722.

[13]

T. Caraco, S. Glavanakov, G. Chen, J. E Flaherty, T. K Ohsumi and B. K. Szymanski, Stage-structured infection transmission and a spatial epidemic: A model for Lyme disease,, Am. Nat., 160 (2002), 348. doi: 10.1086/341518.

[14]

Center for Disease Control and Prevention (CDC), Lyme disease-United States, 2003-2005,, MMWR Morb. Mortal. Wkly. Rep., 56 (2007), 573.

[15]

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 populations,, J. Math. Biol., 28 (1990), 365. doi: 10.1007/BF00178324.

[16]

Z. Feng and V. Vealsco-Hernández, Competitive exclusion in a vector-host model for the dengue fever,, J. Math. Biol., 35 (1997), 523. doi: 10.1007/s002850050064.

[17]

H. D. Gaff and L. J. Gross, Modelling tick-borne disease: A metapopulation model,, Bull. Math. Biol., 69 (2007), 265. doi: 10.1007/s11538-006-9125-5.

[18]

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

[19]

S. A. Gourley, R. Liu and J. Wu, Spatiotemporal distributions of migratory birds: Patchy models with delay,, SIAM J. Appl. Dyn. Syst., 9 (2010), 589. doi: 10.1137/090767261.

[20]

J. M. Heffernan, R. J Smith and L. M Wahl, Perspectives on the basic reproductive ratio,, J. R. Soc. Interface, 2 (2005), 281. doi: 10.1098/rsif.2005.0042.

[21]

M. W. Hirsch, H. L. Smith and X.-Q. Zhao, Chain transitivity, attractivity, and strong repellors for semidynamical systems,, J. Dynam. Differential Equations, 13 (2001), 107. doi: 10.1023/A:1009044515567.

[22]

G. R. Hosack, P. A. Rossignol and P. van den Driessche, The control of vector-borne disease epidemics,, J. Theor. Biol., 255 (2008), 16. doi: 10.1016/j.jtbi.2008.07.033.

[23]

H. Inaba, The Malthusian parameter and $R_0$ for heterogeneous populations in periodic environments,, Math. Biosci. Eng., 9 (2012), 313. doi: 10.3934/mbe.2012.9.313.

[24]

J. G. Kingsolver, Mosquito host choice and the epidemiology of malaria,, Am. Nat., 130 (1987), 811. doi: 10.1086/284749.

[25]

K. Kurtenbach, K. Hanincová, J. I. Tsao, G. Margos, D. Fish and H. Nicholas, Fundamental processes in the evolutionary ecology of Lyme borreliosis,, Nat. Rev. Microbiol., 4 (2006), 660. doi: 10.1038/nrmicro1475.

[26]

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,, Proc. Natl. Acad. Sci. USA, 100 (2003), 567. doi: 10.1073/pnas.0233733100.

[27]

Y. Lou and X.-Q. Zhao, The periodic Ross-Macdonald model with diffusion and advection,, Appl. Anal., 89 (2010), 1067. doi: 10.1080/00036810903437804.

[28]

N. H. Ogden, M. Bigras-Poulin, K. Hanincová, A. Maarouf A, C. J. O'Callaghan, K. Kurtenbach, Projected effects of climate change on tick phenology and fitness of pathogens transmitted by the North American tick Ixodes scapularis,, J. Theor. Biol., 254 (2008), 621. doi: 10.1016/j.jtbi.2008.06.020.

[29]

N. H. Ogden, M. Bigras-Poulin, C. J. O'Callaghan, I. K. Barker, L. R. Lindsay, A. Maarouf, K. E. Smoyer-Tomic, D. Waltner-Toews and D. Charron, A dynamical population model to investigate effects of climate on geographic range and seasonality of the tick Ixodes scapularis,, International Journal of Parasitology, 35 (2005), 375. doi: 10.1016/j.ijpara.2004.12.013.

[30]

N. H. Ogden, L. R. Lindsay, K. Hanincova, I. K. Barker, M. Bigras-Poulin, D. F. Charron, A. Heagy, C. M. Francis, C. J. O'Callaghan, I. Schwartz and R. A. Thompson, Role of Migratory Birds in Introduction and Range Expansion of Ixodes scapularis Ticks and of Borrelia burgdorferi and Anaplasma phagocytophilum in Canada,, Appl. Environ. Microbiol., 74 (2008), 1780. doi: 10.1128/AEM.01982-07.

[31]

N. H. Ogden, L. R. Lindsay, M. Morshed, P. N. Sockett and H. Artsob, The emergence of Lyme disease in Canada,, CMAJ, 180 (2009), 1221. doi: 10.1503/cmaj.080148.

[32]

R. S. Ostfeld, The ecology of Lyme-disease risk,, Am. Sci., 85 (1997), 338.

[33]

S. E. Randolph, Epidemiological uses of a population model for the tick Rhipicephalus appendiculatus,, Trop. Med. Int. Health., 4 (1999).

[34]

S. E. Randolph and D. J. Rogers, A generic population model for the African tick Rhipicephalus appendiculatus,, Parasitology, 115 (1997), 265.

[35]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,, Math. Surveys Monogr., (1995).

[36]

H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations,, J. Math. Biol., 30 (1992), 755. doi: 10.1007/BF00173267.

[37]

C. Thompson, A. Spielman and P. J. Krause, Coinfecting deer-dssociated zoonoses: Lyme disease, babesiosis, and ehrlichiosis,, Clin. Infect. Dis., 33 (2001), 676.

[38]

J. van Buskirk and R. S. Ostfeld, Controlling Lyme disease by modifying the density and species composition of tick hosts,, Ecol. Appl., 5 (1995), 1133. doi: 10.2307/2269360.

[39]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 180 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6.

[40]

W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments,, J. Dynam. Differ. Equations, 20 (2008), 699. doi: 10.1007/s10884-008-9111-8.

[41]

C. L. Wesley and L. J. S. Allen, The basic reproduction number in epidemic models with periodic demographics,, J. Biol. Dynam., 3 (2009), 116. doi: 10.1080/17513750802304893.

[42]

M. L. Wilson, A. M. Ducey, T. S. Litwin, T. A. Gavin and A. Spielman, Microgeographic distribution of immature Ixodes dammini correlated with that of deer,, Med. Vet. Entomol., 4 (1990), 151. doi: 10.1111/j.1365-2915.1990.tb00273.x.

[43]

X. Wu, V. R. Duvvuri, Y. Lou, N. H. Ogden, Y. Pelcat and J. Wu, Developing a temperature-driven map of the basic reproductive number of the emerging tick vector of Lyme disease Ixodes Scapularis in Canada,, J. Theo. Biol., 319 (2013), 50. doi: 10.1016/j.jtbi.2012.11.014.

[44]

X.-Q. Zhao, Global attractivity in monotone and subhomogeneous almost periodic systems,, J. Differ. Equations, 187 (2003), 494. doi: 10.1016/S0022-0396(02)00054-2.

[45]

X.-Q. Zhao, Dynamical Systems in Population Biology,, Springer-Verlag, (2003). doi: 10.1007/978-0-387-21761-1.

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