September  2016, 21(7): 2169-2191. doi: 10.3934/dcdsb.2016042

Disease outbreaks in plant-vector-virus models with vector aggregation and dispersal

1. 

Department of Mathematics, Lower Columbia College, Longview, WA 98632, United States

2. 

Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1042, United States

Received  June 2015 Revised  April 2016 Published  August 2016

While feeding on host plants, viruliferous insects serve as vectors for viruses. Successful viral transmission depends on vector behavior. Two behaviors that impact viral transmission are vector aggregation and dispersal. Vector aggregation may be due to chemical or visual cues or feeding preferences. Vector dispersal can result in widespread disease outbreaks among susceptible host plants. These two behaviors are investigated in plant-vector-virus models. Deterministic and stochastic models are formulated to account for stages of infection, vector aggregation and local dispersal between adjacent crops. First, models for a single crop are studied with aggregation included implicitly through the acquisition and inoculation rates. Second, models with aggregation and dispersal of vectors are studied when one field contains a disease-sensitive crop and another a disease-resistant crop. Analytical expressions are computed for the basic reproduction number in the deterministic models and for the probability of disease extinction in the stochastic models. These two expressions provide useful measures to assess effects of aggregation and dispersal on the rate of disease spread within and between crops and the potential for an outbreak. The modeling framework is based on cassava mosaic virus that causes significant damage in cassava crops in Africa.
Citation: Mary P. Hebert, Linda J. S. Allen. Disease outbreaks in plant-vector-virus models with vector aggregation and dispersal. Discrete and Continuous Dynamical Systems - B, 2016, 21 (7) : 2169-2191. doi: 10.3934/dcdsb.2016042
References:
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O. J. Alabi, P. L. Kumar and R. A. Naidu, Cassava mosaic disease: A curse to food security in sub-Saharan Africa, Online. APS net Features, 2011.

[2]

L. J. S. Allen and E. J. Allen, Deterministic and stochastic SIR epidemic models with power function transmission and recovery rates, in Mathematics of Continuous and Discrete Dynamical Systems (ed. A. Gumel), Amer. Math. Soc. Providence, R. I., 618 (2014), 1-15. doi: 10.1090/conm/618/12329.

[3]

L. J. S. Allen and G. E. Lahodny, Jr., Extinction thresholds in deterministic and stochastic models, Journal of Biological Dynamics, 6 (2012), 590-611. doi: 10.1080/17513758.2012.665502.

[4]

L. J. S. Allen and P. van den Driessche, Relations between deterministic and stochastic thresholds for disease extinction in continuous- and discrete-time infectious disease models, Mathematical Biosciences, 243 (2013), 99-108. doi: 10.1016/j.mbs.2013.02.006.

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F. Brauer and J. A. Nohel, The Qualitative Theory of Ordinary Differential Equations: An Introduction. Courier Dover Publications, Mineola, New York, 1989.

[6]

D. N. Byrne, R. Isaacs and K. H. Veenstra, Local dispersal and migration by insect pests and their importance in IPM strategies, in Radcliffe's IPM World Textbook (eds. E. B. Radcliffe and W. D. Hutchison), URL: http://ipmworld.umn.edu, University of Minnesota, St. Paul, MN, 2013.

[7]

D. N. Byrne, R. J. Rathman, T. V. Orum and J. C. Palumbo, Localized migration and dispersal by the sweet potato whitefly, Bemisia tabaci, Oecologia, 105 (1996), 320-328.

[8]

M. Chan and M. J. Jeger, An analytical model of plant virus disease dynamics with roguing and replanting, Journal of Applied Ecology, 31 (1994), 413-427. doi: 10.2307/2404439.

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K. S. Dorman, J. S. Sinsheimer and K. Lange, In the garden of branching processes, SIAM Review, 46 (2004), 202-229. doi: 10.1137/S0036144502417843.

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D. Fargette, C. Fauquet, E. Grenier and J. M. Thresh, The spread of African cassava mosaic virus into and within cassava fields, Journal of Phytopathology, 130 (1990), 289-302. doi: 10.1111/j.1439-0434.1990.tb01179.x.

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C. Fauquet and E. Fargette, African cassava mosaic virus: Etiology, epidemiology, and control, Plant Disease, 74 (1990), 404-411. doi: 10.1094/PD-74-0404.

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R. S. Ferriss and P. H. Berger, A stochastic simulation model of epidemics of arthropod-vectored plant viruses, Phytopathology, 83 (1993), 1269-1278. doi: 10.1094/Phyto-83-1269.

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K. F. Harris, O. P. Smith and J. E. Duffus, Virus-Insect-Plant Interactions, Academic Press, San Diego, 2001.

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T. Harris, The Theory of Branching Processes, Springer-Verlag, Berlin, 1963.

[15]

M. Hebert, Plant-vector-virus Model with Vector Aggregation of Cassava Mosaic Virus, Master's Thesis, Texas Tech University, Lubbock, Texas, 2014.

[16]

J. A. P. Heesterbeek and M. G. Roberts, The type-reproduction number T in models for infectious disease control, Mathematical Biosciences, 206 (2007), 3-10. doi: 10.1016/j.mbs.2004.10.013.

[17]

J. M. Heffernan, R. J. Smith and L. M. Wahl, Perspectives on the basic reproductive ratio, Journal of the Royal Society Interface, 2 (2005), 281-293. doi: 10.1098/rsif.2005.0042.

[18]

M. E. Hochberg, Non-linear transmission rates and the dynamics of infectious disease, Journal of Theoretical Biology, 153 (1991), 301-321. doi: 10.1016/S0022-5193(05)80572-7.

[19]

J. Holt, J. Colvin and V. Muniyappa, Identifying control strategies for tomato leaf curl virus using an epidemiological model, Journal of Applied Ecology, 36 (1999), 625-633. doi: 10.1046/j.1365-2664.1999.00432.x.

[20]

J. Holt, M. J. Jeger, J. M. Thresh and G. W. Otim-nape, An epidemiological model incorporating vector population dynamics applied to African cassava mosaic virus disease, Journal of Applied Ecology, 34 (1997), 793-806.

[21]

L. L. Ingwell, S. D. Eigenbrode and N. A. Bosque Pérez, Plant viruses alter insect behavior to enhance their spread, Scientific Reports, 2 (2012), p578. doi: 10.1038/srep00578.

[22]

P. Jagers, Branching Processes with Biological Applications, Wiley Series in Probability and Mathematical Statistics. Applied Probability and Statistics, Wiley-Interscience, London, 1975.

[23]

M. J. Jeger, J. Holt, F. van den Bosch and L. V. Madden, Epidemiology of insect-transmitted plant viruses: Modelling disease dynamics and control interventions, Physiological Entomology, 29 (2004), 291-304. doi: 10.1111/j.0307-6962.2004.00394.x.

[24]

M. J. Jeger, F. van den Bosch, L. V. Madden and J. Holt, A model for analysing plant-virus transmission characteristics and epidemic development, Mathematical Medicine and Biology, 15 (1998), 1-18. doi: 10.1093/imammb/15.1.1.

[25]

M. J. Jeger, F. van den Bosch and N. McRoberts, Modelling transmission characteristics and epidemic development of the tospovirus-thrip interaction, Arthropod-Plant Interactions, 9 (2015), 107-120. doi: 10.1007/s11829-015-9363-2.

[26]

R. J. Kryscio and N. C. Severo, Some properties of an extended simple stochastic epidemic model involving two additional parameters, Mathematical Biosciences, 5 (1969), 1-8. doi: 10.1016/0025-5564(69)90031-5.

[27]

G. E. Lahodny Jr. and L. J. S. Allen, Probability of a disease outbreak in stochastic multi-patch epidemic models, Bulletin of Mathematical Biology, 75 (2013), 1157-1180. doi: 10.1007/s11538-013-9848-z.

[28]

J. P. LaSalle and S. Lefschetz, Stability by Lyapunov's Second Method with Applications, Academic Press, New York, 1961.

[29]

S. Legarrea, A. Barman, W. Marchant, S. Diffie and R. Srinivasan, Temporal effects of a Begomovirus infection and host plant resistance on the preference and development of an insect vector, Bemisia tabaci, and implications for epidemics, PLoS ONE, 10 (2015), e0142114. doi: 10.1371/journal.pone.0142114.

[30]

W. Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, Journal of Mathematical Biology, 23 (1986), 187-204. doi: 10.1007/BF00276956.

[31]

L. V. Madden, G. Hughes and F. van den Bosch, The Study of Plant Disease Epidemics, American Phytopathological Society Press, St. Paul, MN, 2007.

[32]

L. V. Madden, M. J. Jeger and F. van den Bosch, A theoretical assessment of the effects of vector-virus transmission mechanism on plant virus disease epidemics, Phytopathology, 90 (2000), 576-594. doi: 10.1094/PHYTO.2000.90.6.576.

[33]

P. McElhany, L. A. Real and A. G. Power, Vector preference and disease dynamics: A study of barley yellow dwarf virus, Ecology, 76 (1995), 444-457. doi: 10.2307/1941203.

[34]

R. E. Mickens, An exactly solvable model for the spread of disease, The College Mathematics Journal, 43 (2012), 114-121. doi: 10.4169/college.math.j.43.2.114.

[35]

A. Moreno-Delafuente, E. Garzo, A. Moren and A. Fereres, A plant virus manipulates the behavior of its whitefly vector to enhance its transmission efficiency and spread, PLoS ONE, 8 (2013), e61543. doi: 10.1371/journal.pone.0061543.

[36]

F. Nakasuji, K. Kiritani and E. Tomida, A computer simulation of the epidemiology of the rice dwarf virus, Researches on Population Ecology, 16 (1975), 245-251. doi: 10.1007/BF02511064.

[37]

A. G. Power, Insect transmission of plant viruses: A constraint on virus variability, Current Opinion in Plant Biology, 3 (2000), 336-340. doi: 10.1016/S1369-5266(00)00090-X.

[38]

M. G. Roberts and J. A. P. Heesterbeek, A new method to estimate the effort required to control an infectious disease, Proceedings of the Royal Society London, Series B, 270 (2003), 1359-1364.

[39]

S. E. Seal, M. J. Jeger and F. van den Bosch, Factors influencing begomovirus evolution and their increasing global significance: Implications for sustainable control, Critical Reviews In Plant Sciences, 25 (2006), 23-46. doi: 10.1080/07352680500365257.

[40]

N. C. Severo, Generalizations of some stochastic epidemic models, Mathematical Biosciences, 4 (1969), 395-402. doi: 10.1016/0025-5564(69)90019-4.

[41]

Z. Shuai, J. A. P. Heesterbeek and P. van den Driessche, Extending the type reproduction number to infectious disease control targeting contacts between types, Journal of Mathematical Biology, 67 (2013), 1067-1082. doi: 10.1007/s00285-012-0579-9.

[42]

Z. Shuai and P. van den Driessche, Global stability of infectious disease models using Lyapunov functions, SIAM J. Applied Mathematics, 73 (2013), 1513-1532. doi: 10.1137/120876642.

[43]

M. S. Sisterson, Effect of insect-vector preference for healthy or infected plants on pathogen spread: Insights from a model, Journal of Economic Entomology, 101 (2008), 1-8. doi: 10.1093/jee/101.1.1.

[44]

G. L. Teetes, Plant Resistance to Insects: A Fundamental Component of IPM, in Radcliffe's IPM World Textbook (eds. E. B. Radcliffe and W. D. Hutchison), URL: http://ipmworld.umn.edu, University of Minnesota, St. Paul, MN, 2013.

[45]

J. M. Thresh and R. J. Cooter, Strategies for controlling cassava mosaic virus disease in Africa, Plant Pathology, 54 (2005), 587-614. doi: 10.1111/j.1365-3059.2005.01282.x.

[46]

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.

[47]

P. van den Driessche and J. Watmough, Chapter 6: Further notes on the basic reproduction number, in Mathematical Epidemiology (eds. F Brauer, P van den Driessche and J. Wu), Lecture Notes in Math, Springer-Verlag, Berlin, 1945 (2008), 159-178. doi: 10.1007/978-3-540-78911-6_6.

[48]

E. B. Wilson and J. Worcester, The law of mass action in epidemiology, PNAS, 31 (1945), 24-34.

[49]

E. B. Wilson and J. Worcester, The law of mass action in epidemiology, II, PNAS, 31 (1945), 109-116.

[50]

X. S. Zhang, J. Holt and J. Colvin, A general model of plant-virus disease infection incorporating vector aggregation, Plant Pathology, 49 (2000), 435-444. doi: 10.1046/j.1365-3059.2000.00469.x.

show all references

References:
[1]

O. J. Alabi, P. L. Kumar and R. A. Naidu, Cassava mosaic disease: A curse to food security in sub-Saharan Africa, Online. APS net Features, 2011.

[2]

L. J. S. Allen and E. J. Allen, Deterministic and stochastic SIR epidemic models with power function transmission and recovery rates, in Mathematics of Continuous and Discrete Dynamical Systems (ed. A. Gumel), Amer. Math. Soc. Providence, R. I., 618 (2014), 1-15. doi: 10.1090/conm/618/12329.

[3]

L. J. S. Allen and G. E. Lahodny, Jr., Extinction thresholds in deterministic and stochastic models, Journal of Biological Dynamics, 6 (2012), 590-611. doi: 10.1080/17513758.2012.665502.

[4]

L. J. S. Allen and P. van den Driessche, Relations between deterministic and stochastic thresholds for disease extinction in continuous- and discrete-time infectious disease models, Mathematical Biosciences, 243 (2013), 99-108. doi: 10.1016/j.mbs.2013.02.006.

[5]

F. Brauer and J. A. Nohel, The Qualitative Theory of Ordinary Differential Equations: An Introduction. Courier Dover Publications, Mineola, New York, 1989.

[6]

D. N. Byrne, R. Isaacs and K. H. Veenstra, Local dispersal and migration by insect pests and their importance in IPM strategies, in Radcliffe's IPM World Textbook (eds. E. B. Radcliffe and W. D. Hutchison), URL: http://ipmworld.umn.edu, University of Minnesota, St. Paul, MN, 2013.

[7]

D. N. Byrne, R. J. Rathman, T. V. Orum and J. C. Palumbo, Localized migration and dispersal by the sweet potato whitefly, Bemisia tabaci, Oecologia, 105 (1996), 320-328.

[8]

M. Chan and M. J. Jeger, An analytical model of plant virus disease dynamics with roguing and replanting, Journal of Applied Ecology, 31 (1994), 413-427. doi: 10.2307/2404439.

[9]

K. S. Dorman, J. S. Sinsheimer and K. Lange, In the garden of branching processes, SIAM Review, 46 (2004), 202-229. doi: 10.1137/S0036144502417843.

[10]

D. Fargette, C. Fauquet, E. Grenier and J. M. Thresh, The spread of African cassava mosaic virus into and within cassava fields, Journal of Phytopathology, 130 (1990), 289-302. doi: 10.1111/j.1439-0434.1990.tb01179.x.

[11]

C. Fauquet and E. Fargette, African cassava mosaic virus: Etiology, epidemiology, and control, Plant Disease, 74 (1990), 404-411. doi: 10.1094/PD-74-0404.

[12]

R. S. Ferriss and P. H. Berger, A stochastic simulation model of epidemics of arthropod-vectored plant viruses, Phytopathology, 83 (1993), 1269-1278. doi: 10.1094/Phyto-83-1269.

[13]

K. F. Harris, O. P. Smith and J. E. Duffus, Virus-Insect-Plant Interactions, Academic Press, San Diego, 2001.

[14]

T. Harris, The Theory of Branching Processes, Springer-Verlag, Berlin, 1963.

[15]

M. Hebert, Plant-vector-virus Model with Vector Aggregation of Cassava Mosaic Virus, Master's Thesis, Texas Tech University, Lubbock, Texas, 2014.

[16]

J. A. P. Heesterbeek and M. G. Roberts, The type-reproduction number T in models for infectious disease control, Mathematical Biosciences, 206 (2007), 3-10. doi: 10.1016/j.mbs.2004.10.013.

[17]

J. M. Heffernan, R. J. Smith and L. M. Wahl, Perspectives on the basic reproductive ratio, Journal of the Royal Society Interface, 2 (2005), 281-293. doi: 10.1098/rsif.2005.0042.

[18]

M. E. Hochberg, Non-linear transmission rates and the dynamics of infectious disease, Journal of Theoretical Biology, 153 (1991), 301-321. doi: 10.1016/S0022-5193(05)80572-7.

[19]

J. Holt, J. Colvin and V. Muniyappa, Identifying control strategies for tomato leaf curl virus using an epidemiological model, Journal of Applied Ecology, 36 (1999), 625-633. doi: 10.1046/j.1365-2664.1999.00432.x.

[20]

J. Holt, M. J. Jeger, J. M. Thresh and G. W. Otim-nape, An epidemiological model incorporating vector population dynamics applied to African cassava mosaic virus disease, Journal of Applied Ecology, 34 (1997), 793-806.

[21]

L. L. Ingwell, S. D. Eigenbrode and N. A. Bosque Pérez, Plant viruses alter insect behavior to enhance their spread, Scientific Reports, 2 (2012), p578. doi: 10.1038/srep00578.

[22]

P. Jagers, Branching Processes with Biological Applications, Wiley Series in Probability and Mathematical Statistics. Applied Probability and Statistics, Wiley-Interscience, London, 1975.

[23]

M. J. Jeger, J. Holt, F. van den Bosch and L. V. Madden, Epidemiology of insect-transmitted plant viruses: Modelling disease dynamics and control interventions, Physiological Entomology, 29 (2004), 291-304. doi: 10.1111/j.0307-6962.2004.00394.x.

[24]

M. J. Jeger, F. van den Bosch, L. V. Madden and J. Holt, A model for analysing plant-virus transmission characteristics and epidemic development, Mathematical Medicine and Biology, 15 (1998), 1-18. doi: 10.1093/imammb/15.1.1.

[25]

M. J. Jeger, F. van den Bosch and N. McRoberts, Modelling transmission characteristics and epidemic development of the tospovirus-thrip interaction, Arthropod-Plant Interactions, 9 (2015), 107-120. doi: 10.1007/s11829-015-9363-2.

[26]

R. J. Kryscio and N. C. Severo, Some properties of an extended simple stochastic epidemic model involving two additional parameters, Mathematical Biosciences, 5 (1969), 1-8. doi: 10.1016/0025-5564(69)90031-5.

[27]

G. E. Lahodny Jr. and L. J. S. Allen, Probability of a disease outbreak in stochastic multi-patch epidemic models, Bulletin of Mathematical Biology, 75 (2013), 1157-1180. doi: 10.1007/s11538-013-9848-z.

[28]

J. P. LaSalle and S. Lefschetz, Stability by Lyapunov's Second Method with Applications, Academic Press, New York, 1961.

[29]

S. Legarrea, A. Barman, W. Marchant, S. Diffie and R. Srinivasan, Temporal effects of a Begomovirus infection and host plant resistance on the preference and development of an insect vector, Bemisia tabaci, and implications for epidemics, PLoS ONE, 10 (2015), e0142114. doi: 10.1371/journal.pone.0142114.

[30]

W. Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, Journal of Mathematical Biology, 23 (1986), 187-204. doi: 10.1007/BF00276956.

[31]

L. V. Madden, G. Hughes and F. van den Bosch, The Study of Plant Disease Epidemics, American Phytopathological Society Press, St. Paul, MN, 2007.

[32]

L. V. Madden, M. J. Jeger and F. van den Bosch, A theoretical assessment of the effects of vector-virus transmission mechanism on plant virus disease epidemics, Phytopathology, 90 (2000), 576-594. doi: 10.1094/PHYTO.2000.90.6.576.

[33]

P. McElhany, L. A. Real and A. G. Power, Vector preference and disease dynamics: A study of barley yellow dwarf virus, Ecology, 76 (1995), 444-457. doi: 10.2307/1941203.

[34]

R. E. Mickens, An exactly solvable model for the spread of disease, The College Mathematics Journal, 43 (2012), 114-121. doi: 10.4169/college.math.j.43.2.114.

[35]

A. Moreno-Delafuente, E. Garzo, A. Moren and A. Fereres, A plant virus manipulates the behavior of its whitefly vector to enhance its transmission efficiency and spread, PLoS ONE, 8 (2013), e61543. doi: 10.1371/journal.pone.0061543.

[36]

F. Nakasuji, K. Kiritani and E. Tomida, A computer simulation of the epidemiology of the rice dwarf virus, Researches on Population Ecology, 16 (1975), 245-251. doi: 10.1007/BF02511064.

[37]

A. G. Power, Insect transmission of plant viruses: A constraint on virus variability, Current Opinion in Plant Biology, 3 (2000), 336-340. doi: 10.1016/S1369-5266(00)00090-X.

[38]

M. G. Roberts and J. A. P. Heesterbeek, A new method to estimate the effort required to control an infectious disease, Proceedings of the Royal Society London, Series B, 270 (2003), 1359-1364.

[39]

S. E. Seal, M. J. Jeger and F. van den Bosch, Factors influencing begomovirus evolution and their increasing global significance: Implications for sustainable control, Critical Reviews In Plant Sciences, 25 (2006), 23-46. doi: 10.1080/07352680500365257.

[40]

N. C. Severo, Generalizations of some stochastic epidemic models, Mathematical Biosciences, 4 (1969), 395-402. doi: 10.1016/0025-5564(69)90019-4.

[41]

Z. Shuai, J. A. P. Heesterbeek and P. van den Driessche, Extending the type reproduction number to infectious disease control targeting contacts between types, Journal of Mathematical Biology, 67 (2013), 1067-1082. doi: 10.1007/s00285-012-0579-9.

[42]

Z. Shuai and P. van den Driessche, Global stability of infectious disease models using Lyapunov functions, SIAM J. Applied Mathematics, 73 (2013), 1513-1532. doi: 10.1137/120876642.

[43]

M. S. Sisterson, Effect of insect-vector preference for healthy or infected plants on pathogen spread: Insights from a model, Journal of Economic Entomology, 101 (2008), 1-8. doi: 10.1093/jee/101.1.1.

[44]

G. L. Teetes, Plant Resistance to Insects: A Fundamental Component of IPM, in Radcliffe's IPM World Textbook (eds. E. B. Radcliffe and W. D. Hutchison), URL: http://ipmworld.umn.edu, University of Minnesota, St. Paul, MN, 2013.

[45]

J. M. Thresh and R. J. Cooter, Strategies for controlling cassava mosaic virus disease in Africa, Plant Pathology, 54 (2005), 587-614. doi: 10.1111/j.1365-3059.2005.01282.x.

[46]

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.

[47]

P. van den Driessche and J. Watmough, Chapter 6: Further notes on the basic reproduction number, in Mathematical Epidemiology (eds. F Brauer, P van den Driessche and J. Wu), Lecture Notes in Math, Springer-Verlag, Berlin, 1945 (2008), 159-178. doi: 10.1007/978-3-540-78911-6_6.

[48]

E. B. Wilson and J. Worcester, The law of mass action in epidemiology, PNAS, 31 (1945), 24-34.

[49]

E. B. Wilson and J. Worcester, The law of mass action in epidemiology, II, PNAS, 31 (1945), 109-116.

[50]

X. S. Zhang, J. Holt and J. Colvin, A general model of plant-virus disease infection incorporating vector aggregation, Plant Pathology, 49 (2000), 435-444. doi: 10.1046/j.1365-3059.2000.00469.x.

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