American Institute of Mathematical Sciences

Advanced
2016, 13(5): 969-980. doi: 10.3934/mbe.2016025

Modeling the spread of bed bug infestation and optimal resource allocation for disinfestation

Ali Gharouni 1, and  Lin Wang 1,

1. 

Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB, E3B 5A3, Canada

Received  September 2015 Revised  March 2016 Published  July 2016

A patch-structured multigroup-like $SIS$ epidemiological model is proposed to study the spread of the common bed bug infestation. It is shown that the model exhibits global threshold dynamics with the basic reproduction number as the threshold parameter. Costs associated with the disinfestation process are incorporated into setting up the optimization problems. Procedures are proposed and simulated for finding optimal resource allocation strategies to achieve the infestation free state. Our analysis and simulations provide useful insights on how to efficiently distribute the available exterminators among the infested patches for optimal disinfestation management.
Keywords: Bed bug infestation, threshold dynamics, optimal resource allocation, disinfestation., $SIS$ model.
Mathematics Subject Classification: Primary: 92B05, 34D23; Secondary: 34C1.
Citation: Ali Gharouni, Lin Wang. Modeling the spread of bed bug infestation and optimal resource allocation for disinfestation. Mathematical Biosciences & Engineering, 2016, 13 (5) : 969-980. doi: 10.3934/mbe.2016025
References:
[1]

L. J. S. Allen, F. Brauer, P. van den Driessche and J. Wu, Mathematical Epidemiology, Springer-Verlag, Berlin Heidelberg, 2008. doi: 10.1007/978-3-540-78911-6.

[2]

C. Boase, Bedbugs-back from the brink, Pestic. Outlook, 12 (2001), 159-162. doi: 10.1039/b106301b.

[3]

C. Castillo-Chevez and H. R. Thieme, Asymptotically autonomous epidemic models, Mathematical Population Dynamics: Analysis of Heterogeneity Vol. One: Theory of Epidemics, (eds. O. Arino, D. Axelrod, M. Kimmel, M. Langlais), Wuerz (1995), 33-50.

[4]

CBC News, Bedbug outbreaks hit Saint John, Sept. 22, 2010, http://www.cbc.ca/news/canada/new-brunswick/bedbug-outbreaks-hit-saint-john-1.870474 (Lastly accessed on February 09, 2015).

[5]

CBC News, Saint John hospital hit by bed bugs, Sept. 17, 2010, http://www.cbc.ca/news/canada/new-brunswick/saint-john-hospital-hit-by-bed-bugs-1.870475 (Lastly accessed on February 09, 2015).

[6]

S. L. Doggett and R. C. Russell, {Bed bugs-latest trends and developments, Synopsis Aust. Environ. Pest Manag. Assoc. Natl. Conf., Pacific Bay Resort, Coffs Harbour Australia, (2007), 22-37.

[7]

S. L. Doggett and A. E. P.t Managers Association, A Code of Practice for the Control of Bed Bug Infestations in Australia, {Westmead Hospital, Department of Medical Entomology}, Australia, 2011.

[8]

S. L. Doggett, D. E. Dwyer, P. F. Peñas and R. C. Russell, Bed bugs: clinical relevance and control options, Clin. Microbiol. Rev. 25 (2012), 164-192. doi: 10.1128/CMR.05015-11.

[9]

S. L. Doggett, M. J. Geary and R. C. Russell, The resurgence of bed bugs in Australia: With notes on their ecology and control, Environmental Health, 4 (2004), 30-38.

[10]

S. L. Doggett, C. J. Orton, D. G. Lilly and R. C. Russell, {Bed bugs-a growing problem worldwide. Australian and international trends update and causes for concern, Aust. Environ. Pest Manag. Assoc. NSW Conf. 2011, Sess. 2A, 2 (2011), 1-24.

[11]

S. L. Doggett and R. Russell, {Bed bugs: What the GP needs to know, Aust. Fam. Physician, 38 (2009), 880-884.

[12]

S. L. Doggett and R. C. Russell, The resurgence of bed bugs, Cimex spp. (Hemiptera: Cimicidae) in Australia, Proc. Sixth Int. Conf. Urban Pests, 6, OOK-Press Kft, Pápai út, Hungary, 37 (2008), 407-425.

[13]

P. Georgescu and G. Morosanu, Pest regulation by means of impulsive controls, Appl. Math. Comput., 190 (2007), 790-803. doi: 10.1016/j.amc.2007.01.079.

[14]

H. J. Harlan, Bed bugs 101: The basics of Cimex lectularius, Am. Entomol., 52 (2006), 99-101.

[15]

S. W. Hwang, T. J. Svoboda, I. J. De Jong, K. J. Kabasele and E. Gogosis, Bed bug infestations in an urban environment, Emerg. Infect. Dis., 11 (2005), 533-538. doi: 10.3201/eid1104.041126.

[16]

L. Krueger, Features-don't get bitten by the resurgence of bed bugs-properly identifying a bed bug infestation is the key to quick control, Pest Control, 68 (2000), 58-64.

[17]

Y. Kang and C. Castillo-Chavez, Dynamics of SI models with both horizontal and vertical transmissions as well as Allee effects, Math. Biosci., 248 (2014), 97-116. doi: 10.1016/j.mbs.2013.12.006.

[18]

M. P. Lehnert, R. M. Pereira, P. G. Koehler, W. Walker and M. S. Lehnert, Control of Cimex lectularius using heat combined with dichlorvos resin strips, Med. Vet. Entomol., 25 (2011), 460-464.

[19]

S. M. Moghadas and A. B. Gumel, Global stability of a two-stage epidemic model with generalized non-linear incidence, Math. Comput. Simulat., 60 (2002), 107-118. doi: 10.1016/S0378-4754(02)00002-2.

[20]

R. K. McCormack and L. J. S. Allen, Disease emergence in multi-host epidemic models, Mathematical Medicine and Biology, 24 (2007), 17-34.

[21]

D. J. Moore and D. M. Miller, Field evaluations of insecticide treatment regimens for control of the common bed bug, Cimex lectularius (L.), Pest Manag. Sci., 65 (2009), 332-338.

[22]

J. D. Murray, Mathematical Biology I: An Introduction, vol. 17 of Interdisciplinary Applied Mathematics, Springer, New York, NY, USA, 2002.

[23]

J. Paul and J. Bates, Is infestation with the common bedbug increasing, BMJ, 320 (2000), 1141-1141. doi: 10.1136/bmj.320.7242.1141.

[24]

C. Paulhus and X.-S. Wang, Global stability analysis of a delayed susceptible-infected-susceptible epidemic model, J. Biol. Dyn., 9 (2014), 45-50. doi: 10.1080/17513758.2014.931474.

[25]

M. Pfiester, P. G. Koehler and R. M. Pereira, Effect of population structure and size on aggregation behavior of Cimex lectularius (Hemiptera: Cimicidae), J. Med. Entomol., 46 (2009), 1015-1020.

[26]

L. J. Pinto, R. Cooper and S. K. Kraft, Bed Bug Handbook: The Complete Guide to Bed Bugs and Their Control, MD: Pinto & Associates, Mechanicsville, ON Canada, 2008.

[27]

K. Reinhardt and M. T. Siva-Jothy, Biology of the bed bugs (Cimicidae), Annu. Rev. Entomol., 52 (2007), 351-374.

[28]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, AMS Math. Surveys and Monographs, Vol. 41, American Mathematical Society, Providence RI, 1995.

[29]

Statistics Canada, 2011 Census http://www12.statcan.gc.ca/census-recensement/2011/dp-pd/index-eng.cfm (Lastly accessed on February 09,2015).

[30]

H. Shu, D. Fan and J. Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission, Nonlinear Anal. Real World Appl., 13 (2012), 1581-1592. doi: 10.1016/j.nonrwa.2011.11.016.

[31]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.

[32]

E. L. Vargo, W. Booth, V. Saenz, R. G. Santangelo, C. Schal, W. H. Robinson and A. E. de Carvalho Campos, Genetic analysis of bed bug infestations and populations, 7th Int. Conf. Urban Pests, Ouro Preto, Brazil, 7 (2011), 319-323.

[33]

C. Wang and X. Wen, Bed bug infestations and control practices in China: Implications for fighting the global bed bug resurgence}, Insects, 2 (2011), 83-95. doi: 10.3390/insects2020083.

[34]

L. Wang and B. Wood, An epidemiological approach to model the viral propagation of memes, Appl. Math. Model., 35 (2011), 5442-5447. doi: 10.1016/j.apm.2011.04.035.

[35]

Z. Xiang, Y. Li and X. Song, Dynamic analysis of a pest management SEI model with saturation incidence concerning impulsive control strategy, Nonlinear Anal. Real World Appl., 10 (2009), 2335-2345. doi: 10.1016/j.nonrwa.2008.04.017.

[36]

Z. Yuan and L. Wang, Global stability of epidemiological models with group mixing and nonlinear incidence rates, Nonlinear Anal. Real World Appl., 11 (2010), 995-1004. doi: 10.1016/j.nonrwa.2009.01.040.

[37]

X.-Q. Zhao and Z.-J. Jing, Global asymptotic behavior in some cooperative systems of functional differential equations, Canad. Appl. Math. Quart., 4 (1996), 421-444.

show all references

References:
[1]

L. J. S. Allen, F. Brauer, P. van den Driessche and J. Wu, Mathematical Epidemiology, Springer-Verlag, Berlin Heidelberg, 2008. doi: 10.1007/978-3-540-78911-6.

[2]

C. Boase, Bedbugs-back from the brink, Pestic. Outlook, 12 (2001), 159-162. doi: 10.1039/b106301b.

[3]

C. Castillo-Chevez and H. R. Thieme, Asymptotically autonomous epidemic models, Mathematical Population Dynamics: Analysis of Heterogeneity Vol. One: Theory of Epidemics, (eds. O. Arino, D. Axelrod, M. Kimmel, M. Langlais), Wuerz (1995), 33-50.

[4]

CBC News, Bedbug outbreaks hit Saint John, Sept. 22, 2010, http://www.cbc.ca/news/canada/new-brunswick/bedbug-outbreaks-hit-saint-john-1.870474 (Lastly accessed on February 09, 2015).

[5]

CBC News, Saint John hospital hit by bed bugs, Sept. 17, 2010, http://www.cbc.ca/news/canada/new-brunswick/saint-john-hospital-hit-by-bed-bugs-1.870475 (Lastly accessed on February 09, 2015).

[6]

S. L. Doggett and R. C. Russell, {Bed bugs-latest trends and developments, Synopsis Aust. Environ. Pest Manag. Assoc. Natl. Conf., Pacific Bay Resort, Coffs Harbour Australia, (2007), 22-37.

[7]

S. L. Doggett and A. E. P.t Managers Association, A Code of Practice for the Control of Bed Bug Infestations in Australia, {Westmead Hospital, Department of Medical Entomology}, Australia, 2011.

[8]

S. L. Doggett, D. E. Dwyer, P. F. Peñas and R. C. Russell, Bed bugs: clinical relevance and control options, Clin. Microbiol. Rev. 25 (2012), 164-192. doi: 10.1128/CMR.05015-11.

[9]

S. L. Doggett, M. J. Geary and R. C. Russell, The resurgence of bed bugs in Australia: With notes on their ecology and control, Environmental Health, 4 (2004), 30-38.

[10]

S. L. Doggett, C. J. Orton, D. G. Lilly and R. C. Russell, {Bed bugs-a growing problem worldwide. Australian and international trends update and causes for concern, Aust. Environ. Pest Manag. Assoc. NSW Conf. 2011, Sess. 2A, 2 (2011), 1-24.

[11]

S. L. Doggett and R. Russell, {Bed bugs: What the GP needs to know, Aust. Fam. Physician, 38 (2009), 880-884.

[12]

S. L. Doggett and R. C. Russell, The resurgence of bed bugs, Cimex spp. (Hemiptera: Cimicidae) in Australia, Proc. Sixth Int. Conf. Urban Pests, 6, OOK-Press Kft, Pápai út, Hungary, 37 (2008), 407-425.

[13]

P. Georgescu and G. Morosanu, Pest regulation by means of impulsive controls, Appl. Math. Comput., 190 (2007), 790-803. doi: 10.1016/j.amc.2007.01.079.

[14]

H. J. Harlan, Bed bugs 101: The basics of Cimex lectularius, Am. Entomol., 52 (2006), 99-101.

[15]

S. W. Hwang, T. J. Svoboda, I. J. De Jong, K. J. Kabasele and E. Gogosis, Bed bug infestations in an urban environment, Emerg. Infect. Dis., 11 (2005), 533-538. doi: 10.3201/eid1104.041126.

[16]

L. Krueger, Features-don't get bitten by the resurgence of bed bugs-properly identifying a bed bug infestation is the key to quick control, Pest Control, 68 (2000), 58-64.

[17]

Y. Kang and C. Castillo-Chavez, Dynamics of SI models with both horizontal and vertical transmissions as well as Allee effects, Math. Biosci., 248 (2014), 97-116. doi: 10.1016/j.mbs.2013.12.006.

[18]

M. P. Lehnert, R. M. Pereira, P. G. Koehler, W. Walker and M. S. Lehnert, Control of Cimex lectularius using heat combined with dichlorvos resin strips, Med. Vet. Entomol., 25 (2011), 460-464.

[19]

S. M. Moghadas and A. B. Gumel, Global stability of a two-stage epidemic model with generalized non-linear incidence, Math. Comput. Simulat., 60 (2002), 107-118. doi: 10.1016/S0378-4754(02)00002-2.

[20]

R. K. McCormack and L. J. S. Allen, Disease emergence in multi-host epidemic models, Mathematical Medicine and Biology, 24 (2007), 17-34.

[21]

D. J. Moore and D. M. Miller, Field evaluations of insecticide treatment regimens for control of the common bed bug, Cimex lectularius (L.), Pest Manag. Sci., 65 (2009), 332-338.

[22]

J. D. Murray, Mathematical Biology I: An Introduction, vol. 17 of Interdisciplinary Applied Mathematics, Springer, New York, NY, USA, 2002.

[23]

J. Paul and J. Bates, Is infestation with the common bedbug increasing, BMJ, 320 (2000), 1141-1141. doi: 10.1136/bmj.320.7242.1141.

[24]

C. Paulhus and X.-S. Wang, Global stability analysis of a delayed susceptible-infected-susceptible epidemic model, J. Biol. Dyn., 9 (2014), 45-50. doi: 10.1080/17513758.2014.931474.

[25]

M. Pfiester, P. G. Koehler and R. M. Pereira, Effect of population structure and size on aggregation behavior of Cimex lectularius (Hemiptera: Cimicidae), J. Med. Entomol., 46 (2009), 1015-1020.

[26]

L. J. Pinto, R. Cooper and S. K. Kraft, Bed Bug Handbook: The Complete Guide to Bed Bugs and Their Control, MD: Pinto & Associates, Mechanicsville, ON Canada, 2008.

[27]

K. Reinhardt and M. T. Siva-Jothy, Biology of the bed bugs (Cimicidae), Annu. Rev. Entomol., 52 (2007), 351-374.

[28]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, AMS Math. Surveys and Monographs, Vol. 41, American Mathematical Society, Providence RI, 1995.

[29]

Statistics Canada, 2011 Census http://www12.statcan.gc.ca/census-recensement/2011/dp-pd/index-eng.cfm (Lastly accessed on February 09,2015).

[30]

H. Shu, D. Fan and J. Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission, Nonlinear Anal. Real World Appl., 13 (2012), 1581-1592. doi: 10.1016/j.nonrwa.2011.11.016.

[31]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.

[32]

E. L. Vargo, W. Booth, V. Saenz, R. G. Santangelo, C. Schal, W. H. Robinson and A. E. de Carvalho Campos, Genetic analysis of bed bug infestations and populations, 7th Int. Conf. Urban Pests, Ouro Preto, Brazil, 7 (2011), 319-323.

[33]

C. Wang and X. Wen, Bed bug infestations and control practices in China: Implications for fighting the global bed bug resurgence}, Insects, 2 (2011), 83-95. doi: 10.3390/insects2020083.

[34]

L. Wang and B. Wood, An epidemiological approach to model the viral propagation of memes, Appl. Math. Model., 35 (2011), 5442-5447. doi: 10.1016/j.apm.2011.04.035.

[35]

Z. Xiang, Y. Li and X. Song, Dynamic analysis of a pest management SEI model with saturation incidence concerning impulsive control strategy, Nonlinear Anal. Real World Appl., 10 (2009), 2335-2345. doi: 10.1016/j.nonrwa.2008.04.017.

[36]

Z. Yuan and L. Wang, Global stability of epidemiological models with group mixing and nonlinear incidence rates, Nonlinear Anal. Real World Appl., 11 (2010), 995-1004. doi: 10.1016/j.nonrwa.2009.01.040.

[37]

X.-Q. Zhao and Z.-J. Jing, Global asymptotic behavior in some cooperative systems of functional differential equations, Canad. Appl. Math. Quart., 4 (1996), 421-444.

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