September  2012, 17(6): 1831-1840. doi: 10.3934/dcdsb.2012.17.1831

Optimal treated mosquito bed nets and insecticides for eradication of malaria in Missira

1. 

Department of Mathematics and Computer Science, Grambling State University, Grambling, LA 71245, United States

2. 

Department of Mathematics, Howard University, Washington, DC 20059

Received  May 2011 Revised  July 2011 Published  May 2012

We extend the deterministic mathematical malaria model framework of Dembele et al. and use it to study the impact of protecting humans from mosquito bites and mass killing of mosquito vectors on malaria incidence in Missira, a village in Mali. As a case study, we fit our model to Missira malaria incidence data. Using the fitted model, we compute the optimal proportion of protected human population from infected mosquito bites and optimal proportion of killed moquitoes that would lead to the eradication of malaria in Missira.
Citation: Bassidy Dembele, Abdul-Aziz Yakubu. Optimal treated mosquito bed nets and insecticides for eradication of malaria in Missira. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1831-1840. doi: 10.3934/dcdsb.2012.17.1831
References:
[1]

A. D. Allen and M. Cisse, Determination of the frequency and correlation between glucose 6-phosphate dehydrogenase deficiency and sickle cell anemia (HbS) in a west African village (Mali), a malaria endemic region, Technical Report, Department of Biology (Howard Hughes Program), Howard University, 2010.

[2]

N. T. J. Bailey, "The Mathematical Theory of Epidemics," Hafner Publishing Co., New York, 1957.

[3]

A. Bekessey, L. Molineaux and J. Storey, Estimation of incidence and recovery rates of Plasmodium falciparum parasitaemia from longitudinal data, Bull. World Health Organ, 54 (1976), 685-693.

[4]

P. Carnevale, J. Mouchet, M. Coosemans, J. Julvez, S. Manguin, R. D. Lenoble and S. Sircoulou, "Biodiversit du Paludisme dans le Monde," John Libbey Eurotext, Paris, 2004.

[5]

D. Coulibaly, D. Diallo, M. Thera, A. Dicko, A. Guindo, A. Kone, Y. Cissoko, S. Coulibaly, A. Djimde, K. Lyke, O. Doumbo and C. Plowe, Impact of preseason treatment on incidence of falciparum malaria and parasite density at a site for testing malaria vaccines in Bandiagara, Mali, Am. J. Trop. Med. Hyg., 67 (2002), 604-610.

[6]

R. Carter, K. N. Mendis and D. Roberts, Spatial targeting of interventions against malaria, Bulletin of the World Health Organization, 78 (2000), 1401-1411.

[7]

Centers for Disease Control and Prevention, Malaria. Availabe from: http://www.cdc.gov/malaria.

[8]

B. Dembele, A. Friedman and A.-A. Yakubu, Malaria model with periodic mosquito birth and death rates, J. Biol. Dyn., 3 (2009), 430-445.

[9]

B. Dembele, A. Friedman and A.-A. Yakubu, Mathematical model for optimal use of sulfadoxine-pyrimethamine as a temporary malaria vaccine, Bulletin of Mathematical Biology, 72 (2010), 914-930. doi: 10.1007/s11538-009-9476-9.

[10]

K. Dietz, Mathematical models for malaria in different ecological, zones, Presented to the 7th International Biometric Conference, Hannover, August 1970, 1621.

[11]

K. Dietz, Mathematical models for transmission and control of malaria, in "Principles and Practice of Malariology" (eds. W. H. Wernsdorfer and I. McGregor), Churchill Livingstone, London, (1988), 1091-1113.

[12]

A. N. Gideon, and W. S. Shu, A Mathematical Model for Endemic Malaria with Variable Human and Mosquito populations, United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency, IC, (1999), 127. Available from: http://www.ictp.trieste.it/puboff.

[13]

N. J. Govella, F. O. Okumu and F. Killeen. Gerry, Insecticide-treated nets can reduce malaria transmission by mosquitoes which feed outdoors, Am. J. Trop. Med. Hyg., 82 (2010), 415-419. doi: 10.4269/ajtmh.2010.09-0579.

[14]

N. C. Grassly and C. Fraser, Seasonal infectious disease epidemiology, Proc. R. Soc. B, 273 (2006), 2541-2550. doi: 10.1098/rspb.2006.3604.

[15]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics , Proc. R. Soc. A, 115 (1927), 700-721. doi: 10.1098/rspa.1927.0118.

[16]

G. F. Killeen, A. Seyoum and B. G. J. Knols, Rationalizing historical successes of malaria control in Africa in terms of mosquito resource availability management, Am. J. Trop. Med. Hyg., 71 (2004), 87-93.

[17]

A. J. Lokta, Contributions to the analysis of malaria epidemiology, Am. J. Hyg., 3 (1923), 11-21.

[18]

G. Macdonald, The analysis of infection rates in diseases in which supperinfection occurs, Trop. Dis. Bull., 47 (1950), 907-915.

[19]

E. Martini, "Berechnungen und Beobachtungen zur Epidemiologie und Bekam pfung der Malaria," Gente, Hamburg, 1921.

[20]

F. E. McKenzie and H. W. Bossert, An integrated model of Plasmodium falciparum dynamics, J. Theor. Biol., 232 (2005), 411-426. doi: 10.1016/j.jtbi.2004.08.021.

[21]

A. L. Menach, E. F. McKenzie, A. Flahault and D. L. Smith, The unexpected importance of mosquito oviposition behaviour for malaria: Non-productive larval habitats can be sources for malaria transmission, Malaria Journal, (2005), 4-23.

[22]

National Institute of Allergy and Infectious Diseases, Malaria, Publication No. 02-7139, 2002.

[23]

T. J. Norman and M. A. Baley, "The Biomathematics of Malaria," Oxford University Press, London, 1982.

[24]

R. Ross, "The Prevention of Malaria," 2nd edition, With Addendum on the Theory of Happenings, Murray, London, 1911.

[25]

N. Sagoba, S. Doumbia, P. Vounatsou, I. Baber, M. Keita, M. Maiga, S. Toure, G. Dolo, T. Smith and J. M. C. Ribeiro, Monitoring of larval habitats and mosquito densities in the Sudan savanna of Mali: Implications of malaria vector control, Am. J. Trop. Med. Hyg., 77 (2007), 82-88.

show all references

References:
[1]

A. D. Allen and M. Cisse, Determination of the frequency and correlation between glucose 6-phosphate dehydrogenase deficiency and sickle cell anemia (HbS) in a west African village (Mali), a malaria endemic region, Technical Report, Department of Biology (Howard Hughes Program), Howard University, 2010.

[2]

N. T. J. Bailey, "The Mathematical Theory of Epidemics," Hafner Publishing Co., New York, 1957.

[3]

A. Bekessey, L. Molineaux and J. Storey, Estimation of incidence and recovery rates of Plasmodium falciparum parasitaemia from longitudinal data, Bull. World Health Organ, 54 (1976), 685-693.

[4]

P. Carnevale, J. Mouchet, M. Coosemans, J. Julvez, S. Manguin, R. D. Lenoble and S. Sircoulou, "Biodiversit du Paludisme dans le Monde," John Libbey Eurotext, Paris, 2004.

[5]

D. Coulibaly, D. Diallo, M. Thera, A. Dicko, A. Guindo, A. Kone, Y. Cissoko, S. Coulibaly, A. Djimde, K. Lyke, O. Doumbo and C. Plowe, Impact of preseason treatment on incidence of falciparum malaria and parasite density at a site for testing malaria vaccines in Bandiagara, Mali, Am. J. Trop. Med. Hyg., 67 (2002), 604-610.

[6]

R. Carter, K. N. Mendis and D. Roberts, Spatial targeting of interventions against malaria, Bulletin of the World Health Organization, 78 (2000), 1401-1411.

[7]

Centers for Disease Control and Prevention, Malaria. Availabe from: http://www.cdc.gov/malaria.

[8]

B. Dembele, A. Friedman and A.-A. Yakubu, Malaria model with periodic mosquito birth and death rates, J. Biol. Dyn., 3 (2009), 430-445.

[9]

B. Dembele, A. Friedman and A.-A. Yakubu, Mathematical model for optimal use of sulfadoxine-pyrimethamine as a temporary malaria vaccine, Bulletin of Mathematical Biology, 72 (2010), 914-930. doi: 10.1007/s11538-009-9476-9.

[10]

K. Dietz, Mathematical models for malaria in different ecological, zones, Presented to the 7th International Biometric Conference, Hannover, August 1970, 1621.

[11]

K. Dietz, Mathematical models for transmission and control of malaria, in "Principles and Practice of Malariology" (eds. W. H. Wernsdorfer and I. McGregor), Churchill Livingstone, London, (1988), 1091-1113.

[12]

A. N. Gideon, and W. S. Shu, A Mathematical Model for Endemic Malaria with Variable Human and Mosquito populations, United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency, IC, (1999), 127. Available from: http://www.ictp.trieste.it/puboff.

[13]

N. J. Govella, F. O. Okumu and F. Killeen. Gerry, Insecticide-treated nets can reduce malaria transmission by mosquitoes which feed outdoors, Am. J. Trop. Med. Hyg., 82 (2010), 415-419. doi: 10.4269/ajtmh.2010.09-0579.

[14]

N. C. Grassly and C. Fraser, Seasonal infectious disease epidemiology, Proc. R. Soc. B, 273 (2006), 2541-2550. doi: 10.1098/rspb.2006.3604.

[15]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics , Proc. R. Soc. A, 115 (1927), 700-721. doi: 10.1098/rspa.1927.0118.

[16]

G. F. Killeen, A. Seyoum and B. G. J. Knols, Rationalizing historical successes of malaria control in Africa in terms of mosquito resource availability management, Am. J. Trop. Med. Hyg., 71 (2004), 87-93.

[17]

A. J. Lokta, Contributions to the analysis of malaria epidemiology, Am. J. Hyg., 3 (1923), 11-21.

[18]

G. Macdonald, The analysis of infection rates in diseases in which supperinfection occurs, Trop. Dis. Bull., 47 (1950), 907-915.

[19]

E. Martini, "Berechnungen und Beobachtungen zur Epidemiologie und Bekam pfung der Malaria," Gente, Hamburg, 1921.

[20]

F. E. McKenzie and H. W. Bossert, An integrated model of Plasmodium falciparum dynamics, J. Theor. Biol., 232 (2005), 411-426. doi: 10.1016/j.jtbi.2004.08.021.

[21]

A. L. Menach, E. F. McKenzie, A. Flahault and D. L. Smith, The unexpected importance of mosquito oviposition behaviour for malaria: Non-productive larval habitats can be sources for malaria transmission, Malaria Journal, (2005), 4-23.

[22]

National Institute of Allergy and Infectious Diseases, Malaria, Publication No. 02-7139, 2002.

[23]

T. J. Norman and M. A. Baley, "The Biomathematics of Malaria," Oxford University Press, London, 1982.

[24]

R. Ross, "The Prevention of Malaria," 2nd edition, With Addendum on the Theory of Happenings, Murray, London, 1911.

[25]

N. Sagoba, S. Doumbia, P. Vounatsou, I. Baber, M. Keita, M. Maiga, S. Toure, G. Dolo, T. Smith and J. M. C. Ribeiro, Monitoring of larval habitats and mosquito densities in the Sudan savanna of Mali: Implications of malaria vector control, Am. J. Trop. Med. Hyg., 77 (2007), 82-88.

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