# American Institute of Mathematical Sciences

2011, 8(4): 889-914. doi: 10.3934/mbe.2011.8.889

## Controlling malaria with indoor residual spraying in spatially heterogenous environments

 1 Department of Mathematics, The University of Ottawa, 585 King Edward Ave Ottawa, ON K1N 6N5, Canada 2 Department of Mathematics and Faculty of Medicine, The University of Ottawa, 585 King Edward Ave Ottawa, ON K1N 6N5, Canada

Received  December 2010 Revised  March 2011 Published  August 2011

Indoor residual spraying – spraying insecticide inside houses to kill mosquitoes – has been one of the most effective methods of disease control ever devised, being responsible for the near-eradication of malaria from the world in the third quarter of the twentieth century and saving tens of millions of lives. However, with malaria resurgence currently underway, it has received relatively little attention, been applied only in select physical locations and not always at regular intervals. We extend a time-dependent model of malaria spraying to include spatial heterogeneity and address the following research questions: 1. What are the effects of spraying in different geographical areas? 2. How do the results depend upon the regularity of spraying? 3. Can we alter our control strategies to account for asymmetric phenomena such as wind? We use impulsive partial differential equation models to derive thresholds for malaria control when spraying occurs uniformly, within an interior disc or under asymmetric advection effects. Spatial heterogeneity results in an increase in the necessary frequency of spraying, but control is still achievable.
Citation: Mo'tassem Al-Arydah, Robert Smith?. Controlling malaria with indoor residual spraying in spatially heterogenous environments. Mathematical Biosciences & Engineering, 2011, 8 (4) : 889-914. doi: 10.3934/mbe.2011.8.889
##### References:
 [1] N. Asmer, “Partial Differential Equations with Fourier Series and Boundary Value Problems,” Pearson Preatice Hall, USA, 2005. [2] D. D. Bainov and P. S. Simeonov, “Systems with Impulsive Effect,” Ellis Horwood Ltd, Chichester, 1989. [3] D. D. Bainov and P. S. Simeonov, “Impulsive Differential Equations: Periodic Solutions and Applications,” Longman Scientific and Technical, Burnt Mill, 1993. [4] D. D. Bainov and P. S. Simeonov, “Impulsive Differential Equations: Asymptotic Properties of the Solutions,” World Scientific, Singapore, 1995. [5] P. F. Beales, V. S. Orlov and R. L. Kouynetsov, eds., “Malaria and Planning for its Control in Tropical Africa,” Moscow, WHO and UNDP, 1989. [6] J. G. Breman, M. S. Alilio and A. Mills, Conquering the intolerable burden of malaria: What's new, what's needed: A summary, Am. J. Trop. Med. Hyg., 71 (2004), 1-15. [7] R. Carter, K. N. Mendis and D. Roberts, Spatial targeting of interventions against malaria, Bulletin of the World Health Organization, 78 (2000), 1401-1411. [8] J. De Zuletta, G. W. Kafuko, A. W. R. McCrae, et al., A malaria eradication experiment in the highlands of Kigezi (Uganda), East African Medical Journal, 41 (1964), 102-120. [9] R. El Attar, “Special Functions and Orthogonal Polynomials,” Lula Press, USA, 2006. [10] M. Finkel, Malaria: Stopping a global killer, National Geographic, July 2007. [11] W. A. Foster, Mosquito sugar feeding and reproductive energetics, Annu. Rev. Entomol., 40 (1995), 443-474. [12] C. Garrett-Jones, Prognosis for interruption of malaria transmission through assessment of the mosquito's vectorial capacity, Nature, 204 (1964), 1173-1174. [13] T. A. Ghebreyesus, M. Haile, K. H. Witten, A. Getachew, M. Yohannes, S. W. Lindsay and P. Byass, Household risk factors for malaria among children in the Ethiopian highlands, Trans. R. Soc. Trop. Med. Hyg., 94 (2000), 17-21. [14] , “Global Malaria Programme: Indoor Residual Spraying,” Report of the World Health Organization, 2006. Available from: http://whqlibdoc.who.int/hq/2006/WHO_HTM_MAL_2006.1112_eng.pdf. [15] G. Hasibeder and C. Dye, Population dynamics of mosquito-borne disease: Persistence in a completely heterogeneous environment, Theor. Pop. Biol., 33 (1988), 31-53. [16] J. Keiser, J. Utzinger, M. Caldas de Castro, T. A. Smith, M. Tanner and B. H. Singer, Urbanization in sub-Saharan Africa and implication for malaria control, Am. J. Trop. Med. Hyg., 71 (2 Suppl) (2004), 118-127. [17] R. I. Kouznetsov, Malaria control by application of indoor spraying of residual insecticides in tropical Africa and its impact on population health, Tropical Doctor, 7 (1977), 81-91. [18] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, “Theory of Impulsive Differential Equations,” World Scientific, Singapore, 1989. [19] A. D. Lopez, C. D. Mathers, M. Ezzati, D. T. Jamison and C. J. Murray, Global and regional burden of disease and risk factors, 2001: Systematic analysis of population health data, Lancet, 367 (2006), 1747-1757. doi: 10.1016/S0140-6736(06)68770-9. [20] M. L. Mabaso, B. Sharp and C. Lengeler, Historical review of malarial control in southern Africa with emphasis on the use of indoor residual house-spraying, Trop. Med. Int. Health, 9 (2004), 846-856. [21] K. Macintyre, J. Keating, Y. B. Okbaldt, M. Zerom, S. Sosler, T. Ghebremeskel and T. P. Eisele, Rolling out insecticide treated nets in Eritrea: Examining the determinants of possession and use in malarious zones during the rainy season, Trop. Med. Int. Health, 11 (2006), 824-833. [22] N. Maidana and H. Yang, A spatial model to describe the dengue propagation, TEMA Tend. Mat. Apl. Comput., 8 (2007), 83-92. [23] E. A. C. Newton and P. Rieter, A model of the transmission of Dengue Fever with an evaluation of the impact of Ultra-Low Volume (ULV) insecticide applications on Dengue epidemics, Am. J. Trop. Med. Hyg., 47 (1992), 709-720. [24] A. Polyanin and A. Manzhirov, “Handbook of Integral Equations,” 2nd edition, Chapman and Hall/CRC, 2008. doi: 10.1201/9781420010558. [25] K. D. Silué, G. Raso, A. Yapi, P. Vounatsou, M. Tanner, E. K. Ńgoran and J. Utzinger, Spatially-explicit risk profiling of Plasmodium falciparum infections at a small scale: A geostatistical modelling approach, Malar J., 7 (2008), 111. doi: 10.1186/1475-2875-7-111. [26] R. J. Smith?, Could low-efficacy malaria vaccines increase secondary infections in endemic areas?, in “Mathematical Modeling of Biological Systems" [27] R. J. Smith? and S. D. Hove-Musekwa, Determining effective spraying periods to control malaria via indoor residual spraying in sub-Saharan Africa, Journal of Applied Mathematics and Decision Sciences, 2008 (2008), 19 pp. [28] R. J. Smith? and E. J. Schwartz, Predicting the potential impact of a cytotoxic T-lymphocyte HIV vaccine: How often should you vaccinate and how strong should the vaccine be?, Math. Biosci., 212 (2008), 180-187. [29] R. W. Snow, C. A. Guerra, A. M. Noor, H. Y. Myint and S. I. Hay, The global distribution of clinical episodes of Plasmodium falciparum malaria, Nature, 434 (2005), 214-217. [30] P. I. Trigg and A. V. Kondrachine, Commentary: Malaria control in the 1990s, Bull. World Health Organ., 76 (1998), 11-16. [31] , “Using Geographical Information Systems for Indoor Residual Spray Area Mapping,” Training Report for the Zambia Ministry of Health, 2007, Available from: http://www.macepalearningcommunity.org/files/IRSGISTrainingReportZambia.pdf.

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##### References:
 [1] N. Asmer, “Partial Differential Equations with Fourier Series and Boundary Value Problems,” Pearson Preatice Hall, USA, 2005. [2] D. D. Bainov and P. S. Simeonov, “Systems with Impulsive Effect,” Ellis Horwood Ltd, Chichester, 1989. [3] D. D. Bainov and P. S. Simeonov, “Impulsive Differential Equations: Periodic Solutions and Applications,” Longman Scientific and Technical, Burnt Mill, 1993. [4] D. D. Bainov and P. S. Simeonov, “Impulsive Differential Equations: Asymptotic Properties of the Solutions,” World Scientific, Singapore, 1995. [5] P. F. Beales, V. S. Orlov and R. L. Kouynetsov, eds., “Malaria and Planning for its Control in Tropical Africa,” Moscow, WHO and UNDP, 1989. [6] J. G. Breman, M. S. Alilio and A. Mills, Conquering the intolerable burden of malaria: What's new, what's needed: A summary, Am. J. Trop. Med. Hyg., 71 (2004), 1-15. [7] R. Carter, K. N. Mendis and D. Roberts, Spatial targeting of interventions against malaria, Bulletin of the World Health Organization, 78 (2000), 1401-1411. [8] J. De Zuletta, G. W. Kafuko, A. W. R. McCrae, et al., A malaria eradication experiment in the highlands of Kigezi (Uganda), East African Medical Journal, 41 (1964), 102-120. [9] R. El Attar, “Special Functions and Orthogonal Polynomials,” Lula Press, USA, 2006. [10] M. Finkel, Malaria: Stopping a global killer, National Geographic, July 2007. [11] W. A. Foster, Mosquito sugar feeding and reproductive energetics, Annu. Rev. Entomol., 40 (1995), 443-474. [12] C. Garrett-Jones, Prognosis for interruption of malaria transmission through assessment of the mosquito's vectorial capacity, Nature, 204 (1964), 1173-1174. [13] T. A. Ghebreyesus, M. Haile, K. H. Witten, A. Getachew, M. Yohannes, S. W. Lindsay and P. Byass, Household risk factors for malaria among children in the Ethiopian highlands, Trans. R. Soc. Trop. Med. Hyg., 94 (2000), 17-21. [14] , “Global Malaria Programme: Indoor Residual Spraying,” Report of the World Health Organization, 2006. Available from: http://whqlibdoc.who.int/hq/2006/WHO_HTM_MAL_2006.1112_eng.pdf. [15] G. Hasibeder and C. Dye, Population dynamics of mosquito-borne disease: Persistence in a completely heterogeneous environment, Theor. Pop. Biol., 33 (1988), 31-53. [16] J. Keiser, J. Utzinger, M. Caldas de Castro, T. A. Smith, M. Tanner and B. H. Singer, Urbanization in sub-Saharan Africa and implication for malaria control, Am. J. Trop. Med. Hyg., 71 (2 Suppl) (2004), 118-127. [17] R. I. Kouznetsov, Malaria control by application of indoor spraying of residual insecticides in tropical Africa and its impact on population health, Tropical Doctor, 7 (1977), 81-91. [18] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, “Theory of Impulsive Differential Equations,” World Scientific, Singapore, 1989. [19] A. D. Lopez, C. D. Mathers, M. Ezzati, D. T. Jamison and C. J. Murray, Global and regional burden of disease and risk factors, 2001: Systematic analysis of population health data, Lancet, 367 (2006), 1747-1757. doi: 10.1016/S0140-6736(06)68770-9. [20] M. L. Mabaso, B. Sharp and C. Lengeler, Historical review of malarial control in southern Africa with emphasis on the use of indoor residual house-spraying, Trop. Med. Int. Health, 9 (2004), 846-856. [21] K. Macintyre, J. Keating, Y. B. Okbaldt, M. Zerom, S. Sosler, T. Ghebremeskel and T. P. Eisele, Rolling out insecticide treated nets in Eritrea: Examining the determinants of possession and use in malarious zones during the rainy season, Trop. Med. Int. Health, 11 (2006), 824-833. [22] N. Maidana and H. Yang, A spatial model to describe the dengue propagation, TEMA Tend. Mat. Apl. Comput., 8 (2007), 83-92. [23] E. A. C. Newton and P. Rieter, A model of the transmission of Dengue Fever with an evaluation of the impact of Ultra-Low Volume (ULV) insecticide applications on Dengue epidemics, Am. J. Trop. Med. Hyg., 47 (1992), 709-720. [24] A. Polyanin and A. Manzhirov, “Handbook of Integral Equations,” 2nd edition, Chapman and Hall/CRC, 2008. doi: 10.1201/9781420010558. [25] K. D. Silué, G. Raso, A. Yapi, P. Vounatsou, M. Tanner, E. K. Ńgoran and J. Utzinger, Spatially-explicit risk profiling of Plasmodium falciparum infections at a small scale: A geostatistical modelling approach, Malar J., 7 (2008), 111. doi: 10.1186/1475-2875-7-111. [26] R. J. Smith?, Could low-efficacy malaria vaccines increase secondary infections in endemic areas?, in “Mathematical Modeling of Biological Systems" [27] R. J. Smith? and S. D. Hove-Musekwa, Determining effective spraying periods to control malaria via indoor residual spraying in sub-Saharan Africa, Journal of Applied Mathematics and Decision Sciences, 2008 (2008), 19 pp. [28] R. J. Smith? and E. J. Schwartz, Predicting the potential impact of a cytotoxic T-lymphocyte HIV vaccine: How often should you vaccinate and how strong should the vaccine be?, Math. Biosci., 212 (2008), 180-187. [29] R. W. Snow, C. A. Guerra, A. M. Noor, H. Y. Myint and S. I. Hay, The global distribution of clinical episodes of Plasmodium falciparum malaria, Nature, 434 (2005), 214-217. [30] P. I. Trigg and A. V. Kondrachine, Commentary: Malaria control in the 1990s, Bull. World Health Organ., 76 (1998), 11-16. [31] , “Using Geographical Information Systems for Indoor Residual Spray Area Mapping,” Training Report for the Zambia Ministry of Health, 2007, Available from: http://www.macepalearningcommunity.org/files/IRSGISTrainingReportZambia.pdf.
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