The Use of Non-Standard Finite Difference Schemes to Solve the DAMP and SIT Models

Joshua A Mwasunda¹, Eunice W Mureithi² and Nyimvua Shaban^2,

¹Department of Mathematics, Mkwawa University College of Education, Tanzania

²Department of Mathematics, University of Dar es Salaam, Tanzania

Cite this paper:
Joshua A Mwasunda, Eunice W Mureithi and Nyimvua Shaban. The Use of Non-Standard Finite Difference Schemes to Solve the DAMP and SIT Models. Journal of Mathematical Sciences and Applications. 2015; 3(2):25-32. doi: 10.12691/jmsa-3-2-2

Abstract

Sterile insect technique (SIT) is a method of biological control that uses sterile male insects to reduce the reproductive rate of a species of target insect. The method relies on the release of sterile or treated males in order to reduce the native population of insects. We propose the model that governs the dynamics of the anopheles mosquito population, and then modify to incorporate the sterile insect technique as an intervention to curtail the reproduction of mosquitoes. The nonstandard finite difference numerical schemes and simulations for these models are provided. The results indicate that sterile technique with frequent and high rate of release can be an alternative to chemical control tools in the fight against malaria.

Keywords:
sterile insect technique finite difference scheme reproduction number

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References:

[1]	World Health Organization (WHO): Media Centre Malaria Factsheet, No.94.October, 2011 http://www.who.int/ mediacentre / factsheets/fs094/en/ [Accessed October 22, 2011].
[2]	World Health Organization (WHO), Media centre – Malaria Factsheet, No.94. April 2010. http: // www.who.int/mediacentre/ factsheets/ fs094/en/[Accessed August 28, 2011].
[3]	Dorta,D.M., Vasuki, V and Rajavel, A, Evaluation of organophosphorus and synthetic pyrethroid insecticides against six vectormosquitoes species. Revistade SaúdePública 1993, 27: 391-7.
[4]	Knipling, E.F, Sterile insect technique as a screwworm control measure: the concept and its development, in: O.H.Graham (Ed.), Symposium on Eradication of the Screwworm from the United States and Mexico, 62, Misc. Publ.Entomol. Soc. America, College Park, MD, p. 4, 1985.
[5]	Bartlett,A.C and Staten, R.T, The sterile release method and other genetic control strategies, in: E.B. Radcliffe, W.D.Hutchison (Eds.), Radcliffe’s IPM World Textbook, University of Minesota, St. Paul, MN, Available at: http://ipmword.umn.edu, 1996.
[6]	Anguelov, R.,Dumont, Y andLubuma,J, Mathematical modelling of sterile insect technology for control of anopheles mosquito, Computers and Mathematics with Applications.
[7]	Steinau, R, Tips from a real pest Control expert-Sterile Insect Technique: www.asktheexterminator.com/Do_It_ Yourself_Pest_Control/Sterile_Insect_ Technique.shtml, 2007.
[8]	Mwasunda, JA, Modelling the Effect of Sterile Insect Technology for control of Anopheles mosquito population in Tanzania, MSc. Dissertation , University of Dar es Salaam, 2012.
[9]	Esteva, Land Yang, H.M, Mathematical model to assess the control of Aedesaegypti mosquitoes by the sterile insect technique, Mathematical Biosciences, 2005, 198, 132-147.
[10]	Esteva, L and Yang, H. M, Control of Dengue Dengue Vector by the Sterile Insect Technique Considering Logistic Recruitment, TEMA Tend. Mat. Apl. Comput., 7, No. 2, 259-268, 2006.
[11]	Dobromir, T. D and Hristo, V. K, Nonstand-ard finite-difference methods for predator–prey models with general functional response, J. Mathematics and Computers in Simulation 2007, 78 1-11.
[12]	Mickens, R.E Advances in the applications of nonstandard finite difference schemes.World Scientific, Singapore, 1994.
[13]	Dumont, Y and Lubuma, J Non-standard finite difference methods for vibro-impact problems, Proc.R. Soc. London, 461A, 2005, 1927-1950.
[14]	Mickens, R..E, Advances in the applications of nonstandard finite difference schemes. World Scientific, Singapore. 1994.
[15]	Mickens, R, Non-standard Finite Difference models of Differential Equation. World Scientific, Singapore 2005.
[16]	Kasim, M. A, Stability of Real-Time Systems, Computer Engineering Department, Philadelphia University, 2011.