×

Optimal control of Aedes aegypti mosquitoes by the sterile insect technique and insecticide. (English) Zbl 1180.92058

Summary: We present a mathematical model to describe the dynamics of a mosquito population when sterile male mosquitoes (produced by irradiation) are introduced as a biological control, besides the application of insecticides. In order to analyze the minimal effort to reduce the fertile female mosquitoes, we search for optimal control considering the cost of insecticide application, the cost of the production of irradiated mosquitoes and their delivery as well as the social cost (proportional to the number of fertilized females mosquitoes). The optimal control is obtained by applying the Pontryagin’s maximum principle.

MSC:

92C60 Medical epidemiology
49N90 Applications of optimal control and differential games
92D30 Epidemiology

Software:

COLNEW; bvp4c; Matlab
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Gubler, D. J., Dengue, (Monath, T. P., The Arboviruses: Epidemiology and Ecology, vol. II (1986), CRC: CRC Boca Raton, FL), 213
[2] Wearing, H. J., Ecological and immunological determinants of dengue epidemics, Proc. Natl. Acad. Sci. USA, 103, 11802-11807 (2006)
[3] Maidana, N. A.; Yang, H. M., Describing the geographic propagation of dengue disease by travelling waves, Math. Biosci., 215, 64-77 (2008) · Zbl 1156.92037
[4] Knipling, E. F., Possibilities of insect control or eradication through the use of sexually sterile males, J. Econ. Entomol., 48, 459-462 (1955)
[5] E.F. Knipling, The basic principles of insect population suppression and management, Agriculture Handbook 512, U.S. Dept. of Agriculture, Washington, DC, 1979.; E.F. Knipling, The basic principles of insect population suppression and management, Agriculture Handbook 512, U.S. Dept. of Agriculture, Washington, DC, 1979.
[6] E.F. Knipling, 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, Misc. Publ. Entomol. Soc. America, 62, College Park, MD, 1985, pp. 4-7.; E.F. Knipling, 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, Misc. Publ. Entomol. Soc. America, 62, College Park, MD, 1985, pp. 4-7.
[7] Bartlett, A. C., Insect sterility, insect genetics, and insect control, (Pimentel, D., Handbook of Pest Management in Agriculture, vol. II (1990), CRC Press: CRC Press Boca Raton, FL), 279-287
[8] Costello, W. G.; Taylor, H. M., Mathematical models of the sterile male technique of insect control, (Charnes, A.; Lynn, W. R., Mathematical Analysis of Decision Problems in Ecology. Mathematical Analysis of Decision Problems in Ecology, Lecture Notes Biomathematics, vol. 5 (1975), Springer-Verlag: Springer-Verlag Berlin), 318-359
[9] Dietz, K., The effect of immigration on genetic control, Theor. Popul. Biol., 9, 58-67 (1976) · Zbl 0335.92015
[10] Ito, Y., A model of sterile insect release for eradication of the melon fly Dacus cucurbitae Coquillett, Appl. Ent. Zool., 12, 303-312 (1977)
[11] Prout, T., The joint effects of the release of sterile males and immigration of fertilized females on a density regulated population, Theor. Popul. Biol., 13, 40-71 (1978)
[12] Barclay, H. J., Pest population stability under sterile release, Res. Pop. Ecol., 24, 405-416 (1982)
[13] Plant, R. E.; Mangel, M., Modeling and simulation in agricultural pest management, SIAM Rev., 29, 235-261 (1987) · Zbl 0612.92014
[14] Barclay, H. J., Models for the sterile insect release method with the concurrent release of pesticides, Ecol. Model., 11, 167-178 (1980)
[15] Harrison, G. W.; Barclay, H. J.; van den Driesche, P., Analysis of a sterile insect release model with predation, J. Math. Biol., 16, 33-44 (1982) · Zbl 0541.92019
[16] Esteva, L.; Yang, H. M., Mathematical model to assess the control of Aedes aegypti mosquitoes by the sterile insect technique, Math. Biosci., 198, 132-147 (2005) · Zbl 1090.92048
[17] Reiter, P.; Amador, M. A.; Anderson, R. M.; Clark, G. G., Short report: dispersal of Aedes aeypti in an urban area after blood feeding as demonstrated by rubidium-marked eggs, Am. J. Trop. Med. Hyg., 52, 2, 177-179 (1995)
[18] Caetano, M. A.L.; Yoneyama, T., Optimal end sub-optimal control in Dengue epidemics, Optim. Control Appl. Meth., 22, 63-73 (2001) · Zbl 1069.92519
[19] Culshaw, R. V.; Ruan, S.; Spiteri, R. J., Optimal HIV treatment by maximizing immune response, J. Math. Biol., 48, 545-562 (2004) · Zbl 1057.92035
[20] Kirschner, D.; Lenhart, S.; Serbin, S., Optimal control of chemotherapy of HIV, J. Math. Biol., 35, 775-792 (1997) · Zbl 0876.92016
[21] Joshi, H. R., Optimal control of an HIV immunology model, Optim. Control Appl. Meth., 23, 199-213 (2002) · Zbl 1072.92509
[22] Stengel, R. F.; Ghiglizza, R.; Kulkarni, N.; Laplace, O., Optimal control of innate immune response, Optim. Control Appl. Meth., 23, 91-104 (2002) · Zbl 1072.92510
[23] Fleming, W.; Rishel, R., Deterministic and Stochastic Optimal Control (1975), Springer-Verlag: Springer-Verlag New York · Zbl 0323.49001
[24] Leitao, A., Cálculo Variacional e Controle Ótimo, 23º Colquio Brasileiro de Matemtica (2001), IMPA: IMPA Rio de Janeiro
[25] Ascher, U. M.; Spiteri, R. J., Collocation software for boundary value differential-algebraic equations, SIAM J. Sci. Comput., 15, 938-952 (1994) · Zbl 0804.65080
[26] L.F. Shampine, M.W. Reichelt, J. Kierzenka, Solving Boundary Value Problems for Ordinary Differential Equations in MATLAB with bvp4c. Available from <URL: http://www.mathworks.com/bvp_tutorial; L.F. Shampine, M.W. Reichelt, J. Kierzenka, Solving Boundary Value Problems for Ordinary Differential Equations in MATLAB with bvp4c. Available from <URL: http://www.mathworks.com/bvp_tutorial
[27] Ascher, U. M.; Mattheij, R. M.M.; Russell, R. D., Numerical solution of boundary value problems for ordinary differential equations, SIAM (1995) · Zbl 0843.65054
[28] H.B. Keller, Numerical solution of two-point boundary value problems, Regional Conference Series in Applied Mathematics No. 24, SIAM, Philadelphia, 1976.; H.B. Keller, Numerical solution of two-point boundary value problems, Regional Conference Series in Applied Mathematics No. 24, SIAM, Philadelphia, 1976.
[29] Yang, H. M.; Macoris, M. L.G.; Galvani, K. C.; Andrighetti, M. T.M.; Wanderley, D. M.V., Assessing the effects of temperature on the population of Aedes aegypti, vector of dengue, Epidem. Infect., 137, 8, 1188-1202 (2009)
[30] Yang, H. M., Modelling vaccination strategy against directly transmitted diseases using a series of pulses, J. Biol. Syst., 6, 2, 187-212 (1998)
[31] Pio, C.; Yang, H. M.; Esteva, L., Assessing the suitability of sterile insect technique applied to Aedes aegypti, J. Biol. Syst., 16, 565-577 (2008)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.