×

Optimal control and stability analysis of malaria disease: a model based approach. (English) Zbl 1479.92008

Summary: In this paper we have proposed a three dimensional mathematical model on malaria disease by considering two distinct classes namely susceptible and infected human population and infected mosquito population. Basic reproductive number of the system has been obtained and its relation regarding the behavior of the system has been established. Two control parameters, namely treatment control on infected human population and insecticide control on mosquito populations are applied in the present system. We formulate and solve the optimal control problem considering treatment and insecticide as the control variables. All the theoretical results are verified by some computer simulation works.

MSC:

92D30 Epidemiology
92D25 Population dynamics (general)
49N90 Applications of optimal control and differential games
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] P. B. Allen, F. Calegari, A. Caraiani, T. Gee, D. Helm, B. V. Le Hung, J. Newton, P. Scholze, R. Taylor, and J. A. Thorne. Potential automorphy over CM fields. arXiv:1812.09999 (December 2018)
[2] P. B. B. Allen, C. Khare, and J. A. Thorne. Modularity of GL 2 ( p )-representations over CM fields. arXiv:1910.12986 (October 2019)
[3] J. Arthur, Unipotent automorphic representations: Conjectures. 171-172, 13-71 (1989) · Zbl 0728.22014
[4] T. Barnet-Lamb, D. Geraghty, M. Harris and R. Taylor, A family of Calabi-Yau varieties and potential automorphy II. Publ. Res. Inst. Math. Sci. 47, 29-98 (2011) · Zbl 1264.11044
[5] B. Bhatt, A. Caraiani, K. S. Kedlaya and J. Weinstein, Perfectoid spaces. Mathematical Surveys and Monographs 242, American Mathematical Society, Providence, RI (2019)
[6] A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups. Second ed., Mathematical Sur-veys and Monographs 67, American Mathematical Society, Providence, RI (2000) · Zbl 0980.22015
[7] G. Boxer, F. Calegari, T. Gee, and V. Pilloni. Abelian Surfaces over totally real fields are Potentially Modular. arXiv:1812.09269 (December 2018)
[8] P. Boyer, Sur la torsion dans la cohomologie des variétés de Shimura de Kottwitz-Harris-Taylor. J. Inst. Math. Jussieu 18, 499-517 (2019) · Zbl 1443.11080
[9] C. Breuil, B. Conrad, F. Diamond and R. Taylor, On the modularity of elliptic curves over Q: Wild 3-adic exercises. J. Amer. Math. Soc. 14, 843-939 (2001) · Zbl 0982.11033
[10] F. Calegari and M. Emerton, Completed cohomology -A survey. In Non-abelian fundamental groups and Iwasawa theory, London Math. Soc. Lecture Note Ser. 393, Cambridge Univ. Press, Cambridge, 239-257 (2012) · Zbl 1288.11056
[11] F. Calegari and D. Geraghty, Modularity lifting beyond the Taylor-Wiles method. Invent. Math. 211, 297-433 (2018) · Zbl 1476.11078
[12] A. Caraiani, Local-global compatibility and the action of monodromy on nearby cycles. Duke Math. J. 161, 2311-2413 (2012) · Zbl 1405.22028
[13] A. Caraiani, Monodromy and local-global compatibility for l = p. Algebra Number Theory 8, 1597-1646 (2014) · Zbl 1310.11061
[14] A. Caraiani, D. R. Gulotta, C.-Y. Hsu, C. Johansson, L. Mocz, E. Reinecke and S.-C. Shih, Shimura varieties at level Γ 1 (p ∞ ) and Galois representa-tions. Compos. Math. 156, 1152-1230 (2020) · Zbl 1452.11063
[15] A. Caraiani, D. R. Gulotta, and C. Johansson. Vanishing theorems for Shimura varieties at unipotent level. arXiv:1910.09214 (October 2019)
[16] A. Caraiani and P. Scholze, On the generic part of the cohomology of compact unitary Shimura varieties. Ann. of Math. (2) 186, 649-766 (2017) · Zbl 1401.11108
[17] A. Caraiani and P. Scholze, On the generic part of the cohomology of compact unitary Shimura varieties. arXiv:1909.01898 (September 2019)
[18] L. Clozel, Purity reigns supreme. Int. Math. Res. Not. IMRN 328-346 (2013) · Zbl 1370.11063
[19] L. Clozel, M. Harris and R. Taylor, Automorphy for some l-adic lifts of au-tomorphic mod l Galois representations. Publ. Math. Inst. Hautes Études Sci. 1-181 (2008) · Zbl 1169.11020
[20] M. Emerton, Completed cohomology and the p-adic Langlands program. In Proceedings of the International Congress of Mathematicians -Seoul 2014. Vol. II, Kyung Moon Sa, Seoul, 319-342 (2014) · Zbl 1373.11039
[21] M. Emerton. Langlands reciprocity: L-functions, automorphic forms, and Diophantine equations. To appear in The Genesis of the Langlands program (2020)
[22] N. Freitas, B. V. Le Hung and S. Siksek, Elliptic curves over real quadratic fields are modular. Invent. Math. 201, 159-206 (2015) · Zbl 1397.11086
[23] M. Harris, K.-W. Lan, R. Taylor and J. Thorne, On the rigid cohomology of certain Shimura varieties. Res. Math. Sci. 3, Paper No. 37, 308 (2016) · Zbl 1410.11040
[24] M. Harris, N. Shepherd-Barron and R. Taylor, A family of Calabi-Yau vari-eties and potential automorphy. Ann. of Math. (2) 171, 779-813 (2010) · Zbl 1263.11061
[25] M. Harris and R. Taylor, The geometry and cohomology of some sim-ple Shimura varieties. Annals of Mathematics Studies 151, Princeton University Press, Princeton, NJ (2001) · Zbl 1036.11027
[26] N. M. Katz and P. Sarnak, Random matrices, Frobenius eigenvalues, and monodromy. American Mathematical Society Colloquium Publications 45, American Mathematical Society, Providence, RI (1999) · Zbl 0958.11004
[27] C. B. Khare and J. A. Thorne, Potential automorphy and the Leopoldt conjecture. Amer. J. Math. 139, 1205-1273 (2017) · Zbl 1404.11076
[28] K.-W. Lan and J. Suh, Vanishing theorems for torsion automorphic sheaves on compact PEL-type Shimura varieties. Duke Math. J. 161, 1113-1170 (2012) · Zbl 1296.11072
[29] K.-W. Lan and J. Suh, Vanishing theorems for torsion automorphic sheaves on general PEL-type Shimura varieties. Adv. Math. 242, 228-286 (2013) · Zbl 1276.11103
[30] EMS MAGAZINE 119 (2021)
[31] R. P. Langlands, Problems in the theory of automorphic forms. In Lectures in modern analysis and applications, III, 18-61. Lecture Notes in Math., Vol. 170 (1970) · Zbl 0225.14022
[32] W.-C. W. Li, The Ramanujan conjecture and its applications. Philos. Trans. Roy. Soc. A 378, 20180441, 14 (2020) · Zbl 1462.11033
[33] E. Mantovan, On the cohomology of certain PEL-type Shimura varieties. Duke Math. J. 129, 573-610 (2005) · Zbl 1112.11033
[34] S. Morel, Construction de représentations galoisiennes de torsion [d’après Peter Scholze]. · Zbl 1470.14003
[35] P. Sarnak, Notes on the generalized Ramanujan conjectures. In Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc. 4, Amer. Math. Soc., Providence, RI, 659-685 (2005) · Zbl 1146.11031
[36] P. Scholze, On torsion in the cohomology of locally symmetric varieties. Ann. of Math. (2) 182, 945-1066 (2015) · Zbl 1345.14031
[37] P. Scholze, p-adic geometry. In Proceedings of the International Congress of Mathematicians -Rio de Janeiro 2018. Vol. I. Plenary lectures, World Sci. Publ., Hackensack, NJ, 899-933 (2018) · Zbl 1441.14001
[38] J.-P. Serre, Abelian l-adic representations and elliptic curves. McGill Uni-versity lecture notes written with the collaboration of Willem Kuyk and John Labute, W. A. Benjamin, Inc., New York-Amsterdam (1968) · Zbl 0186.25701
[39] S. W. Shin, A stable trace formula for Igusa varieties. J. Inst. Math. Jussieu 9, 847-895 (2010) · Zbl 1206.22011
[40] S. W. Shin, Galois representations arising from some compact Shimura varieties. Ann. of Math. (2) 173, 1645-1741 (2011) · Zbl 1269.11053
[41] A. V. Sutherland, Sato-Tate distributions. In Analytic methods in arith-metic geometry, Contemp. Math. 740, Amer. Math. Soc., Providence, RI, 197-248 (2019) · Zbl 1440.11176
[42] R. Taylor, Galois representations. Ann. Fac. Sci. Toulouse Math. (6) 13, 73-119 (2004) · Zbl 1074.11030
[43] R. Taylor, Automorphy for some l-adic lifts of automorphic mod l Galois representations. II. Publ. Math. Inst. Hautes Études Sci. 183-239 (2008) · Zbl 1169.11021
[44] R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras. Ann. of Math. (2) 141, 553-572 (1995) · Zbl 0823.11030
[45] R. Taylor and T. Yoshida, Compatibility of local and global Langlands correspondences. J. Amer. Math. Soc. 20, 467-493 (2007) · Zbl 1210.11118
[46] J. Weinstein, Reciprocity laws and Galois representations: Recent break-throughs. Bull. Amer. Math. Soc. (N.S.) 53, 1-39 (2016) · Zbl 1330.11071
[47] A. Wiles, Modular elliptic curves and Fermat’s last theorem. Ann. of Math. (2) 141, 443-551 (1995) · Zbl 0823.11029
[48] Ana Caraiani is a Royal Society University Research Fellow and Reader in Pure Mathematics at Imperial College London. a.caraiani@imperial.ac.uk
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.