Research and Innovation Articles
Mathematical modelling for malaria under resistance and population movement
Published 2020-11-20
Keywords
- Insecticides,
- antimalarial drug,
- qualitative analysis,
- stability,
- bifurcation
How to Cite
Montoya, C., & Romero Leiton, J. P. (2020). Mathematical modelling for malaria under resistance and population movement. Revista Integración, Temas De matemáticas, 38(2), 133–163. https://doi.org/10.18273/revint.v38n2-2020006
Copyright (c) 2020 Revista Integración, temas de matemáticas
This work is licensed under a Creative Commons Attribution 4.0 International License.
Abstract
In this work, two mathematical models for malaria under resistance are presented. More precisely, the first model shows the interaction between humans and mosquitoes inside a patch under infection of malaria when the human population is resistant to antimalarial drug and mosquitoes population is resistant to insecticides. For the second model, human–mosquitoes population movements in two patches is analyzed under the same malaria transmission dynamic established in a patch. For a single patch, existence and stability conditions for the equilibrium solutions in terms of the local basic reproductive number are developed. These results reveal the existence of a forward bifurcation and the global stability of disease–free equilibrium. In the case of two patches, a theoretical and numerical framework on sensitivity analysis of parameters is presented. After that, the use of antimalarial drugs and insecticides are incorporated as control strategies and an optimal control problem is formulated. Numerical experiments are carried out in both models to show the feasibility of our theoretical results.
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References
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Prosper O., Ruktanonchai N. and Martcheva M., “Assessing the role of spatial heterogeneity and human movement in malaria dynamics and control”, J. of Theoretical Biology, 303 (2012), 1-14. doi: 10.1016/j.jtbi.2012.02.010
Romero-Leiton J. and Ibargüen-Mondragón E., “Stability analysis and optimal control intervention strategies of a malaria mathematical model”, Appl. Sci., 21 (2019), 184-218.
Smith S.J., Kamara A.R., Sahr F., Samai M., Swaray A.S., Menard D. and Warsame M., “Efficacy of artemisinin-based combination therapies and prevalence of molecular markers associated with artemisinin, piperaquine and sulfadoxine-pyrimethamine resistance in sierra leone”, Acta Trop, 185 (2018), 363-370. doi: 10.1016/j.actatropica.2018.06.016
Tchuenche J.M., Chiyaka C., Chan D., Matthews A. and Mayer G., “A mathematical model for antimalarial drug resistance”, Math. Med. Biol., 28 (2011), No. 4, 335-355. doi: 10.1093/imammb/dqq017
Van den Driessche P. and Watmough J., “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission”, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025 5564(02)00108 6
Wei D., Luo X. and Qin Y., “Controlling bifurcation in power system based on lasalle invariant principle”, Nonlinear Dynam., 63 (2011), No. 3, 323-329. doi: 10.1007/s11071- 010-9806-3
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Zhang J., Cosner C. and Zhu H., “Two-patch model for the spread of West Nile virus”, Bull. Math. Biol, 80 (2018), No. 4, 840-863, doi: 10.1007/s11538-018-0404-8
Aldila D., Nuraini N., Soewono E. and Supriatna A., “Mathematical model of temephos resistance in aedes aegypti mosquito population”, AIP Conference Proceedings, vol. 1589, 2014, 460-463. doi: 10.1063/1.4868843
Alphey N, Coleman P. G., Donnelly C. A. and Alphey L., “Managing insecticide resistance by mass release of engineered insects”, J. Econom. Entomology, 100 (2007), No. 5, 1642-1649. doi: 10.1603/0022 0493(2007)100[1642:MIRBMR]2.0.CO;2
Anderson B., Jackson J. and Sitharam M., “Descartes’ rule of signs revisited”, The American Mathematical Monthly, 105 (1998), No. 5, 447-451. doi: 10.2307/3109807
Aneke S., “Mathematical modelling of drug resistant malaria parasites and vector populations”, Math. Methods Appl. Sci., 25 (2002), No. 4, 335-346. doi: 10.1002/mma.291
Bacaër N. and Sokhna C., “A reaction-diffusion system modeling the spread of resistance to an antimalarial drug”, Math. Biosci. Eng., 2 (2005), No. 2, 227-238. doi: 10.3934/mbe.2005.2.227
Barrios E., Lee S. and Vasilieva O., “Assessing the effects of daily commuting in two-patch dengue dynamics: A case study of cali, colombia”, J. Theoret. Biol., 453 (2018), 14-39. doi: 10.1016/j.jtbi.2018.05.015
Bichara D. and Iggidr A., “Multi-patch and multi-group epidemic models: a new framework”, J. Theoret. Biol., 77 (2018), No. 1, 107-134. doi: 10.1007/s00285-017-1191-9
Bloland P.B., Drug resistance in malaria, World Health Organization, Geneva, 2001.
Bock W. and Jayathunga Y., “Optimal control and basic reproduction numbers for a compartmental spatial multipatch dengue model”, Math. Methods Appl. Sci. 41 (2018), No. 9, 3231-3245. doi: 10.1002/mma.4812
Castillo-Chavez C. and Song B., “Dynamical models of tuberculosis and their applications”, Math. Biosci. Eng., 1 (2004), No. 2, 361-404. doi: 10.3934/mbe.2004.1.361
Chitnis N., Hyman J.M. and Cushing J.M., “Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model”, Bull. Math. Biol., 70 (2008), No. 5, 1272-1296. doi: 10.1007/s11538-008 9299-0
De Jesus E.X. and Kaufman C., “Routh-hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations”, Phys. Rev. A (3), 35 (1987), No. 12, 5288-5290. doi: 10.1103/PhysRevA.35.5288
Esteva L., Gumel A.B. and De León C.V., “Qualitative study of transmission dynamics of drug-resistant malaria”, . Comput. Modelling, 50 (2009), No. 3-4, 611-630. doi: 10.1016/j.mcm.2009.02.012
Gao D. and Ruan S., “A multipatch Malaria model with logistic growth populations”, SIAM J. Appl. Math., 72 (2012), No. 3, 819-841. doi: 10.1137/110850761
Gourley S.A., Liu R. and Wu J., “Slowing the evolution of insecticide resistance in mosquitoes: a mathematical model”, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467 (2011), No. 2132, 2127-2148. doi: 10.1098/rspa.2010.0413
Guinovart C., Navia M., Tanner M. and Alonso P., “Malaria: burden of disease”, Current Molecular Medicine, 6 (2006), No. 2, 137-140. doi: 10.2174/156652406776055131
Hasler J., Stochastic and deterministic multipatch epidemic models, ProQuest LLC, Ann Arbor, MI, 2016.
Koella J. and Antia R., “Epidemiological models for the spread of anti-malarial resistance”, Malar J, 2 (2003), 3. doi: 10.1186/1475-2875-2-3
Lee S. and Castillo-Chavez C., “The role of residence times in two-patch dengue transmission dynamics and optimal strategies”, J. Theoret. Biol., 374 (2015), 152-164. doi: 10.1016/j.jtbi.2015.03.005
Lenhart S. and Workman J.T., Optimal control applied to biological models, Chapman Hall/CRC, Boca Raton, FL, 2007
Luz P., Codeco C, Medlock J., Struchiner C., Valle D. and Galvani A., “Impact of insecticide interventions on the abundance and resistance profile of aedes aegypti”, Epidemiology and Infection, 137 (2009), No. 8, 1203-1215. doi: 10.1017/S0950268808001799
Mishra A. and Gakkhar S., “Non-linear dynamics of two-patch model incorporating secondary dengue infection”, Int. J. Appl. Comput. Math., 4 (2018), No. 1, Paper No. 19, 22. doi: 10.1007/s40819-017-0460-z
Okosun K. and Makinde O.D., “Modelling the impact of drug resistance in malaria transmission and its optimal control analysis”, Int. J. Phys. Sci., 6 (2011), 6479-6487. doi: 10.5897/IJPS10.542
Palomino M., Villaseca P., Cárdenas F., Ancca J. and Pinto M., “Eficacia y residualidad de dos insecticidas piretroides contra triatoma infestans en tres tipos de viviendas: Evaluación de campo en Arequipa, Perú”, Rev. perú. med. exp. salud publica, 25 (2008), 9-16.
Prosper O., Ruktanonchai N. and Martcheva M., “Assessing the role of spatial heterogeneity and human movement in malaria dynamics and control”, J. of Theoretical Biology, 303 (2012), 1-14. doi: 10.1016/j.jtbi.2012.02.010
Romero-Leiton J. and Ibargüen-Mondragón E., “Stability analysis and optimal control intervention strategies of a malaria mathematical model”, Appl. Sci., 21 (2019), 184-218.
Smith S.J., Kamara A.R., Sahr F., Samai M., Swaray A.S., Menard D. and Warsame M., “Efficacy of artemisinin-based combination therapies and prevalence of molecular markers associated with artemisinin, piperaquine and sulfadoxine-pyrimethamine resistance in sierra leone”, Acta Trop, 185 (2018), 363-370. doi: 10.1016/j.actatropica.2018.06.016
Tchuenche J.M., Chiyaka C., Chan D., Matthews A. and Mayer G., “A mathematical model for antimalarial drug resistance”, Math. Med. Biol., 28 (2011), No. 4, 335-355. doi: 10.1093/imammb/dqq017
Van den Driessche P. and Watmough J., “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission”, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025 5564(02)00108 6
Wei D., Luo X. and Qin Y., “Controlling bifurcation in power system based on lasalle invariant principle”, Nonlinear Dynam., 63 (2011), No. 3, 323-329. doi: 10.1007/s11071- 010-9806-3
World Health Organization, Community involvement in rolling back malaria, World Health Organization, Geneva, 2002
Zhang J., Cosner C. and Zhu H., “Two-patch model for the spread of West Nile virus”, Bull. Math. Biol, 80 (2018), No. 4, 840-863, doi: 10.1007/s11538-018-0404-8