Modelamiento Matemático para malaria bajo resistencia y movimiento poblacional

• Autores: Cristhian Montoya, Jhoana Romero Leiton
• Localización: Integración: Temas de matemáticas, ISSN 0120-419X, Vol. 38, Nº. 2, 2020, págs. 133-163
• Idioma: español
• DOI: 10.18273/revint.v38n2-2020006
• Enlaces
  • Texto completo (pdf)
• Resumen
  • español

    En este artículo se presentan dos modelos matemáticos para la enfermedad de la malaria bajo la hipótesis de resistencia. Más precisamente, el primer modelo muestra la interacción entre humanos y mosquitos de una región con presencia de infección, considerando que los humanos son resistentes a la droga antimalárica y los mosquitos resistentes a los insecticidas. En el segundo modelo, se consideran las mismas hipótesis del modelo anterior, y adicionalmente movimiento de ambas poblaciones entre regiones. Para el primer modelo, se establecen condiciones de existencia y estabilidad para las soluciones de equilibrio en términos del número básico de reproducción. Estos resultados revelan la existencia de una bifurcación hacia adelante y la estabilidad global del equilibrio libre de enfermedad (DFE por sus siglas en inglés). Para el segundo modelo, se presenta un enfoque teórico y numérico de análisis de sensibilidad de parámetros. Además, se incorporan el uso de droga antimalárica e insecticidas como estrategias de control, con lo cual se formula un problema de control óptimo. A lo largo de este trabajo, los resultados teóricos se validan mediante simulaciones numéricas usando datos reportados en la literatura

  • English

    In this work, two mathematical models for malaria under resistanceare presented. More precisely, the first model shows the interaction betweenhumans and mosquitoes inside a patch under infection of malaria when thehuman population is resistant to antimalarial drug and mosquitoes popula-tion is resistant to insecticides. For the second model, human–mosquitoespopulation movements in two patches is analyzed under the same malariatransmission dynamic established in a patch. For a single patch, existenceand stability conditions for the equilibrium solutions in terms of the local ba-sic reproductive number are developed. These results reveal the existence ofa forward bifurcation and the global stability of disease–free equilibrium. Inthe case of two patches, a theoretical and numerical framework on sensitivityanalysis of parameters is presented. After that, the use of antimalarial drugsand insecticides are incorporated as control strategies and an optimal controlproblem is formulated. Numerical experiments are carried out in both modelsto show the feasibility of our theoretical results.

• Referencias bibliográficas
  • Referencias Agusto F. B.,“Malaria drug resistance: The impact of human movement and spatial heterogeneity”, Bull. Math. Biol., 76 (2014),...
  • Aldila D., Nuraini N., Soewono E. and Supriatna A., “Mathematical model of temephos resistance in aedes aegypti mosquito population”, AIP...
  • Alphey N, Coleman P. G., Donnelly C. A. and Alphey L., “Managing insecticide resistance by mass release of engineered insects”, J. Econom....
  • Aneke S., “Mathematical modelling of drug resistant malaria parasites and vector populations”, Math. Methods Appl. Sci., 25 (2002), No. 4,...
  • Bacaër N. and Sokhna C., “A reaction-diffusion system modeling the spread of resistance to an antimalarial drug”, Math. Biosci. Eng., 2 (2005),...
  • Barrios E., Lee S. and Vasilieva O., “Assessing the effects of daily commuting in two-patch dengue dynamics: A case study of cali, colombia”,...
  • 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:...
  • 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...
  • Castillo-Chavez C. and Song B., “Dynamical models of tuberculosis and their applications”, Math. Biosci. Eng., 1 (2004), No. 2, 361-404. doi:...
  • Chitnis N., Hyman J.M. and Cushing J.M., “Determining important parameters in the spread of malaria through the sensitivity analysis of a...
  • De Jesus E.X. and Kaufman C., “Routh-hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations”,...
  • Esteva L., Gumel A.B. and De León C.V., “Qualitative study of transmission dynamics of drug-resistant malaria”, . Comput. Modelling, 50 (2009),...
  • 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....
  • Guinovart C., Navia M., Tanner M. and Alonso P., “Malaria: burden of disease”, Current Molecular Medicine, 6 (2006), No. 2, 137-140. doi:...
  • 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....
  • 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...
  • Mishra A. and Gakkhar S., “Non-linear dynamics of two-patch model incorporating secondary dengue infection”, Int. J. Appl. Comput. Math.,...
  • Palomino M., Villaseca P., Cárdenas F., Ancca J. and Pinto M., “Eficacia y residualidad de dos insecticidas piretroides contra triatoma infestans...
  • Prosper O., Ruktanonchai N. and Martcheva M., “Assessing the role of spatial heterogeneity and human movement in malaria dynamics and control”,...
  • Romero-Leiton J. and Ibargüen-Mondragón E., “Stability analysis and optimal control intervention strategies of a malaria mathematical model”,...
  • 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...
  • Tchuenche J.M., Chiyaka C., Chan D., Matthews A. and Mayer G., “A mathematical model for antimalarial drug resistance”, Math. Med. Biol.,...
  • Van den Driessche P. and Watmough J., “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission”,...
  • Wei D., Luo X. and Qin Y., “Controlling bifurcation in power system based on lasalle invariant principle”, Nonlinear Dynam., 63 (2011), No....
  • 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