Artículos científicos
Diferentes tipos de bifurcación hacia atrás a causa de una mejora en la eficiencia del tratamiento
Publicado 2018-07-18
Palabras clave
- Bifurcación hacia atrás,
- bifurcación hacia adelante,
- tratamiento,
- modelo SIR
Cómo citar
Osuna Castro, C. O., Guerrero-Flores, S., & Villavicencio-Pulido, G. (2018). Diferentes tipos de bifurcación hacia atrás a causa de una mejora en la eficiencia del tratamiento. Revista Integración, Temas De matemáticas, 36(1), 21–35. https://doi.org/10.18273/revint.v36n1-2018002
Resumen
Comprender por qué existen estados endémicos múltiples cuando R0 < 1 ha sido una de las principales motivaciones para analizar la existencia de una bifurcación hacia atrás en modelos epidemiológicos. La existencia de estados endémicos múltiples está asociada usualmente a ramas de puntos de
equilibrio del sistema, las cuales pueden surgir ya sea desde el equilibrio libre de enfermedad si R0 = 1, o desde un equilibrio no trivial si R0 > 1. En este trabajo se analiza un modelo del tipo SIR con una tasa de tratamiento densodependiente. Se explica la naturaleza del punto de donde surge la bifurcación hacia atrás en función de la velocidad de la tasa de tratamiento per cápita. Se propondrán estrategias para el control o erradicación de la enfermedad en función de la eficiencia del tratamiento.
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Referencias
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incidence rate and treatment rate”, Nonlinear Anal. 9 (2008), No. 5, 2302–2312.
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bifurcation”, J. Math. Biol. 40 (2000), No. 6, 525–540.
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sir model with a saturated treatment function”, Mathematical and Computer Modelling 55
(2012), No. 3, 710–720.
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[3] Ghosh M., Li X. and Li W., “Stability and bifurcation of an epidemic model with nonlinear incidence and treatment”, Appl. Math. Comput. 210 (2009), No. 1, 141–150.
[4] Gumel A.B., Elbasha E.H., Watmough J., Sharom O. and Podder C.N., “Role of incidence function in vaccine-induced backward bifurcation in some HIV models”, Math. Biosci. 210 (2007), No. 2, 436–463.
[5] Gumel A.B. and Sharomi O., “Re-infection-induced backward bifurcation in the transmission dynamics of chlamydia trachomatis”, J. Math. Anal. Appl. 356 (2009), No. 1, 96–118.
[6] Hethcote H., “The mathematics of infectious diseases”, SIAM Rev. 42 (2000), No. 4, 599–653.
[7] Lacitignola D. and Buonomo B., “On the backward bifurcation of a vaccination model with nonlinear incidence”, Nonlinear Anal. Model. Control 16 (2011), No. 1, 30–46.
[8] Liu X. and Zhang X., “Backward bifurcation of an epidemic model with saturated treatment function”, J. Math. Anal. Appl. 348 (2008), No. 1, 433–443.
[9] Lou J., Song B. and and Du W., “Different types of backward bifurcation due to densitydependent treatments”, Math. Biosci. Eng. 10 (2013), No. 5-6, 1651–1668.
[10] Moghadas S.M. and Alexander M.E., “Bifurcation analysis of an sirs epidemic model with generalized incidence”, SIAM J. Appl. Math. 65 (2005), No. 5, 1794–1816.
[11] Perelson A.S., Reluga T.C. and Medlock J., “Backward bifurcations and multiple equilibria in epidemic models with structured immunity”, J. Theor. Biol. 252 (2008), No. 1, 155–165.
[12] Song B. and Castillo-Chávez C., “Dynamical models of tuberculosis and their applications”, Math. Biosci. Eng. 1 (2004), No. 2, 361–404.
[13] Sun Z. and Liu Y., “Backward bifurcation in an hiv model with two target cells”, in Proceedings of the 29th Chinese Control Conference, Beijing, China, 1397–1400, July, 2010.
[14] Van den Driessche P., Arino J. and McCluskey C.C., “Global results for an epidemic model with vaccination that exhibits backward bifurcation”, SIAM J. Appl. Math. 64 (2003), No. 1, 260–276.
[15] Van den Driessche P. and Hadeler K.P., “Backward bifurcation in epidemic control”, Math. Biosci. 146 (1997), No. 1, 15–35.
[16] Velasco-Hernandez J.X. and Kribs-Zaleta C.M., “A simple vaccination model with multiple endemic states”, Math. Biosci. 164 (2000), No. 2, 183–201.
[17] Wan H., Cui J. and Mu X., “Saturation recovery leads to multiple endemic equilibria and backward bifurcation”, J. Theor. Biol. 254 (2008), No. 2, 275–283.
[18] Wang W., “Backward bifurcation of an epidemic model with treatment”, Math. Biosci. 201 (2006), No. 1-2, 58–71.
[19] Wang H., Hu Z. and Liu S., “Backward bifurcation of an epidemic model with standard
incidence rate and treatment rate”, Nonlinear Anal. 9 (2008), No. 5, 2302–2312.
[20] Watmough J. and Van den Driessche P., “A simple sis epidemic model with a backward
bifurcation”, J. Math. Biol. 40 (2000), No. 6, 525–540.
[21] Zheng B., Takeuchi Y., Wang J. and Liu S., “Qualitative and bifurcation analysis using an
sir model with a saturated treatment function”, Mathematical and Computer Modelling 55
(2012), No. 3, 710–720.