Volume 5, Issue 6, December 2019, Page: 94-99
Global Stability of Critical Points for Type SIS Epidemiological Model
Edgar Ali Medina, Department of Science, Nueva Esparta Nucleus, Universidad de Oriente, Guatamare, Venezuela
Manuel Vicente Centeno-Romero, Department of Mathematics, School of Sciences, Sucre Nucleus, Universidad de Oriente, Cumaná, Venezuela
Fernando José Marval López, Department of Mathematics, School of Sciences, Sucre Nucleus, Universidad de Oriente, Cumaná, Venezuela
Received: Jul. 20, 2019;       Accepted: Aug. 19, 2019;       Published: Dec. 2, 2019
DOI: 10.11648/j.ijtam.20190506.13      View  66      Downloads  23
Abstract
The construction of mathematical models is one of the tools used today for the study of problems in Medicine, Biology, Physiology, Biochemistry, Epidemiology, and Pharmacokinetics, among other areas of knowledge; its primary objectives are to describe, explain and predict phenomena and processes in these areas. The simulation, through mathematical models, allows exploring the impact of the application of one or several control measures on the dynamics of the transmission of infectious diseases, providing valuable information for decision-making with the objective of controlling or eradicating them. The mathematical models in Epidemiology are not only descriptive but also predictive, helping to prevent pandemics (epidemics that spread through large areas and populations) or by intervening in vaccination and drug acquisition policies. In this article we study the existence of periodic orbits and the general stability of the equilibrium points for a susceptible-infected-susceptible model (SIS), with a non-linear incidence rate. This type of model has been studied in many articles with a very particular incidence rate, here the novelty of the problem is that the aforementioned incidence rate is very general, in this sense this research provides a solution to an open problem. The methodology used is the Dulac technique, proceeding by reduction to the absurdity of the statement to the main test. It shows that the only point of equilibrium is asymptotically stable global. It can be noted that this problem may be subject to discretion or for equations in timescales. This can generate other research.
Keywords
Periodic Orbits, Global Stability, Equilibrium Points
To cite this article
Edgar Ali Medina, Manuel Vicente Centeno-Romero, Fernando José Marval López, Global Stability of Critical Points for Type SIS Epidemiological Model, International Journal of Theoretical and Applied Mathematics. Vol. 5, No. 6, 2019, pp. 94-99. doi: 10.11648/j.ijtam.20190506.13
Copyright
Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Reference
[1]
I. Sanz. Modelos epimediológicos basados en ecuaciones diferenciales. Trabajo de Fin de Grado. Universidad De La Rioja. Logroño. España. 2016.
[2]
R. Lorente-Antañozas, J. L. Varona, F. Antañozas y J. Rejas-Gutiérrez. (2016). La vacunación anti-neumocócica con la vacuna conjugada 13-Valente en población inmunocompetente de 65 años: análisis del impacto presupuestario en España aplicando un modelo de trnasmisión dinámica, Rev. Esp. Salud Pública 90, 1-12.
[3]
R. Pradas, A. Gil-De Miguel, A. Álvaro, R. Gil-Prieto, R. Lorente, C. Méndez, P. Guijarro y F. Antañozas. (2013). Budget impact analysis of a pneumococcal vaccination programme in the 65-years-old Spanish cohort using a dynamic model. Bio. Med. Central 13, 1-8.
[4]
R. P. Velasco, F. A. Villar y J. Mar. Modelos matemáticos para la evaluación económica: los modelos dinámicos basados en ecuaciones diferenciales, http://www.gacetasanitaria.org/es/modelos-matematicos-evaluacion-economica-los/articulo/S0213911109002349/, Gaceta Sanitaria. 2008, 23, 473-478.
[5]
G. Alpízar-Brenes. (2016). Análisis de un modelo SIS para el estudio de la dinámica de propagación de la Enfermedad al aplicar medidas de control. Tecnología en Marcha. Edición especial. Matemática Aplicada, 42-50.
[6]
R. Welte, R. Leidl, W. Greiner and M. Postma. 2010. “Health economics of infectious diseases”, in Modern Infectious Disease Epidemiology. Springer. A. Krämer, M. Kretzschmar and K. Krickeberg (Eds.), New York, 2010, pp. 253-279.
[7]
WHO (World Health Organization). (2009). New Influenza A(H1N1): Number of laboratory confirmed cases and deaths as reported to WHO. Obtenido de http://www.who.int/csr/don/GlobalSubnationalMaster_ 20090507_1800a. png?ua=1
[8]
F. Brauer, C. Castillo-Chávez, E. De la Pava, C. Castillo-Garsow, D. Chowell, B. Espinoza, P. González, C. Hernández and V. Moreno. (2014). Modelos de la propagación de enfermedades infecciosas. Universidad Autónoma de Occidente. Calí, Colombia.
[9]
H. W. Hethcote. (1976). Qualitative Analysis of communicable disease models. Match. Biosciences 28, 335-356.
[10]
E. A. Medina. Dinámica global de ciertos modelos epidemiológicos. Tesis de Maestría. UCV. Venezuela. 1997.
[11]
G. Carrero. Pattern formation in a SIS epidemiological model. Notas de Matemáticas. Venezuela. 2004.
[12]
A. Carvalho, Sistema dinámicos nao-linares, ICMC-USP, Sao Carlos, 2012.
[13]
H. Leiva, E. Medina and N. Merentes. (2015). Relative Asymptotic Equivalence between difference Equations. Journal of Difference Equations and Applications. 21 (5), 418-436.
[14]
J. Aguilera and M. Lizana. Estudio cualitativo de Ecuaciones Diferenciales Ordinarias. Editorial UCV. 1990.
[15]
H. W. Hirseh and S. Male. Differential equations, Dynamical Systems, and linear Algebra. Academic Press. 1974.
[16]
C. Castillo-Chávez and A. Yakubu. 2002. “Intraspecific competition, dispersal and disease dynamics in discrete-time patchy environments”, in Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction to Models, Methods and Theory, 125, 165-181. Castillo-Chávez C., van den Driessche P., Kirschner D. and Yakubu A.-A. (Eds.), Sringer, New York, 2002.
[17]
G. Alpízar. Estrategias de control óptimas en la propagación de enfermedades en poblaciones acopladas. Tesis de Maestría, Universidad de Puerto Rico, Recinto Mayagüez. 2012.
[18]
J. Arino and P. Driessche. 2006. Disease spread in metapopulations. Fields Institute Communucations, 48, 1-13.
[19]
M. Lizana and J. Rivero. 1996. Multiparametric bifurcation for a model in epidemiology. Jour of Math. Biol. 35 (1), 21-36.
[20]
I. Bendixson. 1901, Sur les courbes définies par des équations différentielles, Acta Mathematica, Springer Netherlands, 24 (1), 1–88.
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