Periodic orbits in a seasonal SIRS model with both incidence and treatment generalized rates

• Shaday Guerrero-Flores ^[1] ; Osvaldo Osuna ^[3] ; José Geiser Villavicencio Pulido ^[2]
  1. [1] Universidad Nacional Autónoma de México

     Universidad Nacional Autónoma de México

     México

     
  2. [2] Universidad Autónoma Metropolitana

     Universidad Autónoma Metropolitana

     México

     
  3. [3] Universidad Michoacana, Ciudad Universitaria

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• Localización: Revista Colombiana de Matemáticas, ISSN-e 0034-7426, Vol. 57, Nº. 1, 2023, págs. 19-36
• Idioma: varios idiomas
• Títulos paralelos:
  • Órbitas periódicas en un modelo SIRS estacional con tasas generalizadas de incidencia y tratamiento
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• Resumen
  • español

    En este trabajo, nosotros probamos que un modelo SIRS estacional denso-dependiente con tasas generalizadas de incidencia y tratamiento tiene soluciones periódicas. Este modelo generalizado es analizado usando teoría de grado de Leray-Schauder para probar la existencia de órbitas periódicas. Finalmente, se muestran simulaciones numéricas para ilustrar los resultados teóricos.

  • Multiple

    In this work, we prove that a seasonal-dependent SIRS model with general incidence and treatment rates has periodic solutions. This generalized model is analyzed using Leray-Schauder degree theory to prove the existence of a periodic solution. Finally, numerical simulations are shown to illustrate the theoretical results.

• Referencias bibliográficas
  • R.M Anderson and R.M May, Infectious diseases of humans: dynamics and control, Oxford University Press, Oxford, 1991.
  • Z. Bai and Y. Zhou, Existence of two periodic solutions for a non-autonomous sir epidemic model, Appl. Math. Model 35 (2011), 382-391.
  • R. Brown, A topological introduction to nonlinear analysis, Birkhauser second edition, Boston, 2013.
  • V. Capasso and G. Serio, A generalisation of the kermack-mckendrick deterministic epidemic model, Math. Biosci. 42 (1978), 43-61.
  • A. Cervantes-Pérez and E. Avila-Vales, Global stability for sirs epidemic models with general incidence rate and transfer from infectious...
  • J. Cui, Y. Sun, and H. Zhu, The impact of media on the control of infectious diseases, J. Dyn. Differ. 20 (2008), no. 1, 31-53.
  • W. Cui and X. Mu, Saturation recovery leads to multiple endemic equilibria and backward bifurcation, J. Theor. Biol. 254 (2008), 275-283.
  • A. Dénes and G. Róst, Global stability for sir and sirs models with nonlinear incidence and removal terms via dulac functions, Discrete and...
  • J. Dushoff, J. B. Plotkin, S.A. Levin, and D. J. D Earn, Dynamical resonance can account for seasonality of influenza epidemics, Proc. Natl...
  • X. Ghosh and X. W. Li, Stability and bifurcation of an epidemic model with nonlinear incidence and treatment, Applied Mathematics and Computation...
  • L. R. González, O. Osuna, and G. Villavicencio, Oscillations in seasonal sir epidemic models with saturated treatment, Revista Integración...
  • N. C. Grassly and C. Fraser, Seasonal infectious disease epidemiology, Proc. R. Soc. B 273 (2006), 2541-2550.
  • S. Guerrero-Flores, O. Osuna, and C. Vargas de León, Periodic solutions for seasonal siqrs models with non linear infection terms, Electronic...
  • H. Hethcote, M. Zhien, and L. Shengbing, Effects of quarantine in six endemic models for infectious diseases, Math. Biosci. 180 (2002), 141-160.
  • L. Jódar, J. R. Villanueva, and A. Arenas, Modeling the spread of seasonal epidemiological diseases: Theory and applications, Math. and Comp....
  • T. Kar and A. Batabyal, Modeling and analysis of an epidemic model with non-monotonic incidence rate under treatment, J. of Math. Research...
  • G. Katriel, Existence of periodic solutions for periodically forced sir model, J. Math. Sc. 201 (2014), no. 3, 335-342.
  • Z. Liu L.Wang, X. Zhang, An seir epidemic model with relapse and general non-linear incidence rate with application to media, Qual. Th. Dyn....
  • L. Li, Y. Bai, and Z. Jin, Periodic solutions of an epidemic model with saturated treatment, Nonlinear Dynam. 76 (2014), 1099-1108.
  • W. Liu, H. Hethcote, and S. Levin, Dynamical behaviour of epidemiological models with nonlinear incidence rates, J. Math. Biol. 25 (1987),...
  • W. Liu, S. Levin, and Y. Iwasa, Influence of nonlinear incidence rates upon the behaviour of sirs epidemiological models, J. Math. Biol. 23...
  • O. Osuna and J. G. Villavicencio-Pulido, Seasonal treatment of an infectious disease is a social driver of sustained oscillations in the disease...
  • L. Song and W. Du, Different types of backward bifurcation due to density-dependent treatments, Mathematical Biosciences and Engineering 10...
  • P. van den Driessche and J. Watmough, A simple sis epidemic model with a backward bifurcation, J. Math. Biol. 40 (2000), 525-540.
  • P. van den Driessche and James Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,...
  • H. Wang and S. Liu, Backward bifurcation of an epidemic model with standard incidence rate and treatment rate, Nonlinear Analysis 9 (2008),...
  • W. Wang, Backward bifurcation of an epidemic model with treatment, Math. Biosci 201 (2006), 58-71.
  • D. G. Williams and C. Dye, Infectious disease persistence when transmission varies seasonally, Mathematical Biosciences 145 (2012), 77-88.
  • D. Xiao and S. Ruan, Global analysis of an epidemic model with non monotone incidence rate, Math. Biosci. 208 (2007), 419-429.
  • X. Zhang and X. Liu, Backward bifurcation of an epidemic model with saturated treatment function, J. Math. Anal. Appl. 348 (2008), 433-443.
  • Y. Zheng, J. Takeuchi, and S. Liu, Qualitative and bifurcation analysis using an sir model with a saturated treatment function, Mathematical...