March  2007, 7(2): 365-375. doi: 10.3934/dcdsb.2007.7.365

Uniqueness and stability of positive periodic numerical solution of an epidemic model

1. 

Department of Mathematics, Inha University, Incheon 402-751, South Korea

Received  January 2006 Revised  October 2006 Published  December 2006

An age structured $s$-$i$-$s$ epidemic model with random diffusion is studied. The model is described by the system of nonlinear and nonlocal integro-differential equations. Finite differences along the characteristics in age-time domain combined with Galerkin finite elements in spatial domain are used in the approximation. It is shown that a positive periodic solution to the discrete system resulting from the approximation can be generated, if the initial condition is fertile. It is proved that the endemic periodic solution is globally stable once it exists.
Citation: Mi-Young Kim. Uniqueness and stability of positive periodic numerical solution of an epidemic model. Discrete and Continuous Dynamical Systems - B, 2007, 7 (2) : 365-375. doi: 10.3934/dcdsb.2007.7.365
[1]

Mikhail Dokuchaev, Guanglu Zhou, Song Wang. A modification of Galerkin's method for option pricing. Journal of Industrial and Management Optimization, 2022, 18 (4) : 2483-2504. doi: 10.3934/jimo.2021077

[2]

Runchang Lin, Huiqing Zhu. A discontinuous Galerkin least-squares finite element method for solving Fisher's equation. Conference Publications, 2013, 2013 (special) : 489-497. doi: 10.3934/proc.2013.2013.489

[3]

Sertan Alkan. A new solution method for nonlinear fractional integro-differential equations. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1065-1077. doi: 10.3934/dcdss.2015.8.1065

[4]

Wafa Hamrouni, Ali Abdennadher. Random walk's models for fractional diffusion equation. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2509-2530. doi: 10.3934/dcdsb.2016058

[5]

Lars Grüne, Peter E. Kloeden, Stefan Siegmund, Fabian R. Wirth. Lyapunov's second method for nonautonomous differential equations. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 375-403. doi: 10.3934/dcds.2007.18.375

[6]

Darya V. Verveyko, Andrey Yu. Verisokin. Application of He's method to the modified Rayleigh equation. Conference Publications, 2011, 2011 (Special) : 1423-1431. doi: 10.3934/proc.2011.2011.1423

[7]

Armando Majorana. A numerical model of the Boltzmann equation related to the discontinuous Galerkin method. Kinetic and Related Models, 2011, 4 (1) : 139-151. doi: 10.3934/krm.2011.4.139

[8]

David Blázquez-Sanz, Juan J. Morales-Ruiz. Lie's reduction method and differential Galois theory in the complex analytic context. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 353-379. doi: 10.3934/dcds.2012.32.353

[9]

Liu Liu. Uniform spectral convergence of the stochastic Galerkin method for the linear semiconductor Boltzmann equation with random inputs and diffusive scaling. Kinetic and Related Models, 2018, 11 (5) : 1139-1156. doi: 10.3934/krm.2018044

[10]

Na An, Chaobao Huang, Xijun Yu. Error analysis of discontinuous Galerkin method for the time fractional KdV equation with weak singularity solution. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 321-334. doi: 10.3934/dcdsb.2019185

[11]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4907-4926. doi: 10.3934/dcdsb.2020319

[12]

Mikhail B. Sevryuk. Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 569-595. doi: 10.3934/dcds.2007.18.569

[13]

Matthias Eller. A remark on Littman's method of boundary controllability. Evolution Equations and Control Theory, 2013, 2 (4) : 621-630. doi: 10.3934/eect.2013.2.621

[14]

Christopher M. Kellett. Classical converse theorems in Lyapunov's second method. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : 2333-2360. doi: 10.3934/dcdsb.2015.20.2333

[15]

Christopher Bose, Rua Murray. The exact rate of approximation in Ulam's method. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 219-235. doi: 10.3934/dcds.2001.7.219

[16]

Bernd Hofmann, Barbara Kaltenbacher, Elena Resmerita. Lavrentiev's regularization method in Hilbert spaces revisited. Inverse Problems and Imaging, 2016, 10 (3) : 741-764. doi: 10.3934/ipi.2016019

[17]

Maolin Cheng, Yun Liu, Jianuo Li, Bin Liu. Nonlinear Grey Bernoulli model NGBM (1, 1)'s parameter optimisation method and model application. Journal of Industrial and Management Optimization, 2022, 18 (3) : 2017-2032. doi: 10.3934/jimo.2021054

[18]

Maika Goto, Kazunori Kuwana, Yasuhide Uegata, Shigetoshi Yazaki. A method how to determine parameters arising in a smoldering evolution equation by image segmentation for experiment's movies. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 881-891. doi: 10.3934/dcdss.2020233

[19]

Zhengyong Zhou, Bo Yu. A smoothing homotopy method based on Robinson's normal equation for mixed complementarity problems. Journal of Industrial and Management Optimization, 2011, 7 (4) : 977-989. doi: 10.3934/jimo.2011.7.977

[20]

Egil Bae, Xue-Cheng Tai, Wei Zhu. Augmented Lagrangian method for an Euler's elastica based segmentation model that promotes convex contours. Inverse Problems and Imaging, 2017, 11 (1) : 1-23. doi: 10.3934/ipi.2017001

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (60)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]