# American Institute of Mathematical Sciences

• Previous Article
Lifting in equation-free methods for molecular dynamics simulations of dense fluids
• DCDS-B Home
• This Issue
• Next Article
Global existence and internal stabilization for a reaction-diffusion system posed on non coincident spatial domains
June  2009, 11(4): 823-853. doi: 10.3934/dcdsb.2009.11.823

## Convergence of a finite volume scheme for nonlocal reaction-diffusion systems modelling an epidemic disease

 1 Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción 2 CI2MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile

Received  March 2008 Revised  January 2009 Published  April 2009

A finite-volume scheme for a nonlocal three-component reaction-diffusion system modeling an epidemic disease with susceptible, infected, and recovered, individuals is analyzed. For this SIR model, the existence of solutions to the finite volume scheme and its convergence to a weak solution of the PDE is established. The convergence proof is based on deriving a series of apriori estimates and by using a general $L^p$ compactness criterion. Finally, numerical simulations from the finite volume scheme are given.
Citation: Mostafa Bendahmane, Mauricio Sepúlveda. Convergence of a finite volume scheme for nonlocal reaction-diffusion systems modelling an epidemic disease. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 823-853. doi: 10.3934/dcdsb.2009.11.823
 [1] Haomin Huang, Mingxin Wang. The reaction-diffusion system for an SIR epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2039-2050. doi: 10.3934/dcdsb.2015.20.2039 [2] Keng Deng. On a nonlocal reaction-diffusion population model. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 65-73. doi: 10.3934/dcdsb.2008.9.65 [3] Keng Deng. Asymptotic behavior of an SIR reaction-diffusion model with a linear source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5945-5957. doi: 10.3934/dcdsb.2019114 [4] José-Francisco Rodrigues, João Lita da Silva. On a unilateral reaction-diffusion system and a nonlocal evolution obstacle problem. Communications on Pure & Applied Analysis, 2004, 3 (1) : 85-95. doi: 10.3934/cpaa.2004.3.85 [5] Bang-Sheng Han, Zhi-Cheng Wang. Traveling wave solutions in a nonlocal reaction-diffusion population model. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1057-1076. doi: 10.3934/cpaa.2016.15.1057 [6] Jia-Feng Cao, Wan-Tong Li, Meng Zhao. On a free boundary problem for a nonlocal reaction-diffusion model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4117-4139. doi: 10.3934/dcdsb.2018128 [7] Nora Aïssiouene, Marie-Odile Bristeau, Edwige Godlewski, Jacques Sainte-Marie. A combined finite volume - finite element scheme for a dispersive shallow water system. Networks & Heterogeneous Media, 2016, 11 (1) : 1-27. doi: 10.3934/nhm.2016.11.1 [8] Pavol Kútik, Karol Mikula. Diamond--cell finite volume scheme for the Heston model. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 913-931. doi: 10.3934/dcdss.2015.8.913 [9] Shu-Xiang Huang, Fu-Cai Li, Chun-Hong Xie. Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1519-1532. doi: 10.3934/dcds.2003.9.1519 [10] Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2133-2156. doi: 10.3934/cpaa.2017106 [11] Huimin Liang, Peixuan Weng, Yanling Tian. Threshold asymptotic behaviors for a delayed nonlocal reaction-diffusion model of mistletoes and birds in a 2D strip. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1471-1495. doi: 10.3934/cpaa.2016.15.1471 [12] Michele V. Bartuccelli, S.A. Gourley, Y. Kyrychko. Comparison and convergence to equilibrium in a nonlocal delayed reaction-diffusion model on an infinite domain. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 1015-1026. doi: 10.3934/dcdsb.2005.5.1015 [13] Joaquin Riviera, Yi Li. Existence of traveling wave solutions for a nonlocal reaction-diffusion model of influenza a drift. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 157-174. doi: 10.3934/dcdsb.2010.13.157 [14] Thomas I. Seidman. Interface conditions for a singular reaction-diffusion system. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 631-643. doi: 10.3934/dcdss.2009.2.631 [15] Jorge Ferreira, Hermenegildo Borges de Oliveira. Parabolic reaction-diffusion systems with nonlocal coupled diffusivity terms. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2431-2453. doi: 10.3934/dcds.2017105 [16] Samira Boussaïd, Danielle Hilhorst, Thanh Nam Nguyen. Convergence to steady state for the solutions of a nonlocal reaction-diffusion equation. Evolution Equations & Control Theory, 2015, 4 (1) : 39-59. doi: 10.3934/eect.2015.4.39 [17] Zhiting Xu, Yingying Zhao. A reaction-diffusion model of dengue transmission. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2993-3018. doi: 10.3934/dcdsb.2014.19.2993 [18] Feng-Bin Wang. A periodic reaction-diffusion model with a quiescent stage. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 283-295. doi: 10.3934/dcdsb.2012.17.283 [19] Shin-Ichiro Ei, Toshio Ishimoto. Effect of boundary conditions on the dynamics of a pulse solution for reaction-diffusion systems. Networks & Heterogeneous Media, 2013, 8 (1) : 191-209. doi: 10.3934/nhm.2013.8.191 [20] Shin-Ichiro Ei, Kota Ikeda, Masaharu Nagayama, Akiyasu Tomoeda. Reduced model from a reaction-diffusion system of collective motion of camphor boats. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 847-856. doi: 10.3934/dcdss.2015.8.847

2018 Impact Factor: 1.008