Abstract
A mathematical model based on cellular automata on graphs to simulate a general epidemic spreading is presented in this paper. Specifically, it is a SIR-type model where the population is divided into susceptible, infected and recovered individuals.
Keywords: Cellular Automaton, Complete Graph, Lyme Disease, Cellular Automaton Model, Epidemic Spreading
Contributor Information
Ildar Batyrshin, Email: batyr1@gmail.com.
Miguel González Mendoza, Email: mgonza@itesm.mx.
Maria Jose Fresnadillo Martínez, Email: jofrema@usal.es.
Enrique García Merino, Email: engarme@gmail.com.
Enrique García Sánchez, Email: engarsan@usal.es.
Jose Elias García Sánchez, Email: joegas@usal.es.
Angel Martín del Rey, Email: delrey@usal.es.
Gerardo Rodríguez Sánchez, Email: gerardo@usal.es.
References
- 1.Ahmed E., Agiza H.N. On modelling epidemics including latency, incubation and variable susceptibility. Physica A. 1998;253:247–352. doi: 10.1016/S0378-4371(97)00665-1. [DOI] [Google Scholar]
- 2.Ahmed E., Elgazzar A.S. On some applications of cellular automata. Physica A. 2001;296:529–538. doi: 10.1016/S0378-4371(01)00182-0. [DOI] [Google Scholar]
- 3.Beauchemin C., Samuel J., Tuszynski J. A simple cellular automaton model for influenza A viral infections. J. Theor. Biol. 2005;232:223–234. doi: 10.1016/j.jtbi.2004.08.001. [DOI] [PubMed] [Google Scholar]
- 4.Boccara N., Cheong K. Critical behavior of a probablistic automata network SIS model for the spread of an infectious disease in a population of moving individuals. J. Phys. A-Math. Gen. 1993;26:3707–3717. doi: 10.1088/0305-4470/26/15/020. [DOI] [Google Scholar]
- 5.Boccara N., Cheong K., Oram M. A probabilistic automata network epidemic model with birds and deaths exhibiting cyclic behaviour. J. Phys A-Math. Gen. 1994;27:1585–1597. doi: 10.1088/0305-4470/27/5/022. [DOI] [Google Scholar]
- 6.Diekmann, O., Heesterbeek, J.O.P.: Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. Wiley (2000)
- 7.Fuentes M.A., Kuperman M.N. Cellular automata and epidemiological models with spatial dependence. Physica A. 1999;267:471–486. doi: 10.1016/S0378-4371(99)00027-8. [DOI] [Google Scholar]
- 8.Fuks H., Lawniczak A.T. Individual-based lattice model for spatial spread of epidemics. Discrete Dyn. Nat. Soc. 2001;6:191–200. doi: 10.1155/S1026022601000206. [DOI] [Google Scholar]
- 9.Hamer W.H. Epidemic disease in England. Lancet. 1906;1:733–739. [Google Scholar]
- 10.Martín del Rey A., Hoya White S., Rodríguez Sánchez G. A Model Based on Cellular Automata to Simulate Epidemic Diseases. In: El Yacoubi S., Chopard B., Bandini S., editors. Cellular Automata; Heidelberg: Springer; 2006. pp. 304–310. [Google Scholar]
- 11.Hoya White S., Martín del Rey A., Rodríguez Sánchez G. Modeling epidemics using cellular automata. Appl. Math. Comput. 2007;186:193–202. doi: 10.1016/j.amc.2006.06.126. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 12.Keeling, M.J., Rohani, P.: Modeling Infectious Diseases in Humans and Animals. Princeton University Press (2008)
- 13.Kermack W.O., McKendrick A.G. Contributions to the mathematical theory of epidemics, part I. Proc. Roy. Soc. Edin. A. 1927;115:700–721. doi: 10.1098/rspa.1927.0118. [DOI] [Google Scholar]
- 14.Mansilla R., Gutierrez J.L. Deterministic site exchange cellular automata model for the spread of diseases in human settlements. Complex Systems. 2001;13:2. [Google Scholar]
- 15.Fresnadillo M.J., García E., García J.E., Martín Á., Rodríguez G. A SIS Epidemiological Model Based on Cellular Automata on Graphs. In: Omatu S., Rocha M.P., Bravo J., Fernández F., Corchado E., Bustillo A., Corchado J.M., editors. Distributed Computing, Artificial Intelligence, Bioinformatics, Soft Computing, and Ambient Assisted Living. Heidelberg: Springer; 2009. pp. 1055–1062. [Google Scholar]
- 16.Molisson D. The dependence of epidemic and population velocities on basic parameters. Math. Biosci. 1991;107:255–287. doi: 10.1016/0025-5564(91)90009-8. [DOI] [PubMed] [Google Scholar]
- 17.Quan-Xing L., Zhen J. Cellular automata modeling of SEIRS. Chinese Phys. 2005;14:1370–1377. doi: 10.1088/1009-1963/14/7/018. [DOI] [Google Scholar]
- 18.Ramani A., Carstea A.S., Willox R., Grammaticos G. Oscillating epidemics: a discrete-time model. Physica A. 2004;333:278–292. doi: 10.1016/j.physa.2003.10.051. [DOI] [Google Scholar]
- 19.Ross, R.: The prevention of malaria, 2nd edn. Murray (1911)
- 20.Rousseau G., Giorgini R., Livi H., Chaté H. Dynamical phases in a cellular automaton model for epidemic propagation. Physica D. 1997;103:554–563. doi: 10.1016/S0167-2789(96)00285-0. [DOI] [Google Scholar]
- 21.Sirakoulis G.C., Karafyllidis I., Thanailakis A. A cellular automaton model for the effects of population movement and vaccination on epidemic propagation. Ecol. Model. 2000;133:209–223. doi: 10.1016/S0304-3800(00)00294-5. [DOI] [Google Scholar]
- 22.Toffoli, T., Margolus, N.: Cellular Automata Machines: A New Environment for Modeling. The MIT Press (1987)
- 23.Vynnycky, E., White, R.G.: An Introduction to Infectious Disease Modelling. Oxford University Press (2010)
- 24.Willox R., Grammaticos B., Carstea A.S., Ramani A. Epidemic dynamics: discrete-time and cellular automaton models. Physica A. 2003;328:13–22. doi: 10.1016/S0378-4371(03)00552-1. [DOI] [Google Scholar]
- 25.Wolfram, S.: A New Kind of Science. Wolfram Media Inc. (2002)
