Abstract
Mathematical models are important instruments in epidemiology to assist in analyzing epidemiological dynamics as well as possible dissemination controls. Classical model uses differential equations to describe dynamics of population over time. A widely used example is susceptible-infected-recovered (SIR) compartmental model. Such model has been used to obtain optimum control policies in different scenarios. This model has been enhanced to include dynamics of reinfection of disease including a new compartment, known as susceptible-infected-recovered-cross-immune (SIRC). An alternative model is to consider each individual as a string or vector of characteristic data and simulate the contagion and recovery processes by computational means. This type of model, referred in literature as individual based model (IBM) has advantage of being flexible as characteristics of each individual can be quite complex, involving, for instance, age, sex, pre-existing health conditions, environmental factors, social, and habits. However, it was not found in literature equivalence in an IBM model for SIRC model. Some works have shown the possibility of equivalence between IBM and SIR models, in order to simulate similar scenarios with models of different natures, in deterministic and stochastic case respectively. In this context, this work proposes implementation of an IBM stochastic model equivalent to SIRC model. Results show that equivalence is also possible only with the proper configuration of parameters of IBM model. Accuracy of equivalent model showed better with reduction of time step end increase the size of population.
References
Filho, N.A., Rouquayrol, M.Z.: Introduo epidemiologia. Guanabara Koogan, Rio de Janeiro (2006)
Iacoviello, D., Stasio, N.: Optimal control for SIRC epidemic outbreak. Comput. Methods Programs Biomed. 110, 333–342 (2013)
Capone, F., De Cataldis, V., De Luca, R.: On the nonlinear stability of an epidemic SEIR reaction-diffusion model. Ricerche di Matematica 62(1), 161–181 (2013)
Behncke, H.: Optimal control of deterministic epidemics. Optim. Control Appl. Methods 21(6), 269–285 (2000)
Cisternas, J., William, G.C., Levin, S., Kevrekidis, I.G.: Equation-free modelling of evolving diseases: coarse-grained computations with individual-based models. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 460, pp. 2761–2779. The Royal Society (2004)
Jódar, L., et al.: Nonstandard numerical methods for a mathematical model for influenza disease. Math. Comput. Simul. 79(3), 622–633 (2008)
Nie, L.-F., Teng, Z.-D., Guo, B.-Z.: A state dependent pulse control strategy for a SIRS epidemic system. Bull. Math. Biol. 75(10), 1697–1715 (2013)
Anderson, R.M., May, R.M.: Infectious Diseases of Humans: Dinamics and Control. Oxford University Press, Oxford (1992)
El-Shahed, M., Alsaedi, A.: The fractional SIRC model and influenza A 2011, 9 (2011)
Wu, M., Wang, L., Li, M., Long, H.: An approach based on the SIR epidemic model and a genetic algorithm for optimizing product feature combinations in feature fatigue analysis (2013)
Ministério da Saúde Brasil. Boletim informativo de influenza (2012). http://portalsaude.saude.gov.br/portalsaude/noticia/6651/785/boletim-informativo-de-influenza:-semana-epidemiologica-32.html. acessado 1-maio-2013
Giancotti, K.H.O., de Assis Dias, F., Teixeira, W.W.M., Nepomuceno, E.G., Kurcbart, S.M.: Anlise da estrutura do MBI: sensibilidade da taxa de infeco e da populao. In: Anais do XVIII Congresso Brasileiro de Automtica, Bonito, MS, In Em (2010)
Samanta, G.P.: Global dynamics of a nonautonomous SIRC model for influenza A with distributed time delay. Differ. Equ. Dyn. Syst. 18(4), 341–362 (2010)
Casagrandi, R., et al.: The SIRC model and influenza A. Math. Biosci. 200(2), 156–169 (2006)
Dormand, J.R., Prince, P.J.: A family of embedded Runge-Kutta formulae. J. Comput. Appl. Math. 6(1), 19–26 (1980)
Galvão Filho, A.R., Galvão, R.K.H., Yoneyama, T.: Otimização da alocação temporal de recursos para combate a epidemias com transmissão sazonal atraves de metodos de barreira. In: Proceeding Series of the Brazilian Society of Computational and Applied Mathematics, pp. 1–6 (2013)
Galvão Filho, A.R., Galvão, R.K.H., Yoneyama, T., Arruda, F.: Programação paralela cuda para simulação de modelos epidemiológicos baseados em indivíduos. In: Anais do X Simpósio Brasileiro de Automação Inteligente, pp. 241–246 (2011)
Galvão Filho, A.R., et al.: CUDA parallel programming for simulation of epidemiological models based on individuals. Math. Methods Appl. Sci. 39(3), 405–411 (2016)
Morton, R., Wickwire, K.H.: On the optimal control of a deterministic epidemic. Adv. Appl. Probab. 6(4), 622–635 (1974)
Arif, S., Olariu, S.: Efficient solution of a stochastic SI epidemic system. J. Supercomput. 62, 1385–1403 (2012)
Grenfell, B.T., Bjørnstad, O.N., Finkenstädt, B.F.: Dynamics of measles epidemics: scaling noise, determinism, and predictability with the TSIR model. Ecol. Monogr. 72(2), 185–202 (2002)
Kermack, W.O., Mckendrick, A.G.: A contribution to the mathematical theory of epidemics. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 115, no. 772, pp. 700–721 (1927). The Royal Society
World Health Organization. Influenza (seasonal): Fact sheet n211 (2009). http://www.who.int/mediacentre/factsheets/fs211/en/. acessado 13-maio-2013
Yoshida, N., Hara, T.: Global stability of a delayed SIR epidemic model with density dependent birth and death rates. J. Comput. Appl. Math. 201(2), 339–347 (2007)
Lu, Z., Chi, X., Chen, L.: The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission 36, 1039–1057 (2002)
Acknowledgments
Authors would like to thank the brazilian research agencies CAPES, FAPEG and CNPq for the financial support provided to this work.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Galvão Filho, A.R., de Lima, T.W., da Silva Soares, A., Coelho, C.J. (2017). A Stochastic Approach of SIRC Model Using Individual-Based Epidemiological Models. In: Oliveira, E., Gama, J., Vale, Z., Lopes Cardoso, H. (eds) Progress in Artificial Intelligence. EPIA 2017. Lecture Notes in Computer Science(), vol 10423. Springer, Cham. https://doi.org/10.1007/978-3-319-65340-2_63
Download citation
DOI: https://doi.org/10.1007/978-3-319-65340-2_63
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-65339-6
Online ISBN: 978-3-319-65340-2
eBook Packages: Computer ScienceComputer Science (R0)