Abstract
In this paper, we present a fuzzy approach to the Reed–Frost model for epidemic spreading taking into account uncertainties in the diagnostic of the infection. The heterogeneities in the infected group is based on the clinical signals of the individuals (symptoms, laboratorial exams, medical findings, etc.), which are incorporated into the dynamic of the epidemic. The infectivity level is time-varying and the classification of the individuals is performed through fuzzy relations. Simulations considering a real problem with data of the viral epidemic in a children daycare are performed and the results are compared with a stochastic Reed–Frost generalization.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abbey, H., 1952. An examination of the Reed-Frost theory of epidemics. Hum. Biol. 24, 201–02.
Barros, L.C., Bassanezi, R.C., Tonelli, P.A., 2000. Fuzzy modeling in population dynamics. Ecol. Model. 128(1), 27–3.
Barros, L.C., Leite, M.B.F., Bassanezi, R.C., 2003. The SI epidemiological models with a fuzzy transmission parameter. Comput. Math. Appl. 45(10–1), 1619–628.
Bellau-Pujol, S., Vabret, A., Legrand, L., Dina, J., Gouarin, S., Petitjean-Lecherbonnier, J., Pozzetto, B., Ginevra, C., Freymuth, F., 2005. Development of three multiplex RT-PCR assays for the detection of 12 respiratory RNA viruses. J. Virol. Methods 126, 53–3.
Coutinho, F.A.B., Burattini, M.N., Lopez, L.F., Massad, E., 2005. An approximate threshold condition for non-autonomous system: An application to a vector-borne infection. Math. Comput. Simul. 70(3), 149–58.
Coutinho, F.A.B., Burattini, M.N., Lopez, L.F., Massad, E., 2006. Threshold conditions for a non-autonomous epidemic system describing the population dynamics of dengue. Bull. Math. Biol. 68(8), 2263–282.
Jafelice, R.M., Barros, L.C., Bassanezi, R.C., Gomide, F., 2004. Fuzzy modeling in symptomatic HIV virus infected population. Bull. Math. Biol. 66(6), 1597–620.
Lefévre, C., Picard, P., 1990. A non-standard family of polynomials and the final size distribution of Reed-Frost epidemic processes. Adv. Appl. Probab. 22, 25–8.
Lopez, L.F., Coutinho, F.A.B., Burattini, M.N., Massad, E., 2007. A schematic age-structured compartment model of the impact of antiretroviral therapy on HIV incidence and prevalence. Math. Comput. Simul. 71(2), 131–48.
Maia, J.O.C., 1952. Some mathematical developments on the epidemic theory formulated by Reed and Frost. Hum. Biol. 24, 167–00.
Massad, E., Azevedo-Neto, R.S., Burattini, M.N., Zanetta, D.M.T., Coutinho, F.A.B., Yang, H.M., Moraes, J.C., Panutti, C.S., Souza, V.A.U.F., Silveira, A.S.B., Struchiner, C.J., Oselka, G.W., Camargo, M.C.C., Omoto, T.M., Passos, S.D., 1995. Assessing the efficacy of a mixed vaccination strategy against rubella in São Paulo, Brazil. Int. J. Epidemiol. 24(4), 842–50.
Massad, E., Burattini, M.N., Ortega, N.R.S., 1999. Fuzzy logic and measles vaccination: designing a control strategy. Int. J. Epidemiol. 28, 550–57.
Massad, E., Ortega, N.R.S., Struchiner, C.J., Burattini, M.N., 2003. Fuzzy epidemiology. Artif. Intell. Med. 29, 241–59.
Massad, E., Burattini, M.N., Lopez, L.F., Coutinho, F.A.B., 2005a. Forecasting versus projection models in epidemiology: The case of the SARS epidemics. Med. Hypotheses 65(1), 17–2.
Massad, E., Coutinho, F.A.B., Burattini, M.N., Lopez, L.F., Struchiner, C.J., 2005b. Yellow fever vaccination: How much is enough? Vaccine 23(30), 3908–914.
Massad, E., Burattini, M.N., Coutinho, F.A.B., Lopez, L.F., 2007. The 1918 influenza A epidemic in the city of São Paulo, Brazil. Med. Hypotheses 68(2), 442–45.
Menezes, R.X., Ortega, N.R.S., Massad, E., 2004. A Reed-Frost model taking into account uncertainties in the diagnostic of the infection. Bull. Math. Biol. 66, 689–06.
Nikravesh, M., Zadeh, L.A., Korotkikh, V., 2004. Fuzzy Partial Differential Equations and Relational Equations. Studies in Fuzziness and Soft Computing, vol. 142. Springer, Berlin.
Ortega, N.R.S., Sallum, P.C., Massad, E., 2000. Fuzzy dynamical systems in epidemic modelling. Kybernetes 29(1–), 201–18.
Ortega, N.R.S., Barros, L.C., Massad, E., 2003. Fuzzy gradual rules in epidemiology. Kybernetes 32(3–), 460–77.
Pearson, D.W., 1997. A property of linear fuzzy differential equations. Appl. Math. Lett. 10(3), 99–03.
Pedrycz, W., Gomide, F., 1998. An Introduction of Fuzzy Sets: Analysis and Design. MIT, USA.
Pereira, J.C., Tonelli, P.A., Barros, L.C., Ortega, N.R.S., 2004. Association and prediction under fuzzy sets theory. Brazilian J. Med. Biol. Res. 37, 701–09.
Picard, P., Lefévre, C., 1991. The dimension of Reed-Frost epidemic models with randomized susceptibility levels. Math. Biosci. 107, 225–33.
Reis, M.A.M., Ortega, N.R.S., Silveira, P.S.P., 2004. Fuzzy expert system in the prediction of neonatal resuscitation. Brazilian J. Med. Biol. Res. 37, 755–64.
Sanchez, E., 1977. Solutions in composite fuzzy relation equations: application to medical diagnosis in Brouwerian logic. In: Gupta, M.M. et al. (Eds.), Fuzzy Automata and Decision Processes. North-Holland, Amsterdam.
Seikkala, S., 1987. On the fuzzy initial value problem. Fuzzy Sets Syst. 24, 319–30.
Zanetta, D.M.T., Cabrera, E.M.S., Azevedo, R.S., Burattini, M.N., Massad, E., 2003. Seroprevalence of rubella antibodies in the State of São Paulo, Brazil, 8 years after the introduction of vaccine. Vaccine 21(25–6), 3795–800.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ortega, N.R.S., Santos, F.S., Zanetta, D.M.T. et al. A Fuzzy Reed–Frost Model for Epidemic Spreading. Bull. Math. Biol. 70, 1925–1936 (2008). https://doi.org/10.1007/s11538-008-9332-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11538-008-9332-3