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Epidemic models with random coefficients. (English) Zbl 1205.60127
Summary: Mathematical models are very important in epidemiology. Many of the models are given by differential equations and most consider that the parameters are deterministic variables. But in practice, these parameters have large variability that depends on the measurement method and its inherent error, on differences in the actual population sample size used, as well as other factors that are difficult to account for. In this paper the parameters that appear in SIR and SIRS epidemic model are considered random variables with specified distributions. A stochastic spectral representation of the parameters is used, together with the polynomial chaos method, to obtain a system of differential equations, which is integrated numerically to obtain the evolution of the mean and higher-order moments with respect to time.

60H30 Applications of stochastic analysis (to PDEs, etc.)
92D30 Epidemiology
37N25 Dynamical systems in biology
34F05 Ordinary differential equations and systems with randomness
Full Text: DOI
[1] Hethcote, H.W., The mathematics of infectious diseases, SIAM rev., 42, 599-653, (2000) · Zbl 0993.92033
[2] Soong, T., Probabilistic modeling and analysis in science and engineering, (1992), Wiley New York
[3] Oksendal, B., Stochastic differential equations, (2003), Springer-Verlag Heidelberg
[4] Metropolis, N.; Ulam, S., The Monte Carlo method, J. amer. statist. assoc., 44, 335-341, (1949) · Zbl 0033.28807
[5] Fishman, G.S., Monte Carlo: concepts, algorithms, and applications, (1995), Springer Verlag New York
[6] Grigoriu, M.; Soong, T., Random vibration of mechanical and structural systems, (1993), Prentice Hall New Jersey · Zbl 0788.73005
[7] Soong, T., Random differential equations in science and engineering, (1973), Academic Press New York · Zbl 0348.60081
[8] Xiu, D.; Karniadakis, G.E., The wiener – askey polynomial chaos for stochastic differential equations, SIAM J. sci. comput., 24, 619-664, (2002) · Zbl 1014.65004
[9] Stanescu, D.; Chen-Charpentier, B., Random coefficient differential equation models for bacterial growth, Math. comput. modelling, 50, 885-895, (2009) · Zbl 1185.34075
[10] R.W. Walters, L. Huyse, Uncertainty quantification for fluid mechanics with applications, ICASE Report No. 2002-1, NASA Langley Research Center, Hampton Va, 2002.
[11] Xiu, D.; Karniadakis, G.E., Modeling uncertainty in flow simulations via generalized polynomial chaos, J. comput. phys., 187, 137-167, (2003) · Zbl 1047.76111
[12] Tornatore, E.; Buccellato, S.M.; Vetro, P., Stability of a stochastic SIR system, Physica A, 354, 111-126, (2005)
[13] Dangerfield, C.E.; Ross, J.V.; Keeling, M.J., Integrating stochasticity and network structure into an epidemic model, J. R. soc. interface, 6, 761-774, (2009)
[14] Murray, J.D., Mathematical biology I, (2002), Springer-Verlag Berlin
[15] Ross, S., A first course in probability, (2002), Prentice Hall New Jersey
[16] Ghanem, R.; Spanos, P.D., Stochastic finite elements: A spectral approach, (1991), Dover Publications Mineola, NJ · Zbl 0722.73080
[17] Wiener, N., The homogeneous chaos, Amer. J. math., 60, 897-936, (1938) · JFM 64.0887.02
[18] Kallianpur, G., Stochastic filtering theory, (1980), Springer Berlin · Zbl 0458.60001
[19] Cameron, R.; Martin, W., The orthogonal development of nonlinear functionals in series of fourier – hermite functionals, Ann. of math., 48, 385-392, (1947) · Zbl 0029.14302
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