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Verified solution method for population epidemiology models with uncertainty. (English) Zbl 1300.92098

Summary: Epidemiological models can be used to study the impact of an infection within a population. These models often involve parameters that are not known with certainty. Using a method for verified solution of nonlinear dynamic models, we can bound the disease trajectories that are possible for given bounds on the uncertain parameters. The method is based on the use of an interval Taylor series to represent dependence on time and the use of Taylor models to represent dependence on uncertain parameters and/or initial conditions. The use of this method in epidemiology is demonstrated using the SIRS model, and other variations of Kermack-McKendrick models, including the case of time-dependent transmission.

MSC:

92D30 Epidemiology
93C41 Control/observation systems with incomplete information
93C10 Nonlinear systems in control theory
93C15 Control/observation systems governed by ordinary differential equations

Software:

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References:

[1] Allen, L. J. S. and Burgin, A. M. (2000). Comparison of deterministic and stochastic SIS and SIR models in discrete time, Mathematical Biosciences 163(1): 1-33. · Zbl 0978.92024 · doi:10.1016/S0025-5564(99)00047-4
[2] Anderson, R. M. and May, R. M. (1979). Population biology of infectious diseases: Part 1, Nature 280(5721): 361-367.
[3] Berz, M. and Makino, K. (1998). Verified integration of ODEs and flows using differential algebraic methods on highorder Taylor models, Reliable Computing 4(4): 361-369. · Zbl 0976.65061 · doi:10.1023/A:1024467732637
[4] Corliss, G. F. and Rihm, R. (1996). Validating an a priori enclosure using high-order Taylor series, in G. Alefeld, A. Frommer and B. Lang , Scientific Computing and Validated Numerics, Akademie Verlag, Berlin, pp. 228-238. · Zbl 0851.65054
[5] de Jong, M. C. M., Diekmann, O. and Heesterbeek, H. (1995). How does transmission of infection depend on population size?, in D. Mollison , Epidemic Models: Their Structure and Relation to Data, Cambridge University Press, Cambridge, pp. 84-94. · Zbl 0850.92042
[6] Dushoff, J., Plotkin, J. B., Levin, S. A. and Earn, D. J. D. (2004). Dynamical resonance can account for seasonality of influenza epidemics, Proceedings of the National Academy of Sciences 101(48): 16915-16916.
[7] Edelstein-Keshet, L. (2005). Mathematical Models in Biology, SIAM, Philadelphia, PA. · Zbl 1100.92001
[8] Fan, M., Li, M. Y. and Wang, K. (2001). Global stability of an SEIS epidemic model with recruitment and a varying total population size, Mathematical Biosciences 170(2): 199-208. · Zbl 1005.92030 · doi:10.1016/S0025-5564(00)00067-5
[9] Greenhalgh, D. (1997). Hopf bifurcation in epidemic models with a latent period and nonpermanent immunity, Mathematical and Computer Modelling 25(2): 85-107. · Zbl 0877.92023 · doi:10.1016/S0895-7177(97)00009-5
[10] Hansen, E. R. and Walster, G. W. (2004). Global Optimization Using Interval Analysis, Marcel Dekker, New York, NY. · Zbl 1103.90092
[11] Hethcote, H. W. (1976). Qualitative analysis of communicable disease models, Mathematical Biosciences 28(4): 335-356. · Zbl 0326.92017 · doi:10.1016/0025-5564(76)90132-2
[12] Jaulin, L., Kieffer, M., Didrit, O. and Walter, É. (2001). Applied Interval Analysis, Springer-Verlag, London. · Zbl 1023.65037
[13] Kearfott, R. B. (1996). Rigorous Global Search: Continuous Problems, Kluwer, Dordrecht. · Zbl 0876.90082
[14] Kermack, W. O. and McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society of London, Part A 115(772): 700-721. · JFM 53.0517.01
[15] Li, M. Y., Graef, J. R., Wand, L. and Karsai, J. (1999). Global dynamics of a SEIR model with varying total population size, Mathematical Biosciences 160(2): 191-215. · Zbl 0974.92029 · doi:10.1016/S0025-5564(99)00030-9
[16] Lin, Y. and Stadtherr, M. A. (2007). Validated solutions of initial value problems for parametric ODEs, Applied Numerical Mathematics 57(10): 1145-1162. · Zbl 1121.65084 · doi:10.1016/j.apnum.2006.10.006
[17] Liu, W., Levin, S. A. and Iwasa, Y. (1986). Influence of non-linear incidence rates upon the behavior of SIRS epidemiological models, Journal of Mathematical Biology 23(2): 187-204. · Zbl 0582.92023 · doi:10.1007/BF00276956
[18] Lohner, R. J. (1992). Computations of guaranteed enclosures for the solutions of ordinary initial and boundary value problems, in J. Cash and I. Gladwell , Computational Ordinary Differential Equations, Clarendon Press, Oxford, pp. 425-435. · Zbl 0767.65069
[19] Makino, K. and Berz, M. (1996). Remainder differential algebras and their applications, in M. Berz, C. Bishof, G. Corliss and A. Griewank , Computational Differentiation: Techniques, Applications, and Tools, SIAM, Philadelphia, PA, pp. 63-74. · Zbl 0867.68062
[20] Makino, K. and Berz, M. (1999). Efficient control of the dependency problem based on Taylor model methods, Reliable Computing 5(1): 3-12. · Zbl 0936.65073 · doi:10.1023/A:1026485406803
[21] Makino, K. and Berz, M. (2003). Taylor models and other validated functional inclusion methods, International Journal of Pure and Applied Mathematics 4(4): 379-456. · Zbl 1022.65051
[22] Nedialkov, N. S., Jackson, K. R. and Corliss, G. F. (1999). Validated solutions of initial value problems for ordinary differential equations, Applied Mathematics and Computation 105:(1): 21-68. · Zbl 0934.65073 · doi:10.1016/S0096-3003(98)10083-8
[23] Nedialkov, N. S., Jackson, K. R. and Pryce, J. D. (2001). An effective high-order interval method for validating existence and uniqueness of the solution of an IVP for an ODE, Reliable Computing 7(6): 449-465. · Zbl 1003.65077 · doi:10.1023/A:1014798618404
[24] Neher, M., Jackson, K. R. and Nedialkov, N. S. (2007). On Taylor model based integration of ODEs, SIAM Journal on Numerical Analysis 45(1): 236-262. · Zbl 1141.65056 · doi:10.1137/050638448
[25] Neumaier, A. (1990). Interval Methods for Systems of Equations, Cambridge University Press, Cambridge. · Zbl 0715.65030 · doi:10.1017/CBO9780511526473
[26] Neumaier, A. (2003). Taylor forms-Use and limits, Reliable Computing 9(1): 43-79. · Zbl 1071.65070 · doi:10.1023/A:1023061927787
[27] Pugliese, A. (1990). An SEI epidemic model with varying population size, in S. Busenberg and M. Martelli , Differential Equations Models in Biology, Epidemiology and Ecology, Lecture Notes in Computer Science, Vol. 92, Springer, Berlin, pp. 121-138. · Zbl 0735.92022
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