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Algorithmic methods for investigating equilibria in epidemic modeling. (English) Zbl 1120.92034
Summary: The calculation of threshold conditions for models of infectious diseases is of central importance for developing vaccination policies. These models are often coupled systems of ordinary differential equations, in which case the computation of threshold conditions can be reduced to the question of stability of the disease-free equilibrium. This paper shows how computing threshold conditions for such models can be done fully algorithmically using quantifier elimination for real closed fields and related simplification methods for quantifier-free formulas. Using efficient quantifier elimination techniques for special cases that have been developed by Weispfenning and others, we can also compute whether there are ranges of parameters for which sub-threshold endemic equilibria exist.

MSC:
92D30 Epidemiology
68U99 Computing methodologies and applications
03C10 Quantifier elimination, model completeness, and related topics
Software:
REDLOG
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References:
[1] Anderson, R.M.; May, R.M., Infectious diseases of humans — dynamics and control, (1992), Oxford University Press Oxford, GB
[2] Beuter, A.; Glass, L.; Mackey, M.C.; Titcombe, M.S., Nonlinear dynamics in physiology and medicine, (2003), Springer · Zbl 1050.92013
[3] Blower, S.M.; Small, P.M.; Hopewell, P.C., Control strategies for tuberculosis epidemics: new models for old problems, Science, 273, 497-500, (1996)
[4] Brown, C.W., Simple CAD construction and its applications, Journal of symbolic computation, 31, 5, 521-547, (2001) · Zbl 0976.65023
[5] Brown, C.W., 2002. The SLFQ system. URL http://www.cs.usna.edu/ qepcad/SLFQ/Home.html
[6] Diekmann, O.; Heesterbeek, J.A.P., Mathematical epidemiology of infectious diseases, (2000), Wiley · Zbl 0997.92505
[7] Dolzmann, A.; Sturm, T., Simplification of quantifier-free formulae over ordered fields, In: applications of quantifier elimination. journal of symbolic computation, 24, 2, 209-231, (1997), (special issue) · Zbl 0882.03030
[8] Dolzmann, A., Sturm, T., 1999. Redlog User Manual. FMI, Universität Passau, 94030 Passau, Germany. URL http://www.fmi.uni-passau.de/ redlog/
[9] Earn, D.J.; Rohani, P.; Bolker, B.M.; Grenfell, B.T., A simple model for complex dynamical transitions in epidemics, Science, 287, 5453, 667-670, (2000), Jan
[10] El Kahoui, M.; Weber, A., Deciding Hopf bifurcations by quantifier elimination in a software-component architecture, Journal of symbolic computation, 30, 2, 161-179, (2000), URL · Zbl 0965.65137
[11] Hadeler, K.P.; van den Driessche, P., Backward bifurcation in epidemic control, Mathematical biosciences, 146, 1, 15-35, (1997) · Zbl 0904.92031
[12] Hethcote, H.W., The mathematics of infectious diseases, SIAM review, 42, 4, 599-653, (2000) · Zbl 0993.92033
[13] Hong, H., Quantifier elimination for formulas constrained by quadratic equations, (), 264-274, URL · Zbl 0964.68596
[14] Hong, H.; Liska, R.; Steinberg, S., Testing stability by quantifier elimination, Journal of symbolic computation, 24, 2, 161-187, (1997), URL · Zbl 0886.65087
[15] Jäger, W.; So, J.W.H.; Tang, B.; Waltman, P., Competition in the gradostat, Journal of mathematical biology, 25, 23-42, (1987) · Zbl 0643.92021
[16] Jirstrand, M., 1996. Applications of quantifier elimination to equilibrium calculations in nonlinear aircraft dynamics. In: Proceedings on the 5th International Student Olympiad. pp. 72-79
[17] Jirstrand, M., Nonlinear control system design by quantifier elimination, Journal of symbolic computation, 24, 2, 137-152, (1997), URL · Zbl 0882.93022
[18] Kalivianakis, M.; Mous, S.L.J.; Grasman, J., Reconstruction of the seasonally varying contact rate for measles, Mathematical biosciences, 214, 2, 225-234, (1994) · Zbl 0818.92021
[19] Kribs-Zaleta, C.M.; Velasco-Hernandez, J.X., A simple vaccination model with multiple endemic states, Mathematical biosciences, 164, 183-201, (2000) · Zbl 0954.92023
[20] Lin, X.-D.; van den Driessche, P., A threshold result for an epidemiological model, Journal of mathematical biology, 30, 6, 647-654, (1992) · Zbl 0763.92009
[21] Liu, W.-M.; van den Driessche, P., Epidemiological models with varying population size and dose-dependent latent period, Mathematical biosciences, 128, 57-69, (1995) · Zbl 0832.92023
[22] May, R., Epidemiology: enhanced: simple rules with complex dynamics, Science, 287, 601-602, (2000)
[23] Novotni, D.; Weber, A., A stochastic method for solving inverse problems in epidemic modelling, (), 467-473
[24] Olsen, L.F.; Schaffer, W.M., Chaos versus noisy periodicity: alternative hypotheses for childhood epidemics, Science, 249, 4968, 499-504, (1990)
[25] ()
[26] Smith, H.L.; Tang, B.; Waltman, P., Competition in an \(n\)-vessel gradostat, SIAM journal of applied mathematics, 51, 5, 1451-1471, (1991) · Zbl 0749.92024
[27] van den Driessche, P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical biosciences, 180, 29-48, (2002) · Zbl 1015.92036
[28] Voit, E.O., Computational analysis of biochemical systems, (2000), Cambridge University Press
[29] Weber, A.; Weber, M.; Milligan, P., Modeling epidemics caused by respiratory syncytial virus (RSV), Mathematical biosciences, 172, 2, 95-113, (2001), URL · Zbl 0988.92025
[30] Weispfenning, V., The complexity of linear problems in fields, Journal of symbolic computation, 5, 1-2, 3-27, (1988) · Zbl 0646.03005
[31] Weispfenning, V., Quantifier elimination for real algebra—the cubic case, (), 258-263 · Zbl 0919.03030
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