Jump to ContentJump to Main Navigation
Show Summary Details
More options …

International Journal of Applied Mathematics and Computer Science

Journal of University of Zielona Gora and Lubuskie Scientific Society

4 Issues per year

IMPACT FACTOR 2016: 1.420
5-year IMPACT FACTOR: 1.597

CiteScore 2016: 1.81

SCImago Journal Rank (SJR) 2016: 0.524
Source Normalized Impact per Paper (SNIP) 2016: 1.440

Mathematical Citation Quotient (MCQ) 2016: 0.08

Open Access
See all formats and pricing
More options …
Volume 19, Issue 3 (Sep 2009)


Verified Solution Method for Population Epidemiology Models with Uncertainty

Joshua Enszer
  • Department of Chemical and Biomolecular Engineering, University of Notre Dame, Notre Dame, IN 46556, USA
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Mark Stadtherr
  • Department of Chemical and Biomolecular Engineering, University of Notre Dame, Notre Dame, IN 46556, USA
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2009-09-24 | DOI: https://doi.org/10.2478/v10006-009-0040-4

Verified Solution Method for Population Epidemiology Models with Uncertainty

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.

Keywords: nonlinear dynamics; epidemiology; interval analysis; verified computing; ordinary differential equations

  • 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.Google Scholar

  • Anderson, R. M. and May, R. M. (1979). Population biology of infectious diseases: Part 1, Nature 280(5721): 361-367.Google Scholar

  • 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.CrossrefGoogle Scholar

  • 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 (Eds.), Scientific Computing and Validated Numerics, Akademie Verlag, Berlin, pp. 228-238.Google Scholar

  • de Jong, M. C. M., Diekmann, O. and Heesterbeek, H. (1995). How does transmission of infection depend on population size?, in D. Mollison (Ed.), Epidemic Models: Their Structure and Relation to Data, Cambridge University Press, Cambridge, pp. 84-94.Google Scholar

  • 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.Google Scholar

  • Edelstein-Keshet, L. (2005). Mathematical Models in Biology, SIAM, Philadelphia, PA.Google Scholar

  • 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.Google Scholar

  • Greenhalgh, D. (1997). Hopf bifurcation in epidemic models with a latent period and nonpermanent immunity, Mathematical and Computer Modelling 25(2): 85-107.CrossrefGoogle Scholar

  • Hansen, E. R. and Walster, G. W. (2004). Global Optimization Using Interval Analysis, Marcel Dekker, New York, NY.Google Scholar

  • Hethcote, H. W. (1976). Qualitative analysis of communicable disease models, Mathematical Biosciences 28(4): 335-356.CrossrefGoogle Scholar

  • Jaulin, L., Kieffer, M., Didrit, O. and Walter, É. (2001). Applied Interval Analysis, Springer-Verlag, London.Google Scholar

  • Kearfott, R. B. (1996). Rigorous Global Search: Continuous Problems, Kluwer, Dordrecht.Google Scholar

  • 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.Google Scholar

  • 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.Google Scholar

  • Lin, Y. and Stadtherr, M. A. (2007). Validated solutions of initial value problems for parametric ODEs, Applied Numerical Mathematics 57(10): 1145-1162.Web of ScienceCrossrefGoogle Scholar

  • 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.CrossrefGoogle Scholar

  • Lohner, R. J. (1992). Computations of guaranteed enclosures for the solutions of ordinary initial and boundary value problems, in J. Cash and I. Gladwell (Eds.), Computational Ordinary Differential Equations, Clarendon Press, Oxford, pp. 425-435.Google Scholar

  • Makino, K. and Berz, M. (1996). Remainder differential algebras and their applications, in M. Berz, C. Bishof, G. Corliss and A. Griewank (Eds.), Computational Differentiation: Techniques, Applications, and Tools, SIAM, Philadelphia, PA, pp. 63-74.Google Scholar

  • Makino, K. and Berz, M. (1999). Efficient control of the dependency problem based on Taylor model methods, Reliable Computing 5(1): 3-12.CrossrefGoogle Scholar

  • 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.Google Scholar

  • 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.Google Scholar

  • 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.CrossrefGoogle Scholar

  • 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.CrossrefGoogle Scholar

  • Neumaier, A. (1990). Interval Methods for Systems of Equations, Cambridge University Press, Cambridge.Google Scholar

  • Neumaier, A. (2003). Taylor forms—Use and limits, Reliable Computing 9(1): 43-79.CrossrefGoogle Scholar

  • Pugliese, A. (1990). An SEI epidemic model with varying population size, in S. Busenberg and M. Martelli (Eds.), Differential Equations Models in Biology, Epidemiology and Ecology, Lecture Notes in Computer Science, Vol. 92, Springer, Berlin, pp. 121-138.Google Scholar

About the article

Published Online: 2009-09-24

Published in Print: 2009-09-01

Citation Information: International Journal of Applied Mathematics and Computer Science, ISSN (Print) 1641-876X, DOI: https://doi.org/10.2478/v10006-009-0040-4.

Export Citation

This content is open access.

Comments (0)

Please log in or register to comment.
Log in