Jump to ContentJump to Main Navigation
Show Summary Details

International Journal of Applied Mathematics and Computer Science

Journal of University of Zielona Gora and Lubuskie Scientific Society

4 Issues per year

IMPACT FACTOR 2015: 1.037
5-year IMPACT FACTOR: 1.151
Rank 83 out of 254 in category Applied Mathematics in the 2015 Thomson Reuters Journal Citation Report/Science Edition

SCImago Journal Rank (SJR) 2015: 1.025
Source Normalized Impact per Paper (SNIP) 2015: 1.674
Impact per Publication (IPP) 2015: 1.648

Mathematical Citation Quotient (MCQ) 2014: 0.07

Open Access
See all formats and pricing
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
/ Mark Stadtherr
  • Department of Chemical and Biomolecular Engineering, University of Notre Dame, Notre Dame, IN 46556, USA
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.

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

  • 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. [Crossref]

  • 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.

  • 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.

  • 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.

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

  • 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.

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

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

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

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

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

  • 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.

  • 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.

  • 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 Science] [Crossref]

  • 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. [Crossref]

  • 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.

  • 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.

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

  • 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.

  • 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.

  • 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. [Crossref]

  • 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. [Crossref]

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

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

  • 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.

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