×

Numerical solution of threshold problems in epidemics and population dynamics. (English) Zbl 1311.65088

Summary: A new algorithm is proposed for the numerical solution of threshold problems in epidemics and population dynamics. These problems are modeled by the delay-differential equations, where the delay function is unknown and has to be determined from the threshold conditions. The new algorithm is based on embedded pair of continuous Runge-Kutta method of order \(p = 4\) and discrete Runge-Kutta method of order \(q = 3\) which is used for the estimation of local discretization errors, combined with the bisection method for the resolution of the threshold condition. Error bounds are derived for the algorithm based on continuous one-step methods for the delay-differential equations and arbitrary iteration process for the threshold conditions. Numerical examples are presented which illustrate the effectiveness of this algorithm.

MSC:

65L03 Numerical methods for functional-differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
65R20 Numerical methods for integral equations
92D25 Population dynamics (general)
92D30 Epidemiology

Software:

bvp4c
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Jackiewicz, Z.; Zubik-Kowal, B., Discrete variable methods for delay-differential equations with threshold-type delays, J. Comput. Appl. Math., 228, 514-523 (2009) · Zbl 1175.65077
[2] Hoppensteadt, F. C.; Waltman, P., A problem in the theory of epidemics, Math. Biosci., 9, 71-91 (1970) · Zbl 0212.52105
[3] Hoppensteadt, F. C.; Jackiewicz, Z., Numerical solution of a problem in the theory of epidemics, Appl. Numer. Math., 56, 533-543 (2006) · Zbl 1085.92035
[4] Butcher, J. C.; Jackiewicz, Z., Construction of diagonally implicit general linear methods of type 1 and 2 for ordinary differential equations, Appl. Numer. Math., 21, 385-415 (1996) · Zbl 0865.65056
[5] Wright, W. M., The construction of order 4 DIMSIMs for ordinary differential equations, Numer. Algorithms, 26, 123-130 (2001) · Zbl 0974.65074
[6] Butcher, J. C.; Chartier, P.; Jackiewicz, Z., Nordsieck representation of DIMSIMs, Numer. Algorithms, 16, 209-230 (1997) · Zbl 0920.65043
[7] Owren, B.; Zennaro, M., Derivation of efficient, continuous, explicit Runge-Kutta methods, SIAM J. Sci. Stat. Comput., 13, 1488-1501 (1992) · Zbl 0760.65073
[8] Thompson, S.; Shampine, L. F., A note on Zdzislaw’s epidemic model
[9] Shampine, L. F.; Gladwell, I.; Thompson, S., Solving ODEs with MATLAB (2003), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1079.65144
[10] Gourley, S. A.; Kuang, Y., A stage structured predator-prey model and its dependence on maturation delay and death rate, J. Math. Biol., 49, 188-200 (2004) · Zbl 1055.92043
[11] Gourley, S. A.; Kuang, Y., A stage structured predator-prey model: importance of maturation delay, (Talk at the Third International Conference on the Numerical Solution of Volterra and Delay Equations (2004), Arizona State University), May 18-21 · Zbl 1055.92043
[12] Aiello, W. G.; Freedman, H. I., A time-delay model of single species growth with stage structure, Math. Biosci., 101, 139-153 (1990) · Zbl 0719.92017
[13] Jackiewicz, Z., Convergence of multistep methods for Volterra functional differential equations, Numer. Math., 32, 307-332 (1979) · Zbl 0388.65028
[14] Jackiewicz, Z., One-step methods of any order for neutral functional differential equations, SIAM J. Numer. Anal., 21, 486-510 (1984) · Zbl 0562.65056
[15] Hale, J. K.; Verduyn Lunel, S. M., Introduction to Functional Differential Equations (1993), Springer-Verlag · Zbl 0787.34002
[16] Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press: Academic Press Boston · Zbl 0777.34002
[17] Smith, H. L., (An Introduction to Delay Differential Equations with Applications to the Life Sciences. An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics (2011), Springer: Springer Berlin) · Zbl 1227.34001
[18] Brunner, H.; van der Houwen, P. J., The Numerical Solution of Volterra Equations (1986), North-Holland: North-Holland Amsterdam, New York · Zbl 0611.65092
[19] Hairer, E.; Nørsett, S. P.; Wanner, G., Solving Ordinary Differential Equations I. Nonstiff Problems (1993), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York · Zbl 0789.65048
[20] Zennaro, M., Natural continuous extensions of Runge-Kutta methods, Math. Comp., 46, 119-133 (1986) · Zbl 0608.65043
[21] Zennaro, M., Natural Runge-Kutta methods and projection methods, Numer. Math., 53, 423-438 (1988) · Zbl 0651.65055
[22] Owren, B.; Zennaro, M., Order barriers for continuous explicit Runge-Kutta methods, Math. Comp., 56, 645-661 (1991) · Zbl 0718.65051
[23] Bellen, A.; Zennaro, M., (Numerical Methods for Delay Differential Equations. Numerical Methods for Delay Differential Equations, Oxford Science Publications (2003), Clarendon Press: Clarendon Press Oxford) · Zbl 0749.65042
[24] Gladwell, I.; Shampine, L. F.; Brankin, R. W., Automatic selection of the initial stepsize for ODE solver, J. Comput. Appl. Math., 18, 175-192 (1987) · Zbl 0623.65080
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.