# zbMATH — the first resource for mathematics

Splitting methods for the numerical approximation of some models of age-structured population dynamics and epidemiology. (English) Zbl 0912.92025
We study a class of schemes for the numerical approximation of some models arising from age-structured population dynamics. Namely, we consider the basic nonlinear Gurtin-MacCamy system $\partial u/\partial t+\partial u/\partial a+ \mu(a,p(t))u= 0,\qquad u(a,0)=u_0(a),\qquad 0\leq a< a_†,\quad t>0,$ $u(0,t)= \int^{a_†}_0 \beta(a,p(t)) u(a,t)da,\quad t>0,\qquad p(t)= \int^{a_†}_0 u(a,t)da,\quad t\geq 0,$  and some related epidemic models that we will introduce later. This system describes the evolution of the age-density $$u(a,t)$$ of a population, with maximum age $$a_†< \infty$$, whose growth is regulated by the vital rates $$\beta(a,p)$$ and $$\mu(a,p)$$.
The paper is organized as follows. First we introduce splitting algorithms based on the method of characteristics. We derive (uniform) stability estimates for the schemes and show convergences to the true solutions by the approximations to be at first-order rates in the maximum norm. Finally, we present the results from numerical experiments with some concluding remarks.

##### MSC:
 92D25 Population dynamics (general) 65M25 Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs 92D30 Epidemiology
Full Text:
##### References:
  Gurtin, M.; MacCamy, R.C., Non-linear age-dependent population dynamics, Archs ration. mech. analysis, 54, 281-300, (1974) · Zbl 0286.92005  Iannelli, M., (), Applied Mathematics Monographs  Webb, G.F., Theory of nonlinear age-dependent population dynamics, (1985), Marcel Dekker, Inc New York · Zbl 0555.92014  Kim, M.-Y.; Park, E.-J., An upwind scheme for a nonlinear model in age-structured population dynamics, Comput. math. applic., 30, 8, 5-17, (1995) · Zbl 0836.92015  Abia, L.M.; López-Marcos, J.C., Runge-Kutta methods for age-structured population models, Appl. numer. math., 17, 1, 1-17, (1995) · Zbl 0822.65070  Milner, F.A.; Rabbiolo, G., Rapidly converging numerical algorithms for models of population dynamics, J. math. biol., 30, 733-753, (1992) · Zbl 0795.92022  M.-Y. Kim and E.-J. Park, Long-time behavior of numerical solutions to Gurtin-MacCamy equations by method of characteristics, preprint.  Barbu, V.; Iannelli, M., Approximating some non-linear equations by a fractional step scheme, Differential and integration equations, 6, 15-26, (1993) · Zbl 0773.34047  Busenberg, S.; Cooke, K.; Iannelli, M., Endemic thresholds and stability in a class of age-structured epidemics, SIAM J. appl. math., 48, 1379-1395, (1988) · Zbl 0666.92013  Busenberg, S.; Iannelli, M.; Thieme, H., Global behavior of an age structured epidemic model, SIAM J. math. anal., 22, 1065-1080, (1991) · Zbl 0741.92015  Iannelli, M.; Milner, F.; Pugliese, A., Analytical and numerical results for the age structured SIS epidemic model with mixed inter-intracohort transmission, SIAM J. math. anal., 23, 662-688, (1992) · Zbl 0776.35032  Inaba, H., Threshold and stability results for an age-structured epidemic model, J. math. biol., 28, 411-434, (1990) · Zbl 0742.92019  M.-Y. Kim, Qualitative behaviors of numerical solutions to an SIS epidemic model, preprint.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.