Splitting methods for the numerical approximation of some models of age-structured population dynamics and epidemiology.

*(English)*Zbl 0912.92025We 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.

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 |

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\textit{M. Iannelli} et al., Appl. Math. Comput. 87, No. 1, 69--93 (1997; Zbl 0912.92025)

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##### References:

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