An approximating algorithm for the solution of an integral equation from epidemics. (English) Zbl 1205.65334
Summary: The following delay integral equation
\[
x(t)=\int_{t-\tau}^{t}f(s,x(s))ds,\quad t\in \mathbb{R},
\]
has been proposed by Cooke and Kaplan to describe the spread of certain infectious diseases with periodic contact rate that varies seasonally. This mathematical model can also be interpreted as an evolution equation of a single species population. The purpose of this paper is to present an approximating algorithm for the continuous positive solution of this integral equation from the theory of epidemics. This algorithm is obtained by applying the successive approximations method and the rectangle formula, used for the calculation of the approximate value of integrals which appear in the right-hand-side of the terms of the sequence of successive approximations. In order to establish this approximating algorithm, we will suppose that this integral equation has a unique solution. The main result contains also the error of approximation of the solution obtained by applying this approximating algorithm.
MSC:
| 65R20 | Numerical methods for integral equations |
| 45G10 | Other nonlinear integral equations |
| 92D30 | Epidemiology |
| 47H10 | Fixed-point theorems |
| 45L05 | Theoretical approximation of solutions to integral equations |
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