# Analysis of nonlinear integro-differential equations arising in age-dependent epidemic models

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Elsevier

## Abstract

PII: 0362-546X(87)90060-5 Nonlinear Adysu. Theoryy. Methods & Application, Vol. Il. No. 8. pp. 913-937. 1987 Pmted in Great Briram. 036?-545X/87 \$3.00 c .OO Pergamon Journals Ltd. ANALYSIS OF NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS ARISING IN AGE-DEPENDENT EPIDEMIC MODELS M. EL-DOMA University of Michigan-Dearborn, Department of Mathematics and Statistics, 4901 Evergreen, Dearborn, Michigan 48128, U.S.A. (Received 16 December 1985; received for publication 2 June 1986) Key wordsandphrases: Asymptotic behaviour, age-dependence, vertical transmission, separable models, almost separable models. 1. INTRODUCTION BASICALLY, there are two modes for directly transmitting an infectious disease within a single population: vertical transmission and horizontal transmission. Vertical transmission is defined as the direct transfer of infection from a parent organism to its offsprings. Horizontal trans- mission is any transfer of infection except that which is vertically transmitted. For example AIDS is both vertically and horizontally transmitted while malaria is horizontally transmitted. Vertically transmitted diseases have seldom been considered in mathematical models of epidemics. Examples of previous such models are found in Anderson and May [I]. Cooke and Busenberg [7], Busenberg and Cooke [3], Busenberg, Cooke and Pozio [4], Fine [lo] and Re’gniere [ 171. Likewise age-dependent diseases has been presented by Cooke and Busenberg [7] and Dietz [8]. Age-dependence introduces a coupling of age-structure and vertical transmission which can produce novel dynamic behavior. In this paper, a system of nonlinear integro-differential equations which model an age- dependent epidemic of a disease with vertical transmission is investigated. This model treats the simple S-, I type of epidemic in this new setting. Existence and uniqueness are proved under suitable hypotheses and the asymptotic behavior of the system is determined. A renewal theorem is used to study t

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