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

Abstract

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 transmission 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 agedependent 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 the behavior of the model equations in various pertinent parameter ranges. A numerical method for integrating this system of equations is developed and is used to obtain approximations of its solutions for some special cases which illustrate the results obtained via analytical means. Moreover, numerical integrations of the equations are used to study some phenomena that were not treated analytically.