Analysis of an age‐dependent SI epidemic model with disease‐induced mortality and proportionate mixing assumption: the case of vertically transmitted diseases

M El-Doma

doi:10.1155/s1110757x0430118x

Abstract

An SI epidemic model for a vertically as well as horizontally transmitted disease is investigated when the fertility, natural mortality, and disease‐induced mortality rates depend on age and the force of infection corresponds to a special form of intercohort transmission called proportionate mixing. We determine the steady states and obtain explicitly computable threshold conditions, and then perform stability analysis.

Highlights

We study an age-structured SI epidemic model, where age is assumed to be the chronological age, that is, the time since birth
Vertical transmission is the passing of infection from parents to newborn or unborn offspring, for example, AIDS, Chagas, and hepatitis B are vertically transmitted diseases
We study an SI age-structured epidemic model with vertical transmission and disease-induced mortality rate

Summary

Introduction

We study an age-structured SI epidemic model, where age is assumed to be the chronological age, that is, the time since birth. We study an SI age-structured epidemic model with vertical transmission and disease-induced mortality rate. In [12, 17], a McKendrick-Von Foerster type equation for an SI age-structured epidemic model with disease-induced mortality, but without assuming vertical transmission, is studied. In [18], an SI age-structured epidemic model with vertical transmission as well as horizontal transmission is studied when the disease-induced mortality rate is constant, and a fraction of the offspring of infected mothers die of AIDS effectively at birth and the remaining fraction survive. We note that several recent papers have dealt with age-structured epidemic models with vertical transmission, but without disease-induced mortality, for example, see [8, 10, 11, 13]. The organization of this paper is as follows: in Section 2, we describe the model and obtain the model equations; in Section 3, we reduce the model equations to several subsystems; in Section 4, we determine the steady states; and in Section 5, we perform stability analysis

The model

Reduction of the model

The steady states

Stability of the steady states