STABILITY ANALYSIS FOR THE HIV/AIDS EPIDEMIC MODEL WITH INFLUENCE OF THE AGE GROUP AND POPULATION DENSITY

Abstract

The issue of  HIV/AIDS is a serious public health problem. AIDS patients are generally dominated by a age group of  adolescent and a productive age. The general pattern of the spread of  infection occur through sexual contact (sexually transmitted diseases = STD). The threat of of  HIV/AIDS epidemics were seen through the data of cases of HIV/AIDS continues to rise.

This paper examines and implementation a deterministic mathematical model of a simple SI (susceptible-infected) model to analyze the stability of the HIV/AIDS by age group and population density. The population is divided into two subpopulations, namely subpopulation of juvenile and adults. Subpopulation of adults who are sexually active is assumed produce both susceptible newborns and infected newborns. The local and global stability for the equilibrium point of the model were analyzed using a combination of analysis of eigenvalues of Jacobian matrix and the Lyapunov-LaSalle’s invariant principle or using a threshold values of the susceptible reproduced ratio (), the infected reproduced ratio (), and  the  infection contact rate ( ) .

For the case of data of HIV/AIDS in Indonesia with  initial population of 2009, the threshold values of the susceptible reproduction ratio,   the infected reproduction ratio,  and  the infection contact rate,  The model of the HIV/AIDS has a unique disease-free equilibrium point,  The disease-free equilibrium point is globally asymptotically stable, namely if parameter values not change then there no infected individual and subpopulation of juvenile and adults susceptible tend to constant positive value.

Key words: HIV/AIDS model, SI model, the reproduction ratio, the equilibrium point,  

                     Global stability