### International Journal ofPhysical Sciences

• Abbreviation: Int. J. Phys. Sci.
• Language: English
• ISSN: 1992-1950
• DOI: 10.5897/IJPS
• Start Year: 2006
• Published Articles: 2557

## A deterministic model of HIV/AIDS with vertical transmission in the presence of infected immigrants

##### Isack Kibona1*, Wilson Mahera1, Daniel Makinde2 and John Mango3
1Departmentof Science and Business Management, Mbeya Institute of Science and Technology, P. O. Box 131, Mbeya, Tanzania. 2Institute for Advance Research in Mathematical Modelling and Computations, Cape Peninsula University of Technology, P. O. Box 1906, Bellville 7535, South Africa. 3Department of Mathematics, University of Dar Es Salaam, P. O. Box 35062, Dar Es Salaam, Tanzania. 4Department of Mathematics, Makerere University, P. O.  Box 7062 Kampala, Uganda.
Email: [email protected]

•  Accepted: 27 May 2011
•  Published: 09 October 2011

Abstract

A non-linear mathematical model for HIV epidemic that spreads in a variable size population through both horizontal and vertical transmission in the presence of HIV infected immigrants is studied. The equilibrium points of the model are found and the stability is investigated. The model exhibits two equilibria, namely, the disease-free and the endemic equilibrium. It is shown that if the basic reproduction number, the disease-free equilibrium is always locally asymptotically stable and in such a case the endemic equilibrium does not exist. If , a unique endemic equilibrium exists, which is locally asymptotically stable and becomes globally asymptotically stable under certain conditions. This shows that the disease becomes endemic due to constant immigration of both HIV infected and non infected individuals into the community. Using stability theory and computer simulation, it is shown that by controlling the rate of vertical transmission, the spread of the disease can be reduced significantly and consequently the equilibrium values of infected population can be maintained at desired levels. A numerical study of the model is also used to investigate the influence of certain other key parameters on the spread of the disease and how to control their influence.

Key words: AIDS epidemic, vertical transmission, immigration, stability, simulation.