ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.16(2013) No.4,pp.313-322 Modeling the Spread of HIV/AIDS with Infective Immigrants and Time Delay Agraj Tripathi 1 ∗ , Ram Naresh 2 , Jean M.Tchuenche 3 , Dileep Sharma 2 1 Department of Mathematics, Bhabha Institute of Technology, Kanpur-209204, India 2 Department of Mathematics, H.B. Technological Institute, Kanpur-208002, India 3 Department of Mathematics, University of Dar es Salaam, P.O. Box. 35062, Dar es Salaam, Tanzania (Received 16 September 2012, accepted 13 September 2013) Abstract: In this paper, we propose and analyze a nonlinear mathematical model to study the spread of HIV/AIDS in a variable population incorporating the roles of immigration, treatment and the effect of time delay. The model exhibits a unique endemic steady state which is globally asymptotically stable if a certain threshold quantity, the treatment-induced basic reproduction number R(λ), which depends solely on the pa- rameters of the model, is greater than unity. Under certain conditions, it has a locally asymptotically stable disease-free equilibrium. The constant immigration of infective as well as susceptible individuals, however, makes the disease more endemic. The public health implication of this is that the disease-free equilibrium is not feasible when recruitment of new asymptomatic infectives into the population is allowed and consequent- ly, making control of the epidemic more difficult. Keywords: HIV/AIDS; time delay; global stability; Lyapunov’s function 1 Introduction The impact of HIV/AIDS on humans is so devastating, since its inception in 1981, that HIV epidemic is widely acknowl- edged to be the most severe health crisis of the modern times. HIV continues to spread at alarming rates through many parts of the world, and there have been few victories in the efforts to contain it. This is true despite remarkable advances in our understanding of the epidemiology of the disease, the molecular biology of the virus and its effects on the body- advances that have led to major therapeutic discoveries in the second decade of the epidemic. For those who are able to obtain treatment with antiretroviral drugs, HIV infection has been transformed from a fatal illness into a chronic condition [17]. This has led to dramatic reductions in mortality and morbidity from the illness (which at present will eventually kill). However, despite these advances on the biomedical front, the epidemic continues to spread and treatment remains unavailable to the overwhelming majority of those who require it. The Human Immuno-deficiency Virus (HIV) infection which causes the Acquired Immuno-deficiency Syndrome popularly known as AIDS, has shown a very high degree of prevalence in both the developed and developing world. Mathematical models (deterministic, stochastic, delay) play an important role to study the transmission dynamics of infectious diseases, and in some sense, delay models give better compatibility with reality, as they capture the dynamics for the time of infection to the infectiousness [13, 29]. Most biological systems have time delays inherent in them; yet, few scientists formulate models with time lags due to the complexity they introduce and also for mathematical convenience and tractability. It is recognized that time delays are natural components of the dynamic process of biology, ecology, physiology, economics, epidemiology and mechanics [10]. A brief comment on recent works with delay provides the context of this paper. In recent years, many studies have been made to model the transmission of infectious diseases and different issues have been addressed which affect the spread of the disease [2-3, 6-8, 10, 14, 17-18, 28, 30-31]. In particular, Hethcote and van den Driessche [8] developed an SIS epidemic model with delay corresponding to the infectious period and introduced the disease related death. Khan and Krishnan [10] examined a SIR model by introducing time delay in the recruitment of infected persons, and show that the introduction of a time delay into the transmission term can destabilize the system and periodic solutions can arise by Hopf bifurcation. Greenhalgh et al. [6] presented an SIRS epidemic model with * Corresponding author. E-mail address: agrajtripathi@gmail.com Copyright c ⃝World Academic Press, World Academic Union IJNS.2013.12.30/773