A nonlinear HIV/AIDS model with contact tracing Ram Naresh a , Agraj Tripathi b,⇑ , Dileep Sharma a a Department of Mathematics, Harcourt Butler Technological Institute, Kanpur 208002, India b Department of Mathematics, Bhabha Institute of Technology, Kanpur D. 209204, India article info Keywords: HIV/AIDS Screening Contact tracing Reproduction number Stability analysis Numerical simulation abstract A nonlinear mathematical model is proposed and analyzed to study the effect of contact tracing on reducing the spread of HIV/AIDS in a homogeneous population with constant immigration of susceptibles. In modeling the dynamics, the population is divided into four subclasses of HIV negatives but susceptibles, HIV positives or infectives that do not know they are infected, HIV positives that know they are infected and that of AIDS patients. Susceptibles are assumed to become infected via sexual contacts with (both types of) infectives and all infectives move with constant rates to develop AIDS. The model is analyzed using the stability theory of differential equations and numerical sim- ulation. The model analysis shows that contact tracing may be of immense help in reduc- ing the spread of AIDS epidemic in a population. It is also found that the endemicity of infection is reduced when infectives after becoming aware of their infection do not take part in sexual interaction. Ó 2011 Elsevier Inc. All rights reserved. 1. Introduction India is facing one of the biggest public health challenges in its history. Still, India has the second highest number of peo- ple living with HIV/AIDS in the world after South Africa, though the current HIV prevalence in India is declining. The first HIV infection case in India was detected in Chennai in 1986 [18]. This signaled the start of AIDS epidemic in the country, and since then, HIV infection has been reported in all states and union territories. India responded to the AIDS epidemic imme- diately after the first ever HIV/AIDS case was reported in the country. Recognizing the seriousness of the situation, the gov- ernment constituted a high-power committee in 1986 under Ministry of Health and Family Welfare. Subsequently, a National AIDS Control Programme was launched in India. The main objectives of NACO are to reduce the spread of HIV infec- tion and to strengthen India’s capacity to respond to HIV/AIDS on a long term basis. The total number of people living with HIV/AIDS in India is estimated at 24 lakhs in 2009. Children (<15 years) account for 3.5% of all infections, while 83% are the in age-group 15–49 years. Of all HIV infections, 39% (9.3 lakhs) are among women. The four high prevalence states of India (Andhra Pradesh – 5 lakhs, Maharashtra – 4.2 lakhs, Karnataka – 2.5 lakhs, Tamil Nadu – 1.5 lakhs) account for 55% of all HIV infections in the country. In India the heterosexual contact is the predominant mode of transmission of HIV [18]. In Indian society there may be many people at risk of HIV infection but do not know that they are infected. Mathematical models of transmission dynamics of HIV play an important role in better understanding of epidemiological patterns for disease control as they provide short and long term prediction of HIV and AIDS incidence. In recent decades or so various modeling studies have been conducted to describe the transmission dynamics of HIV [1–3,6–9,11–15,17,22,23]. In 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.04.033 ⇑ Corresponding author. E-mail addresses: agraj_hbti@yahoo.co.in, agrajtripathi@gmail.com (A. Tripathi). Applied Mathematics and Computation 217 (2011) 9575–9591 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc