APPLICATIONES MATHEMATICAE 36,3 (2009), pp. 263–278 Marek Bodnar, Urszula Foryś and Zuzanna Szymańska (Warszawa) MODEL OF AIDS-RELATED TUMOUR WITH TIME DELAY Abstract. We present and compare two simple models of immune system and cancer cell interactions. The first model reflects simple cancer disease progression and serves as our “control” case. The second describes the pro- gression of a cancer disease in the case of a patient infected with the HIV-1 virus. 1. Introduction. Human immunodeficiency virus (HIV) is a retrovirus often leading to a disease called acquired immunodeficiency syndrome (AIDS) which is estimated to be responsible for killing more than 25 million people since its first recognition in 1981 up to 2005 (cf. data in [12]). AIDS is characterized by deeply impaired functionality of the immune system and by various clinical expressions. The first target for the virus is the dendritic cells present in the mucous membrane. After being infected, the dendritic cells move to surrounding lymph nodes, where during the antigen presentation the T cells become infected. Cells which are actively producing virus are T helper cells, which are in turn destroyed both by HIV and by T cytotoxic cells. Patients with diagnosed AIDS have fewer than 200 T CD4 + cells/mm 3 , while in healthy persons this value is estimated to be between 800 and 1200 T CD4 + cells/mm 3 [8]. Basically four stages of the disease can be distinguished: incubation period, acute infection, latency stage and AIDS. The initial incubation period upon infection is asymptomatic and usually lasts between two and four weeks. The second stage, acute infection, lasts an average of 28 days and can include symptoms similar to those of influenza or infectious mononucleosis. The third, latency stage, that occurs later, shows few or no symptoms and can last anywhere from two weeks to twenty years and beyond. And finally, the fourth stage of HIV infection 2000 Mathematics Subject Classification : 34K60, 34K20, 37N25, 34K28. Key words and phrases : delay differential equations, asymptotic stability, tumour, HIV, immune response. DOI: 10.4064/am36-3-2 [263] c  Instytut Matematyczny PAN, 2009