Cybernetics and Systems Analysis, Vol. 40, No. 5, 2004 CONSTRUCTION AND STUDY OF STABILITY OF AN ANTITUMORAL IMMUNITY MODEL V. P. Martsenyuk UDC 517.977.5, 519.7 A simplified model of antitumor immunity is proposed. The model is based on the Marchuk immune protection model and Gompertz model. Sufficient conditions of its asymptotic equilibrium stability are obtained and formulated in terms of characteristic quasipolynomial coefficients. Keywords: antitumor immunity, Gompertz dynamics, Marchuk model of the immune system, stability, quasipolynomial. INTRODUCTION The main function of immunity is the control of processes of proliferation of cellular differentiations and elimination of mutant cells [1]. The achievements of antitumor immunology during the last 30 years add support to F. Bernet’s idea that was first formulated in 1959 and stated that the antimicrobic action is only a partial manifestation of immunity. Thus, the infectious immunology became the base of creation of a new domain of scientific knowledge, namely, noninfectious immunology whose important trend is the study of antitumor immunity. This immunity depends on the cause of a tumor (viruses, carcinogenic chemical substances, or “spontaneous” tumors). Immunity is specific to the viruses that induce a tumor (DNA- or RNA-containing viruses). Immunity is developed in some days or even hours after the introduction of viruses and is supported during months. The immunity to tumors induced by carcinogens is weaker than the immunity to tumors induced by viruses and is weaker than that to tumor cells that “spontaneously” arise [2]. The objective of this article is the construction of a simplified model of antitumor immunity and also the obtaining of the sufficient stability conditions for this model. This investigation is based on the model of immune protection that is proposed in [6]. The stability of the latter model was investigated in [3]. A tumor is described in terms of the Gompertz dynamics [7, 8]. CONSTRUCTION OF A SIMPLIFIED MODEL OF ANTITUMOR IMMUNITY The model that is described below is based on a simplified representation of antitumor immunity [1]. The immune system induces an immune response (cellular by means of cytotoxic T-lymphocytes and humoral with factors such as antibodies, for example, specific IgG and IgM) to a growing tumor. Immune reactions are induced by a specific tumor antigen that can be found in different parts of a tumor cell (usually on its surface). We assume that the simplified model satisfies the following conditions. 1. The populations of cancerous cells, antibodies, and plasma cells are homogeneous. 2. The change in the population of cancerous cells obeys the laws of the Gompertz dynamics. 3. The immune reaction is induced by a tumor antigen and antibodies of one kind. 4. The concentration of tumor antigens at a moment of time t is proportional to the number of tumor cells Lt ( ). 5. Cancerous cells suppress the increase in the population of antibodies. 778 1060-0396/04/4005-0778 © 2004 Springer Science+Business Media, Inc. I. Ya. Gorbachevskii Ternopol State Medical Academy, Ternopol, Ukraine marceniuk@yahoo.com. Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 177-182, September-October 2004. Original article submitted October 31, 2003.