tailieunhanh - MODELLING AIDS EPIDEMIC AND TREATMENT WITH DIFFERENCE EQUATIONS K. M. TAMIZHMANI, A. RAMANI, B.

MODELLING AIDS EPIDEMIC AND TREATMENT WITH DIFFERENCE EQUATIONS K. M. TAMIZHMANI, A. RAMANI, B. GRAMMATICOS, AND A. S. CARSTEA Received 15 October 2003 and in revised form 10 March 2004 We propose two models for the description of the dynamics of an AIDS epidemic and of the effect of a combined-drugs AIDS treatment based on difference equations. We show that our interacting population model, despite its extreme simplicity, describes adequately the evolution of an AIDS epidemic. A cellular-automaton analogue of the discrete system of equations is presented as well. In the case of drug treatment, we identify two different regimes corresponding to. | MODELLING AIDS EPIDEMIC AND TREATMENT WITH DIFFERENCE EQUATIONS K. M. TAMIZHMANI A. RAMANI B. GRAMMATICOS AND A. S. CARSTEA Received 15 October 2003 and in revised form 10 March 2004 We propose two models for the description of the dynamics of an AIDS epidemic and of the effect of a combined-drugs AIDS treatment based on difference equations. We show that our interacting population model despite its extreme simplicity describes adequately the evolution of an AIDS epidemic. A cellular-automaton analogue of the discrete system of equations is presented as well. In the case of drug treatment we identify two different regimes corresponding to efficient and inefficient medication. The effect of the discreteness of the equations is also studied. 1. Introduction The modelling of biological systems goes back in time to an era when the very word modelling in the present acceptation was unknown. The use of mathematical modelling which has met with such a great success in physics was extended to the description of the behavior of living organisms under various conditions. The advantage of this paperware rather than wetware laboratory approach is clear. Mathematical models allow us to explore the effect of changes of various parameters in biological systems in an easy fast and inexpensive way while the real experiment may be sometimes unfeasible to say nothing of the ethical issues . Moreover the explicit construction of the mathematical model constrains the modeller to a detailed analysis of the mechanisms involved which leads to a better understanding of the whole process. The mathematical models 8 of biological processes can be classified into two broad categories stochastic and deterministic models. In the first one is interested in the behavior of small samples where fluctuations can play an important role and probabilistic answers are usually sought. In the second one deals with larger samples and the model is usually expressed in terms of differential equations 13 . In