ABSTRACT
In order to obtain a comprehensive form of mathematical models describing non-linear phenomena such as HIV infection processes, a general class of time-dependent evolution equations is introduced in such a way that the nonlinear operator is decomposed into the sum of a relatively good operator and a perturbation which is nonlinear in general and satisfies no global continuity condition. An attempt is then made to combine the implicit approach (usually adapted for convective diffusion operators) and explicit approach (more suited to treat continuous-type operators representing various physiological and reaction processes), resulting in a semi-implicit product formula. Decomposing the operators in this way and considering their individual properties, it is seen that approximation-solvability of the original equation is verified under suitable conditions. Once appropriate terms are formulated to describe treatment by antiretroviral therapy, the time dependence of the reaction terms appears, and such product formula is useful for generating approximate numerical solutions to the original equation. With this knowledge, a continuous model for HIV infection processes is formulated and physiological interpretations are provided. The abstract theory is then applied to show existence of unique solutions to the continuous model describing the behavior of the HIV virus in the human body and its reaction to treatment by antiretroviral therapy. The product formula suggests appropriate discrete models describing the HIV infection mechanism and to perform numerical simulations based on the model of the HIV infection processes. Finally, the results of our numerical simulations are visualized, and it is observed that our results agree with medical aspects in a qualitative way and on a physiologically fundamental level.
