Journal of Mahani Mathematical Research
Volume:5 Issue: 2, Summer and Autumn 2017

In this article, a mathematical model describing the growth or terminating myelogenous leukemia blood cancer's cells against naive T-cell and e ective T-cell population of body, presented by fractional di erential equations. We use this model to analyze the stability of the dynamics, which occur in the local interaction of e ector-immune cell and tumor cells. We will also investigate the optimal control of combined chemo-immunotherapy. We claim that our fractional di erential equations model is superior to its ordinary di erential equations counterpart in facilitating understanding of the natural immune interactions to tumor and of the detrimental side e ects which chemotherapy may have on a patient's immune system.

In this paper at rst, a history of mathematical models is given. Next, some basic information about random variables, stochastic processes and Markov chains is introduced. As follows, the entropy for a discrete time Markov process is mentioned. After that, the entropy for SIS stochastic models is computed, and it is proved that an epidemic will be disappeared after a long time.

This paper provides a review on major ergodic features of semi- independent hyper MV {algebra dynamical systems. Theorems are presented to make contribution to calculate the entropy. Particularly, it is proved that the total entropy of those semi-independent hyper MV {algebra dynamical systems that have a generator can be calculated with respect to their generator rather than considering all the partitions.

ýIn the following text for arbitrary $X$ with at least two elementsý, ýnonempty countable set $\Gamma$ý ýwe make a comparative study on the collection of generalized shift dynamical systems like $(X^\Gamma,\sigma_\varphi)$ where $\varphi:\Gamma\to\Gamma$ is an arbitrary self mapý.
ýWe pay attention to sub-systems and combinations of generalized shifts with counterexamples regarding Devaneyý, ýexact Devaneyý, ýLi-Yorkeý, ýe-chaoticity and P chaoticityý.