The Mean Drift: Tailoring the Mean Field Theory of Markov Processes for Real-World Applications

The statement of the mean field approximation theorem in the mean field theory of Markov processes particularly targets the behaviour of population processes with an unbounded number of agents. However, in most real-world engineering applications one faces the problem of analysing middle-sized systems in which the number of agents is bounded. In this paper we build on previous work in this area and introduce the mean drift. We present the concept of population processes and the conditions under which the approximation theorems apply, and then show how the mean drift is derived through a systematic application of the propagation of chaos. We then use the mean drift to construct a new set of ordinary differential equations which address the analysis of population processes with an arbitrary size

Analysis of Petri Net Models through Stochastic Differential Equations

It is well known, mainly because of the work of Kurtz, that density dependent Markov chains can be approximated by sets of ordinary differential equations (ODEs) when their indexing parameter grows very large. This approximation cannot capture the stochastic nature of the process and, consequently, it can provide an erroneous view of the behavior of the Markov chain if the indexing parameter is not sufficiently high. Important phenomena that cannot be revealed include non-negligible variance and bi-modal population distributions. A less-known approximation proposed by Kurtz applies stochastic differential equations (SDEs) and provides information about the stochastic nature of the process. In this paper we apply and extend this diffusion approximation to study stochastic Petri nets. We identify a class of nets whose underlying stochastic process is a density dependent Markov chain whose indexing parameter is a multiplicative constant which identifies the population level expressed by the initial marking and we provide means to automatically construct the associated set of SDEs. Since the diffusion approximation of Kurtz considers the process only up to the time when it first exits an open interval, we extend the approximation by a machinery that mimics the behavior of the Markov chain at the boundary and allows thus to apply the approach to a wider set of problems. The resulting process is of the jump-diffusion type. We illustrate by examples that the jump-diffusion approximation which extends to bounded domains can be much more informative than that based on ODEs as it can provide accurate quantity distributions even when they are multi-modal and even for relatively small population levels. Moreover, we show that the method is faster than simulating the original Markov chain

Anticancer and cytotoxic effects of troxerutin on HeLa cell line: an in-vitro model of cervical cancer

Cervical cancer is one of the grave uterine tumors which leads to death in women worldwide. Troxerutin (TRX) as a bioflavonoid compound has many pharmacological effects such as anti-neoplastic, radioprotective, and anti-cancer. The present study was designed to examine the cytotoxic effect of TRX on human HeLa tumor cells. Human HeLa cells were cultured and treated with different doses of TRX (20�640 mg/ml) to evaluate the effective half-maximal inhibitory concentration (IC50) after 24 h. MTT (3-(4,5-dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide) test was used for cell proliferation assay. Also, the Bax, Bcl-2, cleaved caspase-3, and tumor necrosis factor-α (TNF-α) protein expression levels were detected with immunoblotting analysis. The malondialdehyde (MDA) concentration, glutathione peroxidase (GPx) and superoxide dismutase (SOD) activity levels were measured via their commercial kits. Data were analyzed using one-way ANOVA. The result showed that TRX at 320 mg/ml concentration (IC50) has a growth inhibitory effect against HeLa cells at 24 h treatment (P � 0.01). Moreover, it increased the MDA concentration and also decreased the GPx and SOD activity levels at 320 mg/ml concentration versus control (P < 0.001). Also, TRX significantly up-regulated the Bax, cleaved caspase-3 and TNF-α proteins expression levels (P < 0.01) and down-regulated the Bcl-2 protein expression in HeLa tumor cells at 320 mg/ml concentration compared to control (P < 0.05). Our study showed that 24 h of treatment with TRX (320 mg/ml) has apoptotic and growth inhibitory effects against HeLa cells. It can induce inflammation (at least via up-regulating the TNF-α protein expression) and oxidative stress in human HeLa cells. © 2020, Springer Nature B.V