Probability of Incipient Spanning Clusters in Critical Square Bond Percolation

The probability of simultaneous occurence of at least k spanning clusters has been studied by Monte Carlo simulations on the 2D square lattice at the bond percolation threshold $p_c=1/2$. It is found that the probability of k and more Incipient Spanning Clusters (ISC) has the values $P(k>1) \approx 0.00658(3)$ and $P(k>2) \approx 0.00000148(21)$ provided that the limit of these probabilities for infinite lattices exists. The probability $P(k>3)$ of more than three ISC could be estimated to be of the order of 10^{-11} and is beyond the possibility to compute a such value by nowdays computers. So, it is impossible to check in simulations the Aizenman law for the probabilities when $k>>1$. We have detected a single sample with 4 ISC in a total number of about 10^{10} samples investigated. The probability of single event is 1/10 for that number of samples.Comment: 7 pages, 1 table, 5 figures (1PS+4*Latex),uses epsf.sty Int.J.Mod.Phys. C (submitted to

Real-world study of efficacy, risk management and reasons for discontinuation of natalizumab for treatment of multiple sclerosis in Russia.

BackgroundNTZ is approved in Russia for the treatment of highly active relapsing remitting multiple sclerosis and is reimbursed via federal budget program. However, no data about NTZ treatment in Russia and the effect of federal reimbursement have been performed so far.ObjectiveTo characterize the population of patients receiving natalizumab and assess the efficacy and risk-management plan (RMP) implementation of NTZ therapy in routine clinical practice in Russia.MethodsWe analyzed data for 334 patients, who received at least one infusion of NTZ. Relapse rate, MRI activity, NEDA-3 status after 2 years were assessed. Anti-JC virus antibodies status and RMP implementation were evaluated. Drop-out rate and reasons for therapy discontinuation were analyzed.ResultsPatients switched to natalizumab in Russia are mainly female (63%), with median EDSS score of 3.5 and high disease activity: 93% had at least 1 relapse and 58% had both T1Gd+ and new T2 lesion a year before therapy initiation. Introduction of federal reimbursement allowed patients with less relapses to start therapy with natalizumab. The only predictor of 6-month progression was EDSS score at the baseline of therapy (HR = 2.1375, 95%CI 1.0026-4.5570, p = 0.0492). 82% patients reached NEDA-3 at 24 month of therapy. 25% of patients discontinued NTZ for reasons: tolerability (14.5%), JCV antibody status (61%), and patient's decision (17%). RMP was implemented in only 36% patients.ConclusionNatalizumab appeared to have high efficacy in Russian clinical practice. Federal reimbursement allowed less active patients to start natalizumab. More efforts should be done to improve RMP implementation