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In modeling the bilateral selection of states of the process, Dynkin (Dokl Akad Nauk USSR 185:241–288, 1969) proposed a two-person game in which players use stopping moments as strategies. The purpose of this work is to present a model of the game in which the players have different information about the process itself, as well as various laws to stop the process and accept its state. The game model uses the stochastic process apparatus, in particular, the ability to create different filters for the same process. The sets of stopping moments based on different filters are not identical, which allows us to model different sets of strategies for players. We show that the follower, by observing the behavior of a rational leader, can recover information that is lost due to the lack of complete observation of the state of the process. In the competition of two opponents for the maximum of the i.i.d. sequence, one of whom has access to full information and the other only knows their relative ranks, we found the generalized Stackelberg equilibrium. If the priority of a player observing the relative ranks is less than 50%, then that player modifies his strategy based on the behavior of the second player. For a player with full information, information about the behavior of the player observing the relative ranks is useless.

Senility has been identified among the strongest risk predictors for unfavorable COVID-19-outcome. However, even in the elderly population, the clinical course of infection in individual patients remains unpredictable. Hence, there is an urgent need for developing a simple tool predicting adverse COVID-19-outcomes. We assumed that the C2HEST-score could predict unfavorable clinical outcomes in the elderly subjects with COVID-19-subjects. Methods: We retrospectively analyzed 1047 medical records of patients at age > 65 years, hospitalized at the medical university center due to COVID-19. Subsequently, patients were divided into three categories depending on their C2HEST-score result. Results: We noticed significant differences in the in-hospital and 3-month and 6-month mortality-which was the highest in high-risk-C2HEST-stratum reaching 35.7%, 54.4%, and 65.9%, respectively. The medium-risk-stratum mortalities reached 24.1% 43.4%, and 57.6% and for low-risk-stratum 14.4%, 25.8%, and 39.2% respectively. In the C2HEST-score model, a change from the low to the medium category increased the probability of death intensity approximately two-times. Subsequently, transfer from the low-risk to the high-risk-stratum raised all-cause-death-intensity 2.7-times. Analysis of the secondary outcomes revealed that the C2HEST-score has predictive value for acute kidney injury, acute heart failure, and cardiogenic shock. Conclusions: C2HEST-score analysis on admission to the hospital may predict the mortality, acute kidney injury, and acute heart failure in elderly subjects with COVID-19.

Background: Diabetes mellitus is among the most frequent comorbidities worsening COVID-19 outcome. Nevertheless, there are no data regarding the optimal risk stratification of patients with diabetes and COVID-19. Since individual C2HEST components reflect the comorbidities, we assumed that the score could predict COVID-19 outcomes. Material and Methods: A total of 2184 medical records of patients hospitalized for COVID-19 at the medical university center were analyzed, including 473 diabetic patients and 1666 patients without any glucose or metabolic abnormalities. The variables of patients’ baseline characteristics were retrieved to calculate the C2HEST score and subsequently the diabetic and non-diabetic subjects were assigned to the following categories: low-, medium- or high-risk. The measured outcomes included: in-hospital mortality; 3-month and 6-month all-cause mortality; non-fatal end of hospitalization (discharged home/sudden-deterioration/rehabilitation) and adverse in-hospital clinical events. Results: A total of 194 deaths (41%) were reported in the diabetic cohort, including 115 in-hospital deaths (24.3%). The 3-month and 6-month in-hospital mortality was highest in the high-risk C2HEST stratum. The C2HEST score revealed to be more sensitive in non-diabetic-group. The estimated six-month survival probability for high-risk subjects reached 0.4 in both cohorts whereas for the low-risk group, the six-month survival probability was 0.7 in the diabetic vs. 0.85 in the non-diabetic group—levels which were maintained during whole observation period. In both cohorts, receiver operating characteristics revealed that C2HEST predicts the following: cardiogenic shock; acute heart failure; myocardial injury; and in-hospital acute kidney injury. Conclusions: We demonstrated the usefulness and performance of the C2HEST score in predicting the adverse COVID-19 outcomes in hospitalized diabetic subjects.

Senility has been identified among the strongest risk predictors for unfavorable COVID-19-outcome. However, even in the elderly population, the clinical course of infection in individual patients remains unpredictable. Hence, there is an urgent need for developing a simple tool predicting adverse COVID-19-outcomes. We assumed that the C2HEST-score could predict unfavorable clinical outcomes in the elderly subjects with COVID-19-subjects.We retrospectively analyzed 1047 medical records of patients at age65 years, hospitalized at the medical university center due to COVID-19. Subsequently, patients were divided into three categories depending on their C2HEST-score result.We noticed significant differences in theC2HEST-score analysis on admission to the hospital may predict the mortality, acute kidney injury, and acute heart failure in elderly subjects with COVID-19.

Background: The COVID-GRAM is a clinical risk rating score for predicting the prognosis of hospitalized COVID-19 infected patients. Aim: Our study aimed to evaluate the use of the COVID-GRAM score in patients with COVID-19 based on the data from the COronavirus in the LOwer Silesia (COLOS) registry. Material and methods: The study group (834 patients of Caucasian patients) was retrospectively divided into three arms according to the risk achieved on the COVID-GRAM score calculated at the time of hospital admission (between February 2020 and July 2021): low, medium, and high risk. The Omnibus chi-square test, Fisher test, and Welch ANOVA were used in the statistical analysis. Post-hoc analysis for continuous variables was performed using Tukey’s correction with the Games–Howell test. Additionally, the ROC analysis was performed over time using inverse probability of censorship (IPCW) estimation. The GRAM-COVID score was estimated from the time-dependent area under the curve (AUC). Results: Most patients (65%) had a low risk of complications on the COVID-GRAM scale. There were 113 patients in the high-risk group (13%). In the medium- and high-risk groups, comorbidities occurred statistically significantly more often, e.g., hypertension, diabetes, atrial fibrillation and flutter, heart failure, valvular disease, chronic kidney disease, and obstructive pulmonary disease (COPD), compared to low-risk tier subjects. These individuals were also patients with a higher incidence of neurological and cardiac complications in the past. Low saturation of oxygen values on admission, changes in C-reactive protein, leukocytosis, hyperglycemia, and procalcitonin level were associated with an increased risk of death during hospitalization. The troponin level was an independent mortality factor. A change from low to medium category reduced the overall survival probability by more than 8 times and from low to high by 25 times. The factor with the strongest impact on survival was the absence of other diseases. The medium-risk patient group was more likely to require dialysis during hospitalization. The need for antibiotics was more significant in the high-risk group on the GRAM score. Conclusion: The COVID-GRAM score corresponds well with total mortality. The factor with the strongest impact on survival was the absence of other diseases. The worst prognosis was for patients who were unconscious during admission. Patients with higher COVID-GRAM score were significantly less likely to return to full health during follow-up. There is a continuing need to develop reliable, easy-to-adopt tools for stratifying the course of SARS-CoV-2 infection.

We consider a sequence of independent random variables with the known distribution observed sequentially. The observation $n$ is assumed to be a value of one order statistics such as s:n-th, where 1 is less than s is less than n. It the instances following the $n$th observation it may remain of the s:m or it will be the value of the order statistics r:m (of m> n observations). Changing the rank of the observation, along with expanding a set of observations there is a random phenomenon that is difficult to predict. From practical reasons it is of great interest. Among others, we pose the question of the moment in which the observation appears and whose rank will not change significantly until the end of sampling of a certain size. We also attempt to answer which observation should be kept to have the "good quality observation" as long as possible. This last question was analysed by Ferguson, Hardwick and Tamaki (1991) in the abstract form which they called the problem of duration. This article gives a systematical presentation of the known duration models and a new modifications. We collect results from different papers on the duration of the extremal observation in the no-information (denoted as rank based) case and the full-information case. In the case of non-extremal observation duration models the most appealing are various settings related to the two extremal order statistics. In the no-information case it will be the maximizing duration of owning the relatively best or the second best object. The idea was formulated and the problem was solved by Szajowski and Tamaki (2006). The full-information duration problem with special requirement was presented by Kurushima and Ano (2010).