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In this paper, we build the Brin-Katok's formula of conditional entropy with respect to invariant, decreasing and a large family of increasing measurable partitions by replacing the Bowen metric with the Feldman-Katok metric. [ABSTRACT FROM AUTHOR]

Background: Coronavirus disease 2019 (COVID-19) is a severe infectious illness. It has been reported that COVID-19 has an effect on thyroid function. However, the association between thyroid function and prognosis of COVID-19 is still unclear. Methods: This retrospective study included patients with COVID-19 admitted to Tongji Hospital in Wuhan from January 28 to April 4, 2020. Demographic, epidemiological, clinical, laboratory, treatment, and outcome data were collected from patients with laboratory-confirmed COVID-19. Patients without history of thyroid disease who had a thyroid function test at admission were enrolled in the final analysis. Risk factors of in-hospital death were explored using univariable and multivariable Cox regression analyses. Survival differences were assessed with Kaplan–Meier curves and log-rank test. Results: A total of 127 patients were included in this study, with 116 survivors and 11 non-survivors. The serum levels of thyroid stimulating hormone (TSH) [0.8 (0.5–1.7) vs. 1.9 (1.0–3.1) μIU/mL, P =.031] and free triiodothyronine (FT3) [2.9 (2.8–3.1) vs. 4.2 (3.5–4.7) pmol/L, P <.001] were lower in non-survivors than in survivors, and a low FT3 state (defined as FT3 < 3.1 pmol/L) at admission accounted for a higher proportion in non-survivors than in survivors (72.7% vs. 11.2%, P <.001). Univariate Cox regression analysis showed that FT3 level (HR 0.213, 95% CI: 0.101–0.451, P <.001) and the low FT3 state (HR 14.607, 95% CI: 3.873–55.081, P <.001) were negatively and positively associated with the risk of in-hospital death, respectively. Furthermore, multivariate Cox regression analysis revealed that a low FT3 state was associated with an increased risk of in-hospital death after adjusting for confounding factors (HR 13.288, 95% CI: 1.089–162.110, P =.043). Moreover, Kaplan-Meier curves indicated a lower survival probability in COVID-19 patients with a low FT3 status. Conclusion: Serum FT3 level is lower in non-survivors among moderate-to-critical patients with COVID-19, and the low FT3 state is associated with an increased risk of in-hospital mortality of COVID-19. [ABSTRACT FROM AUTHOR]