Model daily change in the epidemic
We started our modeling analysis with data of cumulative number of diagnosed COVID-19 infections per day. Let xi =diagnosed new cases at day i, i =(1, 2, …t), the cumulative number of diagnosed new cases F(x) can be mathematically described as below: 1 \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym} 				\usepackage{amsfonts} 				\usepackage{amssymb} 				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$ F(x)={\int}_{i=1}^t{x}_i=\sum \limits_{i=1}^t{x}_i. $$\end{document}Fx=∫i=1txi=∑i=1txi.
Results of F(x) provide information most useful for resource allocation to support the prevention and treatment; however F(x) is very insensitive to changes in the epidemic. To better monitor the epidemic, the first derivative of F(x) can be used: 2 \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym} 				\usepackage{amsfonts} 				\usepackage{amssymb} 				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$ F^{\prime }(x)={\int}_{i=1}^{\left(t+1\right)}{x}_i-{\int}_{i=1}^t{x}_i=\sum \limits_{i=1}^{t+1}{x}_i-\sum \limits_{i=1}^t{x}_i $$\end{document}F′x=∫i=1t+1xi−∫i=1txi=∑i=1t+1xi−∑i=1txi
Information provided by the first derivative F ′ (x) will be more sensitive than F(x), thus can be used to gauge the epidemic. Practically, F ′ (x) is equivalent to the newly diagnosed cases every day. A further analysis indicates that F ′ (x), although measuring the transmission speed of the epidemic, provides no information about the acceleration of the epidemic, which will be more sensitive than F ′ (x). We thus used the second derivative F″(x): 3 \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym} 				\usepackage{amsfonts} 				\usepackage{amssymb} 				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$ {F}^{{\prime\prime} }(x)={F}^{\prime}\left({x}_{\mathrm{i}+1}\right)-{F}^{\prime}\left({x}_i\right) $$\end{document}F″x=F′xi+1−F′xi
Mathematically, F′′(x) measures the acceleration of the epidemic or changes in new infections each day. Therefore, F′′(x) ≈ 0 is an early indication of neither acceleration nor deceleration of the epidemic; F′′(x) > 0 presents an early indication of acceleration of the epidemic; while F′′(x) < 0 represents an early indication of deceleration.