Theoretical analysis, quantitative analysis and simulation are needed for the prediction of various infectious diseases. The above analysis cannot be carried out without models established for various infectious diseases.
Infectious disease transmission is a complicated diffusion process occurring in the crowd. Models can be established for this process to analyze and study the transmission process of infectious diseases theoretically 4 , so that we can accurately predict the future development trend of infectious diseases 5 .
Therefore, in order to control or reduce the harm of infectious diseases, the research and analysis of infectious disease prediction models have become a hot research topic 6 .
Traditional infectious disease prediction models mainly include differential equation prediction models and time series prediction models based on statistics and random processes.
The differential equation prediction models are to establish a differential equation that can reflect the dynamic characteristics of infectious diseases according to the characteristics of population growth, the occurrence of diseases and the laws of transmission within the population. Through qualitative and quantitative analysis and numerical simulation of the model dynamics, the occurrence process of diseases is displayed, the transmission laws are revealed, the change and development trends are predicted, the causes and key factors of disease transmission are analyzed, the optimal strategies for prevention and control are sought, and the theoretical basis and quantitative basis are provided for people to make prevention and control decisions. Common models for predicting infectious disease dynamics differential equations have ordinary differential systems, which directly reflect the relationship between the instantaneous rate of change of individuals in each compartment and the corresponding time of all compartments. Partial differential system is a common model system when considering age structure.
Delay differential system is a kind of differential system that appears when the structure of the stage is considered (e.g. the infected person has a definite infectious period, the latent person has a definite incubation period, the immunized person has a definite immune period, etc. The currently widely studied and applied models include SI model, SIS model, SIR model and SEIR model, etc 7 . System individuals are divided into different categories, and each category is in a state, respectively: S (Susceptible), E(Exposed), I (Infected) and R (Remove).
The classical differential equation prediction model assumes that the total number of people in a certain area is a constant, which can prompt the natural transmission process of infectious diseases, describe the evolution relationship of different types of nodes with time, and reveal the overall information transmission law. However, in practice, the population is changing over time. There will always be some form of interaction with other populations in terms of food, resources and living space. The connection between individuals is random, and the difference between spreading individuals is ignored, thus limiting the application scope of the model. The infectious disease prediction model established by this method relies on curve fitting and parameter estimation of available time series data, so it is difficult to apply it to a large number of irregular data.
Infectious disease surveillance research based on the Internet has begun to rise since the mid-1990s 9 . It can provide information services for public health management institutions, medical workers and the public. After analyzing and processing, it can provide users with early warning and situational awareness information of infectious diseases 10 .
In the early research, traditional Web page web information (for example, related news topics, authoritative organizations, etc.) was the main data source. However, with the development of the Internet, research has begun to expand data sources to social media (such as Twitter, Facebook, microblog, etc.) and multimedia information in recent years. Due to the global spread of the Internet, people use Internet search engines, social networks and online map tools to track the frequency and location information of query keywords, strengthen the integration of information on social, public focus and hot issues, realize disease monitoring based on search engines and social media, and predict the incidence of infectious diseases, which can provide important reference for the decision and management of infectious disease prevention and control 11 .
In theory, Internet search tracking is efficient, and can reflect the real-time status of infectious diseases. Therefore, the infectious disease prediction models based on Internet and search engine are good supplement to the traditional infectious disease prediction models 12 Compared with the traditional prediction models, the Internet-based infectious disease prediction models have the advantages of real-time and fast, which can predict the incidence trend of infectious diseases as early as possible, and are suitable for data analysis of a large number of people. However, the sensitivity, spatial resolution and accuracy of its prediction need to be further improved. So Internet-based infectious disease prediction models cannot replace the traditional prediction models, and they can just be used as an extension of the traditional infectious disease prediction model 16 .
This paper will use 2003 SARS data to verify three mathematical models (Logistic model, Bertalanffy model and Gompertz model) to predict the development trend of the virus, and then use these three models to fit and analyze the epidemic trend of COVID-19 in Wuhan, mainland China and non-Hubei areas, including the total number of confirmed cases, the number of deaths and the end time of the epidemic.
In short, machine learning is to learn more useful information from a large amount of data using its own algorithm model for specific problems. Machine learning spans a variety of fields, such as medicine, computer science, statistics, engineering technology, psychology, etc 17 . For example, neural network, a relatively mature machine learning algorithm, can simulate any high-dimensional non-linear optimal mapping between input and output by imitating the processing function of the biological brain's nervous system. When faced with complex data relations, the traditional statistical method is not such effective, which may not receive accurate results as the neural network 18 .
Since most new infectious diseases occurring in human beings are of animal origin (animal infectious diseases), it is an effective prerequisite to predict diseases by determining the common intrinsic characteristics of species and environmental conditions that lead to the overflow of new infections. By analyzing the intrinsic characteristics of wild species through machine learning, new reservoirs (mammals) and carriers (insects) of zoonotic diseases can be accurately predicted 19 . The overall goal of machine learning-based approach is to extend causal inference theory and machine learning to identify and quantify the most important factors that cause zoonotic disease outbreaks, and to generate visual tools to illustrate the complex causal relationships of animal infectious diseases and their correlation with zoonotic diseases 20 . However, the highly nonlinear and complex problems to be analyzed in the early prediction model of infectious diseases based on machine learning usually lead to local minima and global minima, leading to some limitations of the machine learning model. Infectious disease prediction models mainly include differential equation prediction models based on dynamics and time series prediction models based on statistics and random processes, Internet-based infectious disease prediction model and machine learning methods. Some models are too complicated and too many factors are considered, which often leads to over-fitting. In this paper, Logistic model, Bertalanffy model and Gompertz model, which are relatively simple but accord with the statistical law of epidemiology, are selected to predict the epidemic situation of COVID-19. After the model is selected, the least square method is used for curve fitting. Least square method is a mathematical optimization technique. It finds the best function match of data by minimizing the sum of squareed errors. Using the least square method, unknown data can be easily obtained, and the sum of squares of errors between these obtained data and actual data is minimized.
(1) Logistic model Logistic model is mainly used in epidemiology. It is commonly to explore the risk factors of a certain disease, and predict the probability of occurrence of a certain disease according to the risk factors. We can roughly predict the development and transmission law of epidemiology through logistic regression analysis,.
Qt is the cumulative confirmed cases (deaths); a is the predicted maximum of confirmed cases (deaths).
b and c are fitting coefficients. t is the number of days since the first case. t0 is the time when the first case occurred.
(2) Bertalanffy model Bertalanffy model is often used as a growth model. It is mainly used to study the factors that control and affect the growth. It is used to describe the growth characteristics of fish. Other species can also be used to describe the growth of animals, such as pigs, horses, cattle, sheep, etc. and other infectious diseases.
The development of infectious diseases is similar to the growth of individuals and populations. In this paper, Bertalanffy model is selected to describe the spread law of infectious diseases and to study the factors that control and affect the spread of COVID-19.
Qt is the cumulative confirmed cases (deaths); a is the predicted maximum of confirmed cases (deaths).
b and c are fitting coefficients. t is the number of days since the first case. t0 is the time when the first case occurred.
(3) Gompertz model
The model was originally proposed by Gomperts (Gompertz,1825) as an animal population growth model to describe the extinction law of the population. The development of infectious diseases is similar to the growth of individuals and populations. In this paper, Gompertz model is selected to describe the spread law of infectious diseases and to study the factors that control and affect the spread of COVID-19.
Qt is the cumulative confirmed cases (deaths); a is the predicted maximum of confirmed cases (deaths).
b and c are fitting coefficients. t is the number of days since the first case. t0 is the time when the first case occurred.
The regression coefficient (R 2 ) is used to evaluate the fitting ability of various methods and can be obtained by the following equation.
is the actual cumulative confirmed COVID-19 cases; � is the predicted cumulative confirmed COVID-19 cases; � is the average of the actual cumulative confirmed COVID-19 cases. The closer the fitting coefficient is to 1, the more accurate the prediction.
As COVID-19 and SARS virus are both coronaviruses, the infection pattern may be similar. Firstly, we used SARS data to verify the rationality of our model.
The 
The cumulative number of confirmed cases of the novel coronavirus (hereinafter referred to as COVID- Figure 3 . The number of confirmed cases has dramatically increased in China since the first case was confirmed. The epidemic spreads to other parts of Hubei and the whole country with Wuhan as the center. Since the main confirmed cases were in Wuhan, the development trend of new confirmed cases in the whole country are basically the same with Wuhan. Judging from the prediction results, the three models can predict the epidemic situation of COVID-19 well in the later stage of the epidemic.
Among them, Logistic model is better than the other two models in fitting all the data in Wuhan, while Gompertz model is better in fitting the data outside Wuhan.
Due to various reasons, It is worth noting that the number of confirmed cases suddenly increased by 13,332 on February 12, 2020.Obviously, the mutation of this data does not originate from the mechanism of the virus, so our treatment method is to remove the impact of this part of data mutation (13, 332 people).Then, the impact of the sudden increase confirmed cases will be considered in the later fitting analysis.
According to the daily real-time updated data of COVID-19, we used the above three mathematical models (Logistic model, Bertalanffy model and Gompertz model) to carry out fitting analysis on the epidemic of COVID-19. The prediction results are shown in Table 1 in which a is the prediction of cumulative confirmed number (the final predicted cumulative confirmed number = a+13332); b and c are fitting coefficients; t is the number of days since the first case. R 2 (C) means the fitting goodness of cumulative confirmed cases, R 2 (N) means the fitting goodness of new confirmed cases (c) Three models for predicting COVID-19 cases in non-Hubei areas since January 15,2020
In order to predict the turning point, we use the above three models to compare the new confirmed cases in Wuhan, China mainland and non-Hubei areas. As can be seen in Figure 4 , the turning points in W Wuhan, China mainland and non-Hubei areas are February 9, February 6 and February 2, 2020 respectively. From the results, for the prediction of newly confirmed cases, the three models can predict the COVID-19 epidemic well in the early and late stages of the epidemic. Among them, the Logistic model is better than the other two models in fitting all the data. 
The fitting parameters of each model can be seen in Table 2 in which a is the prediction of the death toll;
b and c are fitting coefficients; t is the number of days since the first case. R 2 (DC) means the fitting goodness of cumulative deaths.
According to the available data, the death toll predicted by the three models is Wuhan: 2502-5108; non- 
The stronger, the support for various resources across the country is getting stronger, and the ability to refine management and treatment is getting stronger. These factors are likely to rapidly reduce the mortality rate of COVID-19. The above factors may also have some influence on the cumulative number of confirmed cases, but due to the large number of confirmed cases, the influence of these favorable human factors on the cumulative number of confirmed cases may be small.
At present, there are only a few papers on the prediction of COVID-19 epidemic. We have collected some COVID-19 epidemic predictions of other researchers, as shown in Table 3 . It can be seen from Table 3 