results of the transmission according to the medical information. Later, the extended models, such as the susceptible-exposed-infectious-recovered (SEIR) model 7 , are proposed. However, these models may give inconsistent predictions because of the hypotheses and reality variety. With neural networks [8] [9] [10] [11] , researchers can generate predictions with the training data 9 , and the artificial intelligence (AI) methodology has deeply changed the modeling methods but avoiding the overfitting problems is another important topic.
In this work, we present three model-free predictive methods and propose an ODE combined predictive neural network, multi-model ODEs neural network (MMODEs-NN), to predict the ending of the transmission. To lay a basis for further discussion, we first get the 2019-nCoV daily transmission data 12 from the Chinese Center for Disease Control and Prevention (CDC) and the CDCs of each province in mainland China from January 10 to February 5, 2020. The data contain the confirmed infected amount, recovered amount and death amount. Then, we also collect the population, density and transportation data of each province during the Chinese SFTR 2 . The three model-free methods, which are based on the sigmoid function 10, 14 , Gaussian function, and Poisson distribution 15 , respectively, only require the daily data. The sigmoid function method requires the total amount of the infected patients, whereas the other two methods require the daily new confirmed data. The MMODEs-NN is a feedforward neural network 16 , which takes all the gathered data as the factors and uses the last 4 days data as the test set. The neurons in this neural network are activated by the proposed susceptible-exposedinfected-recovered-susceptible-death (SEIRSD) ODEs, and each of them stores the variable states. Similar to the recurrent neural networks 17 , the network can calculate the states for each time step recurrently, and the neuron links can simulate the interprovincial disease transmission in neuron wide propagation and population change according to the transportation data. With certain error evaluation and optimization methods 18 , the parameters of the model, such as the disease spreading parameters, can be learned. For more details on the methods and data, please refer to the Method and Supplementary Information sections.
Here we present the prediction results of the 2019-nCoV transmission. Let us denote January 10, 2020, as the 1st day in the training and prediction intervals. When using model-free methods, the results are dependent on the original data. The predictions of these methods are shown in Figs. 1-3, respectively. From these results, one can see that the earliest ending would be the end of February. For the MMODEs-NN, the results are less dependent on the data, so we can train under hypotheses to avoid overfitting and data problems. In general, one has known that the central government of China has taken stronger policies than those in 2003, and we assume that the CDCs in China can count the confirmed patients with no delay. With the help of conjugate gradient method on optimizing the parameters of the disease and transportations, the simulation shows that the ratios between the contact population and the population-area density in other provinces are 8 × 10 !" , while the ratio in Hubei Province is 0.086 (108x). In this situation, the countrywide total confirmed count would be around 40000. The prediction results are depicted in Fig. 4 and imply that the slowdown would begin on February 7, and the ending would exist before March. However, every disease has a certain time for the researchers to identify. From the latest data, one can see that the data of Hubei Province's total confirmed amount has a boom since January 28, which means the possible existence of this delay. Figs. 5 and 6 suggest the delay situation. The ratio in Hubei Province would be 0.103 (129x), and the countrywide confirmed count would be at the scale of 46000 if the statistical delay is 1 day, while the ratio of Hubei Province would be 0.135 (169x) and the countrywide confirmed count would be at the scale of 58000 if a 2-day delay exists. All the prediction results are listed in Table S7 . In these situations, overestimations are acceptable and common in the beginning, and the deceleration date of the transmission would be around February 10, and the ending would happen in March.
What is worth noticing is that the government of Wuhan City declared policies to close the transportation exits of the city 19 on January 23, and later, some other cities in China also announced the similar statements to decelerate the transmission speed. The central government of China also declared other policies to help people and find potential patients under the difficulties, such as prolonging the Spring Festival vacation and conducting strict body temperature measurements in blocks 19 . We consider these policies and code them with the proposed network. According to transportation data in the Supplementary Information, the traffic during the 2020 Spring Festival in China decreases by 90% compared with that in 2019. These policies help to control the disease and shrink the total confirmed amounts in each province. We also conduct another simulation without these policies, and the results plotted in Fig. 7 show that the final infected patient amount would reach 450000. In this situation, the most optimistic deceleration date is February 26, and the ending would be around April 28. Thus, these policies work, and their effects are undoubtedly obvious.
In summary, the above numerical prediction results give the expectations on the ending of the transmission. From the experiments, one can see that the model-free methods and the proposed MMODEs-NN can give consistent and confident predictions on the disease and the transmission of the coronavirus would be under control soon. However, every predictive method has its corresponding advantages and shortcomings. The model-free methods show statistical predictions by learning the general form of the developing data, while the proposed MMODEs-NN generates predictions from the simulation by synthesizing the data independency of the ODEs dynamic system and the fitting power of neural networks. For model-free methods, due to the dependency of the data, they are fast to learn the data pattern but would be more likely to make errors if the real-time data are incorrect or have latencies; for the ODEs-NN combined model, it requires more time to train and generate overfitting problems but it can find the human-like errors in the training data and provide more independent predictions. Both of them have strengths, and the predictions would be more robust and more possible to happen if their results are consistent. Additionally, a variety of uncertain random events happen in real life. For example, the boom in Zhejiang is related to clustering and weak related to the concerned factors in the simulations of MMODEs-NN. However, for the total confirmed amount, our models are relatively accurate, even if the CDC has lowered the standard of judging the confirmed patients since February 9. Thus, we summarize the predicted endings in Table 1 and conservatively estimate that the transmission of the 2019-nCoV would slow down before February 18 and the transmission of this year would finally come to the ending before April. For more detailed data, please refer to the Supplementary Information.
In conclusion, this study has presented comprehensive predictions for the 2019-nCoV transmission ending, providing a guideline for policymaking and a panorama view on the disease future development. Besides the ending, the long treatment cycle, the potential multicycle transmission shown in Fig. 1 , and the potential high death rate shown in Figs. 4-7 are also worth and important for the medical staff to care about. Based on this work, we expect to do more and deeper predictive researches on the geographical spreading problems with modelfree methods and the proposed SEIRSD MMODEs-NN. For it is the first time to apply the comprehensive methods to predict the 2019-nCoV transmission procedure, we anticipate our work to be a starting point for prediction researches of the 2019-nCoV. To reduce the count of the function parameters, we adopt the tanh function to fit the data. One can see from the fitting results that the potential transmission ending of the 2019-nCoV would be February 28, 2020, and the final total confirmed amount would be 46000.
We take the Gaussian function as the target function type. By using new confirmed data, the fitting results can be obtained and indicate that the potential transmission ending of the 2019-nCoV would be March 10, 2020. shows the potential total confirmed amount would be 46000. shows the potential total confirmed amount would be around 58000. 
In this study, we present three model-free methods to predict the 2019-nCoV transmission ending. They are based on the sigmoid function, Gaussian function, and Poisson distribution, respectively. Here we present their technical details.
Sigmoid functions, belonging to a class of S-shaped functions 10, 14 , include the sigmoid function and tanh function. In the epidemiological models, such as SR and SI models, the results are sigmoid-like functions. In this work, we adopt the tanh function to fit the provincial data. The definition of tanh function is
where t denotes the time, the parameters a, b, c, and d refer to the amplification, scale, initial phase, bias, respectively. By using the confirmed data in Table S1 , we can get the fitting results for each province. The countrywide confirmed predictions are shown in Fig. 1 , and the detailed data are depicted in Table S2 .
Gaussian distribution is a common existed distribution in real life. From the aspect of time, it means that the probabilities of a certain-theme event will follow a corresponding Gaussian distribution in continuous time. In this work, we assume the probability of an infected patient going to the hospital follows a Gaussian function, which can be written as
where t denotes the time, a denotes the amplification, b denotes the expectation, and c denotes the standard deviation of the Gaussian distribution. By using the daily newly confirmed data in Table S1 , we can get the ending date for each province, and the prediction results are plotted in Fig. 2 . The detailed data are depicted in Table S3 .
Poisson distribution is frequently used as random events distribution 15 . In this work, we combine the Poisson distribution with Stirling's approximation 20 to get the fitting parameters. The Poisson distribution fitting function is
where t denotes the time, a denotes the amplification, and b denotes the expectation in a unit time step. By using the daily newly confirmed data in Table S1 , we can get the ending date for each province, and the prediction results are plotted in Fig. 3 . The detailed data are depicted in Table S4 .
The SEIRSD denotes the susceptible-exposed-infected-recovered-susceptible-dead loop. A patient may experience a susceptible period and an exposed period and would be confirmed to be infected. The patient would have a chance to recover and a chance to become dead. The recovered patient might have another chance to get in the loop for another time. For the patient, who is in the exposed state, he/she may become recovered at a certain chance. To describe the loop in a more comprehensive way, here we give the ordinary differential equation set of the SEIRSD, that is,
where the capitalized variables mean the amount of the corresponding words, while the parameters , , # , $ , , and denotes the confirmed rate, the transmission rate, the recovery rate of the exposed patients, the recovery rate of the confirmed patients, the death rate, and the recurrence rate, respectively. Like the R0 value for virus transmission 21 , these parameters are related to the disease properties and can be learned with the MMODEs-NN. In general situations, these parameters would not change rapidly, except for the variation. The hypothesis shared by the ODE-oriented model, which is the total count of the system stays unchanged 6 , can be extended 7 by the proposed neural network to be the conservation of the involved people ratio. The collected data are the historical provincial total confirmed amounts, the total recovered amounts, and the death amounts, so another two ordinary differential equations can be added, they are,
where F and C denote the provincial total confirmed amount and recovered amount of the transmission, respectively.
The ODE neural network 16 is a kind of time-series neural network. The neuron activation methods of this kind are ODE or ODEs instead of nonlinear activation functions. The neuron can store the states of each time step, and their cell states can be activated and propagate to the cells in the next layer. In this work, we present a fully connected feedforward SEIRSD ODEsactivated neural network with multi-model techniques. To make a robust simulation, the proposed network is designed to have 31 SEIRSD-activated neurons in each layer which represents the corresponding time step, and the structure is shown in Fig. S1 . The neuron can store some basic information of the corresponding province, such as the population and density. The links between the layers are fully connected to propagate the interprovincial population change during the SFTR. The weights of the links are controlled by the transportation data, which is listed in Table S5 . Besides the SEIRSD model in each neuron, the sub-models that we concerned about are the 2020 SFTR model, the noncontact-contact model, and the statistical where %&' and %&' denote the amount of the moving-out population and exposed patients, respectively, while T and denote the interprovincial transportation ratios and the restriction force, respectively. For the parameters, A and B denote the departure and arrival province, respectively, r denotes a random value, (%&' denotes the exceed rate of the exposed patients, and denotes the shrink rate of the transportation. From the above, one can see that =0.9. The noncontact-contact model means that we assume that there is a part of the population who is hard to get in contact with the potential virus carriers, such as the self-isolated people or the students. The amount of these people can be evaluated with a formula, that is,
where * denotes the virus-contact population, denotes the population density and the parameter * denotes contact ratio. The * will get involved in the SEIRSD ODEs calculation with other variables in the normalized form. As for the statistical delay model, it is used to calculate the delay of patient statistic work. If some confirmed patients do not be found or a period is needed to research the virus, we take the time duration as the statistical delay. In the SEIRSD MMODEs-NN, thanks to the time-flattened structure, we can keep some states of the neurons without dragging the layer wide propagations.
To simplify the calculations, some of the parameters, such as , can be directly obtained from the related statistic works. We assume that Hubei Province might have a statistical delay at the beginning of the 2019-nCoV transmission. Thus, in this model, we have 6 virus-related and 1 sub-model parameters to optimize. The error evaluation methods of the historical provincial total confirmed amount and death amount are the mean absolute errors (MAE). With the help of the conjugate gradient optimizer 18 , the parameters can be learned, and the data trend can be fitted. Figs. S2 and S3 show the visualizations of the provincial transmissions in mainland China on February 28 and March 31. For more data, please refer to Table S6 for the training  details and Table S7 for the predictions.
In the article part, we give a basic view of the SEIRSD model from the ODEs aspect and the corresponding ODEs network. Here we present more detailed illustrations of them. For the SEIRSD model, Fig. S1 (a) displays the structure and the state transition diagram, corresponding to the ODEs of the SEIRSD model that we have shown, which is
For all the variables in the ODEs, we have to change them into the normalized form, and for the parameters,
i.e., , , ! , " , , and , they are assigned within [0, 1] corresponding to the rate of their meanings. To simulate the transportation population, we add the inputs and outputs to the susceptible and exposed population 1 . From the meaning of the state transition diagram, one can see that a person, who is involved with the disease, has a procedure and would be a virus carrier if he/she left the province in the exposed state.
With neuron wide propagation, we can see the final interprovincial transmission results.
For the SEIRSD MMODEs-NN, one can see from the structure depicted in Fig. S1 (b) that the network is similar to the fully connected neural network. To simulate the true environment in the Chinese Spring
Festival travel rush (SFTR), we assume that the interprovincial transportation population during the SFTR is large enough 2 to ignore the geographical distances, and the weights of each neuron link are obtained with the transportation data displayed in Table S5 . What is worth noticing is that each neuron consists of a After the numerical simulations, whose results are shown in the article, we believe our network can predict the patient counts with 10% errors. These errors might be aroused by the random events that cannot be predicted, such as clustering events happened in Zhejiang and Guangdong. Thus, simply judging the deceleration from the descendent of the newly confirmed patient count would be more likely to make errors.
The standard that we judge the starting date of the transmission deceleration is different from the differential methods. We adopt the first day whose new confirmed patient amount is smaller but 1.22 times larger than the amount on the day before, and larger than 1.22 times the amount on the day after. More specifically, the ratio 1.22 is equal to 11/9, which is consistent with the error bounds of the proposed network. By judging from the prediction values, we can make sure the true deceleration dates without making error predictions.
We also visualize the prediction results with the map of China. Figs The short code for the provinces 7 can be obtained from Wikipedia (https://en.wikipedia.org/wiki/Provinces_of_China) or
China Historical GIS. The data are collected from the CCDC 8 and CDCs of each province in mainland China. Table S1 . Historical provincial data of 2019-nCoV confirmed patient amount from January 10 to February 5 (Part 2)   24  25  26  27  28  29  30  31  01  02  03  04  05   AH  15  39  60  70  106  152  200  237  297  340  408  480  530   BJ  36  41  68  80  91  111  121  139  168  191  212  228  253   CQ  27  57  75  110  132  165  206  238  262  275  337  366  389   FJ  10  18  35  59  80  84  101  120  144  159  179  194  205   GD  53  78  111  151  207  277  354  436  535  632  725  813 The short code for the provinces 7 can be obtained from Wikipedia (https://en.wikipedia.org/wiki/Provinces_of_China) or
China Historical GIS. The data are collected from the CCDC 8 and CDCs of each province in mainland China. The data in (row A, column B) mean the ratios between the people departing from A to B and the moving-out people in A. The "E" in each item denotes the base of 10, and the "aEb" means × 10 * . For example, 2E-02 denotes 2 × 10 +, . The transportation data are collected from Baidu. The data in (row A, column B) mean the ratios between the people departing from A to B and the moving-out people in A. The "E" in each item denotes the base of 10, and the "aEb" means × 10 * . For example, 2E-02 denotes 2 × 10 +, . The transportation data are collected from Baidu. 