Data fitting and inference of the time-dependent parameters by TDINN algorithm for the local outbreaks in Xi’an, Guangzhou, and Yangzhou.

(a)-(b), (e)-(f) and (i)-(j) show the fitting results in Xi’an, Guangzhou and Yangzhou, respectively, where the cyan and purple solid dots represent the daily reported data from communities and quarantined population respectively, green solid curves represent the best fitting results by TDINN, the dashed curves represent the corresponding solution curves after substituting various combinations of the family of functions (4) and (5) into model (1). (c)-(d), (g)-(h) and (k)-(m) show the inference and fitting results of the time-dependent contact rate c(t) and quarantined rate q(t) in Xi’an, Guangzhou and Yangzhou, respectively, where the magenta pentagrams represent the inference results of c(t) and q(t) by TDINN and the solid curves represent the fitting results of c(t) and q(t) based on different functions in (4) and (5).</p

Estimated potential and flow matrices for the selected cities on February 6, 2020.

A: The estimated potential versus growth rate, COVID-19 cases and diffusion distance. B: The correlation between estimated potential and other three factors (i.e., growth rate, COVID-19 cases and diffusion distance) which share the same color used in Fig A. C: The reconstructed net flow matrix, and the relationship between outflow and potential and diffusion distance, respectively. The value in each cell (i, j) of the net flow matrix represents the movement rate from city j to city i, where i is the row label and j is the column label. D: The relative flow matrix where the value in cell (i, j) is logarithmic travel flow from city j to city i under the condition that the travel flow from city i to city j is 1. The cities are ranked by numbers of COVID-19 cases.</p

Predictions of reported daily cases in all cities.

The impact of variations in noise and contact rates on our lockdown exit strategy. The simulations in Wuhan are not shown in the graphics window. (EPS)</p

Mobility network among twenty-nine cities with the most active personnel movements in Mainland China in early 2020.

A: The selected cities are ranked by the value of the migration rate from 1 to 22 January. B: The estimated relative outflow from Wuhan to other cities versus the observed aggregate population outflow. C: The mobility network among these cities. The diameter of nodes indicates the size of the resident population in each city, and the gray scale indicates the relative population flow between two cities. The base layer of the map was created using public sources: https://data.humdata.org/dataset/china-administrative-boundaries.</p

Data used in the case study.

Included in this document are the epidemiological data of each city and the geographic distance between the cities. Demographic data can be found in the S1 Text. (XLSX)</p

Testable predictions of the number of daily cases reported in Guangzhou under our lockdown exit strategy.

A: The baseline movements induce a secondary outbreak of COVID-19 in Guangzhou easily. B: The travel pattern we present does not cause a secondary outbreak. C-D: The simulations based on our travel pattern with positive and negative noise, respectively.</p

Schematic diagram of transmission-dynamics-informed neural network.

Different neural networks are used to represent the state variables (green shaded area) and time-dependent parameters (purple shaded area) of model (1). The symbols “σ” and “” represent the activation function and the automatic differentiation operator, respectively.</p

Parameter definitions and estimation for model (1).

Parameter definitions and estimation for model (1).</p

Hyperparameters for the problems in this study.

During the COVID-19 pandemic, control measures, especially massive contact tracing following prompt quarantine and isolation, play an important role in mitigating the disease spread, and quantifying the dynamic contact rate and quarantine rate and estimate their impacts remain challenging. To precisely quantify the intensity of interventions, we develop the mechanism of physics-informed neural network (PINN) to propose the extended transmission-dynamics-informed neural network (TDINN) algorithm by combining scattered observational data with deep learning and epidemic models. The TDINN algorithm can not only avoid assuming the specific rate functions in advance but also make neural networks follow the rules of epidemic systems in the process of learning. We show that the proposed algorithm can fit the multi-source epidemic data in Xi’an, Guangzhou and Yangzhou cities well, and moreover reconstruct the epidemic development trend in Hainan and Xinjiang with incomplete reported data. We inferred the temporal evolution patterns of contact/quarantine rates, selected the best combination from the family of functions to accurately simulate the contact/quarantine time series learned by TDINN algorithm, and consequently reconstructed the epidemic process. The selected rate functions based on the time series inferred by deep learning have epidemiologically reasonable meanings. In addition, the proposed TDINN algorithm has also been verified by COVID-19 epidemic data with multiple waves in Liaoning province and shows good performance. We find the significant fluctuations in estimated contact/quarantine rates, and a feedback loop between the strengthening/relaxation of intervention strategies and the recurrence of the outbreaks. Moreover, the findings show that there is diversity in the shape of the temporal evolution curves of the inferred contact/quarantine rates in the considered regions, which indicates variation in the intensity of control strategies adopted in various regions.</div

The optimal contact/quarantine rates from the family of functions (4) and (5) for Hainan and Xinjiang.

(a, d) Root mean square error(), corresponding to fitting the time-dependent contact rate learned by TDINN algorithm using c1(t), c2(t) and c3(t) in Hainan and Xinjiang. (b, e) Root mean square error(), corresponding to fitting the time-dependent quarantine rate learned by TDINN algorithm using q1(t), q2(t) and q3(t) in Hainan and Xinjiang. (c, f) Average root mean square error (), corresponding to fitting epidemic data using model (1) based on various combinations of the family of functions (4) and (5) in Hainan and Xinjiang.</p