Abstract
At the moment of writing, the future evolution of the COVID19 epidemic is unclear. Predictions of the further course of the epidemic are decisive to deploy targeted disease control measures. We consider a networkbased model to describe the COVID19 epidemic in the Hubei province. The network is composed of the cities in Hubei and their interactions (e.g., traffic flow). However, the precise interactions between cities is unknown and must be inferred from observing the epidemic. We propose the NetworkInferenceBased Prediction Algorithm (NIPA) to forecast the future prevalence of the COVID19 epidemic in every city. Our results indicate that NIPA is beneficial for an accurate forecast of the epidemic outbreak.
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Introduction
In December 2019, the novel coronavirus SARSCoV2 emerged in the Chinese city Wuhan (Munster et al. 2020). The SARSCoV2 virus causes the COVID19 disease. Contrary to initial observations (Cheng and Shan 2020), the COVID19 virus does spread from person to person, as confirmed in Chan et al. (2020). On March 19, 2020, there were more than 215,000 confirmed infections, and more than 8500 people died (World Health Organization 2020; ‘Situation Update Worldwide, as of 18 March 2020’, www.ecdc.europa.eu/en/geographicaldistribution2019nCoVcases, unpublished; ‘Coronavirus (COVID19)’, www.cdc.gov/coronavirus/2019nCoV/index.html, unpublished). Assessing the further spread of the COVID19 epidemic poses a major public health concern.
Many studies aim to estimate the basic reproduction number R_{0} of the COVID19 epidemic (Zhao et al. 2020; Majumder and Mandl 2020; Li et al. 2020; Yang et al. 2020; Imai et al. 2019; Liu et al. 2020; Riou and Althaus 2020; Read et al. 2020; Wu et al. 2020). The basic reproduction number R_{0} is a crucial quantity to evaluate the hostility of a virus (Hethcote 2000; Heesterbeek 2002). The basic reproduction number R_{0} is defined (Diekmann et al. 1990) as “The expected number of secondary cases produced, in a completely susceptible population, by a typical infective individual during its entire period of infectiousness”. The greater the basic reproduction R_{0}, the more individuals are infected in the longterm endemic state of the virus. If R_{0}<1, then the virus dies out. The estimates for the basic reproduction number R_{0} of the COVID19 epidemic range from R_{0}=2.0 to R_{0}=3.77.
The basic reproduction number R_{0} only coarsely assesses the quantitative behaviour of the epidemic. To obtain a more detailed picture of the epidemic, the development of epidemic outbreak prediction methods is focal. A diverse body of research considers the prediction of general epidemics. For instance, prediction methods are based on Kalman filtering (Yang et al. 2014), Bayesian model averaging (Yamana et al. 2017), basic regression (Brooks et al. 2015) and kernel density estimation (Ray and Reich 2018). Recent work focussed on the dependency of population flow and the viral spread (Colizza et al. 2006; Balcan et al. 2009; Belik et al. 2011; Brockmann and Helbing 2013). As shown by (Pei et al. 2018), the spread of influenza can be more accurately predicted by taking the population flow between cities into account. Read et al. (2020) predicted the COVID19 epidemic by using the Official Aviation Guide (OAG) Traffic Analyser dataset. Additionally to the OAG dataset, (Wu et al. 2020) used the Tencent database to predict the COVID19 viral spread.
The population flow clearly has an impact on the evolution of an epidemic. However, the exact population flow is unknown, and epidemic prediction methods must account for inaccuracies of population flow data. In this work, we consider the most extreme case by assuming no prior knowledge of the population flow. To forecast the COVID19 epidemic, we design the networkbased prediction method NIPA that estimates the interactions between cities as an intermediate step. On February 14th, 2020, approximately 75% of the global COVID19 infections are located in the Chinese province Hubei. Thus, we focus on the COVID19 epidemic in Hubei. More precisely, our goal is to predict the COVID19 outbreak for every city in Hubei.
Materials and methods
Data on the COVID19 epidemic outbreak in Hubei
The time series of reported infections in Hubei forms the basis for the epidemic outbreak prediction. Hubei is divided into 17 cities (more precisely, prefecturelevel divisions) and contains the city Wuhan, as illustrated by Fig. 1. We do not consider the city Shennongjia, since the number of infections in Shennongjia is small. We denote the number of considered cities by N=16. The number of newly reported infections for each city in Hubei is openly accessible via the website of the Hubei Province Health Committee (http://www.hubei.gov.cn/, unpublished). The data is updated daily and follows the standard time offset of UTC+08:00. Except for Wuhan, the total number of reported infections is small before January 21, 2020. Hence, we consider the COVID19 epidemic outbreak starting from January 21. From February 13 on, a new diagnosing method on the basis of chest scans has been used for reporting the infections in Hubei (‘Coronavirus Latest: China’s Epicentre Records No New Cases’, www.nature.com/articles/d4158602000154w, unpublished). The new diagnosing method resulted in an erratic spike in the number of reported infections. We focus on predicting the number of infections of the initial diagnosing method, which is based on genetic tests. The number of reported infections of the initial diagnosing method is accessible from (http://www.hubei.gov.cn/, unpublished) until February 14, 2020. Thus, we focus on the COVID19 epidemic in Hubei from January 21 until February 14, 2020.
We denote the discrete time by \(k\in \mathbb {N}\). The difference of time k to k+1 equals one day, and the initial time k=1 corresponds to January 21, 2020. The website (http://www.hubei.gov.cn/, unpublished) states the number of reported infections N_{rep,i}[k] at every time k in every city i=1,...,N. We obtain the population size p_{i} of each city i from the Hubei Statistical Yearbook (Li and Xu 2016). The reported fraction of infected individuals in city i at time k follows as
Supplementary Table S2 states the population size p_{i} and the complete time series of the number of infections N_{rep,i}[k] for each city in Hubei.
Modelling the COVID19 epidemic between cities
We model the spread of the COVID19 virus by the SIRmodel: At any discrete time k, every individual is in either one of the compartments susceptible (healthy), infectious or removed. Susceptible individuals can get infectious due to contact with infectious individuals. Due to curing, hospitalisation, quarantine measures or death, infectious individuals become removed individuals, which cannot infect susceptible individuals any longer. For every city i, we denote the 3×1viral state vector at time k by
The components \(\mathcal {S}_{i}[k], \mathcal {I}_{i}[k]\), and \(\mathcal {R}_{i}[k]\) denote the fraction of susceptible, infectious, and removed individuals, respectively. Thus, it holds that \(\mathcal {S}_{i}[k]+\mathcal {I}_{i}[k]+\mathcal {R}_{i}[k] = 1\) for every city i at every time k. The discretetime SIR model follows from applying Euler’s method to the continuoustime meanfield SIR model of (Youssef and Scoglio 2011):
Definition 1
(SIR Epidemic Model (Youssef and Scoglio 2011; Prasse and Van Mieghem 2020)) For every city i, the viral state \(v_{i}[k] = (\mathcal {S}_{i}[k], \mathcal {I}_{i}[k], \mathcal {R}_{i}[k])^{T}\) evolves in discrete time k=1,2,... according to
and the fraction of susceptible individuals follows as
Here, β_{ij} denotes the infection probability from city j to city i, and δ_{i} denotes the curing probability of city i.
The SIR model (3) assumes that the spreading parameters δ_{i},β_{ij} do not change over time k. The curing probability δ_{i} quantifies the capacity of individuals in city i to cure from the virus. The infection probability β_{ij} specifies the number of contacts of individuals in city j with individuals in city i. We emphasise that β_{ii}≠0 since individuals within one city i do interact with each other. The contact network between cities in Hubei is given by the N×N matrix
whose elements are probabilities 0≤β_{ij}≤1. Neither the curing probabilities δ_{i} nor the infection probabilities β_{ij} are known for the COVID19 epidemic. Potentially, it is possible to state bounds or estimates for the spreading parameters δ_{i} and β_{ij} by making use of the people flow or geographical distances between the respective cities. Nevertheless, there would remain an uncertainty regarding the precise value of the spreading parameters δ_{i} and β_{ij}. In this work, we consider the most extreme case: there is no a priori knowledge on the curing probabilities δ_{i} nor the infection probabilities β_{ij}.
Networkinferencebased prediction algorithm (NIPA)
We propose the NIPA method to predict the outbreak of COVID19 virus, which consists of three steps. First, we preprocess the raw data of the confirmed number of infected individuals to obtain an SIR time series v_{i}[1],...,v_{i}[n] of the viral state for every city i. Here, the number of observations is denoted by n. Second, based on the time series v_{i}[1],v_{i}[2],..., we obtain estimates \(\hat {\delta }_{i}\) and \(\hat {\beta }_{ij}\) of the unknown spreading parameters δ_{i} and β_{ij}. Third, the estimates \(\hat {\delta }_{i}\) and \(\hat {\beta }_{ij}\) result in an SIR model (3), which we iterate for future times k to predict the evolution of the 2019Cov virus. In the following, we give an outline of the first two steps of the prediction method. We refer the reader to Supplementary Information S1 for further details on NIPA.
Data preprocessing
We denote the number of observations by n, which equals the number of days since January 21, 2020. Based on the reported number of infections N_{rep,i}[k], our goal is to obtain an SIR viral state vector \(v_{i}[k]= (\mathcal {S}_{i}[k], \mathcal {I}_{i}[k], \mathcal {R}_{i}[k])^{T}\) for every city i at any time k=1,...,n. The fraction of susceptible individuals follows as \(\mathcal {S}_{i}[k] = 1  \mathcal {I}_{i}[k]  \mathcal {R}_{i}[k]\) at any time k≥1. Thus, it suffices to determine the fraction of infectious individuals \(\mathcal {I}_{i}[k]\) and recovered individuals \(\mathcal {R}_{i}[k]\).
The fraction of infectious individuals \(\mathcal {I}_{i}[k]\) follows from the reported fraction of infections \(\mathcal {I}_{{rep}, i}[k]\). To be precise, the reported data is the number N_{rep,i}[k] of individuals that are detected to be infected by COVID19. Upon detection of the infection, the respective individuals are hospitalised and, hence, not infectious any more to individuals outside of the hospital. We consider the reported fraction of infections \(\mathcal {I}_{{rep}, i}[k]\) as an approximation for the number of infectious individuals \(\mathcal {I}_{i}[k]\). In fact, the reported fraction of infections \(\mathcal {I}_{{rep}, i}[k]\) lowerbounds the true fraction of infected individuals \(\mathcal {I}_{i}[k]\) for two reasons. First, not all infectious individuals are aware that they are infected. Second, the diagnosing capacities in the hospitals are limited, particularly when the number of infections increases rapidly. Hence, not all infectious individuals that arrive at a hospital can be reported timely.
We do not know the fraction of removed individuals \(\mathcal {R}_{i}[k]\). At the initial time k=1, it is realistic to assume that \(\mathcal {R}_{i}[1]=0\) holds for every city i. At any time k≥2, the removed individuals \(\mathcal {R}_{i}[k]\) could be obtained from (3), if the curing probability δ_{i} were known. However, we do not know the curing probability δ_{i}. Hence, we consider 50 equidistant candidate values for the curing probability δ_{i}, ranging from δ_{min}=0.01 to δ_{max}=1. We define the set of candidate values as Ω={δ_{min},...,δ_{max}}. For every candidate value δ_{i}∈Ω, the fraction of removed individuals \(\mathcal {R}_{i}[k]\) follows from (3) at all times k≥2. Thus, we obtain 50 potential sequences \(\mathcal {R}_{i}[1],...,\mathcal {R}_{i}[n]\), each of which corresponding to one candidate value δ_{i}∈Ω. We estimate the curing probability δ_{i}, and hence implicitly the sequence \(\mathcal {R}_{i}[1],...,\mathcal {R}_{i}[n]\), as the element in Ω that resulted in the best fit of the SIR model (3) to the reported number of infections.
The raw time series \(\mathcal {I}_{{rep},i}[1],..., \mathcal {I}_{{rep},i}[n]\) exhibits erratic fluctuations. There is a single outlier in city i=1 (Wuhan) at time k=8 (January 28, 2020), which we replace by \(\mathcal {I}_{{rep},1}[8]= (\mathcal {I}_{{rep},1}[7]+\mathcal {I}_{{rep},1}[9])/2\). (Potentially, the outlier is due to the increase in the maximum number of individuals that can be diagnosed in Wuhan, from 200 to 2000 individuals per day as of January 27th (https://m.chinanews.com/wap/detail/zw/sh/2020/0128/9071697.shtml, unpublished). To reduce the fluctuations, we apply a moving average, provided by the Matlab command smoothdata, to the time series \(\mathcal {I}_{{rep},i}[1],..., \mathcal {I}_{{rep},i}[n]\) of every city i. The preprocessed time series \(\mathcal {I}_{i}[1],..., \mathcal {I}_{i}[n]\) equals the output of smoothdata.
Network inference
For every city i, the curing probability δ_{i} is estimated as one of the candidate values in the set Ω, as outlined above. The remaining task is to estimate the infection probabilities β_{ij}. The goal of network inference (Peixoto 2019; Ma et al. 2019; Di Lauro et al. 2019; Timme and Casadiego 2014; Wang et al. 2016) is to estimate the matrix B of infection probabilities from the SIR viral state observations v_{i}[1],...,v_{i}[n]. The matrix B can be interpreted as a weighted adjacency matrix. We adapt a network inference approach (Prasse and Van Mieghem 2018; 2020), which is based on formulating a set of linear equations and the least absolute shrinkage and selection operator (LASSO) (Tibshirani 1996; Hastie et al. 2015). We remark that the network inference approach (Prasse and Van Mieghem 2020) is also applicable to general compartmental epidemic models (Sahneh et al. 2013), such as the SusceptibleExposedInfectedRemoved (SEIR) epidemic model. The crucial observation from the SIR governing equations (3) is that β_{ij} appears linearly, whereas the state variables \(\mathcal {S}_{i}, \mathcal {I}_{i}\) and \(\mathcal {R}_{i}\) do not. From (3), the infection probabilities β_{ij} satisfy
for all cities i=1,...,N. Here, the (n−1)×1 vector V_{i} and the (n−1)×N matrix F_{i} are given by
and
If the SIR model (3) were an exact description of the evolution of the coronavirus, then the linear system (4) would hold with equality. However, the viral state vector v_{i}[k] in city i does not exactly follow the SIR model (3). Instead, the evolution of the viral state vector v_{i}[k] is described by
where the 3×1 vector f_{SIR}(v_{1}[k],...,v_{N}[k]) denotes the righthand sides of the SIR model (3), and the 3×1 vector w_{i}[k] denotes the unknown model error of city i at time k. Due to the model errors w_{i}[k], the linear system (4) only holds approximately. Thus, we resort to estimating the infection probabilities β_{ij} by minimising the deviation of the left side and the right side of (4). We infer the network by the LASSO (Tibshirani 1996; Hastie et al. 2015) as follows:
The first term in the objective function of (7) measures the deviation of the left side and the right side of (4). The sum in the objective of (7) is an ℓ_{1}–norm regularisation term which avoids overfitting. We choose to not penalise the probabilities β_{ii}, since we expect the infections among individuals within the same city i to be dominant. The regularisation parameter ρ_{i}>0 is set by cross–validation. The LASSO network inference (7) allows for the incorporation of a priori knowledge of the contact network B by adding further constraints to the infection probabilities β_{ij}. We emphasise that an accurate prediction of an SIR epidemic outbreak does not require an accurate network inference (Prasse and Van Mieghem 2020), see also Supplementary Information S1. If the observed viral state sequence v_{i}[1],..., v_{i}[n] is generated by the SIR model (3), then NIPA accurately predicts the infection state \(\mathcal {I}_{i}[k]\). Furthermore, NIPA provides accurate shortterm predictions, also when the viral state v_{i}[k] does not exactly follow the SIR model (3), i.e., in the presence of model errors w_{i}[k]. We refer the reader to Supplementary Information S1 for further details on NIPA.
Logistic regression
The accuracy of NIPA is evaluated by comparison to a simple prediction method. Qualitatively, the virus spread in many epidemiological models follows a sigmoid function, see also (Van Mieghem 2016). A particular sigmoid function is obtained by logistic regression. As a comparison to NIPA, we apply logistic regression on the reported fractions \(\mathcal {I}_{{rep}, i}[1]\),..., \(\mathcal {I}_{{rep}, i}[n]\) of infection individuals, independently for each city i in Hubei. Logistic regression is advantageous because a logistic function is a closedform expression. Moreover, the logistic function is an approximation to the exact solution of some epidemiological models and population growth models (Verhulst 1838; Van Mieghem 2016; Prasse and Van Mieghem 2019).
A logistic curve is given by the following equation
In our formulation, y(t) is the timedependent fraction of infectious individuals, t is the time in days, where January 21 serves as initial condition (t=0), y_{∞} is the fraction of infected individuals when time approaches infinity, K is the logistic growth rate and t_{0} indicates the inflection point of the logistic equation. For each city in Hubei, we have applied the Matlab command lsqcurvefit to fit the reported cumulative fraction
of infected individuals to Eq (8).
Results and discussion
To evaluate the prediction accuracy, we remove the data for a fixed number of days, say m, prior to February 14. The prediction model is determined by the observation from 21 January up to 14−m February, 2020. Then, we predict the course of the disease up to February 14. The course of the disease is shown in Fig. 2 for the removal of m=1,2,3,4 days. For most predictions shown in Fig. 2, the logistic curve appears to underestimate the true fraction of infected individuals, whereas NIPA seems to overestimate the true value.
We quantify the prediction accuracy by the Mean Absolute Percentage Error (MAPE)
at any prediction time k≥n+1. Here, the predicted cumulative fraction of individuals of city i at time k equals
Figure 3 depicts the MAPE prediction error for the data shown in Fig. 2. Two observations are worth mentioning. First, as expected, the prediction error increases when predicting more days ahead. Second, the prediction accuracy of NIPA is almost always better than the logistic regression. In particular, NIPA provides more accurate shortterm predictions.
Lastly, Fig. 4 illustrates the prediction accuracy versus the time that the epidemic outbreak has been observed. As the epidemic evolves over time, the prediction accuracy of both methods increases. For nearly all forecasts, the NIPA method outperforms logistic regression. Also, as expected, forecasting more days ahead always decreases the prediction accuracy for both prediction methods.
Conclusion
We applied a networkbased SIR epidemic model to predict the outbreak of the COVID19 virus for each city in the Chinese province Hubei. The epidemic model allows to explicitly specify the interactions of individuals of different cities, for instance by using traffic patterns between cities. However, the precise interactions between cities is unknown and must be inferred from observing the evolution of the epidemic.
We proposed the NIPA prediction method, which estimates the interactions between cities as an intermediate step. We did not assume any prior knowledge on the interactions between cities. The prediction method is evaluated on past data of the COVID19 outbreak in Hubei. Our results indicate that a networkbased modelling approach may yield more accurate predictions than modelling the epidemic for each city independently. We believe that the prediction accuracy of NIPA could be further improved, e.g., by using traffic flow patterns as prior knowledge.
Availability of data and materials
All data generated or analysed during this study are included in this published article [and its supplementary information files].
Abbreviations
 COVID19:

Coronavirus disease 2019
 LASSO:

Least absolute shrinkage and selection operator
 MAPE:

Mean absolute percentage error
 NIPA:

Network inferencebased prediction algorithm
 OAG:

Official aviation guide
 SARSCoV2:

severe acute respiratory syndrome coronavirus 2
 SIR:

Susceptible infected removed (epidemic model)
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Acknowledgements
We are grateful to Fenghua Wang for helping with collecting the data.
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LM is supported by the China scholarship council.
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BP and MA developed the mathematical and algorithmic framework. MA carried out the simulations. LM has made substantial contributions to the design of the work and collected the epidemic data. PVM initiated and supervised the research. All authors read and approved the manuscript.
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Additional file 1
Appendix S1 – Details of NIPA. The details and pseudocode of the NetworkInferencebased Prediction Algorithm (NIPA). Furthermore, the prediction accuracy of NIPA is evaluated on the SIR epidemic model.
Additional file 2
Table S2 – Data of the COVID19 epidemic outbreak in Hubei. The time series of the reported number of infections and the population size for every city in Hubei.
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Prasse, B., Achterberg, M.A., Ma, L. et al. Networkinferencebased prediction of the COVID19 epidemic outbreak in the Chinese province Hubei. Appl Netw Sci 5, 35 (2020). https://doi.org/10.1007/s41109020002742
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DOI: https://doi.org/10.1007/s41109020002742