A Spatial Stochastic SIR Model for Transmission Networks with Application to COVID-19 Epidemic in China
aa r X i v : . [ q - b i o . P E ] A ug A Spatial Stochastic SIR Model for Transmission Networkswith Application to COVID-19 Epidemic in China ∗ Tatsushi Oka † Wei Wei ‡ Dan Zhu § August 18, 2020
Abstract
Governments around the world have implemented preventive measures against the spreadof the coronavirus disease (COVID-19). In this study, we consider a multivariate discrete-timeMarkov model to analyze the propagation of COVID-19 across 33 provincial regions in China.This approach enables us to evaluate the effect of mobility restriction policies on the spreadof the disease. We use data on daily human mobility across regions and apply the Bayesianframework to estimate the proposed model. The results show that the spread of the disease inChina was predominately driven by community transmission within regions and the lockdownpolicy introduced by local governments curbed the spread of the pandemic. Further, wedocument that Hubei was only the epicenter of the early epidemic stage. Secondary epicenters,such as Beijing and Guangdong, had already become established by late January 2020, and thedisease spread out to connected regions. The transmission from these epicenters substantiallydeclined following the introduction of human mobility restrictions across regions.
Keywords : COVID-19, Infection, Heterogeneity, Spatial Model, Bayesian AnalysisJEL code: C11, C33, C54 ∗ We would like to acknowledge the financial support from the Centre for Development Economics and Sustain-ability (CDES) at Monash University. Yunyun Wang provided superb research assistance. † Department of Econometrics and Business Statistics, Monash University ( [email protected] ). ‡ Department of Econometrics and Business Statistics, Monash University ( [email protected] ). § Department of Econometrics and Business Statistics, Monash University ( [email protected] ). Introduction
The ongoing pandemic of coronavirus disease (COVID-19) poses a threat to public health andhas disrupted economic activities globally. Although there are limited policy tools available tostem the disease spread, restricting human mobility though lockdown or border closure policieswas identified as an effective measure. Simply put, the virus itself cannot move anywhere withoutassistance. In many countries, mobility restriction led to the containment of the virus’s spread.Given the importance of mobility restriction as an effective policy, it is critical to quantify itseffects.In this study, we consider a multivariate discrete-time Markov model to analyze the propagationof COVID-19 across 33 provincial regions of China. Thereby, we allow for heterogeneous diseasetransmission both within and across regions. Our model takes into account human mobility as akey driver of disease transmission across regions and identifies epicenters of disease propagation,as well as the effect of mobility restrictions on infection rates. We extract information on dailyhuman mobility across regions from January 11 to March 15, 2020, from the Baidu-Qianxi (2020)database and apply the Bayesian framework to estimate the model. The sampling period in usefor our analysis exhibits substantial exogenous variations in human mobility rates due to the highnumber of movements around Chinese New Year (January 25) and a sudden decline in movementsafter policy interventions were introduced. We evaluate the effect of mobility restrictions on thedisease spread between regions by comparing outcomes under actual and counterfactual humanmobility, which is extracted from the 2019 data.Our empirical results document substantial heterogeneity in the rate of infection across regions.The results also demonstrate the effectiveness of the lockdown policy in curbing the spread of thepandemic. The transmission mechanism of the disease in China is found to be predominatelycommunity transmission within all regions. Further, our analysis based on the 2019 mobility datasuggests that the external transmission would not have been suppressed if people had continuedto be allowed to move freely across provincial borders as usual. Interestingly, our results showthat Hubei is not the only epicenter of the early epidemic stage. Other epicenters, such as Beijingand Guangdong, had already become established by late January 2020. The pandemic radiatedout to the subordinate regions of these cities with varying degrees of severity. Our approach shedslight on the evolution of the transmission network over time and provides useful insight into theformulation of lockdown policies amid the pandemic.The methodological part of the paper draws on and contributes to several literatures. First,since the outbreak of COVID-19, many studies have provided simulations and predictions using adeterministic susceptible-infective-recovered (SIR) model (Kermack and McKendrick, 1927). TheSIR model divides a well-defined population into three compartments, namely susceptible, infec- China consists of 27 provinces, four special administrative cities in mainland, and two special administrativeregions (Hong Kong and Macao). Through the paper, we use “region” for the provinces and special administrativeregions. Kraemer et al.(2020) use human mobility information from Baidu-Qianxi and analyze the disease spread fromWuhan to other regions between January 1 and February 10, 2020. They predict daily case countsin the early phase of disease spread using three different models: Poisson, negative binomial, andlog-linear regression. Both Wu et al. (2020) and Kraemer et al. (2020) document the significanceof human mobility from Wuhan in causing the spread of the disease in the early phase. Bothauthors also underscore that the effect travel restrictions in Hubei had on containing the spreadof the disease. In our study, we estimate a model that accounts for disease transmission across allregions, using data spanning from the beginning of the epidemic until the end of the first wave inChina. Our result is consistent with the existing findings, in that we show that Hubei is the earliestepicenter. However, we also show that additional regions became epicenters in an early phase ofthe pandemic in China. Thus, our results suggest that local government interventions, such aslockdown in Wuhan, cannot fully explain the containment of the disease. Mobility restrictionacross regions is essential. Our research complements the existing research by providing a more They combine three data sets: 1) the monthly number of domestic and international flight bookings fromWuhan in January to February 2019, 2) the number of daily domestic passengers by train and car, and 3) travelvolumes forecast from and to Wuhan by Wuhan Municipal Transportation Management Bureau.
This section first introduces the variation of the susceptible-infective-recovered (SIR) model ap-plied in this study. Subsequently, it explains the specification of internal and external diseasetransmission in the model.
We apply a variation of stochastic SIR model to describe the evolution of three variables: S jt , I jt and R jt , which denote the number of susceptible, infective, and recovered individuals in region j at time t , respectively. Also, let D jt denote the cumulative number of deaths by t and let N j bethe total population in region j . Then, we have the following identity: N j = S jt + I jt + R jt + D jt . We observe regional panel data of ( I jt , R jt , D jt , N j ) for region j = 1 , . . . , J and time t = 0 , . . . , T with J and T denoting the sample size of regions and time periods, respectively. In what follows,we use F t to denote the available information set at time t .We denote by ∆ Ij,t +1 the number of transitions from susceptible to infected states in region j at time t + 1 . The number of newly infected individuals ∆ Ij,t +1 is assumed to be a random variablefollowing the Poisson distribution conditional on F t , with the conditional mean given by E [∆ Ij,t +1 |F t ] = (cid:16) β jt I jt N j + λ jt (cid:17) S jt . (1)At its core, the equation above follows the Bass model (Bass, 1969), which was originally proposedfor describing the diffusion of new products. The key feature of the Bass model is that the accep-4ance of a new product is driven by either internal influences, such as contagious adopters to whichother individuals are connected, or external influences, such as mass media or commercials. Thedistinction between internal and external influences is adopted by Fibich (2016) in a deterministicSIR model. Similarly, we can interpret the term β jt I jt /N as region j ’s internal infection rate,which depends on the proportion of infected individuals I jt /N and the internal transmission rate β jt . Further, we consider the term λ jt as the external infection rate, which reflects the rate ofinfection attributable to transmission from outside of region j . If the border to region j is closed,the external effect λ jt equals zero and the model becomes the standard stochastic SIR model (e.g.,Allen, 2008).To describe the state transition from the infected state, we use a Markov chain model in whichinfected individuals either remain infected or move to another state: recovery or death. Morespecifically, let ∆ Rj,t +1 := R j,t +1 − R jt and ∆ Dj,t +1 := D j,t +1 − D jt be changes in the number ofrecoveries and deaths, respectively. We assume that the transition probability from the infectedstate at time t follows a multinomial distribution conditional on F t , satisfying that E [∆ Rj,t +1 |F t ] = γI j,t and E [∆ Dj,t +1 |F t ] = δI j,t . Here, the parameters γ and δ are used to represent the recovery anddeath rates, respectively. Given the stochastic transition among all states, the number of infectedand susceptible individuals at time t + 1 are given by the following state equations: I j,t +1 = I jt + ∆ Ij,t +1 − ∆ Rj,t +1 − ∆ Dj,t +1 , (2) S j,t +1 = S jt − ∆ Ij,t +1 . (3) The internal transmission rate β jt measures to what extent contacts between an infected individualand the susceptible population at time t leads to the transmission of the pathogen. Thus, it canbe interpreted as the number of “effective” contacts. We allow for β jt to vary per region and acrosstime. This is because the contact frequency depends on region-specific characteristics, such aspopulation density, as well as time-varying factors, such as policy intervention (e.g. contact tracingand forced quarantine) and behavior changes (e.g. better hygiene practices and social distancing).In China, almost all local governments declared the top-level state of emergency in the early phaseof the pandemic (January 23-25, 2020), which effectively induced changes in individuals’ behavior.Thus, we assume that intervention by local governments affects internal transmission gradually.Specifically, we consider the following specification: log β jt = log β j,t − + α j X j,t − h , (4)where X j,t − h is an observed dummy variable taking the value of 1 if the local government in region j has activated the top-level health emergency response at time t − h and 0 otherwise. This X j,t − h lockdownpolicy . We consider a lag h > to account for lagged effects of the policy intervention and set fourdays ( h = 4 ) for our estimation. The parameter α j is allowed to be heterogeneous across regions,reflecting different measures taken by local governments and regional characteristics. The time-varying parameter β jt in (4) depends on the initial value β j, and the response to the intervention α j . We specify a hierarchical structure for the transmission parameters across regions, by usinga bivariate normal distribution: (log β j, , α j ) ′ ∼ N ( µ, Σ) with mean µ := ( µ β , µ α ) ′ and variancematrix Σ . Under this specification, the average of the internal transmission rate without anycontrol is given by E [ β j, ] = exp( µ β + 1 / ) with Σ denoting the (1,1)-element of Σ , while theeffect of intervention on average is given by E [ α j ] = µ α . Using Baidu’s daily mobility data, we construct a measure of the “intensity” of the disease trans-mission between regions. The mobility data includes an outflux mobility index for all regions anddetails the proportion of travelers between regions. We use M outkt to denote the outflux mobilityindex in region k at time t and we use P kjt to represent the proportion of travelers from region k to region j at time t . The mobility index M outkt represents a relative strength measure of theoutflux, which is scaled by Baidu’s proprietary method, rather than the numbers of outflux. Thisindex is comparable across regions and time. The change of M outkt from its standard level reflects mobility restrictions , which we discuss in the supplementary material. Additionally, we observe theproportion of daily travelers P kjt between the 31 mainland regions in the sample of 33 provincialregions, which means Hong Kong and Macao are excluded. We impute the entries for Hong Kongand Macao based on the radiation model (Simini et al., 2012). In our supplementary material,we demonstrate that the prediction of influx based on the imputed P kjt value traces the index ofhuman influx well. It also outperforms the prediction using only the radiation model.We use M outkt P kjt to measure the (scaled) flux from origin k to destination j and then constructan “intensity” of infected flux from origin k to destination j at time t by M outk,t − h P kj,t − h ( I kt /N k ) witha lag h > in the mobility measure. As there is a time lag between getting infected and showingsymptoms, our formulation takes into account that travelers from origin k at time t − h face casecounts I kt , which are recorded at t . Given the “intensity” of daily infected flux, we consider the In the existing literature the mean incubation period of COVID-19 is estimated as roughly 5 days (see Li et al.,2020; Kraemer et al., 2020, among others). Our specification allows β jt to approach zero in consideration of the draconian measures adopted in China andthe suppression of the disease in the first wave. Alternatively, β jt could be set to approach a non-zero value as inFernández-Villaverde and Jones (2020). Their dynamics can be considered as a special case of the transfer functionmodel in Box and Tiao (1975) for intervention analysis. j at time t as follows: λ jt = θ t N j X k = j M outk,t − h P kj,t − h I kt N k , (5)The time-varying parameter θ t reflects the strength of external transmission and also normalizesthe unit because the index M outkt is a scaled measure. As in the specification for β jt , we allow θ t torespond to policy intervention gradually, i.e., log θ t = log θ t − + ρX j,t − h , where ρ is a parameter. We use the daily data on COVID-19 infection and individuals’ mobility from January 11 to March15, 2020. The daily data of the infection, death, and recovery cases for each region are obtainedfrom the National Health Commission of China and its affiliates. The human mobility data isobtained from Baidu Migration (Baidu-Qianxi, 2020). The data provides a daily outflux index foreach of the 33 regions as well as the destinations of the outflux. For our counterfactual analysis, weuse the mobility data set of 2019 from Baidu-Qianxi matched according to the Chinese New Year.The plots of the outflux in both 2020 and 2019 are shown in Figure 1. The outflux indices beforethe Chinese New Year in both 2019 and 2020 are dominated by provinces such as Guangdong,Zhejiang, and Beijing. It is expected that most workers would be leaving these areas to return fortheir home provinces for the holiday. For Hubei, the outflux was moderate in both years. Theoutflux reduced to a negligible level at the time when the lockdown policy prevailed.
We adopt a Bayesian framework for estimation. Given the information on infection, recovery anddeath cases, we can estimate our model separately for the infection and the recovery and death.In our model, the number of recovered and death cases follows a multinomial distribution. Thus,the likelihood of the parameters of recovery rate γ and death rate δ has an analytic form. We usea standard random-walk Metropolis sampler with uninformative prior.For the new case counts following the Poisson distribution, we simulate the posterior distri-bution using the algorithm in Chib et al. (1998), which is based on data augmentation and aMetropolis-Hastings-within-Gibbs sampler. We divide the set of parameters into J + 2 blocks: { (log β j, , α j ) } Jj =1 , ( µ, Σ) , and (log θ , ρ ) , and then sample sequentially using their conditional pos-teriors. For each block of { (log β j, , α j ) } Jj =1 , we use a multivariate- t proposal density whose meanand covariance are computed from the mode and Hessian of the conditional posterior. For ( µ, Σ) , igure 1: Daily Outflux in 2020 and 2019
Daily outflux in 2020
CNY
BeijingHunanGuangdongHong Kong
Daily outflux in 2020: Hubei
CNY
Daily outflux in 2019
CNY
BeijingHunanGuangdongHong Kong
Daily outflux in 2019: Hubei
CNY
Notes:
Two panels on the left column show daily outflux from all regions in 2020 and 2019. The ones on the rightcolumn show the outflux only in Hubei. In each panel, a dashed vertical line shows the date of the Chinese NewYear (CNY) in 2019 and 2020. we specify a Gaussian-inverse Wishart prior,
N IW ( µ ∗ , κ ∗ , Λ ∗ , ν ∗ ) , with µ ∗ = ( − , − . ′ , κ ∗ = 1 , Λ ∗ = diag (1 , . , and ν ∗ = 10 . This prior is weakly informative in µ and moderately informativein the variance matrix Σ . Lastly, the block (log θ , ρ ) is updated using a Gaussian prior N ( π, Ω) with π = (0 . , − . ′ and Ω = diag (0 . , . . As the posterior of (log θ , ρ ) depends on all J × T observations, the contribution of the prior is minimal. This section first presents the estimation result for the heterogeneous internal infection rate and theeffect of the regional intervention. We then compare the results of internal and external infectionand provide additional findings based on a transmission network between regions.
In Figure 2, we present the estimation result of internal transmission rates. Panel (a) of Figure 2shows the posterior means of the initial transmission parameter, β j, . The significant heterogeneityin the initial infection rate is evident here. Hubei has the highest value with a very tight posterior8redible interval. Panel (b) of Figure 2 reports the transition of posterior means of the internaltransmission rate β j,t , which depicts the effects of policy intervention. The top-level health emer-gency response was activated for January 23-25, 2020, in all regions, except Xizang, which wentinto the state of emergency on January 30, 2020. As in equation (4), the number of new infectionsis shown to be affected by the policy implemented five days before. In Figure 2, the effect of theintervention is evident but not immediate; it shows that for most regions, it took 4 to 7 days for β jt to decrease to half of its original value. The posterior mean of recovery rate γ is 4.15% with a95% credible interval (4.11%,4.18%). The posterior mean of death rate δ and 0.213%, with a 95%credible interval (0.206%,0.220%). Figure 2:
Internal Transmission Rate (a)
Initial Transmission Rate ( β j, ) (b) Internal Transmission Rate ( β j,t ) BeijingHubeiGuangdongHong Kong
Notes:
Panel (a) shows that the posterior mean of the basic reproduction number for each region with the linesegment representing the 95% posterior credible interval. Panel (b) reports the posterior mean of the effectivereproduction number ( R t ) across regions over time. In combination with the lockdown policy, it is important to specifically study human mobilityin the context of the COVID-19 pandemic. Our analysis decomposes the expected number of9nfections into infections resulting from internal and external transmission for all regions. Figure3 presents results for four regions, each of which represents a different region, but all have sim-ilar characteristics. Namely, we consider megacities (Beijing), the neighboring regions of Hubei(Hunan), the secondary epicenters (Guangdong), and the special administrative regions outsidemainland China (Hong Kong).
Figure 3:
The Number of Infected Individuals: External vs Internal Transmission
Beijing
ExternalInternal
Hunan
Guangdong
Hong Kong
Notes:
The blue area represents the expected number of infections due to the external infection and the pink arearepresents the expected number of infections due to internal infection.
Internal transmission in Beijing and Guangdong follows a similar pattern with an exponentialincrease from the beginning of the outbreak, which dominates the external transmission influence.In these regions, there is an initial peak during the Chinese New Year (January 24 - February2, 2020). This finding is empirical evidence that the pandemic had already expanded outsideof Wuhan as early as late January 2020. By this time, other major cities can be considered tohave been suffering from localized outbreaks already. On the other hand, the dominating form oftransmission in Hunan is external until January 27. Similarly, in Hong Kong, external transmissiondominates as the source of infection until February 5. Both internal and external transmissionsubsequently exhibit an exponential decrease due to unprecedented policy interventions, such asstay-at-home instructions and extended public holidays. The only exception to this observation isthe evidence of internal transmission in Hong Kong, which still increased substantially followingFebruary 5 until it stabilized on February 15. 10igure 4 presents the expected number of infections from external transmission in all regions.It is not surprising to see relatively high external infections in Hunan, an area that shares a borderwith Hubei. External transmissions peaked during the holiday periods across most regions andreduced to a negligible level by February 15. Guangdong shows the highest external infection rate,which also peaked later than most regions. This can be attributed to the large numbers of migrantworkers returning from their hometown following the Chinese New Year.
Figure 4:
The Number of Infected Individuals due to External Transmission
BeijingHunanGuangdongHong Kong
Notes:
The expected number of infections due to external transmission is reported for all regions, and lines for thefour representative cities are highlighted.
To shed further light on these observations, we also conduct an analysis that applies 2019mobility data. Specifically, the lockdown effect is maintained but the 2020 mobility data is replacedwith the 2019 data to isolate interactive fixed effects of mobility across regions and time, followingBai (2009). The results are reported in Figure 5. The first conclusion which can be drawn fromthis analysis is that the lockdown policy is effective. This is clear because the exponential decay ofthe external transmission is still evident across all 33 regions. However, if mobility is maintained atits usual level as indicated by the interactive fixed effects, the exponential decay would be delayed,resulting in much more significant external transmission rates across regions, except for HongKong. Hong Kong is an exceptional case because most of its external transmission originated fromGuangdong (its only neighbor). Outflux from Guangdong in 2020 was significantly higher than itsusual level in January. This led to a higher influx in Hong Kong during this time. Consequently,this abnormal influx in 2020 resulted in a high number of external transmissions in Hong Kong atthe early stage of the outbreak. For the counterfactual analysis, we first match the periods of 2019 and 2020 data according to the Chinese NewYear and then extract interactive fixed effects from the 2019 data. This approach decomposes daily human mobilityacross regions into daily and regional factors in a flexible yet parsimonious manner. More details are provided inthe supplementary material. igure 5: External Transmission under Actual and Counterfactual Mobility
Beijing
Counterfactual2020
Hunan
Guangdong
Hong Kong
Notes:
Panels (a)-(d) show the estimated number of infections due to the external transmission in four regions.The dashed (red) line shows the value based on mobility information in 2020, while the solid (blue) line shows thevalue based on interactive fixed effects.
The transmission network between the regions of China is observed to evolve on each day of thepandemic. Kraemer et al. (2020) focuses on the transmission from Wuhan to the rest of China andthey conclude that the propagation of COVID-19 in China during the early stage of the outbreakwas mostly explained by human mobility originated from Wuhan. However, the authors did notconsider the mobility network among the rest of China’s geography, and thus, the scope of analysisof the transmission channels is limited. The main advantage of the model developed in this study isthat it enables the transmission network to be analyzed on a more granular level. This means thatthe sources of external transmission and their respective intensities can be identified. Specifically,based on (5), we can obtain the rate of external transmission from region k to region j , A jkt := θ t N t M outk,t − h P kj,t − h I kt N k S jt . Following the literature on network theory (e.g., Aldous and Wilson, 2003), we can interpret thesquare matrix consisting of A jkt for j, k = 1 , . . . , as an adjacency matrix of a directed graphwith weighted directions A jkt from k to j at time t . The sum P j = k A jkt represents the diseases12ransmissions which originated from region k and moved to the other regions at time t .Figure 6 presents the heatmap of P j = k A jkt , thus clarifying the top 10 most influential regions,that is, the regions which were the source of the most transmissions, over time. Hubei stands out asthe primary exporter of the infection during the Chinese New Year holidays, though results showthat secondary epicenters, such as Beijing, Guangdong, and Shanghai, started being a significantsource of transmission from around January 22. The outflux from epicenters, including the primaryone, Hubei, gradually diminished following the enactment of policy interventions. Figure 6:
Origins of Transmission: the effective infected outflux from each region over time
ZhejiangShanghaiJiangxiJiangsuHunanHubeiHenanGuangdongChongqingBeijingAnhui
Date P r o v i n c e Value
Notes:
The horizontal axis shows days from January 16 to February 4 and the vertical axis shows ten regions,which are the origins of the ten highest external daily transmission to the other regions. The heatmap reportsvalues of P j = k A jkt for each origin k . To examine the transmission network more closely, we present a section of the transmissionnetwork on January 27, 2020, in Figure 7. All the regions in the network are depicted in thefigure according to their geographic location. The arrows reflect transmission directions. Thelines display transmissions with A kjt greater than two, whereby the line width is proportional tothe transmission strength. Figure 7 shows that Hubei is the primary epicenter, particularly forgeographically proximate provinces (e.g., Henan and Hunan), but also that secondary epicenterssuch as Beijing, Guangdong, and Shanghai, have already developed on this date. The dynamicmigration between the secondary epicenters—which are cultural and economic centers—and therest of China accelerated the propagation of the disease. Regions such as Shandong and Guangxiare only influenced by secondary epicenters, whereas regions such as Sichuan evidence diseasetransmission originating from both the primary epicenter and the secondary epicenters.13 igure 7: Transmission Network on January 27, 2020
Notes:
Regions are located geographically. The arrow indicates the direction of transmission, the lines displayexternal transmissions that are greater than two. The line widths are proportional to the external transmission inthe indicated direction and the size of the nodes are proportional to the total export from a region.
We analyze the propagation of COVID-19 among 33 provinces and special administrative regionsin China. We develop a spatial model to estimate the transmission network and evaluate theeffect of the policy interventions of lockdown and mobility restrictions on the disease spread. Ourempirical results suggest that secondary epicenters developed at a very early stage of the epidemicand that mobility restriction across provinces prevented further spread of the disease. Thus, theobserved pandemic propagation in China could be re-contained to localized outbreaks. Communitytransmission was observed to be the primary source of infection, and regional policy interventionstemmed the spread of the disease. Our empirical findings suggest that the coordination of centraland local government policies is essential to suppress the spread of infectious diseases.14 eferences
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