Spatial Interactions and the Spread of COVID-19: A Network Perspective

Abstract

We use unique data on the travel history of confirmed patients at a daily frequency across 31 provinces in China to study how spatial interactions influence the geographic spread of pandemic COVID-19. We develop and simultaneously estimate a structural model of dynamic disease transmission network formation and spatial interaction. This allows us to understand what externalities the disease risk associated with a single place may create for the entire country. We find a positive and significant spatial interaction effect that strongly influences the duration and severity of pandemic COVID-19. And there exists heterogeneity in this interaction effect: the spatial spillover effect from the source province is significantly higher than from other provinces. Further counterfactual policy analysis shows that targeting the key province can improve the effectiveness of policy interventions for containing the geographic spread of pandemic COVID-19, and the effect of such targeted policy decreases with an increase in the time of delay.

Appendix: Bayesian Estimation

Let $\theta ={\left({\Psi }^{{\prime}},{\beta }{^{\prime}},{\beta }_{3},{\beta }_{4},{\sigma }_{\mu }^{2}\right)}^{{\prime}}$ with $\Psi ={\left(\lambda ,\rho \right)}^{{\prime}}$, $\beta ={\left({\beta }_{1}{^{\prime}},{\beta }_{2}{^{\prime}}\right)}{^{\prime}}$ and $\Gamma ={\left( {\Phi }{^{\prime}},{\sigma }_{\varepsilon }^{2} \right)}{^{\prime}}$ with $\Phi ={\left({\alpha }_{0}, {\alpha }_{x}{^{\prime}},{\alpha }_{y},{\alpha }_{d},{\alpha }_{g},{\alpha }_{f},{\alpha }_{v} \right)}{^{\prime}}$. Following Han et al. (2021), our prior distributions are

$$\Phi \,\backsim\, {N}_{6+k}\left({\Phi }_{0},{P}_{0}\right),$$

$$\beta \,\backsim\, {N}_{2k}\left({\beta }_{0},{B}_{0}\right),$$

$${\beta }_{3}\,\backsim\, N\left({\beta }_{30},{B}_{30}\right),$$

$${\beta }_{4}\,\backsim\, N\left({\beta }_{40},{B}_{40}\right),$$

$${\sigma }_{\mu }^{2}\,\backsim\, \mathrm{\rm I}G\left(\frac{{a}_{\mu }}{2},\frac{{b}_{\mu }}{2}\right),$$

$${\sigma }_{\varepsilon }^{2}\,\backsim\, \mathrm{\rm I}G\left(\frac{{a}_{\varepsilon }}{2},\frac{{b}_{\varepsilon }}{2}\right),$$

$$\lambda \,\backsim\, U\left(-\frac{1}{{\Delta }_{G}},\frac{1}{{\Delta }_{G}}\right),$$

$$\rho \,\backsim\, U\left(-1+\left|\lambda \right|{\Delta }_{G},1-\left|\lambda \right|{\Delta }_{G}\right),$$

where ${\Delta }_{G}={max}_{t=\mathrm{1,2},\ldots ,T}{\parallel {G}_{t}\parallel }_{\infty }$ due to the stability conditions: ${S}_{t}\left(\lambda \right)={I}_{J}-\lambda {G}_{t}$ is invertible and ${\parallel {S}_{t}^{-1}\left(\lambda \right)\rho {I}_{J}\parallel }_{\infty }<1$, ${N}_{l}\left(\cdot \right)$, $IG\left(\cdot \right)$ and $U\left(\cdot \right)$ are respectively the multivariate normal, inverse gamma and uniform distribution, and ${\beta }_{4}$ is the coefficient of time fixed effects ${\alpha }_{t}$. We choose the hyperparameters in the prior distributions as follows: ${\Phi }_{0}=0$, ${P}_{0}=100{I}_{6+k}$, ${\beta }_{0}=0$, ${B}_{0}=100{I}_{2k}$, ${\beta }_{30}=0$, ${B}_{30}=100$, ${\beta }_{40}=0$, ${B}_{40}=100$, ${a}_{\mu }=3$, ${b}_{\mu }=2$, ${a}_{\varepsilon }=3$, ${b}_{\varepsilon }=2$.

The latent variables ${V}_{t}$ are modeled by a Markov process with the initial distribution

$$\mathrm{\rm P}\left({V}_{1}|{\sigma }_{{v}_{0}}^{2}\right)=\prod_{i=1}^{J}\phi \left({v}_{i1}|0,{\sigma }_{{v}_{0}}^{2}\right)$$

and the transition equation

$$\mathrm{\rm P}\left({V}_{t}|{{V}_{t-1},\sigma }_{v}^{2}\right)=\prod_{i=1}^{J}\phi \left({v}_{it}|{v}_{i,t-1},{\sigma }_{v}^{2}\right)$$

for $t=2, 3,\ldots ,T$, where $\phi$ denotes the normal density and ${\sigma }_{{v}_{0}}^{2}$ and ${\sigma }_{v}^{2}$ are normalized to be 1.

The MCMC sampling procedure combines the Gibbs sampling and the Metropolis–Hastings (M–H) algorithm. It consists of the following steps:

Step 1: Sample ${v}_{i1}$ from $\mathrm{\rm P}\left({v}_{i1}|{Y}_{1},{G}_{1},{v}_{-i,1},{v}_{i2},\theta ,\Gamma \right)\propto \phi \left({v}_{i1}\right)\times \phi \left({v}_{i2}|{v}_{i1}\right)\times \mathrm{\rm P}\left({Y}_{1},{G}_{1}|{V}_{1},\theta ,\Gamma \right)$ for $i=\mathrm{1,2},\ldots ,J$, using the M–H algorithm.

Step 2: Sample ${v}_{it}$ from $\mathrm{\rm P}\left({v}_{it}|{Y}_{t},{G}_{t},{{v}_{i,t+1},v}_{-i,t},{v}_{i,t-1},\theta ,\Gamma \right)\propto \phi \left({v}_{i,t+1}|{v}_{it}\right)\times \phi \left({v}_{it}|{v}_{i,t-1}\right)\times \mathrm{\rm P}\left({Y}_{t},{G}_{t}|{V}_{t},\theta ,\Gamma \right)$ for $i=1, 2,\ldots ,J$ and $t=2,\ldots ,T-1$, using the M–H algorithm.

Step 3: Sample ${v}_{iT}$ from $\mathrm{\rm P}\left({v}_{iT}|{Y}_{T},{G}_{T},{{v}_{-i,T},v}_{i,T-1},\theta ,\Gamma \right)\propto \phi \left({v}_{iT}|{v}_{i,T-1}\right)\times \mathrm{\rm P}\left({Y}_{T},{G}_{T}|{V}_{T},\theta ,\Gamma \right)$ for $i=\mathrm{1,2},\ldots ,J$, using the M–H algorithm.

Step 4: Sample $\Psi$ from

$\mathrm{\rm P}\left(\Psi |\left\{{Y}_{t}\right\},\left\{{G}_{t}\right\},\left\{{V}_{t}\right\},\beta ,{\beta }_{3},{\beta }_{4},{\sigma }_{\mu }^{2}\right)\propto u\left(\Psi \right)\times \mathrm{\rm P}\left(\left\{{Y}_{t}\right\}|\left\{{G}_{t}\right\},\left\{{V}_{t}\right\},\Psi ,\beta ,{\beta }_{3}{,{\beta }_{4},\sigma }_{\mu }^{2}\right)$ using the M–H algorithm, where $u\left(\cdot \right)$ is the uniform density function.

Step 5: Sample ${\beta }_{3}$ from $\mathrm{\rm P}\left({\beta }_{3}|\left\{{Y}_{t}\right\},\left\{{G}_{t}\right\},\left\{{V}_{t}\right\},\Psi ,\beta ,{\beta }_{4},{\sigma }_{\mu }^{2}\right)\propto \phi \left({T}_{{\beta }_{3}},{\Sigma }_{{\beta }_{3}}\right)$, where ${\Sigma }_{{\beta }_{3}}={\left({\mathrm{B}}_{30}^{-1}+{\sigma }_{\mu }^{-2}\sum_{t=1}^{T}{V}_{t}^{{\prime}}{V}_{t}\right)}^{-1}$ and ${T}_{{\beta }_{3}}={\Sigma }_{{\beta }_{3}}\left({\mathrm{B}}_{30}^{-1}{\upbeta }_{30}+{\sigma }_{\mu }^{-2}\sum_{t=1}^{T}{V}_{t}^{{\prime}}\left({S}_{t}\left(\lambda \right){Y}_{t}-\rho {Y}_{t-1}-{{\varvec{X}}}_{t}\beta -{d}_{t}{\beta }_{4}\right)\right)$ with ${{\varvec{X}}}_{t}=\left({X}_{t},{\overline{G} }_{t}{X}_{t}\right)$ and ${\overline{G} }_{t}$ denotes the row-normalized ${G}_{t}$. We restrict ${\beta }_{3}>0$ to pin down the signs of ${V}_{t}$’s.

Step 6: Sample $\beta$ from $\mathrm{\rm P}\left(\beta |\left\{{Y}_{t}\right\},\left\{{G}_{t}\right\},\left\{{V}_{t}\right\},\Psi ,{\beta }_{3},{\beta }_{4},{\sigma }_{\mu }^{2}\right)\propto {\phi }_{2k}\left({T}_{\beta },{\Sigma }_{\beta }\right)$, where ${\phi }_{2k}\left(\cdot \right)$ is the multivariate $2k$-dimensional normal density function, ${\Sigma }_{\beta }={\left({B}_{0}^{-1}+{\sigma }_{\mu }^{-2}\sum_{t=1}^{T}{{\varvec{X}}}_{t}^{{\prime}}{{\varvec{X}}}_{t}\right)}^{-1}$ and ${T}_{\beta }={\Sigma }_{\beta }\left({B}_{0}^{-1}{\beta }_{0}+{\sigma }_{\mu }^{-2}\sum_{t=1}^{T}{{\varvec{X}}}_{t}^{{\prime}}\left({S}_{t}\left(\lambda \right){Y}_{t}-\rho {Y}_{t-1}-{V}_{t}{\beta }_{3}-{d}_{t}{\beta }_{4}\right)\right)$.

Step 7: Sample ${\sigma }_{\mu }^{2}$ from $\mathrm{\rm P}\left({\sigma }_{\mu }^{2}|\left\{{Y}_{t}\right\},\left\{{G}_{t}\right\},\left\{{V}_{t}\right\},\Psi ,{\beta }_{3},{\beta }_{4},\beta \right)\propto \iota \gamma \left(\frac{{a}_{\mu }^{new}}{2},\frac{{b}_{\mu }^{new}}{2}\right)$, where $\iota \gamma \left(\cdot \right)$ is the inverse gamma density function, ${a}_{\mu }^{new}={a}_{\mu }+T\cdot J$ and ${b}_{\mu }^{new}={b}_{\mu }+\sum_{t=1}^{T}{\mu }_{t}^{{\prime}}{\mu }_{t}$.

Step 8: Sample ${\beta }_{4}$ from $\mathrm{\rm P}\left({\beta }_{4}|\left\{{Y}_{t}\right\},\left\{{G}_{t}\right\},\left\{{V}_{t}\right\},\Psi ,{\beta ,\beta }_{3},{\sigma }_{\mu }^{2}\right)\propto \phi \left({T}_{{\beta }_{4}},{\Sigma }_{{\beta }_{4}}\right)$, where ${\Sigma }_{{\beta }_{4}}={\left({B}_{40}^{-1}+{\sigma }_{\mu }^{-2}\sum_{t=1}^{T}{d}_{t}^{{\prime}}{d}_{t}\right)}^{-1}$ and ${T}_{{\beta }_{4}}={\Sigma }_{{\beta }_{4}}\left({B}_{40}^{-1}{\beta }_{40}+{\sigma }_{\mu }^{-2}\sum_{t=1}^{T}{d}_{t}^{{\prime}}\left({S}_{t}\left(\lambda \right){Y}_{t}-\rho {Y}_{t-1}-{{\varvec{X}}}_{t}\beta -{V}_{t}{\beta }_{3}\right)\right)$.

Step 9: Sample $\Phi$ from $\mathrm{\rm P}\left(\Phi |\left\{{G}_{t}\right\},\left\{{V}_{t}\right\},{\sigma }_{\varepsilon }^{2}\right)\propto { \phi }_{6+k}\left({T}_{\Phi },{\Sigma }_{\Phi }\right)$, where ${\Sigma }_{\Phi }={\left({\mathrm{\rm P}}_{0}^{-1}+{\sigma }_{\varepsilon }^{-2}\sum_{t=1}^{T}{\chi }_{t}^{{\prime}}{\chi }_{t}\right)}^{-1}$, ${T}_{\Phi }={\Sigma }_{\Phi }\left({\mathrm{\rm P}}_{0}^{-1}{\Phi }_{0}+{\sigma }_{\varepsilon }^{-2}\sum_{t=1}^{T}{\chi }_{t}^{{\prime}}{\tilde{G }}_{t}\right)$ with

$${\chi }_{t}=\left(\begin{array}{c}1,\left({X}_{1,t}-{X}_{2,t}\right),\left({Y}_{1,t-1}-{Y}_{2,t-1}\right),{D}_{12,\mathrm{t}},{g}_{12,\mathrm{t}-1},{F}_{12,t-1},\left|{v}_{1,t}-{v}_{2,t}\right|\\ \begin{array}{c}1,\left({X}_{1,t}-{X}_{3,t}\right),\left({Y}_{1,t-1}-{Y}_{3,t-1}\right),{D}_{13,\mathrm{t}},{g}_{13,\mathrm{t}-1},{F}_{13,t-1},\left|{v}_{1,t}-{v}_{3,t}\right|\\ \begin{array}{c}\vdots \\ \begin{array}{c}1,\left({X}_{i,t}-{X}_{j,t}\right),\left({Y}_{i,t-1}-{Y}_{j,t-1}\right),{D}_{ij,\mathrm{t}},{g}_{ij,\mathrm{t}-1},{F}_{ij,t-1},\left|{v}_{i,t}-{v}_{j,t}\right|\\ \begin{array}{c}\vdots \\ 1,\left({X}_{J,t}-{X}_{J-1,t}\right),\left({Y}_{J,t-1}-{Y}_{J-1,t-1}\right),{D}_{J,J-1,\mathrm{t}},{g}_{J,J-1,\mathrm{t}-1},{F}_{J,J-1,t-1},\left|{v}_{J,t}-{v}_{J-1,t}\right|\end{array}\end{array}\end{array}\end{array}\end{array}\right)$$

for $i,j=\mathrm{1,2},\ldots ,J$,$i\ne j$, and let ${G}_{it\backslash i}=\left({{g}_{i1,t},g}_{i2,t},\ldots {{g}_{i,i-1,t},g}_{i,i+1,t},\ldots ,{g}_{iJ,t}\right)$ for $i=\mathrm{1,2},\ldots ,J$, ${\tilde{G }}_{t}={\left({G}_{1t\backslash 1},{G}_{2t\backslash 2},\ldots {,G}_{Jt\backslash J}\right)}^{{\prime}}$.

Step 10: Sample ${\sigma }_{\varepsilon }^{2}$ from $\mathrm{\rm P}\left({\sigma }_{\varepsilon }^{2}|\left\{{G}_{t}\right\},\left\{{V}_{t}\right\},\Phi \right)\propto \iota \gamma \left(\frac{{a}_{\varepsilon }^{new}}{2},\frac{{b}_{\varepsilon }^{new}}{2}\right)$, where ${a}_{\varepsilon }^{new}={a}_{\varepsilon }+T\cdot J$ and ${b}_{\varepsilon }^{new}={b}_{\varepsilon }+\sum_{t=1}^{T}{\varepsilon }_{t}^{{\prime}}{\varepsilon }_{t}$.

We collect the draws from iterating the above steps and compute the posterior mean and the posterior standard deviation as our estimation results.