A spatio-temporal causal network for multi-scale analysis of infectious respiratory diseases transmission

Abstract

Understanding the spatio-temporal transmission characteristics of infectious respiratory diseases is crucial for effective control. However, most existing studies rely on correlation analysis, which obscures the true causal pathways and directionality of infectious respiratory disease transmission, preventing accurate identification of epidemic sources and sinks. To address these challenges, we proposed a novel spatio-temporal causal analysis framework. First, a spatio-temporal causal network is constructed using the Convergent Cross Mapping (CCM) model. This method effectively overcomes the limitations of traditional correlation analysis in identifying spurious correlations and determining causal direction. Subsequently, the weighted k-shell decomposition and Louvain algorithm are applied to analyze the multi-scale structural characteristics of the network, including critical paths, core nodes, and community structures, revealing the multi-scale transmission patterns of the system. We conducted a case study using influenza data from 30 provinces in mainland China from 2010 to 2018. A total of 120 directional transmission pathways were identified, primarily driven by interprovincial population mobility, showing an 83.9 % concordance with the results of the Bayesian phylogenetic analysis. Moreover, provincial importance in transmission was found to be highly correlated with the Hu Huanyong Line. This study provided new insights into the causal relationships and multi-scale structure of infectious disease transmission, offering an important reference for formulating targeted regional prevention and control strategies.

1. Introduction

Infectious respiratory diseases are transmitted through the mobility of people, which spreads an epidemic from local outbreaks to national or global pandemics resulting in very substantial damage (Guo et al., 2022). The critical role of population mobility in disease transmission is well established (Cai et al., 2019; De Jong et al., 2025). Evolving transportation networks have reshaped population mobility patterns (Wei & Wang, 2020; Zhang et al., 2020), creating complex infectious respiratory diseases transmission pathways with multi-scale structure (Buchel et al., 2021; Tan et al., 2021). This structure is characterized not only by dense local clustering but also by long-distance transmission across regions (Yang, 2021; Zhang et al., 2022). Therefore, understanding the inherent structure and propagation mechanism underlying the spatio-temporal spread of infectious respiratory diseases is crucial for developing effective public health interventions.

In the existing study on the spatio-temporal spread of infectious respiratory diseases, approaches that trace transmission pathways based on viral genetic data are constrained by the quality of available genetic sequences and the uncertainties introduced by viral variations (Chen et al., 2025; Cheng et al., 2021; Lei et al., 2022), such as antigenic drift and shift (Li & Yao, 2023). More studies have used population mobility and transportation data within a complex network framework to characterize epidemic transmission (Qiu et al., 2021). For example, by analyzing mobility networks based on Baidu Migration data (Ma et al., 2015), Wei's group have examined the hierarchical and spatial subnetwork structure of national population movement networks, revealing a significant positive correlation between population mobility from Wuhan and the number of confirmed COVlD-19 cases in destination cities (Wei & Wang, 2020). Using the Louvain algorithm (Newman, 2004) on population mobility networks, Buchel's group showed that restricting commutes between high- and low-risk zones is an effective way to curb the multi-location spread of COVID-19 (Buchel et al., 2021). Xu's group constructed a national highway network to show that city-level centrality measures were correlated with epidemiological features (Xu et al., 2019). Using population mobility data, Tan's group simulated migration patterns in China over different time frames. They applied the Louvain algorithm to segment cities into ten communities, with the results being applicable to prevention strategies (Tan et al., 2021). However, a fundamental limitation of these studies is their reliance on correlation analysis to infer transmission networks, which makes it difficult to disentangle true transmission links from ‘spurious correlations’ driven by shared factors such as climate or socioeconomic conditions (Grande et al., 2022). Furthermore, while some studies have revealed the hierarchical and modular characteristics of the network by complex methods (Buchel et al., 2021; Tan et al., 2021), but these correlation-based methods fail to characterize the directionality of transmission (Janse et al., 2021). As a result, they cannot distinguish transmission sources from sinks. While existing studies have offered empirical insights into the transmission of respiratory infectious diseases, they are insufficient in clarifying the spatio-temporal causal relationships and the complex multi-scale structure of transmission networks.

To address these challenges, we proposed a novel framework for constructing and analyzing the spatio-temporal causal network of infectious respiratory diseases transmission. This framework constructed a directed transport network based on the convergent cross-mapping model (Sugihara et al., 2012), and combined complex methods to explore its multi-scale structure. Convergent cross-mapping (CCM) model is based on state-space reconstruction that can detect causal relationships and their directionality between interacting variables, thereby overcoming the limitations of traditional correlation analysis in identifying spurious correlations and determining directionality. Subsequently, we employed the weighted k-shell decomposition (Garas et al., 2012) and Louvain algorithm to reveal the multi-scale structure of the network, such as critical paths, core nodes and community structures. The effectiveness and reliability of the proposed framework were validated by using influenza surveillance data from 30 of mainland China's 31 provinces during 2010–2018. This spatio-temporal causal network framework can provide a valuable reference for studying the spatio-temporal dynamics and informing control strategies for other infectious respiratory diseases and regions.

2. Material and methods

2.1. Influenza data

The Influenza-Like Illness (ILI) samples were obtained from the National Influenza Surveillance Network, which is managed by the Chinese Center for Disease Control and Prevention (CDC). We summarized the number of weekly 31 province-level ILI samples tested for influenza and influenza virus positive isolates between September 27, 2010 and May 27, 2018, after the end of 2009 H1N1 influenza pandemic. It should be noted that surveillance data were not available for Hong Kong, Macao, and Taiwan. Tibet was excluded due to insufficient number of samples, as fewer than 10 specimens were tested in Tibet in 48 % of the reporting weeks. In this study, the weekly influenza positivity rate (weekly number of influenza virus positive isolates/weekly number of ILI samples tested) was used as the core indicator to measure the intensity of influenza activity in each province.

To validate the network from an etiological perspective, we collected nucleotide sequences from human hosts in China between September 27, 2010, and May 27, 2018, from the National Center for Biotechnology Information (NCBI). After removing sequences with unknown provincial origins, the resulting HA gene sequences included 232 influenza A (H1N1) subtype, 175 influenza A (H3N2) subtype, and 79 influenza B virus sequences for subsequent Bayesian system dynamics analysis.

2.2. The influencing factors data

We obtained the Baidu Migration Index from Baidu Migration website (http://qianxi.baidu.com/) for the 30 mainland China provinces, from January 2020 to June 2022. This index reflects the daily volume of out-migration. We aggregated the daily data to measure migration intensity between provinces. Defining inter-provincial mobility variables: For province $i$, its primary source provinces (defined as the set $J$ of provinces with the top 50 % of migration intensity) were selected. Subsequently, for each province $j\in J$, its weekly influenza positivity rate was used as an independent explanatory variable for CCM analysis to test whether influenza activity in province $j$ causally influenced province $i$.

We also collected data on factors influencing influenza activity in 30 provincial-level regions in mainland China between September 27, 2010, and May 27, 2018. The data included four categories: Risk population variables (population aged 0–4 years, population aged 65 years and over, built-up area population density, and net population inflow/outflow); Socioeconomic variables (resident disposable income); Medical variables (number of influenza test specimens submitted, number of medical technicians per 1000 population, and hospital bed utilization rate); Environmental variables (precipitation, sunshine duration, wind speed, temperature, air pressure, humidity, and PM2.5 concentration). Data sources were the National Bureau of Statistics (https://www.stats.gov.cn/), China Health Statistics Yearbook (http://www.nhc.gov.cn), and the National Meteorological Data Center (https://data.cma.cn/).All data were converted to weekly format, and Variance Inflation Factor (VIF) tests were performed on all variables according to provinces. Variables with VIF>5 were eliminated to control multicollinearity (Kim, 2019).

2.3. Establishment of spatio-temporal causal network framework

We applied Convergent Cross Mapping (Sugihara et al., 2012) to examine the causality of influenza among provinces. Subsequently, we constructed a weighted directed network, employing migration intensity as the edge weight, to apply the weighted k-shell decomposition and the Louvain algorithm. Fig. 1 presents the flowchart for the proposed framework.

Fig. 1.

Flowchart of the spatio-temporal causal network framework. The workflow consists of three main stages: multi-source data integration, causal network construction, and multi-scale analysis.

2.3.1. Construction of spatio-temporal causal network

On provincial scale, the weekly influenza positivity rate $Y=\left[Y_1,Y_2,\dots ,Y_L\right]$ were used as the response variable, where $L$ was the length of the time series. Explanatory variables include inter-provincial mobility variables and the provincial risk population, socioeconomic, medical, and environmental variables after screening by the variance inflation factor (VIF ≤5). These variables were denoted as $X^i=\left[X_1^i,X_2^i,\dots ,X_L^i\right]\left(i=1,2,\dots ,n\right)$, where $n$ was the number of explanatory variables. The CCM was used to quantify the causal impact of each variable $X^i$ on influenza activity $Y$.

Based on Takens embedding theorem, the delayed coordinate method (Sugihara et al., 2012) was used to construct the shadow manifolds of $X^i$ and $Y$:

$$M_{x^i}=\left\{x_t|x_t=\left[X_t^i,X_{t-\tau }^i,\dots ,X_{t-\left(E-1\right)\tau }^i\right],t=\left(E-1\right)\tau +1,\dots ,L\right\}\text{,}$$ (1)

$$M_y=\left\{y_t|y_t=\left[Y_t,Y_{t-\tau },\dots ,Y_{t-\left(E-1\right)\tau }\right],t=\left(E-1\right)\tau +1,\dots ,L\right\}\text{,}$$ (2)

where the embedding dimension E is determined by computing the maximum cross-mapping skill ρ and in the range [2, 10]. The time lag as $\tau =1$ (Sun et al., 2024). On the manifold $M_{x^i}$, for each $x_t$, find its $E+1$ nearest neighbors in Euclidean space. The cross mapping that predicts $Y$ from $M_{x^i}$ is defined as:

$${\hat{Y}}_t|M_{x^i}=\sum _{E+1}^{i=1}{w}_i{Y}_{t_i}\text{,}$$ (3)

where:

$$w_i=u_i/\sum _{E+1}^{j=1}{u}_j,with{u}_i=\exp \left\{-d\left(x_t,x_{t_i}\right)/d\left(x_t,x_{t_1}\right)\right\}\text{,}$$ (4)

Where $d\left(\right)$ denotes the Euclidean distance. The cross mapping correlations are defined as:

$${\rho }_{CCM}\left(X^i\to Y\right)=\rho \left(Y^L,{\hat{Y}}^L\right)\text{.}$$ (5)

Where $\rho$ is the Pearson correlation. If $\rho$ gradually converges and increases with $L$, a causal effect is considered to exist (Yu et al., 2022). Statistical significance was assessed using 100 permutation tests, with a threshold of p ≤ 0.05. The strength of the causal effect is measured by the ρ value, which represents the relative contribution of different factors to the spatio-temporal distribution of influenza.

The identified significant inter-provincial mobility variables (p ≤ 0.05) were incorporated into the inter-provincial transmission network as directed edges, ultimately forming a spatio-temporal causal network in China. To validate this network, we used a two-pronged approach: (1) Comparing the network structure with a Bayesian system dynamics network constructed based on viral gene sequences; (2) Pearson correlation coefficient networks and traditional Bayesian networks were constructed based on influenza surveillance data for comparison at the methodological level.

2.3.2. Identification of critical paths and core provinces

Critical transmission pathways were identified by selecting the strongest transmission links (top 30 % by weight) that also formed geographically contiguous routes. We employed the weighted k-shell decomposition to identify core provinces in the transmission network.

The weighted k-shell decomposition (Garas et al., 2012) is used to divide the network into layers. This method is based on the principle of degree centrality and realizes network structure analysis by dividing the importance of network nodes into layers. The weighted degree of a node $i$ is defined as:

$$k_i^{\prime }={\left(k_i^{\alpha }·{\left({\sum }_j{w}_{ij}\right)}^{\beta }\right)}^{1/\left(\alpha +\beta \right)}\text{,}$$ (6)

where $k_i$ is the node degree of province$i$, and ${\sum }_j{w}_{ij}$ is the sum of the weights of the edges connected to province $i$. In this study, we discussed only the case when $\alpha =\beta =1$, which treats the weight and the degree equally. However, to ensure a typical weighted link is seen as unit weight before calculating $k_i^{\prime }$ with equation (6), we followed these procedures. We began by normalizing weights to their mean, divided the normalized weights by the smallest value, and rounded them to the nearest integer, ensuring the minimum link weight is 1(Garas et al., 2012). K-shell as a hierarchical label in network decomposition, its value range starts from 1 and gradually reaches the core node level. The larger the value, the more core the node position in the network. This analysis was performed from three perspectives: undirected (overall influence), in-degree (as transmission sinks), and out-degree (as transmission sources) to evaluate the propagation role of each province in the network.

2.3.3. Identify community structure

Louvain algorithm (Newman, 2004) is a modularity-based approach that works well for large networks and is relatively fast. It divides the network into several communities based on the topological distance between nodes. The nodes within the community are closely connected, while the connections between communities are relatively weak. This study used the "modularity" proposed by Newman to measure the strength of the network community structure. The higher the modularity value, the better the division result. Modularity is defined as (Chen et al., 2023; Newman, 2004):

$$Q=\frac{1}{2m}{\sum }_{ij}\left(A_{ij}-\frac{k_i{k}_j}{2m}\right)\delta \left(C_i,C_j\right)\text{.}$$ (7)

Where $A_{ij}$ is the link weight between province $i$ and province $j$; $k_i$ and $k_j$ are the sum of links to and from province $i$ and $j$. $\delta \left(C_i,C_j\right)$ is equal to 1 if there is a link between nodes $i$ and $j$. Otherwise, it is 0. The $Q$ value ranges from 0 to 1. The closer it is to 1, the more obvious the community structure is.

3. Result

3.1. The spatio-temporal causal network reveals an asymmetric, regionally clustered structure

Our analysis of factors influencing provincial influenza activity revealed that inter-provincial mobility was the dominant driver. Among all variables with significant causal effects, inter-provincial mobility accounted for 61.2 % (120 variables), higher than Environmental variables (33.7 %, 66 variables) and Medical variables (5.1 %, 10 variables). Furthermore, the median cross-mapping skill for inter-provincial mobility variables ($\rho =0.77$) was also higher than Medical variables ($\rho =0.54$) and Environmental variables ($\rho =0.46$) (Fig. S1 Supplementary Material). These findings suggest that inter-provincial mobility variables play a significant role in influenza transmission.

The resulting spatio-temporal causal network comprised 120 directed transmission pathways between provinces (Fig. 2), which exhibited two major structural characteristics: regional clustering and asymmetry. First, the network with 62 pathways between northern provinces and 46 pathways between southern provinces. In contrast, cross-regional transmission was sparse, with only 6 pathways from north to south and 6 pathways from south to north. Second, the causal relationships between most provinces were found to be unidirectional.

Fig. 2.

The spatio-temporal causal network of influenza transmission in China. The dots represent significant causal links between provinces. The bar chart shows the number of provinces acting as sources and targets in the transmission pathway.

To validate the spatio-temporal causal network, we conducted a Bayesian phylogeographic analysis of influenza HA gene sequences using BEAST. Comparison with the resulting Bayesian system dynamics network showed an 83.9 % consistency in identified inter-provincial transmission pathways (Table S1 Supplementary Material). Furthermore, we constructed two traditional networks: the Pearson correlation coefficient-based network (containing 216 paths) and the Bayesian network (containing 44 paths). The results showed that these networks were 74.2 % and 12.5 % consistent respectively with the Bayesian system dynamics network (Fig. S2 Supplementary Material).

3.2. Identification of critical transmission paths and core provincial nodes

By selecting the top 30 % of edges by weight and ensuring geographic continuity, we identified several critical influenza transmission corridors. These included the north-south routes of "Heilongjiang-Jilin-Liaoning-Beijing-Hebei" and "Shanghai-Zhejiang-Fujian-Guangdong," and the east-west routes of "Shanghai-Jiangsu-Anhui-Hubei" and "Shandong-Hebei-Shanxi-Shaanxi-Ningxia-Gansu-Xinjiang."

We used the weighted k-shell decomposition to evaluate the importance of spatio-temporal causal networks from three perspectives: overall, in-degree, and out-degree influence. The results showed that the importance of each province in influenza transmission was highly correlated with the Hu HuanyongLine, with eastern provinces generally having higher k-shell values than western provinces (Fig. 3A–C). The overall network structure (Fig. 3-A) revealed that Shanghai, Jiangsu, Zhejiang, and Anhui possessed the highest k-shell values, indicating their central position in the influenza transmission network. The in-degree analysis (Fig. 3-B), which reflects a province's role as a "receiver," identified Guangdong and Fujian as having the highest k-shell values, suggesting they were primary destinations for influenza importation. The out-degree analysis (Fig. 3-C) identified the "transmitters" within the network, revealing a "hub-and-spoke" network structure where provinces along the Hu HuanyongLine acted as a core, disseminating the virus to surrounding regions. As shown in these, provinces clusters are not only close in geographical location but also with almost the same k-shell values. Thus, it's necessary to specify these clusters.

Fig. 3.

Provincial centrality analysis using weighted k-shell decomposition. (A) Overall centrality based on the undirected network. (B) In-degree centrality, identifying transmission sinks. (C) Out-degree centrality, identifying transmission sources.

3.3. Community detection reveals a multi-core, modular transmission architecture

To further explore the internal structure of influenza transmission, we partitioned the spatio-temporal causal network using the Louvain algorithm. The network was divided into four modular communities (Fig. 4), with a modularity of 0.602, which indicated a clear internal transmission structure. It is shown in k-shell values exist communities centered in Beijing, Shanghai, and Guangzhou. Based on the Louvain algorithm, the identified communities were defined as follows: (1) the Yangtze River Delta community; (2) the Beijing-Tianjin-Hebei community; (3) the South China community; and (4) the Northern China region. The provinces for each community are listed (Table S3 Supplementary Material).

Fig. 4.

Community detection for the spatio-temporal causal network. Communities represent groups of provinces with dense causal transmission links. The main figure shows the four communities in the network, with different colors representing different communities. The four subfigures below show the detailed propagation paths within each community.

Significant differences in k-shell values were observed among these four communities. A one-way ANOVA revealed significant intergroup differences for both overall and outgoing influence (p ≤ 0.05), while the incoming influence was not statistically significant (p = 0.19). For metrics with significant differences, post-hoc Tukey HSD tests were conducted for pairwise comparisons (Fig. 5). The Yangtze River Delta community had the highest mean overall k-shell value (50.20) and the highest mean outgoing k-shell value (23.40), significantly higher than the northern communities (p < 0.05), indicating its core role. The Beijing-Tianjin-Hebei and South China communities also displayed high k-shell values, with no significant difference compared to the Yangtze River Delta community. The Beijing-Tianjin-Hebei community had the highest mean incoming k-shell value (27.38), suggesting it is a major sink for influenza importation, but this trend did not reach statistical significance. The North China community consistently showed the lowest k-shell values across all three metrics. In summary, these findings reveal that the influenza transmission network in China is characterized by a multi-core, modular structure.

Fig. 5.

Comparison of k-shell centrality across the four transmission communities. Boxplots show the distribution of (A) overall, (B) in-degree. Community codes: 1 - South China, 2 - North China, 3 - Beijing-Tianjin-Hebei, 4 - Yangtze River Delta. Asterisks denote significance levels from post-hoc Tukey HSD tests: ∗p < 0.05, ∗∗p < 0.01, ∗∗∗p < 0.001.

4. Discussion

Our work employed the convergent cross mapping method to distinguish true causality from the spurious correlations often found in traditional correlation analyses (Sugihara et al., 2012), and to resolve the issue of directionality in constructing transmission networks (Grande et al., 2022; Janse et al., 2021). We then applied a multi-scale network analysis, combining weighted k-shell decomposition and Louvain community detection, to characterize the network's hierarchical and modular architecture. Our objective was to construct a "backbone network" that is more representative of the actual transmission process (Tian et al., 2020). Using interprovincial influenza transmission in China as a case study, we established a spatio-temporal causal transmission network covering 30 mainland provinces, offering a new perspective on understanding influenza dynamics at the provincial level.

The network constructed based on correlation analysis can only reveal the correlation trends between nodes, rather than their inherent causal relationships. In contrast, CCM based on the principle of spatial reconstruction, assesses whether the time series of one variable can be reliably estimated from the reconstructed state space of another. Its core premise is that cross-mapping skill converges as the sample size increases. True causality is indicated by this stable convergence, a characteristic not exhibited by spurious correlations (Chen et al., 2023; Sun et al., 2021; Tian et al., 2020). Therefore, CCM focuses on information transfer within the dynamical system rather than surface-level coordinated fluctuations, thereby fundamentally overcoming the limitations of traditional correlation analysis.

Our empirical findings revealed that the spatio-temporal causal network exhibits regional clustering and asymmetry, a finding consistent with existing studies on epidemic transmission and viral evolution (Han et al., 2022; Zhang et al., 2019). The identified transmission pathways strongly align with China's major high-speed rail corridors (Fig. S3 Supplementary Material), suggesting that physical transportation networks act as key conduits for virus transmission. This result corroborates the significant impact of transportation modes on influenza spread (Charu et al., 2017; Wei & Wang, 2020) and reinforces the well-established link between population mobility and influenza dissemination (Luo et al., 2025; Tunnicliffe & Warren-Gash, 2022). Compared with the study based on geographical division by Xie et al. (Xie et al., 2022), our data-driven causal inference provides a more detailed and comprehensive inter-provincial transmission path (only one is missing), further proving the effectiveness of the constructed network.

By dissecting the network's multi-scale architecture, we were able to assign distinct functional roles to different regions, moving beyond simplistic hub classifications. The weighted k-shell method focuses on identifying the influence hierarchy of nodes from their network position and global edge weights, effectively distinguishing between core and edge nodes at different diffusion depths within the network. The Louvain algorithm, as an efficient community detection method, can identify the functional community structure within the network based on the principle of modular optimization. By distinguishing the in-degree and out-degree centrality of nodes, a "profile" of the roles of different provinces in the diffusion network was constructed, providing a more nuanced understanding than previous studies. The Yangtze River Delta region exhibited the highest overall and outgoing influence, serving as the "core engine" of the entire network; the Beijing-Tianjin-Hebei region exhibited the strongest incoming influence, acting as a "convergence basin." This provides a more nuanced distinction between roles, compared to some studies based on traffic data or genetic sequences that view Beijing, Shanghai, Guangdong, and other provinces as equally important diffusion hubs (Lei et al., 2022; Li & Yao, 2023).

While our study yielded meaningful findings, several limitations should be acknowledged. First, the provincial scale of our analysis may have obscured more detailed transmission dynamics at the sub-provincial level, such as among cities. Second, the CCM model infers statistical causality, which is not equivalent to the individual-level transmission chains tracked in epidemiology. Nevertheless, the macro-scale validity of our approach was supported by validation against genetic networks.

The findings of our work can inform the development of more precise influenza prevention and control strategies. As demonstrated by Saxena (Saxena et al., 2018), immunization strategies targeting core network nodes are most effective in controlling an epidemic's scale. First, the role distinction between the "core engine" and the "convergence basin" suggests differentiated intervention priorities. Our k-shell analysis shows that the Yangtze River Delta region is the core "outgoing" engine of the network. Therefore, using this region as a "sentinel" for influenza surveillance and conducting early warning and intervention may be the most effective way to delay the spread of the virus to the whole country. As the main "incoming" convergence point, the Beijing-Tianjin-Hebei region should strengthen the monitoring of multi-source virus strains to cope with the complex importation risk. Second, the network's clear community structure provides a theoretical basis for regional coordinated prevention and control. The internal connections within the four major transmission communities we identified are substantially stronger than the connections between them. This finding supports the establishment of joint prevention and control mechanisms among provinces within each community to improve information sharing and resource coordination.

CRediT authorship contribution statement

Xincao Zheng: Writing – original draft, Methodology, Formal analysis, Data curation. Wenjing Yu: Writing – original draft, Methodology, Formal analysis, Data curation. Lu Wang: Methodology. Jiaoe Wang: Supervision. Yilan Liao: Writing – review & editing, Methodology, Conceptualization. LingCai Kong: Supervision.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Handling Editor: Dr. Raluca Eftimie

Appendix A. Supplementary data

The following is the Supplementary data to this article.