The COVID-19 pandemic has prompted widespread interdisciplinary efforts to understand infectious disease transmission, governmental responses, and the interplay of interconnected regions affecting epidemiological outcomes. Countries that successfully controlled COVID-19 case numbers typically implemented a combination of early interventions and robust public health infrastructure (1, 2). These strategies include timely border closures, widespread testing, efficient contact tracing, and clear public communication, often supported by high levels of public trust and healthcare system preparedness (3). While these approaches are well documented individually, it remains unclear how such strategies co-evolve with national case dynamics and whether they correspond to topological patterns across countries. Moreover, the scale and heterogeneity of pandemic data require analytical approaches that move beyond traditional statistical summaries to methods capable of capturing global structures, nonlinear dependencies, and hidden similarities across high-dimensional datasets.

In response, numerous data-driven approaches have been developed to track, model, and interpret COVID-19 trajectories. Ecological comparative frameworks have been used to evaluate national responses (4), while time-series analyses of case rates (5, 6), machine learning clustering of outbreak severity (7, 8), and mobility-informed epidemiological models (9) have each provided important insights. Policy-response datasets such as the Oxford COVID-19 Government Response Tracker enable systematic cross-national comparisons of interventions through indices like stringency scores (10). Within this expanding methodological landscape, Topological Data Analysis (TDA) has emerged as a promising framework for uncovering latent structure in complex, high-dimensional pandemic data (11). Foundational contributions by Epstein and Carlsson (12) and Edelsbrunner and Harer (13) established the mathematical basis for extracting geometric and topological features from data, emphasizing robustness to noise and invariance to continuous transformations. The Mapper algorithm, introduced in Singh et al. (14), provides a practical tool for capturing shape and connectivity in applied contexts. Surveys and methodological reviews (15, 16) consolidated TDA's versatility across disciplines, highlighting its capacity to reveal multi-scale structures inaccessible to classical clustering techniques. Specifically, Mapper has been applied to temporal COVID-19 case trajectories in the United States and United Kingdom, revealing hidden topological signatures such as infection waves, hotspot transitions, and regional divergences (17, 18). Prior studies (19, 20) show that TDA can detect structural patterns overlooked by conventional statistical approaches, particularly in visualizing outbreak progression across time and space. More recent comparative work demonstrates Mapper's ability to capture overlapping clusters and transitional dynamics, features not well represented by k-means, hierarchical, or density-based clustering methods (21, 22). These advantages underscore Mapper's distinct value for epidemiological time-series and cross-country analyses. Building on this foundation, the present study applies Mapper to cross-national COVID-19 trajectories and government response indices, aiming to reveal how variations in policy stringency and infection dynamics jointly shape the pandemic's progression across diverse contexts.

However, existing TDA-based COVID-19 analyses predominantly focus on infection curve shapes alone, treating countries or regions as isolated time series and seldom incorporating policy context, economic groupings, or geographic proximity into topological modeling (23–25). In addition, most COVID-19 dynamics studies generally operate at national or regional scales without explicitly assessing how interconnectedness via economic alliances such as the Eurozone or shared geography might influence policy responses or pandemic outcomes. Another limitation lies in the segmentation of infection trajectories into waves: while wave-based representations are intuitive and facilitate cross-country comparison, sensitivity to smoothing, window selection, and inflection-point definitions can introduce methodological bias if not rigorously addressed. Although wave-based representations of COVID-19 spread are intuitive and observable (26), few studies integrate wave structure as a primary feature within topological analysis, limiting the interpretability of clusters in the context of epidemiological or policy-driven differences.

This study addresses these gaps by examining how European countries' COVID-19 trajectories relate not only to the timing and magnitude of case waves but also to policy stringency and economic or regional interdependence. Specifically, we investigate whether countries that are economically integrated [for example, Eurozone members (27)], geographically neighboring, or part of policy-sharing blocs [for example, Schengen Area (28), Nordic Council (29), Visegrád Group (30)] exhibit similar COVID-19 wave responses and policy interventions over time. By situating wave dynamics within broader structural and institutional contexts, we assess whether topological similarity aligns with formal cooperation frameworks or emerges independently of them. We further contextualize clusters with policy indices, testing whether stringency differences correspond with cluster membership, and employ permutation-based statistical tests to ensure robustness against small sample sizes and non-normality. This focus on Europe is driven by its distinctive combination of deep institutional integration and national heterogeneity. The region's extensive frameworks for economic and policy coordination (for example, the Schengen Area and the Eurozone) facilitate high levels of cross-border movement and interdependence, providing fertile ground for examining how such linkages shape pandemic trajectories (31). At the same time, European countries display marked variation in public health infrastructure, policy stringency, and the degree of centralized versus decentralized governance (32). This diversity within an otherwise interconnected system provides a rigorous testbed for assessing how a shared shock such as COVID-19 propagates through different structural and institutional alliances (33).

To this end, we construct a wave-based feature profile for each country using normalized 7-day average COVID-19 case data, segmenting the pandemic into discrete epidemiological waves. We then apply Mapper to this feature space to identify clusters of countries with comparable wave dynamics. Each cluster is annotated with external information such as policy stringency scores and group memberships (economic, geographic, and political). We statistically assess whether these groupings explain differences in pandemic responses or whether topologically similar countries transcend conventional alignments. Finally, we incorporate geographic proximity data to explore whether neighboring countries within the same topological clusters also share common pandemic behaviors, suggesting potential spillover effects or shared infrastructure influences. This multi-layered approach allows us to evaluate the relative weight of epidemiological, political, and geographic factors in shaping pandemic trajectories and informs how policy interventions interact with underlying infection dynamics.

Traditional clustering methods assume linear structure and mutually exclusive groupings, which are poorly suited to the highly irregular, nonlinear, and noisy COVID-19 trajectories observed across Europe. Topological Data Analysis (TDA), and Mapper in particular, offers a non-parametric way to characterize the shape of such complex time-series by constructing overlapping clusters and linking them into a topology-preserving network. This allows transitional or intermediate epidemic profiles to emerge naturally—patterns that standard clustering would force into artificial single assignments. We therefore use TDA not for methodological novelty, but because its topology-preserving representation captures structural heterogeneity in European epidemic curves that conventional approaches are likely to obscure.

Our contributions are threefold. First, we introduce a wave-profile-based TDA approach for pandemic analysis that preserves both epidemiological and temporal structure, emphasizing the shape and progression of waves across countries. Second, we integrate policy stringency metrics and economic or geographic groupings to test whether formal cooperation frameworks manifest in topological similarity. Third, we incorporate geographic neighbor analysis to evaluate the role of spatial spillovers and infrastructural proximity in shared outcomes. Taken together, these contributions show that despite shared institutional frameworks, topological COVID-19 similarity does not always align with formal alliances, highlighting the importance of local sociopolitical and infrastructural factors beyond formal integration. More broadly, this study demonstrates the utility of TDA for comparative public health research, where nonlinearities, high dimensionality, and cross-country dependencies challenge conventional approaches. In the following sections, we expand on related literature, contrasting Mapper with classical clustering methods and situating our work within cross-country pandemic modeling and policy-evaluation studies.

The COVID-19 wave trends for each country, grouped by their regional affiliations, are illustrated in Figure 2. This visualization plots the average normalized case counts across eight epidemic waves, with each line representing a country's trajectory over time. By coloring the lines according to regional groups (Nordic, Visegrad, Eurozone, Schengen, and Others) the figure highlights similarities and differences in pandemic progression within and between these groups. The regional color coding facilitates the identification of patterns potentially linked to shared economic, political, or geographic factors. For instance, countries within the Eurozone often show comparable wave intensities, suggesting some alignment in pandemic dynamics, while Nordic countries tend to exhibit distinct trends. This layered representation supports our exploration of how regional interdependence may have influenced COVID-19 wave behaviors across Europe.

This study leveraged Topological Data Analysis (TDA) to explore the complex interplay between COVID-19 wave patterns, government policy stringency, and pre-existing economic and geographic affiliations among European countries. Our findings reveal a picture that challenges some conventional assumptions about coordinated responses and the direct impact of policy measures. The Mapper algorithm naturally produces overlapping nodes that capture transitional behaviors and multifaceted epidemic profiles, but certain quantitative comparisons with classical clustering methods require discrete assignments. For these analyses (such as computing the Adjusted Rand Index (ARI) against k-means and hierarchical clustering) each country is temporarily assigned to its smallest node ID when multiple memberships occur. This hardening is solely for validation and does not alter the underlying Mapper graph, which continues to preserve overlapping memberships; countries like Ireland, Germany, and Austria illustrate continuous, transitional epidemic behaviors. This approach enables both quantitative comparison and retention of TDA's ability to capture complex, overlapping epidemic structures.

A key revelation from our Mapper analysis is that topological similarity in pandemic trajectories does not consistently align with formal economic or geographic groupings (Eurozone, Schengen Area, Visegrád Group), a sentiment also found in Loeffler-Wirth et al. (37). This contrasts with the intuitive anticipation of greater epidemiological convergence or policy harmonization within such blocs due to shared governance structures, economic interdependence, or ease of movement. For instance, while some research has highlighted the potential for cross-border transmission and the need for coordinated responses within integrated regions (38), our TDA approach, which focuses on the shape and evolution of epidemic waves, suggests that the intrinsic dynamics of the pandemic within each country may be more strongly influenced by national-level factors than by these broader affiliations. This is particularly evident in the resulting clusters (Figures 6, 7), where countries from diverse regions or economic groups are grouped together. These overlapping and transitional epidemic profiles cannot be adequately captured by classical clustering methods (for example, k-means or hierarchical clustering), which enforce strict and disjoint partitions. This aligns with a growing body of literature emphasizing the heterogeneity of pandemic responses and outcomes, even within seemingly homogeneous regions, pointing to the significant roles of local context, public trust in institutions, and specific national healthcare capacities (39). Our results suggest that while formal ties exist, their influence on the dynamics of the pandemic, as captured by Mapper, might be less direct or overshadowed by other, unmeasured internal factors.

Furthermore, our statistical tests (ANOVA and permutation-based non-parametric tests) revealed no significant variation in average policy stringency between the identified topological clusters. This is a particularly intriguing finding, as it challenges the intuitive expectation that countries exhibiting similar wave profiles would have adopted comparable levels of government intervention, or that highly stringent policies would consistently lead to distinct epidemiological outcomes. Many studies have utilized stringency indices to correlate policy rigor with infection rates or mortality [for example in Hale et al. (10) and Islamaj et al. (40)]. However, our results suggest that the average stringency score alone may not be the primary driver shaping the temporal profile of epidemic waves. Quantitative measures of cluster quality, such as the adjusted Rand index and silhouette scores, indicate that Mapper's overlapping clusters capture transitional dynamics more effectively than classical methods, which tend to produce artificially low silhouette scores when intermediate behaviors are forced into single partitions. This could imply several possibilities: (1) the timing and duration of specific interventions might be more critical than their overall average stringency (41); (2) public adherence and behavioral responses to policies, which are not directly captured by the stringency index, play a substantial role in shaping wave dynamics (42, 43); (3) other confounding factors, such as population density, age demographics, pre-existing immunity, or the prevalence of specific viral variants, could exert a stronger influence on wave shapes (44–46), potentially masking the direct impact of average stringency. This reinforces the idea that the effectiveness of government interventions is multifaceted and cannot be reduced to a single index.

The consistent appearance of certain countries (notably Ireland and Estonia) across multiple clusters and configurations further reinforces the idea of transitional or multifaceted pandemic profiles (47). These overlapping assignments illustrate the interpretability of the Mapper graph, showing continuous relationships between countries that classical clustering cannot represent. Visual inspection of the Mapper network highlights transitional nodes, offering actionable insights into countries with hybrid epidemic dynamics or mixed policy responses. This flexibility in categorization is a strength of the Mapper algorithm, allowing for the identification of overlapping and continuous structures that traditional clustering methods might miss.

The findings indicate that topological similarity in COVID-19 trajectories is often independent of formal economic or geographic groupings, suggesting that relying solely on institutional affiliations for coordinated responses may be insufficient. Policymakers should focus on real-time epidemiological signals (such as growth rates, wave intensity, and transitional cluster membership) when designing interventions, rather than assuming uniformity within economic or regional blocs. The presence of overlapping epidemic profiles highlights the need for adaptive, flexible policies that can be rapidly adjusted as countries move between different topological clusters. Scenario-based policy frameworks, which trigger interventions according to dynamic epidemic indicators rather than static memberships, may improve responsiveness and reduce cross-border transmission risks. Moreover, the decoupling of wave similarity from average policy stringency emphasizes the importance of timing, enforcement, public adherence, and local context in shaping outcomes. Investments in surveillance, data-sharing, and inter-country communication become critical for preemptively identifying transitional dynamics and coordinating timely interventions. In sum, these insights advocate for a more nuanced, data-driven approach to pandemic governance that complements formal institutional mechanisms.

Despite providing valuable insights, this study has several limitations. First, the analysis is limited to 15 European countries, which may restrict generalizability to other regions or a broader global context. COVID-19 case counts vary across countries due to differences in testing intensity, testing policies, and reporting delays, potentially affecting the accuracy of inferred wave dynamics. Second, while we initially considered imputation for missing data, the final datasets were complete with no missing values after truncation at the end of 2022, eliminating the need for imputation. Third, reliance on the Oxford stringency index, which measures de jure policy strictness, does not capture actual compliance or enforcement and may attenuate observed associations. Fourth, the use of wave-averaged stringency could obscure the impact of time-sensitive interventions such as school closures or mask mandates; future research could incorporate more granular policy data, vaccination rates, variant prevalence, and mobility metrics.

Also, while TDA reveals structural relationships among countries' trajectories, it does not establish causality, and the relatively small number of countries warrants caution in interpreting Mapper nodes. These considerations should guide the interpretation of cross-national comparisons and the drawing of policy implications. Even if epidemic peaks occurred at different times across countries, we initially applied uniform calendar-based wave boundaries to facilitate cross-country comparison. To ensure that this choice did not bias results, we performed sensitivity analyses using country-specific wave segmentation based on changepoint detection. The resulting Mapper clusters and topological structures were largely consistent (adjusted Rand index >0.85), indicating that our main findings are robust to differences in wave timing. Nevertheless, we acknowledge that uniform wave boundaries may introduce minor distortions in the representation of individual country trajectories.

Furthermore, we conducted comprehensive parameter sensitivity analyses for the Mapper algorithm, systematically varying cover sizes (10, 12, and 15 cubes) and overlap percentages (40%, 50%, and 60%). Cluster stability assessed via Jaccard similarity showed high consistency (average >0.92) across parameter variations, with the optimal combination (12 cubes, 50% overlap) selected for the final analysis. This confirms that the observed transitional behaviors represent genuine epidemic dynamics rather than parameter-driven instability. Additionally, robustness checks for normalization methods (comparing Min-Max, Z-score, Robust, and logarithmic transformations) demonstrated that linear normalization approaches produce highly consistent feature spaces (Pearson correlations ≥0.95, adjusted Rand index = 1.0), and the use of correlation distance was justified based on its focus on pattern similarity independent of magnitude. Despite these robustness checks, we acknowledge that preprocessing choices and the relatively small sample of countries may affect cluster composition and topological interpretations. Readers should therefore interpret the observed structures as suggestive patterns rather than definitive causal relationships.

The finding that topological similarity in pandemic trajectories does not consistently align with formal economic or geographic affiliations carries significant policy implications, echoing the work in Quaglia and Verdun (31) and Lavery and Schmid (48). It suggests that the efficacy of public health coordination should not rely solely on existing institutional frameworks (like the Eurozone or Schengen Area), but rather on real-time epidemiological alignment [see also Stroisch et al. (49)]. Policymakers must be prepared for dynamic clustering, where nations temporarily share epidemic profiles even without a strong political or economic bond. This mandates the creation of flexible, scenario-based policy playbooks that trigger coordinated responses based on a country's epidemiological state, rather than political membership (50). Furthermore, the observation of transitional epidemic profiles implies that policy measures must be continuously adaptive, allowing for rapid shifts in strategy as a country moves between topological clusters. We acknowledge that cross-country differences in age structure, health-system capacity, vaccination coverage, variant dominance, behavioral compliance, and data quality likely influence epidemic outcomes. Our analysis focuses on epidemic waves and policy stringency, and interpretations of cluster-level differences should be viewed as partial rather than comprehensive explanations.