We employed a retrospective spatial analysis of COVID-19 clusters across the United States by examining how case distributions correlate with population density. Utilizing a Poisson-based spatial scanning model, we identified clusters by employing a cylindrical scanning window to capture areas of heightened risk relative to surrounding regions. The analysis applies likelihood ratio testing to detect significant spatial clusters, with primary and secondary clusters reported based on statistical significance. Monte Carlo simulations were used to validate the results, offering insights into the geographic spread and demographic impact of COVID-19 across different communities in the U.S.

We obtained COVID-19 mortality data for the United States from The New York Times, which provides daily records of COVID-19 deaths from February 2020 onward. This data, publicly available at [44].

The dataset reports daily COVID-19 deaths across all U.S. states and territories. In many states, health departments reconcile these records with death certificates, omitting deaths where COVID-19 was not the cause of death. Following this protocol, non-COVID-19 deaths were removed from the dataset when there was a confirmation that COVID-19 was not the cause, specifically in cases of homicide, suicide, car accidents, or drug overdoses. Details of these adjustments can be found in [44].

To examine significant variations in COVID-19 mortality trends, we focused on data spanning from May 24, 2020, to March 12, 2023. We divided this period into seven time intervals to analyze trends within consistent segments of time: May 24, 2020 − September 13, 2020; September 13, 2020 − March 14, 2021; March 14, 2021 − June 13, 2021; June 13, 2021 − October 31, 2021; October 31, 2021 − March 13, 2022; March 13, 2022 − October 16, 2022; and October 16, 2022 − March 12, 2023. Each interval represents a critical phase in the pandemic, allowing us to study temporal fluctuations in COVID-19-related mortality rates (see Table 1 for a summary). Specifically, these intervals were selected based on the emergence of new SARS-CoV-2 variants, which played a pivotal role in shaping the trajectory of the pandemic. This approach aimed to capture virological and epidemiological shifts corresponding to significant phases of the outbreak. As shown in Figure 2 (a), each interval, except the third, includes at least one peak in COVID-19 mortality within the United States. The third interval corresponds to the emergence of the Alpha variant, which did not result in a mortality peak during this period but contributed to a significant peak in the subsequent interval. While this method highlights variant-driven dynamics, we acknowledge that alternative approaches, such as structural change detection methods [22], could potentially offer a data-driven determination of breakpoints. Incorporating such methods in future studies could further refine the interval selection process and enhance the robustness of predictive modeling.

To analyze the geographical distribution of COVID-19 cases adjusted for population demographics, we applied a Poisson model. Our spatial analysis utilized a cylindrical scanning approach, with a moving circular base representing the spatial dimension, and set a maximum cluster size of 25% of the population at risk to prevent excessively large clusters. This approach created numerous overlapping windows of varying sizes, allowing full coverage of the study area. Each window was considered as a potential cluster. Under the null hypothesis H0, the COVID-19 risk is uniform inside and outside the scanning window, while the alternative hypothesis Ha suggests a higher risk within the window. Each cylinder expanded up to a preset upper limit on cluster size. To identify clusters, we applied a likelihood ratio test.

The likelihood ratio L(C)L0 is calculated by (1) L(C)L0=ncμ(c)ncN-ncN-μ(c)N-ncNμ(T)N, where L(C) is the maximum likelihood for the cylinder based on the observed cases nc and expected cases μ(c) within the window. Here, L0 is the likelihood under the null hypothesis, N is the total cases observed in the U.S. for each time interval, and μ(T) represents the total expected cases. A likelihood ratio greater than one indicated that observed cases exceeded expectations. The cluster with the highest likelihood was labeled the primary cluster, while other clusters were reported if they achieved significance at p<0.05. P-values for space-time clusters were determined through Monte Carlo simulations, using 999 randomized iterations of the data to assess statistical significance [15].

Figure 1 illustrates the significant spatial clusters of COVID-19 mortality across the United States. High-risk clusters, characterized by a relative risk greater than one, are represented by red circles, while low-risk clusters, with a relative risk less than one, are indicated by blue circles. Panel (a) shows the significant spatial clusters for the entire study period; however, it may not adequately reflect the temporal changes across the defined time intervals from 1 to 7. Panels (b) through (h) correspond to the spatial data associated with intervals 1 through 7, respectively. Panel (b) shows that significant high-risk clusters were initially concentrated in the southern regions of the U.S. As shown in panel (c), these high-risk clusters extend across much of the country, with the notable exceptions of certain areas in the East and Northwest, which may be attributed to lower vaccination rates. Panel (d) illustrates the impact of extensive vaccination efforts, which have proven to be substantially effective. The emergence of the Delta variant is highlighted in panel (e), demonstrating its influence on the overall mortality trends. The effectiveness of vaccination updates and booster shots is evident in panel (f) when it is compared with panel (e). Panel (g) indicates that the significance of the Omicron variant has become pronounced in the central regions of the U.S. Finally, panel (h) reveals a marked reduction in the prevalence of significant high-risk clusters, underscoring the impact of vaccination and public health interventions on COVID-19 mortality rates.

Figure 2 shows the dynamics of COVID-19 mortality trends and the geographical distribution of risk clusters in the U.S., highlighting key patterns in mortality fluctuations, spatial risk persistence, and transitions between risk classifications. Panel (a) illustrates the time series of COVID-19-related mortality in the U.S. from May 24, 2020, to March 12, 2023, segmented into seven distinct periods. Panel (b) shows the percentage of the U.S. area covered by significant high-risk clusters (i.e., relative risk greater than 1) and low-risk clusters (i.e., relative risk less than 1) for each of the seven intervals. It suggests that both high-risk and low-risk clusters may gradually reach an equilibrium below 20% of the U.S. area, indicating a stable distribution of risk across the country. Panel (c) examines the persistence of high-risk and low-risk clusters from time interval i to interval i+1. It demonstrates that approximately 45% of areas classified as high-risk and about 40% of areas classified as low-risk maintain their status, particularly from time interval 5 onward, suggesting a notable consistency in spatial risk distribution over time. Panel (d) illustrates the percent transition of clusters from high-risk to low-risk and vice versa between consecutive intervals. This transition appears to stabilize around 15%, indicating a moderate but consistent change in the risk classification of certain areas over the specified periods.

Table 2 presents a detailed summary of spatial high-risk and low-risk clusters of COVID-19 mortality across intervals 1 through 7. The second and third columns list the number of high-risk and low-risk clusters identified in each interval. Columns 4 through 7 provide the percentage of the total U.S. area covered by these clusters. Notably, in interval 4, high-risk clusters encompassed more than 50% of the U.S. area.

Table 3 details the transitions between high-risk and low-risk COVID-19 mortality clusters, as well as the persistence of clusters across intervals from i to i+1. Two percentages are provided: the first represents the overlap of clusters from interval i with those in interval i+1, and the second indicates the overlap of clusters in interval i+1 with interval i. Notably, the values approach steady-state levels over time.

Figure 3 represents the accuracy of identified spatial clusters. Panel (a) shows the observed-to-expected mortality counts plotted against the estimated relative risk for intervals 1–7; panel (b) provides box plots of residuals, indicating low residuals; panel (c) illustrates the mortality data within each identified cluster, giving a localized perspective on mortality distribution; and panel (d) displays the estimated relative risks associated with spatial clusters for each interval, showing the changes in risk levels over time. The observed-to-expected mortality counts and the relative risks of significant spatial clusters are available in the supplementary document Tables 4 and 5, respectively.

In tackling the challenge of integrating the entire cluster search and prediction process into a manageable model grounded in rigorous statistical analysis, we propose that analyzing trends in transition probabilities across observed clusters over k intervals could enhance the accuracy of our predictions for interval k+1. Given that k is relatively small (between 5 and 10), we employ a predictor-corrector method. The predictor utilizes a Markov chain model [29, 47], while the corrector optimally chooses between multiple linear regression and exponential smoothing methods based on the distance between transition matrices, aiming for an effective prediction of clusters in interval k+1.

Our approach starts with the establishment of initial datasets, labeled T1, each representing a specific time interval. As the pandemic progresses, additional datasets T2,T3,T4, and so on are created, forming a sequence that captures different phases of the pandemic. The collection Tk+1 denotes the next set of real data gathered at a subsequent interval. Our goal is to predict the parameters of this new collection Tk+1 using insights from the preceding datasets T1,T2,…,Tk.

Traditionally, predictions for mortality rates have relied solely on the most recent dataset, Tk. However, we recognize that the characteristics of the upcoming dataset Tk+1 may differ from those represented by Tk. To improve predictive accuracy, we advocate for a broader analysis of the trends in transition probabilities observed across the series of datasets T1,T2,…,Tk.

Understanding the patterns and shifts in transition probabilities over time can provide valuable insights into the evolving dynamics of the pandemic. By examining how these probabilities change, we aim to capture underlying trends that may enhance the accuracy of predicting the parameters for the next dataset, Tk+1. This approach contrasts with conventional methods, suggesting that a more nuanced analysis of historical data could lead to a more robust and reliable predictive model for mortality rates in the ongoing fight against COVID-19.

Let Tk+1* represent the prediction for Tk+1. Predicting T1 is impractical because there is no initial data available before T1. Similarly, predicting T2 is challenging due to the presence of only one preceding data set, which does not provide sufficient patterns or trends. However, Tk* can be predicted for k>2 using the data sets Tk-1…T1. This means that historical data can be maintained as prediction chains throughout the data collection process. More importantly, since predictions are made for each dataset as it becomes available, any inaccurate predictions can be discarded, offering valuable feedback on the quality of the predictive process.

Our proposed method corrects the Markov chain prediction of Tk+1 utilizing either multiple linear regression or exponential smoothing at each iteration k. The predictor, denoted as Tk+1*, is formulated as follows: (2) Tk+1*=α*Tk+1-α*Tk*, which is a weighted sum between a “good corrector” T*k and data Tk. The “good corrector” T*k of Tk based on transition matrices from previous datasets T1,T2,…,Tk-1, whereas the predictor is given by α*Tk.

Specifically, let Tˆk be a predictor of Tk determined through multiple linear regression, and let T˜k be a predictor of Tk determined via exponential smoothing. T˜k depends on α*, an optimal value satisfying 0<α*<1, and is determined by (3) ∑i=3kd(T˜i(α*),Ti)=min0<a<1{∑i=3kd(T˜i(α),Ti)}, Tk* is chosen such that (4) dTk*,Tk=mindTˆk,Tk,dT˜k,Tk. where dTk*,Tk denotes a distance from transition matrix Tk* to matrix Tk. We will use a distance measure of the form (5) dTk*,Tk=∑m∑nTk*(m,n)-Tk(m,n)Tk(m,n)+1 Here, m represents the matrix row index, while n denotes the matrix column index. This distance metric accounts for relative differences rather than absolute differences. See Ref. [38] for more details.

As a result, we derive the parameters of Tk+1* by utilizing the previous dataset Tk, which is established with real data, and Tk*, a predictor for Tk that encapsulates historical information from the datasets T1,…,Tk-1. Throughout this process, we eventually obtain the actual Tk+1 and compare it to our estimate Tk+1* using the aforementioned distance measure. This method allows us to assess the accuracy of our predictions with each successive data collection. Importantly, the distance measure can reveal whether certain datasets are outliers for various reasons, enabling us to exclude those data points from our analysis.

The ultimate goal is to determine Tk+1*, for which no corresponding Tk+1 will be available, especially when Tk represents the final data collection before making a prediction. Thus, Tk+1* serves as our prediction for practical use, representing our estimate in the absence of actual observed data for Tk+1.

We employ one of two methods to predict Tk+1* based on the transition matrices T1,T2,…,Tk. The value T˜k is calculated using exponential smoothing, while Tˆk is obtained through multiple linear regression. The method that results in the smaller distance measure, as defined earlier, is chosen for the prediction.

As described in Section 3, the predictor of chain T6 has the form T˜6=αT5+(1−α)T˜5⋅⋅⋅⋅⋅⋅⋅⋅⋅T˜3=αT2+(1−α)T^2T^2=1.568+0.131T1 where Tˆ2 is a predictor of T2, given T1, determined through simple linear regression. T˜6α* is chosen so that ∑i=36dT˜iα*,Ti=min0<a<1∑i=36dT˜i(α),Ti, where the distance d is defined as in Section 3. represents the values of D(α)=∑i=36dT˜i(α),Ti. The minimum occurs for α*=0.02 giving T˜6=0.02T5+0.98Tˆ5 Also, note that dT˜6,T6=∑m=16∑n=16T˜6(m,n)-T6(m,n)Tk(m,n)+1=11.756

Multiple linear regression relates transition matrix T6 to T1,…,T6 in one linear functional form. Results in least squares coefficients are presented in Table 2.

The prediction equation is Tˆ6=0.5T1+6.357T2−0.561T3+12.287T4−8.645T5 d(Tˆ6,T6)=∑m=16∑n=16|(Tˆ6(m,n)−T6(m,n))|Tk(m,n)+1=5.887 which is smaller than dT˜6,T6 The better predictor, therefore, of T6 is Tˆ6 Consequently, the predictor of T7 is (9) T7*=0.02T6+0.98Tˆ6

Figure 4 shows the estimation of the optimal parameter α, and the prediction accuracy of relative risks for interval 7 across different models. Panel (a) shows the determination of the optimal α by minimizing the distance between estimated and observed relative risk values. Panel (b) compares observed versus predicted values (on a log scale) for the proposed Markov chain model and the Exponential Smoothing. The predictions of the relative risks look promising but have room for improvement. The proposed model demonstrates superior predictive performance compared to the Exponential Smoothing method, as evidenced by a lower Mean Squared Error (164.8 vs. 258.7).