The Global Moran’s I index measures the overall spatial autocorrelation across the study area. Its mathematical formulation is as follows (Equation 1) (12):

The Global Moran’s I index ranges from −1 to 1, where n represents the sample size, W_ij denotes the elements of the spatial weight matrix, x_i and x_j are the observed values, and X ¯ is the mean value. Values between 0 and 1 indicate significant positive spatial autocorrelation (clustering of similar values), values between −1 and 0 demonstrate significant negative spatial autocorrelation (clustering of dissimilar values), while a value of 0 suggests a random spatial distribution, with statistical significance determined by the standardized Z-score (13) (Equation 2).

Here, E(I) indicates Moran’s coefficient’s expected value and var.(I) denotes its variance. Spatial aggregation of mumps incidence is indicated when, at the 0.05 significance level, Z > 1.96 (i.e., when Moran’s coefficient has a p-value <0.05).

LISA was employed to identify local spatial heterogeneity patterns, calculated as (Equation 3):

Using the same variable definitions as Global Moran’s I, LISA identifies four spatial patterns: high-high clusters (HH), low-low clusters (LL), high-low outliers (HL), and low-high outliers (LH) (14). These statistically significant patterns are visualized through LISA cluster maps (15).

To further identify statistically significant hotspots and coldspots of mumps incidence, we performed Getis-Ord Gi* analysis. This method calculates a Z-score for each spatial unit by comparing its value with neighboring units defined by the same spatial weight matrix used in Moran’s I analysis. Areas with Z > 1.96 (p < 0.05) were classified as significant hotspots (high-value clustering), while Z < −1.96 (p < 0.05) indicated coldspots (low-value clustering). The results were visualized in Gi* cluster maps, complementing the LISA analysis by emphasizing the intensity of spatial concentration patterns.

The factor detector serves as the core analytical module in the Geodetector method. Based on the spatial stratified heterogeneity theory, this method calculates the PD statistic (also called q-value) to quantify the independent explanatory power of each influencing factor (Equation 4) (20):

The statistical parameters include N_h (sample size of stratum h), σ_h² (variance of stratum h), N (total sample size), and σ² (total variance), with the PD value ranging between 0 and 1, where higher values indicate greater explanatory power of the factor for spatial differentiation.

The Risk Detector is primarily employed to identify and evaluate differences in risk levels across various stratified influencing factors. This method quantifies mean differences between strata using independent samples t-tests (Equation 5) (17).

Where Y ¯ h represents the average incidence rate of layer h, n h is samples, Var represents sample variance, t follows the Student’s-t-test distribution.

The null hypothesis as follows Equation 6:

If the null hypothesis is rejected at significance level α, we conclude that there is a significant difference in the average mumps incidence rate between the two regions.

The ecological detector is an ANOVA-based statistical method designed to assess significant differences in explanatory power between distinct influencing factors on spatial differentiation patterns of geographical phenomena. This method operates on the principle of F-statistic calculation to evaluate the significance of within-strata variance differences for two influencing factors (Equations 7–9).

Where n_x1 and n_x2 represent the samples of two factors x₁ and x₂, respectively. SSW_X1 and SSW_X2 represent the sum of the within-strata variance of x₁ and x₂, respectively; L₁ and L₂ represent the number of layers of x₁ and x₂, respectively.

The null hypothesis states that both factors equally explain the spatial variation. If the calculated F-value exceeds the critical threshold, we reject the null hypothesis, indicating statistically significant differences in explanatory power between the factors.

The Interaction Detector quantitatively assesses interaction types between paired influencing factors and their synergistic effects on spatial differentiation of dependent variables. By comparing single-factor PD-value with interaction PD-value, this module systematically identifies five characteristic interaction patterns (Table 2) (21): nonlinear-weakening, univariate-weakening, bivariate enhancement, independent, and nonlinear-enhancement effects.

Continuous variables in Geodetector analysis require discretization. Table 3 summarizes the five evaluated methods with their mathematical properties and applications.

The optimal method was selected by systematically comparing all approaches across 3–8 classification levels, choosing the combination maximizing the PD value (22). This data-driven strategy ensures objective identification of the most effective discretization for each variable.

The 12 independent variables were categorized using five distinct classification methods (equal interval, natural breaks, quantile, geometric interval, and standard deviation) with category numbers ranging from 3 to 8. For each variable, we calculated the Power of Determinant (PD) values through factor detector analysis, where PD quantifies the strength of association between an independent variable and mumps incidence (with values ranging from 0 to 1, higher values indicating stronger explanatory power). The optimal classification scheme for each variable was selected based on the maximum PD value, as detailed in Table 4.

Two healthcare resource factors—total health expenditure per capita and number of licensed physicians per 1,000 population—demonstrated p-values greater than 0.05, indicating no statistically significant association with mumps incidence.

As shown in Table 4, the PD values ranged from 0.31 to 0.54. The child dependency ratio (defined as the ratio of population aged 0–14 to the working-age population 15–64) showed the strongest association with mumps incidence (PD = 0.54). This suggests that regions with higher proportions of children relative to working-age adults may experience elevated mumps transmission risks, likely due to increased contact rates among school-aged populations. In comparison, healthcare accessibility factors such as the number of community health centers per 100,000 population and general practitioners per 10,000 population exhibited weaker associations (PD = 0.31 and 0.32, respectively).

The risk detector analysis revealed the average incidence rate of mumps across different stratification levels of each factor, along with an assessment of statistically significant differences between these strata. In this study, we take GDP per capita as an illustrative example. As illustrated in Figure 4, the relationship between GDP per capita and mumps incidence demonstrates a non-linear pattern. The lowest mumps incidence rate (5.88) was observed when GDP per capita ranged between ¥88,210.05 and ¥164,889.47, while the peak incidence (16.99) occurred in the ¥50,799.76–¥55,130.94 GDP per capita range. Notably, the minimum GDP per capita did not correspond to the highest mumps incidence, indicating a complex, non-monotonic relationship between economic development and disease prevalence.

Table 5 presents the significance test results for differences in average mumps incidence rates across different per capita GDP layers (Y indicates significant, N indicates non-significant). The fifth layer corresponds to the highest per capita GDP range (88,210.05 to 164,889.47 RMB) in Figure 3, while the first layer represents the lowest range (35994.81 to 50799.75 RMB). The results demonstrated statistically significant differences in mumps incidence rates between the second layer and all other layers.

Furthermore, the risk detector quantified the association between various factors and mumps incidence by identifying high-risk intervals for each determinant. The peak incidence of mumps corresponds to the high-risk intervals of each contributing factor (23). Therefore, Table 6 presents the identified high-risk intervals for all factors. The identification of these high-risk intervals enables public health authorities to implement more targeted surveillance systems and optimize intervention strategies. By focusing monitoring efforts when factors reach these critical ranges, health systems can enhance early detection and timely response to potential outbreaks. Furthermore, these thresholds serve as valuable reference points for evaluating intervention effectiveness and guiding resource allocation decisions in mumps prevention programs.

Epidemiologically, these high-incidence zones can be interpreted as primary impact area for each factor (21). These relationships have been visually represented in Figures 5A–J through geospatial visualization techniques. Figures 5A–J systematically illustrates the spatial characteristics of high-incidence zones for each determinant through geospatial visualization. The maps demonstrate the geographic extent of high-risk areas, providing visual context for the quantitative risk intervals identified in Table 6. These representations enable public health practitioners to correlate statistical risk thresholds with specific geographic settings, supporting the development of localized prevention strategies.

Our ecological detector analysis revealed important insights into the multifactorial nature of mumps transmission. The comparative analysis of potential differences (PD values) among key determinants (Table 7) showed no statistically significant differences between factor pairs (all p > 0.05, denoted by “N”), indicating no single factor demonstrated clear dominance in mumps incidence.

This finding carries three key implications: First, the ecological detector results demonstrate that mumps epidemiology is fundamentally shaped by interconnected networks of environmental, social and biological determinants, rather than any single dominant factor. Second, it underscores the need for integrated prevention strategies that simultaneously address environmental, social and biological determinants. Most significantly, this work establishes an urgent research priority: elucidating the precise mechanisms underlying these factor interactions will be essential for designing context-specific control strategies that account for local variations in mumps incidence.

Our interaction analysis uncovered meaningful patterns in how various factors jointly influence mumps incidence, with significant implications for public health practice. The consistent enhancement effects observed (all interaction PD values >0.5 in Table 8), particularly the strongest interaction between per capita GDP and illiteracy rate (PD = 0.88), demonstrate that combined factors exert greater influence on disease incidence than their individual effects would affect.

The identified nonlinear enhancement effects among economic indicators and healthcare resources (interactions between x₁ and x₁₁, x₁ and x₁₂) suggest a particularly important dynamic: investments in healthcare infrastructure become increasingly effective at reducing mumps incidence when implemented in conjunction with economic development initiatives. This nonlinear relationship implies that coordinated policy interventions addressing both economic conditions and healthcare access could yield disproportionately large public health benefits compared to isolated improvements in either domain alone.

Similarly, the bivariate enhancement effects observed among other factor pairs reveal synergistic relationships, where simultaneous improvements in two areas produce predictable combined benefits for disease control. These patterns collectively underscore that effective mumps prevention requires integrated public health strategies that account for how different determinants interact within specific contexts.