1. Introduction
The COVID-19 pandemic has presented an unprecedented global challenge, characterized by a continuous surge of new infections and the emergence of distinct viral variants. The sheer volume and granularity of data collected during this period — encompassing daily case counts, geographical spread, demographic impact, and mortality rates — provide a rich resource for understanding the intricate dynamics of infectious disease transmission
[1][2]
Different viral strains exhibit varied transmissibility, virulence, and immune evasion properties, leading to the manifestation of epidemic processes at diverse scales. Short-term fluctuations might reflect localized outbreaks driven by a highly transmissible variant, while longer-term trends could be influenced by the introduction of new variants, the impact of public health interventions, or seasonal patterns
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This is where wavelet analysis
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This paper explores the application of wavelet analysis to detailed COVID-19 epidemiological data, demonstrating its efficiency in uncovering intricate patterns and providing a deeper understanding of the multi-scale nature of the pandemic, particularly as influenced by the evolution of its constituent viral strains.
2. Research methods and principles
The
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The core idea behind the wavelet transform is to decompose a signal into a set of basis functions (wavelets) that are translated and scaled versions of a single “mother wavelet.” The output of the wavelet transform is a set of wavelet coefficients, or a field of continuous wavelet transform field. These coefficients represent the degree of similarity between the signal and the chosen wavelet at a particular scale (inversed frequency) and time location. A large coefficient indicates that the wavelet at that specific scale and time is a good representation of the signal at that point.
Continuous wavelet transform for a given signal
[LATEX_FORMULA]CWT[x](\tau,t)=\frac{1}{|a|^2}\int_{-\infty}^{\infty}x(t')\overline{\psi}\left(\frac{t'-t}{\tau}\right)\mathrm{d}t[/LATEX_FORMULA]
Here
The key advantages of wavelet transform are time-frequency localization, multi-resolution analysis, adaptability for different signal types and successful detection of transient features even for purely-resolved digital sampling.
The wavelet transform is connected with a
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[LATEX_FORMULA]TFD[x](f,t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}x(t')e^{i 2\pi ft'}w(t'-t)\mathrm{d}t'[/LATEX_FORMULA]
where
Based on the wavelet transform,
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The WST operates through a series of cascaded layers, each employing a wavelet transform followed by a non-linear operation. This layered structure allows for the progressive capture of increasingly complex signal properties.
At the foundation of the WST lies the zero-order scattering coefficient
represents the coarse, low-frequency component of the signal, essentially capturing its overall structure without fine details.
The first layer of the WST generates the first-order scattering coefficients
A key principle underpinning the WST’s ability to capture multi-scale information is the exponential expansion of scales. The wavelet filters are applied at scales that increase exponentially (or according to a geometric progression). This ensures that the decomposition covers a wide range of temporal or spatial resolutions, from very fine details to very coarse structures. This exponential expansion is crucial for capturing the full spectrum of underlying phenomena, from rapid fluctuations to long-term trends, which is particularly relevant when analyzing signals with diverse temporal characteristics, such as epidemiological data influenced by different viral strains operating at different timescales.
Subsequent layers of the WST build upon the information captured in the previous layers. For instance, the second-order scattering coefficients (
3. Main results
As a data, the new cases distributed by the regions of Russia will be used. An example is represented on Figure 1. The data has typical oscillations with a sharp peak in the beginning of 2022 corresponding the Omicron strain appearance.
Overall behaviour of new cases in Moscow
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Spectrogram for Moscow data
Wavelet scalogram for Moscow
To compare the difference in the behaviour in different regions, the wavelet-coherence
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Wavelet-coherence for Moscow as refercnce region
At the foundation of the WST lies the zero-order scattering coefficient
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Illustration of S0 work and different smoothings for Moscow data
The idea of the application is the following: to compare the local behaviour on the different time scales using
S1-coherence averaged by reference regions
S1-coherence averaged by reference and target regios
To reconstruct the behavior of this multi-component system, we employ Principal Component Analysis (PCA)
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Figure 8 illustrates the evolution of new case data for Moscow, with the trajectory colored by time. The Omicron outbreak is clearly visible as a distinct deviation from the primary trajectory (indicated in light green, corresponding to January 2022). Outside of this period, the system exhibits quasi-cyclic behavior, highlighting the stability of the long-term epidemic trends. Note that axes PC1Missing Mark : sub and PC2 Missing Mark : subdo not have the direct sense because they are the main directions of the PCA which the phase space is projected on. However, it does not interfere the visualisation.
System behaviour dynamic in Moscow visualization with PCA
Overall behaviour dynamic visualization with PCA
4. Conclusion
This study successfully leveraged the power of wavelet analysis to unravel the complex, multi-scale dynamics of COVID-19 epidemiological data. By applying wavelet transforms, we were able to decompose the temporal series of infection rates and other relevant metrics into their constituent frequency components, revealing patterns that operate across distinct temporal scales.
A key contribution of this research lies in the application of wavelet coherence analysis. This powerful technique allowed us to not only identify the presence of correlations between different epidemiological signals (e.g., case counts across regions, or case counts and intervention timelines) but also to pinpoint the specific ranges of temporal scales where these correlations are maximized. This finding is crucial, as it suggests that epidemiological processes, driven by factors such as the emergence and spread of specific viral strains, tend to manifest with similar characteristics at certain scales.
The identification of these “privileged” scales of correlation has significant implications for epidemiological modeling and intervention strategies. It suggests that models developed for understanding disease dynamics in one region may indeed be generalizable and transferable to other regions, provided they are applied at these identified scales. This is because the underlying drivers of epidemic spread that are most strongly correlated across different locations appear to operate within these specific frequency bands. Consequently, understanding the dynamics at these particular scales can offer a more robust foundation for predictive modeling and the development of effective public health responses that are less geographically constrained.
Additionally, the PCA projection of the dynamic system is applied and used for the visualization of the system's behaviour. It is shown that it can be successfully used for detection of the anomal behaviour in the data alongside the wavelet transform.