A Scaling Analysis of Influenza Transmission in Nepal

Over the last few decades, Influenza has emerged as a significant public health threat in Nepal, marked by a consistent rise in infection rates. To develop successful mitigation strategies, it is vital to understand the underlying mechanics of how these outbreaks evolve. One sophisticated way to study these trends is through scaling methods, which utilize mathematical concepts like scaling exponents and self-similarity to map the behavior of the virus. Self-similarity occurs when a structure is composed of smaller parts that statistically mirror the whole. In simpler terms, the pattern looks the same regardless of how much you "zoom in" or "zoom out." In the context of Nepal’s Influenza data, self-similarity suggests that the fluctuations we see on a small scale (daily or weekly case counts) reflect the broader patterns seen on a large scale (seasonal or yearly cycles). By identifying these recurring structures, health officials can:

• Improve Forecasting: Predict future outbreaks by recognizing small-scale signals.
• Optimize Resources: Allocate medical supplies more efficiently based on recognized patterns.
• Refine Interventions: Create strategies that address both sudden spikes and long-term trends.

In epidemiology, memory refers to how much previous data points influence future trends. This is often categorized into two types:

1. Long Memory (Long-Range Dependence)

This occurs when current infection rates are significantly shaped by events from months or even years ago. If Influenza has a "long memory," it suggests that the disease is driven by persistent factors such as:

• Deep-rooted social behaviours.
• Long-standing environmental conditions.
• The lasting impact of historical public health policies.

2. Short Memory (Short-Range Dependence)

In this scenario, current cases are only influenced by the very recent past. The correlation between data points fades quickly. If an outbreak has "short memory," it implies the disease is highly sensitive to immediate changes, such as a sudden weather shift or a recent localized event, rather than long-term trends. Nygård and Glattre (2003) utilized fractal analysis to reveal hidden patterns in epidemiological time series, an approach useful for detecting trends in Influenza case data. Eco-epidemiological analyses by Chakravarti and Kumaria (2005) and Raheel et al. (2011) provided insights into the factors influencing Influenza spread in India, essential for understanding the context of scaling analysis. Dias (2013) discussed self-affinity and self-similarity in ecological contexts, aiding in the modeling of self-similar Influenza case distributions. Saha et al. (2019) examined the relationship between irregularity and scaling indices in self-similar and self-affine signals, offering insights into the irregularities and scaling properties of Influenza trends. Ghosh (2020) applied scaling analysis to COVID-19 cases, demonstrating the effectiveness of these techniques in epidemiological studies. Time series forecasting using machine learning has also been explored by Bodapati et al. (2020) and Lestari et al. (2022), who employed Long Short-Term Memory (LSTM) networks to predict COVID-19 and Influenza cases, respectively. LSTM networks' ability to capture temporal dependencies makes them suitable for predicting future Influenza cases. Abdulwasaa et al. (2021) and El-Dessoky and Khan (2022) applied fractal-fractional models to epidemic forecasting, showing their potential for predicting Influenza outbreaks. Li et al. (2022) enhanced Influenza forecasts by integrating geospatial big data and historical information, highlighting the importance of comprehensive data for accurate predictions. Raubitzek et al. (2023) investigated scaling exponents in time series data through machine learning, providing methodologies relevant to analyzing Influenza trends. These studies collectively offer a robust foundation for using scaling approaches to analyze Influenza trends, demonstrating the applicability of various methods to understand and predict the spread of Influenza, crucial for developing effective public health strategies in Nepal. This research examines the daily fluctuations of Influenza cases in Nepal through a scaling framework, specifically focusing on self-similarity and self-affinity. By utilizing scaling analysis and calculating the Hurst exponent (H), the study identifies latent patterns within the time series that conventional statistical methods often overlook. The Hurst exponent serves as a critical metric to quantify the "memory" and "roughness" of the data, revealing the extent to which past trends influence future outbreaks. Our findings confirm that daily incidence rates follow self-affine patterns, providing a more nuanced understanding of Influenza epidemiology. These insights are then discussed in the context of public health, offering a scientific basis for more robust monitoring and targeted intervention strategies in Nepal.

2. Theory:

2.1 Hurst Exponent

The Hurst exponent (H) measures the memory of a self-similar time series reflecting its tendency either to regress strongly to the mean or to cluster in a certain direction. For a self-similar time series, it ranges between 0 and 1. The following are its implications at different ranges.

H = 0: Represents white noise, where data points are completely uncorrelated.

0 < H < 0.5: Indicates anti-persistent behaviour (negative autocorrelation), where the time series exhibits short memory.

H = 0.5: Indicates a true random walk (Brownian motion), with no memory.

0.5 < H < 1.0: Indicates persistent behaviour (positive autocorrelation), suggesting long-term memory.

H = 1: Represents a perfectly smooth time series.

2.2 Scaling Analysis and Estimation of Hurst Exponent

Scaling analysis investigates how statistical features of data change with the scale of observation. It is employed to comprehend the fundamental patterns and behaviours that hold true throughout various temporal or spatial ranges. For a given finite time series xtn  (where n=1,2, N and t represents time), The finite Variance Scaling Method (FVSM), specifically Standard Deviation Analysis (SDA), is used to estimate H. The sequence of cumulative standard deviations Dtj  associated with the partial time series xtn  (for n=1,2, j) is calculated as:

Dtj?=1jn=1jx2tn-1jn=1jxtn212

For self-similar time series, Dt  follows a power law:

D(t)∝tH

Here, H can be estimated from the slope of the best-fitted straight line in the log-log plot of D(t) versus t.

RESULTS

In this study, we analyze the monthly number of new affected cases of Influenza in the most affected Districts of Nepal: Bhaktapur, Palpa, Bhojpur, Dhading, Makwanpur, Chitawan, Rupandehi, Katmandu. The data for this analysis has been sourced from the Ministry of Health and Population Department of Health Services Epidemiology and Disease Control Division ([14]). Figures 1-8 illustrate the monthly profiles for states namely Bhaktapur, Palpa, Bhojpur, Dhading, Makwanpur, Chitawan, Rupandehi, Katmandu in Nepal from July, 2022 to June, 2024.

The Finite Variance Scaling Method (FVSM) has been utilized on these ten distinct time series. The resulting log-log graphs, which depict the cumulative standard deviation plotted against time (in months), are shown in Figures 9 - 16.

Table 1: Estimation of Hurst exponent of affected cases due to Influenza in mostly affected Districts of Nepal: