Fractional Stochastic Differential Equation Approach for Spreading of Diseases

Area of Science:

• Epidemiology and Mathematical Modeling
• Statistical Analysis of Time Series Data
• Stochastic Processes in Disease Dynamics

Background:

• The COVID-19 pandemic presents complex dynamics, particularly in high-incidence countries like Brazil.
• Understanding the time evolution and future trends of infectious diseases requires robust mathematical and statistical approaches.
• Official case data often exhibits significant daily fluctuations, necessitating methods that can account for this randomness.

Purpose of the Study:

• To apply a nonlinear fractional stochastic differential equation model to analyze COVID-19 infection dynamics.
• To investigate the Hurst parameter (H) and its role in modeling the time evolution of coronavirus cases.
• To project future novel case numbers by analyzing quadratic mean deviation and probability density functions.

Main Methods:

• Utilizing a nonlinear fractional stochastic differential equation framework.
• Incorporating fractional Brownian motion to model random fluctuations in daily case data.
• Employing Rescaled Range (RS) analysis to determine the Hurst index (H) for time series data.
• Performing statistical tests to ascertain the future probability density of novel cases.

Main Results:

• The study successfully applies the fractional stochastic differential equation approach to model COVID-19 case evolution in Brazil.
• The Hurst parameter (H) was determined for the time series of novel cases, providing insights into data persistence.
• The quadratic mean deviation was used for future case projections, offering a statistical estimation of uncertainty.
• Statistical tests were conducted to explore the probability distribution of future case numbers.

Conclusions:

• The nonlinear fractional stochastic differential equation model provides a viable method for studying and projecting infectious disease dynamics, particularly with noisy data.
• The Hurst parameter is a crucial indicator for characterizing the memory and long-term behavior of epidemic time series.
• Future COVID-19 case trends can be statistically projected, aiding in public health preparedness and resource allocation.