Mathematical modeling has become an important tool for understanding and controlling infectious diseases. It allows researchers to simulate and predict the spread of diseases, identify key transmission factors, assess the impact of interventions, and inform decision-making for disease control. Researchers can use mathematical models to explore different scenarios, evaluate the effectiveness of various interventions, and estimate the potential outcomes of different control strategies. This enables policymakers and public health professionals to make informed decisions and implement targeted measures to mitigate the impact of infectious diseases. In addition, mathematical models can be used to examine and evaluate the effectiveness of control strategies such as vaccination, social restrictions, and drug use. The results show that prevention strategies such as population vaccination and social restrictions can significantly help reduce the spread of influenza. This article presents a mathematical model for Influenza A (H1N1), as well as two other models specifically for Influenza A (H1N1) after quarantine and treatment. The purpose of the article is to provide a brief review of these models and compare them.

Air pollution remains a critical public health and environmental issue in megacities, with fine particulate matter (PM2.5) posing significant risks in Tehran, Iran. Traditional modeling methods often fail to capture the non-stationary and multiscale characteristics of urban air pollution dynamics. This study introduces a wavelet-based analytical framework to uncover both long-term seasonal trends and short-lived pollution events using a seven-year dataset (March 2017–March 2024) from nine monitoring stations across Tehran. We applied the Continuous Wavelet Transform (CWT) with Morlet and Daubechies 6 (db6) wavelets to decompose the PM2.5 time series into multiple temporal scales. The Morlet wavelet identified persistent winter peaks associated with temperature inversions and atmospheric stagnation, while db6 revealed abrupt, high-frequency pollution spikes linked to localized traffic emissions and industrial activity. Wavelet coherence analysis further illustrated the relationship between meteorological variables such as temperature, east-west winds, and north-south winds and PM2.5 concentrations, highlighting how these factors modulate air quality across different spatial and temporal scales. While ARIMA demonstrated stronger point-forecasting accuracy, the wavelet-based framework provided superior interpretability by preserving both gradual seasonal cycles and transient anomalies simultaneously structures that conventional models tend to smooth out. This comparison was conducted not to establish forecasting superiority, but to highlight the distinct and complementary role of wavelet analysis as a diagnostic tool for pollution hotspot identification and early-warning systems rather than pure numerical prediction. These findings underscore the value of wavelet analysis as not only an advanced analytical tool but also a decision-support system for real-time air quality monitoring and spatially targeted policy planning. By capturing the full spectrum of PM2.5 dynamics from chronic seasonal exposure to acute pollution spikes, this approach offers a more comprehensive framework for adaptive urban air quality management in complex environments like Tehran.