The COVID-19 pandemic has had a significant impact on countries worldwide, including the United Kingdom (UK). The UK has faced numerous challenges, but its response, including the rapid vaccination campaign, has been noteworthy. While progress has been made, the study of the pandemic is important to enable us to properly prepare for future epidemics. Collaboration, vigilance, and continued adherence to public health measures will be crucial in navigating the path to recovery and building resilience for the future. In this article, we propose an overview of the COVID-19 situation in the UK using both mathematical (a nonlinear differential equation model) and statistical (time series modeling on a moving window) models on the transmission dynamics of the COVID-19 virus from the beginning of the pandemic up until July 2022. This is achieved by integrating a hybrid model and daily empirical case and death data from the UK. We partition this dataset into before and after vaccination started in the UK to understand the influence of vaccination on disease dynamics. We used the mathematical model to present some mathematical analyses and the calculation of the basic reproduction number ($R_0$). Following the sensitivity analysis index, we deduce that an increase in the rate of vaccination will decrease $R_0$. Also, the model was fitted to the data from the UK to validate the mathematical model with real data, and we used the data to calculate time-varying $R_0$. The homotopy perturbation method (HPM) was used for the numerical simulation to demonstrate the dynamics of the disease with varying parameters and the importance of vaccination. Furthermore, we used statistical modeling to validate our model by performing principal component analysis (PCA) to predict the evolution of the spread of the COVID-19 outbreak in the UK on some statistical predictor indicators from time series modeling on a 14-day moving window for detecting which of these indicators capture the dynamics of the disease spread across the epidemic curve. The results of the PCA, the index of dispersion, the fitted mathematical model, and the mathematical model simulation are all in agreement with the dynamics of the disease in the UK before and after vaccination started. Conclusively, our approach has been able to capture the dynamics of the pandemic at different phases of the disease outbreak, and the result presented will be useful to understand the evolution of the disease in the UK and future and emerging epidemics.

Wildfire spread modeling is governed by a complex system of nonlinear partial differential equations (PDEs) that capture the intricatedynamics of wildfire behavior, including heat transfer and moisture interaction. A comprehensive understanding of these dynamics is critical for developing effective management, mitigation, and intervention strategies. In this study, temperature-dependent diffusion and convection terms are incorporated into the volume fraction of moisture, enriching the model framework and improving its accuracy in representing wildfire spread. To ensure the mathematical robustness of the model, the non-linear PDE system is transformed into a dimensionless form using appropriate dimensionless variables, facilitating the analysis of the equations. The model equations describe the dynamics of combustible forest material (CFM) in terms of the volume fractions of dry organic matter, moisture, coke, heat, and oxygen. The conditions for the existence and uniqueness of solutions to the model equations are rigorously established using the Lipschitz continuity criterion. The results confirm that unique solutions exist when the Lipschitz conditions are satisfied.

To demonstrate the dynamics of the Mycobacterium tuberculosis disease population, a mathematical model was presented. The model has five compartments, and the resulting equations were resolved. While multiple cases of illness transmission were simulated using the compartmental model of infectious disease spread for a structured population model, the fundamental reproduction number was found using the next-generation matrix. Based on the results of the simulations, the system’s disease-free and endemic equilibrium was created by presenting, analyzing, and graphing the various subpopulations across time. The Homotopy Perturbation Method (HPM) analytical technique was then used to resolve the model.