In this paper, we develop a deterministic mathematical epidemic model for tuberculosis outbreaks in order to study the disease’s impact in a given population. We develop a qualitative analysis of the model by showing that the solution of the model is positive and bounded. The global stability analysis of the model uses Lyapunov functions and the threshold quantity of the model, which is the basic reproduction number is estimated. The existence and uniqueness analysis for Caputo fractional tuberculosis outbreak model is presented by transforming the deterministic model to a Caputo sense model. The deterministic model is used to predict real data from Uganda and Rwanda to see how well our model captured the dynamics of the disease in the countries considered. Furthermore, the sensitivity analysis of the parameters according to $R_0$ was considered in this study. The normalised forward sensitivity index is used to determine the most sensitive variables that are important for infection control. We simulate the Caputo fractional tuberculosis outbreak model using the Adams–Bashforth–Moulton approach to investigate the impact of treatment and vaccine rates, as well as the disease trajectory. Overall, our findings imply that increasing vaccination and especially treatment availability for infected people can reduce the prevalence and burden of tuberculosis on the human population.

In this study, a mathematical model for studying the dynamics of monkeypox virus transmission with non-pharmaceutical intervention is created, examined, and simulated using real-time data. Positiveness, invariance, and boundedness of the solutions are thus examined as fundamental features of mathematical models. The equilibrium points and the prerequisites for their stability are achieved. The basic reproduction number and thus the virus transmission coefficient ℜ₀ were determined and quantitatively used to study the global stability of the model's steady state. Furthermore, this study considered the sensitivity analysis of the parameters according to ℜ₀. The most sensitive variables that are important for infection control are determined using the normalized forward sensitivity index. Data from the United Kingdom collected between May and August 2022, which also aid in demonstrating the usefulness and practical application of the model to the spread of the disease in the United Kingdom, were used. In addition, using the Caputo–Fabrizio operator, Krasnoselskii's fixed point theorem has been used to analyze the existence and uniqueness of the solutions to the suggested model. The numerical simulations are presented to assess the system dynamic behavior. More vulnerability was observed when monkeypox virus cases first appeared recently as a result of numerical calculations. We advise the policymakers to consider these elements to control monkeypox transmission. Based on these findings, we hypothesized that another control parameter could be the memory index or fractional order.

The rising tide of smoking-related diseases has irreparably damaged the health of both young and old people, according to the World Health Organization. This study explores the dynamics of the age-structure smoking model under fractal-fractional (F-F) derivatives with government intervention coverage. We present a new fractal-fractional model for two-age structure smokers in the Caputo–Fabrizio framework to emphasize the potential of this operator. For the existence-uniqueness criterion of the given model, successive iterative sequences are defined with limit points that are the solutions of our proposed age-structure smoking model. We also use the functional technique to demonstrate the proposed model stability under the Ulam–Hyers condition. The two age-structure smoking models are numerically characterized using the Newton polynomial. We observe that in Groups 1 and 2, a change in the fractal-fractional orders has a direct effect on the dynamics of the smoking epidemic. Moreover, testing the inherent effectiveness of government interventions shows a considerable impact on potential, occasional, and temporary smokers when the fractal-fractional order is 0.95. It is the view that this study will contribute to the applicability of the schemes, the rich dynamics of the fractal, and the fractional perspective of future predictions.

The formulation of mathematical models using differential equations has become crucial in predicting the evolution of viral diseases in a population in order to take preventive and curative measures. In December 2019, a novel variety of Coronavirus (SARS-CoV-2) was identified in Wuhan, Hubei Province, China, which causes a severe and potentially fatal respiratory syndrome. Since then, it has been declared a pandemic by the World Health Organization and has spread around the globe. A reaction–diffusion system is a mathematical model that describes the evolution of a phenomenon subjected to two processes: a reaction process, in which different substances are transformed, and a diffusion process, which causes their distribution in space. This article provides a mathematical study of the Susceptible, Exposed, Infected, Recovered, and Vaccinated population model of the COVID-19 pandemic using the bias of reaction–diffusion equations. Both local and global asymptotic stability conditions for the equilibria were determined using a Lyapunov function, and the nature of the stability was determined using the Routh–Hurwitz criterion. Furthermore, we consider the conditions for the existence and uniqueness of the model solution and show the spatial distribution of the model compartments when the basic reproduction rate ${\mathcal{R}}_0<1$ and ${\mathcal{R}}_0>1$. Thereafter, we conducted a sensitivity analysis to determine the most sensitive parameters in the proposed model. We demonstrate the model’s effectiveness by performing numerical simulations and investigating the impact of vaccination, together with the significance of spatial distribution parameters in the spread of COVID-19. The findings indicate that reducing contact with an infected person and increasing the proportion of susceptible people who receive high-efficacy vaccination will lessen the burden of COVID-19 in the population. Therefore, we offer to the public health policymakers a better understanding of COVID-19 management.

(1) Background: Impact and severity of coronavirus pandemic on health infrastructure vary across countries. We examine the role percentage health expenditure plays in various countries in terms of their preparedness and see how countries improved their public health policy in the first and second wave of the coronavirus pandemic; (2) Methods: We considered the infectious period during the first and second wave of 195 countries with their current health expenditure as gross domestic product percentage (CHE/GDP). An exponential model was used to calculate the slope of the regression line while the ARIMA model was used to calculate the initial autocorrelation slope and also to forecast new cases for both waves. The relationship between epidemiologic and CHE/GDP data was used for processing ordinary least square multivariate modeling and classifying countries into different groups using PC analysis, K-means and hierarchical clustering; (3) Results: Results show that some countries with high CHE/GDP improved their public health strategy against virus during the second wave of the pandemic; (4) Conclusions: Results revealed that countries who spend more on health infrastructure improved in the tackling of the pandemic in the second wave as they were worst hit in the first wave. This research will help countries to decide on how to increase their CHE/GDP in order to properly tackle other pandemic waves of the present COVID-19 outbreak and future diseases that may occur. We are also opening up a debate on the crucial role socio-economic determinants play during the exponential phase of the pandemic modelling.

(1) Background: The estimation of daily reproduction numbers throughout the contagiousness period is rarely considered, and only their sum R₀ is calculated to quantify the contagiousness level of an infectious disease. (2) Methods: We provide the equation of the discrete dynamics of the epidemic’s growth and obtain an estimation of the daily reproduction numbers by using a deconvolution technique on a series of new COVID-19 cases. (3) Results: We provide both simulation results and estimations for several countries and waves of the COVID-19 outbreak. (4) Discussion: We discuss the role of noise on the stability of the epidemic’s dynamics. (5) Conclusions: We consider the possibility of improving the estimation of the distribution of daily reproduction numbers during the contagiousness period by taking into account the heterogeneity due to several host age classes.