Nonstandard finite-difference (NSFD) methods, pioneered by R. E. Mickens, offer accurate and efficient solutions to various differential equation models in science and engineering. NSFD methods avoid numerical instabilities for large time steps, while numerically preserving important properties of exact solutions. However, most NSFD methods are only first-order accurate. This paper introduces two new classes of explicit second-order modified NSFD methods for solving n-dimensional autonomous dynamical systems. These explicit methods extend previous work by incorporating novel denominator functions to ensure both elementary stability and second-order accuracy. This paper also provides a detailed mathematical analysis and validates the methods through numerical simulations on various biological systems.

Mathematical models play a crucial role in predicting disease dynamics and estimating key quantities. Non-autonomous models offer the advantage of capturing temporal variations and changes in the system. In this study, we analyzed the transmission of typhoid fever in a population using a compartmental model that accounted for dynamic changes occurring periodically in the environment. First, we determined the basic reproduction number, ${\mathcal{R}}_0$, for the periodic model and derived the time-average reproduction rate, $\left[{\mathcal{R}}_0\right]$, for the non-autonomous model as well as the basic reproduction number, ${\mathcal{R}}_0^A$, for the autonomous model. We conducted an analysis to examine the global stability of the disease-free solution and endemic periodic solutions. Our findings demonstrated that when ${\mathcal{R}}_0<1$, the disease-free solution was globally asymptotically stable, indicating the extinction of typhoid fever. Conversely, when ${\mathcal{R}}_0>1$, the disease became endemic in the population, confirming the existence of positive periodic solutions. We also presented numerical simulations that supported these theoretical results. Furthermore, we conducted a sensitivity analysis of ${\mathcal{R}}_0^A$, $\left[{\mathcal{R}}_0\right]$ and the infected compartments, aiming to assess the impact of model parameters on these quantities. Our results showed that the human-to-human infection rate has a significant impact on the reproduction number, while the environment-to-human infection rate and the bacteria excretion rate affect long-cycle infections. Moreover, we examined the effects of parameter modifications and how they impact the implementing of efficient control strategies to combat TyF. Although our model is limited by the lack of precise parameter values, the qualitative results remain consistent even with alternative parameter settings.

We present a nonautonomous compartmental model that incorporates vaccination and accounts for the seasonal transmission of typhoid fever. The dynamics of the system are governed by the basic reproductive number ${\mathcal{R}}_0$. This demonstrates the global stability of the disease-free solution if ${\mathcal{R}}_0<1$. On the contrary, if ${\mathcal{R}}_0>1$, the disease persists and positive periodic solutions exist. Numerical simulations validate our theoretical findings. To accurately fit typhoid fever data in Taiwan from 2008 to 2023, we use the model and estimate its parameters using Latin hypercube sampling and least squares techniques. A sensitivity analysis reveals the significant influence of parameters such as infection rates on the reproduction number. Increasing vaccination coverage, despite challenges in developing countries, reduces typhoid cases. Accessible and highly effective vaccines play a critical role in suppressing the epidemic, outweighing concerns about the efficacy of the vaccine. Investigating possible parameter changes in Taiwan highlights the importance of monitoring and managing transmission rates to prevent recurring annual epidemics.

In this paper, we develop a mathematical model to investigate HIV infection dynamics, where we focus on the virus’s dual-target mechanism involving both CD$4^{+}$ T cells and macrophages. Our model is structured as a system of seven nonlinear ordinary differential equations describing the interactions between susceptible, latent, and infected cells, alongside free virus particles. We derive the basic reproduction number, ${\mathcal{R}}_0$, as two components, ${\mathcal{R}}_{01}$ and ${\mathcal{R}}_{02}$, which quantify the respective contributions of CD$4^{+}$ T cells and macrophages to viral spread. It is deduced that the infection-free steady state is globally asymptotically stable once ${\mathcal{R}}_0\le 1$, ensuring viral eradication. For ${\mathcal{R}}_0>1$, a stable endemic steady state emerges, indicating the persistence of the infection. Later, we develop an optimal control strategy to study the impact of reverse transcriptase and protease inhibitors. This analysis identifies a critical drug efficacy threshold, ${ϵ}^{\ast }=1-{\displaystyle \frac{1}{{\mathcal{R}}_0}}$, necessary for viral eradication. The numerical simulations and the sensitivity analysis provide key parameters that drive viral dynamics, offering practical insights for designing targeted therapies, particularly during the early stages of infection.

Smoking is associated with various detrimental health conditions, including cancer, heart disease, stroke, lung illnesses, diabetes, and fatal diseases. Motivated by the application of fractional calculus in epidemiological modeling and the exploration of memory and nonlocal effects, this paper introduces a mathematical model that captures the dynamics of relapse in a smoking cessation context and presents the dynamic behavior of the proposed model utilizing Caputo fractional derivatives. The model incorporates four compartments representing potential, persistent (heavy), temporally recovered, and permanently recovered smokers. The basic reproduction number $R_0$ is computed, and the local and global dynamic behaviors of the free equilibrium smoking point (${\mathcal{Y}}_0$) and the smoking-present equilibrium point (${\mathcal{Y}}^{*}$) are analyzed. It is demonstrated that the free equilibrium smoking point (${\mathcal{Y}}_0$) exhibits global asymptotic stability when $R_0\le 1$, while the smoking-present equilibrium point (${\mathcal{Y}}^{*}$) is globally asymptotically stable when $R_0>1$. Additionally, analytical results are validated through a numerical simulation using the predictor–corrector PECE method for fractional differential equations in Matlab software.

The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is responsible for coronavirus disease-19 (COVID-19). This virus has caused a global pandemic, marked by several mutations leading to multiple waves of infection. This paper proposes a comprehensive and integrative mathematical approach to the third wave of COVID-19 (Omicron) in the Kingdom of Saudi Arabia (KSA) for the period between 16 December 2022 and 8 February 2023. It may help to implement a better response in the next waves. For this purpose, in this article, we generate a new mathematical transmission model for coronavirus, particularly during the third wave in the KSA caused by the Omicron variant, factoring in the impact of vaccination. We developed this model using a fractal-fractional derivative approach. It categorizes the total population into six segments: susceptible, vaccinated, exposed, asymptomatic infected, symptomatic infected, and recovered individuals. The conventional least-squares method is used for estimating the model parameters. The Perov fixed point theorem is utilized to demonstrate the solution’s uniqueness and existence. Moreover, we investigate the Ulam–Hyers stability of this fractal–fractional model. Our numerical approach involves a two-step Newton polynomial approximation. We present simulation results that vary according to the fractional orders ($\gamma$) and fractal dimensions ($\theta$), providing detailed analysis and discussion. Our graphical analysis shows that the fractal-fractional derivative model offers more biologically realistic results than traditional integer-order and other fractional models.