Fawaz K. Alalhareth

A within-host HIV dynamics model incorporating latent reservoirs, distributed time delays, and a B-cell-mediated humoral immune response is developed and analyzed mathematically. The model includes five compartments: uninfected CD4⁺ T cells, latently infected cells, actively infected cells, free virions, and B cells. Four distinct distributed delays are introduced to account for the periods between viral entry and the emergence of latently or actively infected cells, reactivation of latently infected cells, and intracellular virion production. For the non-delayed system, the basic reproduction number ${\mathcal{R}}_0$ is derived using the next-generation matrix method. Using Lyapunov functions and LaSalle’s Invariance Principle, a sharp threshold dynamic is proven: the infection-free equilibrium is globally asymptotically stable (GAS) when ${\mathcal{R}}_0\le 1$, whereas a unique endemic equilibrium is GAS when ${\mathcal{R}}_0>1$. For the full distributed-delay system, a delay-dependent reproduction number ${\mathcal{R}}_0^d$ is defined. The global asymptotic stability of the infection-free equilibrium is established for ${\mathcal{R}}_0^d\le 1$, and the global asymptotic stability of the endemic equilibrium is established for ${\mathcal{R}}_0^d>1$, using suitably constructed Lyapunov functionals that account for the delay history. Numerical simulations validate the analytical threshold behavior. A sensitivity analysis of ${\mathcal{R}}_0^d$ identifies the most influential parameters for potential intervention. A treatment-dependent reproduction number is derived, and the critical drug efficacy required for viral eradication is determined. The intracellular production delay is shown to act as a critical threshold for infection clearance.

This paper presents a mathematical investigation of Guinea worm disease (GWD) dynamics using a two-patch model that incorporates discrete delays and treatment interventions. The model accounts for two distinct host populations sharing a common water source, with dynamics described by a system of delay differential equations. We compute the basic reproduction number $ \mathcal{R}_0^d $ for both the delayed and nondelayed systems and establish threshold conditions for disease eradication and persistence. Analytical results demonstrate that the disease-free equilibrium is globally asymptotically stable (GAS) when $ \mathcal{R}_0^d \leq 1 $, while a unique endemic equilibrium exists and is globally stable when $ \mathcal{R}_0^d > 1 $. Sensitivity analysis identifies key parameters influencing disease transmission, and numerical simulations explore the impact of time delays and treatment efficacy on disease dynamics. Our findings reveal that both treatment interventions and specific time delays can significantly reduce the basic reproduction number, providing valuable insights for designing effective GWD control strategies.

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.

Salmonellosis continues to be a global public health priority in which humans, livestock, and the contaminated environment interact with food to create complex interactions. Here, a new non-autonomous model is proposed to capture seasonal dynamics of Salmonella typhimurium transmission with key compartments that include humans, cattle, and bacteria in environmental and food sources. The model explores how bacterial growth, shedding, and ingestion rates, along with contamination pathways, determine disease dynamics. Some analytical derivations of the basic reproduction number (${\mathcal{R}}_0$) and threshold conditions for disease persistence or extinction are derived by using the spectral radius of a linear operator associated with the monodromy matrix. Parameter estimation for the model was accomplished with the aid of Latin hypercube sampling and least squares methods on Salmonella outbreak data from Saudi Arabia ranging from 2018 to 2021. The model was able to conduct an analysis based on the estimated 0.606 value of ${\mathcal{R}}_0$, and this meant that the model was able to fit reasonably well for both the cumulative and the new individual case data, which in turn, suggests the disease is curable. Predictions indicate a gradual decline in the number of new cases, with stabilization anticipated at approximately 40,000 cumulative cases. Further simulations examined the dynamics of disease extinction and persistence based on ${\mathcal{R}}_0$. When ${\mathcal{R}}_0$ is less than 1, the disease-free equilibrium is stable, resulting in the extinction of the disease. Conversely, when ${\mathcal{R}}_0$ exceeds 1, the disease persists, exhibiting endemic characteristics with recurrent outbreaks. Sensitivity analysis identified several parameters as having a significant impact on the model’s outcomes, specifically mortality and infection rates, along with decay rates. These findings highlight the critical importance of precise parameter estimation in understanding and controlling the transmission dynamics of Salmonella. Sensitivity indices and contour plots were employed to assess the impact of various parameters on the basic reproduction number and provide insights into the factors most influencing disease transmission.

In this paper, we presented a comprehensive mathematical analysis of a chemostat model involving two mutualistic bacterial species competing for nutrients, with the inclusion of leachate recycling. We bridged theoretical ecology and bioreactor design, offering insights into microbial coexistence and system stability. Moreover, we addressed practical challenges in waste-water treatment and bioreactor design by optimizing microbial mutualism and nutrient recycling. The novelty of this study lies in integrating mutualistic interactions, leachate recycling, and optimal control of dilution rates-features seldom combined in chemostat models. We discussed a chemostat model involving two bacterial species that were mutualism competing for two essential nutrients with leachate recycling for one of them. The model was reduced from five dimensions to three, and several equilibrium points were identified: $ E_0 $ (extinction of both species), $ E_1 $ (extinction of species 2), $ E_2 $ (extinction of species 1), and $ E_{12} $ (coexistence of both species). The local stability of these equilibria was analyzed. We proved that the coexistence of both species is conditional to some assumptions on the growth rates of species. The coexistence of the two competing bacteria was demonstrated using the theory of uniform persistence applied to the three-variable reduced system. The sensitivity analysis provided valuable insights into the influence of key parameters (e.g., dilution rate and mutualism coefficients) on system dynamics. The optimal control section extends the model's applicability to bioreactor optimization, which is a significant contribution to the field. Several simulations effectively corroborate theoretical findings, illustrating transitions between equilibria and the impact of parameter variations.

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.

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.