Fractional calculus generalizes well-known differentiation and integration to noninteger orders, allowing a more accurate framework for modeling complex dynamical behaviors. The application of fractional-order systems is quite wide in engineering, biology, and physics because they inherently capture the memory effects and long-range dependencies. Out of these, fractional jerk chaotic systems have gained attention regarding their applications in secure communication, signal processing, and control systems. This work develops a comparative analysis of a fractional jerk system that includes constant- and variable-order derivatives to contribute to chaos–stability analysis. Additionally, this study uncovers novel chaotic behaviors, further expanding our understanding of complex dynamical systems. The results yield new insights into using variable-order dynamics to enable chaotic systems to better adapt to real applications.

Infectious diseases include all diseases caused by the transmission of a pathogenic agent such as bacteria, viruses, parasites, prions, and fungi. They, therefore, cover a wide spectrum of benign pathologies such as colds or angina but also very serious ones such as AIDS, hepatitis, malaria, or tuberculosis. Many epidemic diseases exhibit seasonal peak periods. Studying the population behaviours due to seasonal environment becomes a necessity for predicting the risk of disease transmission and trying to control it. In this work, we considered a five-dimensional system for a fatal disease in a seasonal environment. We studied, in the first step, the autonomous system by investigating the global stability of the steady states. In a second step, we established the existence, uniqueness, positivity, and boundedness of a periodic orbit. We showed that the global dynamics are determined using the basic reproduction number denoted by ${\mathcal{R}}_0$ and calculated using the spectral radius of an integral operator. The global stability of the disease-free periodic solution was satisfied if ${\mathcal{R}}_0<1$, and we show also the persistence of the disease once ${\mathcal{R}}_0>1$. Finally, we displayed some numerical investigations supporting the theoretical findings, where the trajectories converge to a limit cycle if ${\mathcal{R}}_0>1$.

In the present work, we study a simple mathematical model that describes the competition of two bacterial species with an obligate one-way beneficial relationship for a limited substrate in a bioreactor associated with leachate recirculation. The substrate is present into two forms, insoluble and soluble substrates, where the latter is consumed by the two competing bacteria, which have two general nonlinear growth rates. The reduction of the model to a 2D one facilitates the study of the nature of the equilibrium points. The dynamic system admits multiple steady states. We provide necessary and sufficient conditions on the added insoluble and soluble substrates and the dilution rate to guarantee the existence, uniqueness, and local and global stability of such steady states. It is deduced that the coexistence of both bacteria is possible, which contradicts the competitive exclusion principle. In the second step, we propose an optimal control on the leachate recirculation rate that reduces the organic matter inside the reactor. Finally, we provide some numerical examples that corroborate and reinforce the theoretical findings.

In this work, we propose and investigate a model of the dynamical behavior of HIV/AIDS transmission by considering a new compartment of the population with HIV: the latent asymptomatic class. The infection reproduction number that stabilizes the global dynamics of the model is evaluated. We analyze the model’s global asymptotic stability using the Lyapunov function and LaSalle’s invariance principle. To identify the primary factors affecting the dynamics of HIV/AIDS, a sensitivity analysis of the model parameters is conducted. We also examine a fractional-order HIV model using the Caputo fractional differential operator. Through qualitative analysis and applications, we determine the existence and uniqueness of the model’s solutions. We derive some results from the fixed-point theorem and Ulam–Hyers stability. Ultimately, the obtained numerical simulation results are in agreement with the analytical outcomes obtained from the model analysis. Our findings illustrate the efficacy of the fractional model in depicting the dynamics of the HIV/AIDS epidemic and offering critical insights for the formulation of effective control strategies. The results show that early intervention and treatment in the latent phase of infection can decrease the spread of the disease and its progression to AIDS, as well as increase the success of treatment strategies.

This paper presents a comprehensive mathematical analysis of a novel compartmental model describing the dynamics of dispersed water pollutants and their interaction with two distinct host populations. The model is formulated as a system of integro-differential equations that incorporates multiple distributed delays to realistically account for time lags in the infection process and pollutant transport. We rigorously establish the biological well-posedness of the model by proving the non-negativity and ultimate boundedness of solutions, confirming the existence of a positively invariant feasible region. The analysis characterizes the long-term behavior of the system through the derivation of the basic reproduction number ${\mathcal{R}}_0^d$, which serves as a sharp threshold determining the system’s fate. For the model without delays, we prove the global asymptotic stability of the infection-free equilibrium (IFE) when ${\mathcal{R}}_0\le 1$ and of the endemic equilibrium (EE) when ${\mathcal{R}}_0>1$. These stability results are extended to the distributed-delay model by using sophisticated Lyapunov functionals, demonstrating that ${\mathcal{R}}_0^d$ universally governs the global dynamics: the IFE (${\mathcal{E}}_0^d$) is globally asymptotically stable (GAS) if ${\mathcal{R}}_0^d\le 1$, while the EE (${\mathcal{E}}_d^{\ast }$) is GAS if ${\mathcal{R}}_0^d>1$. Numerical simulations validate the theoretical findings and provide further insights. Sensitivity analysis identifies the most influential parameters on ${\mathcal{R}}_0^d$, highlighting the recruitment rate of susceptible individuals, exposure rate, and pollutant shedding rate as key intervention targets. Furthermore, we investigate the impact of control measures, showing that treatment efficacy exceeding a critical value is sufficient for disease eradication. The analysis also reveals the inherent mitigating effect of the maturation delay, demonstrating that a delay longer than a critical duration can naturally suppress the outbreak. This work provides a robust mathematical framework for understanding and managing dispersed water pollution, emphasizing the critical roles of multi-source contributions, time delays, and targeted interventions for environmental sustainability.

In this work, we investigated a simple mathematical model describing the consumption of virus-infected phytoplankton by zooplankton in a chemostat. The system was studied by calculating the basic reproduction number, the equilibrium points, and their local and global stability. A sensitivity analysis was used to identify key chemostat factors that significantly affected the aquatic system. Additionally, we considered an optimal strategy based on the use of the dilution rate as an operating parameter that helps maintain the ecological balance of the aquatic food web.