In this paper, a new kind of mathematical modeling is studied by providing a five-compartmental system of differential equations with respect to new hybrid generalized fractal-fractional derivatives. For the first time, we design a model of giving up smoking to analyze its dynamical behaviors by considering two parameters of such generalized operators; i.e., fractal dimension and fractional order. We apply a special sub-category of increasing functions to investigate the existence of solutions. Uniqueness property is derived by a standard method based on the Lipschitz rule. After proving stability property, the equilibrium points are obtained and asymptotically stable solutions are studied. Finally, we illustrate all analytical results and findings via numerical algorithms and graphs obtained by Lagrangian piece-wise interpolation, and discuss all behaviors of the relevant solutions in the fractal-fractional system.

This study develops and implements an agent-based model (ABM) that endogenizes intellectual property rights (IPR) regimes and examines their impact on innovation dynamics in Namibia and South Africa. By integrating micro-level agent behaviors with macro-level Markov chain regime switching, the study analyzes how institutional persistence and volatility shape innovation trajectories. Using Monte Carlo simulations, variance-based global sensitivity analysis (Sobol), and spectral analysis of estimated transition matrices, the study demonstrates that South Africa exhibits stronger long-run persistence in strong IPR regimes and more stable innovation outcomes,whereas Namibia presents faster convergence but higher volatility. The integrated methodological framework provides both numerical robustness and theoretical validation of the model.

This paper extends a system of first order nonlinear ordinary differential equations (ODEs) for Lassa Hemorrhagic Fever (LHF) to an innovative fractional-order optimal control model. The fractional-order derivatives are incorporated to better reflect memory and anomalous diffusion effects in disease dynamics, whereas the optimal control strategies incorporates four control strategies, representing early intervention through treatment and rodent evacuation measures. Thus the combination of fractional and optimal control strategies is aimed at improving the our approach of capturing the disease's transmission dynamics, long-term effects, as well as the memory-dependent processes. Qualitative analyses, employing well-established mathematical methods, confirm that the extended dynamics are well-posed. This implies that the established effective reproduction number is effective when is less than unity and ineffective otherwise. The sensitivity analysis, conducted using the Partial rank correlation coefficient (PRCC), also identifies the effective contact rates and the rodents' recruitment rate as the most influential parameters, under the extended dynamics. This implies that the extended dynamic is formulated and analyzed, focusing on minimizing the disease burden and intervention costs through the controls strategies. Numerical simulations validate the theoretical findings and demonstrate the influence of fractional dynamics on disease progression and control effectiveness. The results indicate that fractional dynamics provide a more nuanced understanding of the system, enabling the design of optimal control policies that reduce disease prevalence more effectively compared to classical models. The presence of saturation effects and fractional memory processes drives the system toward a fractional-order disease free equilibrium (DFE) under optimal control implementation.

Gambling behaviour in Namibia's Khomas Region may be shaped not only by local behavioural transitions but also by spatial heterogeneity in gambling-related influence, including uneven exposure to gambling venues, advertising, accessibility, and socially engaged populations. Existing reaction--diffusion formulations of spatial gambling dynamics capture local behavioural transitions and undirected diffusive redistribution but do notexplicitly represent directed transport associated with spatially varying gambling-related influence. This study extends the reaction--diffusion framework to an influence-driven reaction--advection--diffusion model formulated within a conservation-law framework. A prescribed gambling influence potential $\Phi(x)$ is constructed from aggregate behaviouralobservations and prescribed spatial influence profiles, and its spatial gradient generates an influence-driven advective flux. Positivity, conservation, $L^1$-boundedness, and stability properties of the resulting dynamical system are analysed. The nonlinear system is solved numerically using a Chebyshev spectral collocation method in space combined with animplicit-explicit (IMEX) Euler scheme in time, with the linear diffusion--advection operator treated implicitly and factorised once for efficient reuse at each time step, while the nonlinear reaction terms are treated explicitly. The model uses aggregate behavioural data from theKhomas Region for model specification and investigates the roles of the influence-sensitivity coefficient $\chi$ and the influence-weighting parameter $\eta$ through numerical sensitivity experiments. Numerical results demonstrate spectral convergence in space and first-order accuracy in time, while the parameter experiments indicate that $\chi$ has a stronger effect on influence-driven spatial redistribution than $\eta$, with changes in $\chi$ producing more pronounced spatial redistribution of gambling engagement. A local sensitivity analysis further identifies the engagement-loss and potential-to-engaged transition rates as parameterswith comparatively strong effects on the model response. The results demonstrate how a prescribed spatial gambling-influence field can extend reaction--diffusion gambling models by providing a mathematically explicit mechanism for influence-driven spatial redistribution.