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.

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 ${\mathcal{R}}_0^d\le 1$, 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.