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Effectively combating mosquito-borne diseases necessitates innovative strategies beyond traditional methods like insecticide spraying and bed nets. Among these strategies, the sterile insect technique (SIT) emerges as a promising approach. Previous studies have utilized ordinary differential equations to simulate the release of sterile mosquitoes, aiming to reduce or eradicate wild mosquito populations. However, these models assume immediate release, leading to escalated costs. Inspired by this, we propose a non-smooth Filippov model that examines the interaction between wild and sterile mosquitoes. In our model, the release of sterile mosquitoes occurs when the population density of wild mosquitoes surpasses a specified threshold. We incorporate a density-dependent birth rate for wild mosquitoes and consider the impact of immigration. This paper unveils the complex dynamics exhibited by the proposed model, encompassing local sliding bifurcation and the presence of bistability, which entails the coexistence of regular equilibria and pseudo-equilibria, as crucial model parameters, including the threshold value, are varied. Moreover, the system exhibits hysteresis phenomena when manipulating the rate of sterile mosquito release. The existence of three types of limit cycles in the Filippov system is ruled out. Our main findings indicate that reducing the threshold value to an appropriate level can enhance the effectiveness of controlling wild insects. This highlights the economic benefits of employing SIT with a threshold policy control to impede the spread of disease-carrying insects while bolstering economic outcomes.

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RESUMEN

Mathematical simulation of drug diffusion is a significant tool for predicting the bio-transport process. Moreover, the reported models in the literature are based on Fick's approach, which leads to an infinite propagation speed. Consequently, it is essential to construct a mathematical model to represent the diffusion processes for estimating drug concentrations at different sites and throughout the circulation. Thus, in this article, the diffusion process is employed to propose three models for estimating the drug release from multi-layer cylindrical tablets. A fractional model is presented based on Fick's approach, while classical and fractional Cattaneo models are presented using the relaxed principle. Various numerical methods are used to solve the specified problem. The numerical scheme's stability and convergence are demonstrated. Drug concentration and mass profiles are presented for the tablet and the external medium and compared with the in vivo plasma profiles. The results show the efficiency and precision of the proposed fractional models based on the fourth-order weighted-shifted Grünwald-Letnikov difference operator approximation. These models are compatible with the in vivo data compared with the classical Fick's one.

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This study presents methods of hygiene and the use of masks to control the disease. The zero basic reproduction number can be achieved by taking the necessary precautionary measures that prevent the transmission of infection, especially from uninfected virus carriers. The existence of time delay in implementing the quarantine strategy and the threshold values of the time delay that keeping the stability of the system are established. Also, it is found that keeping the infected people quarantined immediately is very important in combating and controlling the spread of the disease. Also, for special cases of the system parameters, the time delay can not affect the asymptotic behavior of the disease. Finally, numerical simulations have been illustrated to validate the theoretical analysis of the proposed model.