Our study essentially concerns the dynamic behavior of an SIRS epidemic model in discrete time. Two equilibrium points are obtained; one is disease-free while the other is endemic. We are interested in the endemic fixed point as well as its asymptotic stability. Depending on the parameters which are estimated using the data from US Department of health and SIRS modelling with optimization, two Flip and Transcritical bifurcations appear. We illustrate their diagrams, as well as their bifurcation curves using the method of Carcasses \cite{carcasses1993determination,carcasses1995singularities} for the Flip bifurcation and by an implicit function deduced from such an equation for the Transcritical bifurcation. We use the scanning of the parametric plane to have a global view of the behavior of the model and to highlight the zones of stability of the existing singularities. A superposition of the bifurcation curves with the parametric plane can show the overlap of the curves with the boundaries of the stability domains, which confirms the smooth running of the simulation and its correspondence with the theory, we finish this article with constrained optimal control applied to infection rate and recruitment rate for an SIRS discrete epidemic model. Pontryagin's maximum principle is used to determine these optimal controls. Finally using COVID-19 data in the USA, we obtain results that demonstrate the effectiveness of the proposed control strategy to mitigate the spread of the pandemic.

Mathematical modeling is crucial in investigating the pandemic of the ongoing coronavirus disease (COVID-19). The primary target area of the SARS-CoV-2 virus is epithelial cells in the human lower repertory track. During this viral infection, infected cells can initiate innate and adaptive immune responses to viral infection. Immune response in COVID -19 infection can lead to longer recovery time and more severe secondary complications. We formulate a target cell limited mathematical model by incorporating a saturation term for SARS-CoV-2 infected epithelial cell loss reliant on infected cell levels. Forward and backward bifurcation between disease-free and endemic equilibrium points has been analyzed. Global stability of both disease-free and endemic equilibrium is provided. We have seen that the disease-free equilibrium is globally stable for $R_0<1$, and endemic equilibrium exists and is globally stable for $R_0>1$. Impulsive application of drug dosing has been applied for the treatment of covid-19 patients. Also, the dynamics of the impulsive system are discussed when a patient takes drug holidays. The numerical simulations are performed in support of our analytical findings and for the qualitative analysis of the system's dynamics with and without impulse drug dosing.

Heat pumps are recognized as a key tool in the energy transition toward a carbon-neutral society, enabling the electrification of the heating sector at least for low- and medium-temperature heat demands. In recent years, natural refrigerants have been reconsidered due to their low environmental impact: among them, CO2 is a safe option without impact on the ozone layer and low global warming potential compared to synthetic fluids. However, as a consequence of its thermophysical properties, its thermodynamic cycle is transcritical and is particularly suitable for specific end-user temperature profiles. This paper analyzes in a systematic and thorough way the most significant modifications to the reference cycle that have been proposed in the literature to improve the performance, finding how the optimum configurations change with a change in the rated operating conditions (inlet temperature and temperature glide of the heat demand, and ambient temperature). Exergy analysis explains why there is an optimal gas cooler pressure and why its trend with the average temperature is split into two distinct regions, clearly recognizable in all cycle layouts. The maximum coefficient of performance (COP) of the reference cycle varies in the 1.52–3.74 range, with a second-law efficiency of 6.436.1, for an optimal gas cooler pressure of up to 15.45MPa, depending on ambient temperature and end-user temperature profile. The most effective modification is the cycle with ejector and internal heat exchanger, which raises the COP to 1.84–4.40 (second-law efficiency 8.745.56). The presented results provide an extensive guide to understanding the behavior of a transcritical CO2 cycle and predict its performance in heat pump applications.