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This study investigates the dynamics of an epidemic employing an (Susceptible-Infected-Hospitalized-Recovered-Susceptible (SIHRS) epidemiological model highlighting the crucial importance of hospital bed availability and a non-monotone incidence function, which incorporates the influences of stringent governmental measures, social behavior dynamics and public responses in both autonomous and non-autonomous scenarios. The analysis investigates the conditions for existence of infection-free and endemic steady states based on the basic reproduction number as it surpasses unity. Sensitivity analysis has been conducted to evaluate the impact of different system parameters on disease transmission. This work also investigates alterations in stability of the system caused by transcritical, Hopf and saddle-node bifurcations. Additionally, two-parameter bifurcation identifies the regions where the stability of both the equilibrium points has been examined. Numerical simulations are shown to illustrate all the theoretically obtained results. Also, the model examines the dynamical behavior of the epidemic when the quantity of available hospital beds varies periodically. This aspect of the study highlights the significant impact of hospital bed availability. Such factors are crucial in preventing disease spread during an epidemic. The results provide valuable insights into how dynamic patterns of disease transmission are influenced by healthcare infrastructure and public health interventions. This comprehensive exploration underscores the importance of integrated approaches combining medical resources and societal measures in managing and mitigating the effects of epidemics.

Using a proper choice of the dynamical variables, we show that a non-autonomous dynamical system transforming to an autonomous dynamical system with a certain coordinate transformations can be obtained by solving a general nonlinear first-order partial differential equations. We examine some special cases and provide particular physical examples. Our framework constitutes general machineries in investigating other non-autonomous dynamical systems.

In this paper, following a previous paper ([32] Permanence and extinction of a non-autonomous HIV-1 model with two time delays, preprint) on the permanence and extinction of a delayed non-autonomous HIV-1 within-host model, we introduce and investigate a delayed HIV-1 model including maximum homeostatic proliferation rate of CD4⁺ T-cells and varying coefficients. By applying the asymptotic analysis theory and oscillation theory, we show: (i) the system will be permanent when the threshold value R_* > 1, and for this case we also obtain the explicit estimate of the eventual lower bound of the HIV-1 virus load; (ii) the threshold value R* < 1 implies the extinction of the virus. Furthermore, we obtain that the threshold dynamics is in agreement with that of the corresponding autonomous system, which extends the classic results for the system with constant coefficients. Numerical simulations are also given to illustrate our main results, and in particular, some sensitivity test of R_* is established.

The paper characterizes the global threshold dynamics of an epidemic model of SIQS type in environments with fluctuations, where the quarantine class is explicitly involved. Criteria are established for the permanence and extinction of the infective in environments with time oscillations. In particular, we further consider an environment which varies periodically in time. The global threshold dynamic scenarios i.e. the existence and global asymptotic stability of the disease-free periodic solution, the existence of the endemic periodic solution and the permanence of the infective are completely characterized by the basic reproduction number defined by the spectral radius of an associated linear integral operator.