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We propose new mathematical models based on ordinary differential equations to investigate the transmission dynamics of infectious diseases by adding a hospitalization compartment $H$ into the classic $SIS$ and $SIR$ models. The models incorporate a general incidence rate between susceptibles and infected individuals and a nonlinear recovery rate for hospitalized individuals. We rigorously analyze the existence, local stabilities, and global stabilities of equilibria and derive threshold conditions determining whether the disease dies out or persists. As an application of our $SIH$ model, we perform a case study with an endemic of COVID-19, conduct data fitting and sensitivity analysis for model parameters, and present simulation results to emphasize the role of the hospitalization rate.

Background: Hospital readmissions are associated with several negative health outcomes and higher hospital costs. The HOSPITAL score is one of the tools developed to identify patients at high risk of hospital readmission, but its predictive capacity in more heterogeneous populations involving different diagnoses and clinical contexts is poorly understood. Objective: The aim of this study is to externally validate the HOSPITAL score in a hospitalized Brazilian population. Methods: A retrospective cohort study was carried out with patients over the age of 18 years in a tertiary university hospital. We performed a refitted HOSPITAL score with the same definitions and predictive variables included in the original HOSPITAL score and compared the predictive capacity of both. The receiver operating characteristic was constructed by comparing the performance risk forecasting tools measuring the area under the curve (AUC). Results: Of the 47,464 patients, 50.9% were over 60 years and 58.4% were male. The frequency of 30-day potentially avoidable readmission was 7.70%. The accuracy of original and refitted HOSPITAL scores was close, although statistically different ($p=0.001$), AUC: 0.733 (CI 95%: 0.718, 0.748) and 0.7401 (CI 95%: 0.7256, 0.7547), respectively. The frequency of 60, 90, 180, and 365-days readmissions ranged from 10.60% to 18.30%. Conclusion: The original and refitted HOSPITAL score is a useful tool to identify patients at high risk of 30-day potentially avoidable readmission, in patients with different diagnoses in public tertiary hospitals. In this sense, our study expands and reinforces the usefulness of the HOSPITAL score as a tool that can be used as part of intervention strategies to reduce the rate of hospital readmission.

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Ebola virus disease (EVD) has emerged as a rapidly spreading potentially fatal disease. Several studies have been performed recently to investigate the dynamics of EVD. In this paper, we study the transmission dynamics of EVD by formulating an SEIR-type transmission model that includes isolated individuals as well as dead individuals that are not yet buried. Dynamical systems analysis of the model is performed, and it is consequently shown that the disease-free steady state is globally asymptotically stable when the basic reproduction number, ${{ℛ}}_0$ is less than unity. It is also shown that there exists a unique endemic equilibrium when ${{ℛ}}_0>1$. Using optimal control theory, we propose control strategies, which will help to eliminate the Ebola disease. We use data fitting on models, with and without isolation, to estimate the basic reproductive numbers for the 2014 outbreak of EVD in Liberia and Sierra Leone.