Reinfection and multiple viral strains are among the latest challenges in the current COVID-19 pandemic. In contrast, epidemic models often consider a single strain and perennial immunity. To bridge this gap, we present a new epidemic model that simultaneously considers multiple viral strains and reinfection due to waning immunity. The model is general, applies to any viral disease and includes an optimal control formulation to seek a trade-off between the societal and economic costs of mitigation. We validate the model, with and without mitigation, in the light of the COVID-19 epidemic in England and in the state of Amazonas, Brazil. The model can derive optimal mitigation strategies for any number of viral strains, whilst also evaluating the effect of distinct mitigation costs on the infection levels. The results show that relaxations in the mitigation measures cause a rapid increase in the number of cases, and therefore demand more restrictive measures in the future.

OBJECTIVE To estimate the required number of public beds for adults in intensive care units in the state of Rio de Janeiro to meet the existing demand and compare results with recommendations by the Brazilian Ministry of Health.METHODS The study uses a hybrid model combining time series and queuing theory to predict the demand and estimate the number of required beds. Four patient flow scenarios were considered according to bed requests, percentage of abandonments and average length of stay in intensive care unit beds. The results were plotted against Ministry of Health parameters. Data were obtained from the State Regulation Center from 2010 to 2011.RESULTS There were 33,101 medical requests for 268 regulated intensive care unit beds in Rio de Janeiro. With an average length of stay in regulated ICUs of 11.3 days, there would be a need for 595 active beds to ensure system stability and 628 beds to ensure a maximum waiting time of six hours. Deducting current abandonment rates due to clinical improvement (25.8%), these figures fall to 441 and 417. With an average length of stay of 6.5 days, the number of required beds would be 342 and 366, respectively; deducting abandonment rates, 254 and 275. The Brazilian Ministry of Health establishes a parameter of 118 to 353 beds. Although the number of regulated beds is within the recommended range, an increase in beds of 122.0% is required to guarantee system stability and of 134.0% for a maximum waiting time of six hours.CONCLUSIONS Adequate bed estimation must consider reasons for limited timely access and patient flow management in a scenario that associates prioritization of requests with the lowest average length of stay.

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This paper introduces a mathematical model which describes the dynamics of the spread of HIV in the human body. This model is comprised of a system of ordinary differential equations that involve susceptible cells, infected cells, HIV, immune cells and immune active cells. The distinguishing feature in the proposed model with respect to other models in the literature is that it takes into account cells that represent two distinct mechanisms of the immune system in the defense against HIV: the non-HIV-activated cells and the HIV-activated cells. With a view at minimizing the side effects of a treatment that employs a drug combination designed to attack the HIV at various stages of its life cycle, we introduce control variables that represent the infected patient's medication. The optimal control rule that prescribes the medication for a given time period is obtained by means of Pontryagin's Maximum Principle.

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