A consistent discrete version of a non-autonomous SIRVS model

A family of discrete non-autonomous SIRVS models with general incidence is obtained from a continuous family of models by applying Mickens non-standard discretization method. Conditions for the permanence and extinction of the disease and the stability of disease-free solutions are determined. The consistency of those discrete models with the corresponding continuous model is discussed: if the time step is sufficiently small, when we have extinction (respectively permanence) for the continuous model we also have extinction (respectively permanence) for the corresponding discrete model. Some numerical simulations are carried out to compare the different possible discretizations of our continuous model using real data.Comment: 20 figures, 29 page

A Consistent Discrete Version of a Nonautonomous SIRVS Model

A family of discrete nonautonomous SIRVS models with general incidence is obtained from a continuous family of models by applyingMickens nonstandard discretizationmethod. Conditions for the permanence and extinction of the disease and the stability of disease-free solutions are determined. Concerning extinction and persistence, the consistency of those discrete models with the corresponding continuous model is discussed: if the time step is sufficiently small, when we have extinction (permanence) for the continuous model, we also have extinction (permanence) for the corresponding discrete model. Some numerical simulations are carried out to compare the different possible discretizations of our continuous model using real data.- Unidade de Investigaçãoo para o Desenvolvimento do Interior (UDI), Instituto Politécnico da Guarda, 6300-559 Guarda, Portugal - Centro de Matemática e Aplicações e Departamento de Matemática, Universidade da Beira Interior, 6201-001 Covilhã, Portuga

On a Controlled Se(Is)(Ih)(Iicu)AR Epidemic Model with Output Controllability Issues to Satisfy Hospital Constraints on Hospitalized Patients

An epidemic model, the so-called SE(Is)(Ih)(Iicu)AR epidemic model, is proposed which splits the infectious subpopulation of the classical SEIR (Susceptible-Exposed-Infectious-Recovered) model into four subpopulations, namely asymptomatic infectious and three categories of symptomatic infectious, namely slight infectious, non-intensive care infectious, and intensive care hospitalized infectious. The exposed subpopulation has four different transitions to each one of the four kinds of infectious subpopulations governed under eventually different proportionality parameters. The performed research relies on the problem of satisfying prescribed hospitalization constraints related to the number of patients via control interventions. There are four potential available controls which can be manipulated, namely the vaccination of the susceptible individuals, the treatment of the non-intensive care unit hospitalized patients, the treatment of the hospitalized patients at the intensive care unit, and the transmission rate which can be eventually updated via public interventions such as isolation of the infectious, rules of groups meetings, use of face masks, decrees of partial or total quarantines, and others. The patients staying at the non-intensive care unit and those staying at the intensive care unit are eventually, but not necessarily, managed as two different hospitalized subpopulations. The controls are designed based on output controllability issues in the sense that the levels of hospital admissions are constrained via prescribed maximum levels and the measurable outputs are defined by the hospitalized patients either under a joint consideration of the sum of both subpopulations or separately. In this second case, it is possible to target any of the two hospitalized subpopulations only or both of them considered as two different components of the output. Different algorithms are given to design the controls which guarantee, if possible, that the prescribed hospitalization constraints hold. If this were not possible, because the levels of serious infection are too high according to the hospital availability means, then the constraints are revised and modified accordingly so that the amended ones could be satisfied by a set of controls. The algorithms are tested through numerically worked examples under disease parameterizations of COVID-19.This research received funding from the Spanish Institute of Health Carlos III through Grant COV 20/01213, the Spanish Government and the European Commission through Grant RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE) and the Basque Government for Grant IT1207-19