## 1 Introduction
In particular, SEIADR-type models are considered in [31; 32; 33] which incorporate asymptomatic and dead populations to the typical SEIR models, and which include, in general, vaccination and treatment controls as well as impulsive controls to retire the infective bodies from the streets in third world countries hit by Ebola outbreaks. This paper proposes and investigates an extended SEIR epidemic model with seven subpopulations, namely, the so-called SE(Is)(Ih)(Icicu)AR epidemic model, under the points of view of fulfilling general suitable properties like, for instance, the positivity and stability of the solutions and its output controllability properties for appropriately defined outputs which fix the maximum levels of reasonable presence of hospitalized patients at certain checking time instants. The model contains as integrated subpopulations the susceptible (S) and the recovered (R) subpopulations plus five more infective subpopulations which are the Exposed subpopulation (\(E\)) plus four extra infectious subpopulations. Those infectious subpopulations are the slight symptomatic infectious (\(I_{s}\)) (who do not require hospital care), the seriously symptomatic infectious, or hospitalized, (\(I_{h}\)) (who require hospital care but not intensive care), the very serious infectious at the intensive care unit (\(I_{icu}\)) and the Asymptomatic Infectious (A). It is important to specifically consider both the seriously infectious subpopulation and its subpopulation staying at the intensive care unit as separate subpopulations from the slight infectious individuals due to their high consumption of hospital resources and special staff attention related to the slight infectious individuals. Each individual of exposed subpopulation has a transition either to one of the above mentioned symptomatic infectious subpopulation or to the asymptomatic one. The respective transmission rates are considered to be different in general. The proposed epidemic model is also subject, in the most general framework, to feedback vaccination and treatment controls. It is tested through worked and tested examples under parameterizations related to the recent COVID-19 pandemic which is being exhaustively studied in the available medical and computational background literature. See, for instance, [36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47] and references therein. COVID-19 is a respiratory viral infectious disease which has a very high contagiousness, [48], with respect to the typical influenza or the known common cold. It also exhibits very different symptoms and later secondary effects depending on the particular infected individual running from asymptomatic or very slight symptoms to very serious ones needing extreme hospital care, sometimes producing serious damage in organs like lungs, liver or hearts and sometimes ending in the patient dead, [48; 57]. In particular, the particle swarm optimization algorithm (PSO) has been used to estimate an SEIR model parameterization of COVID-19 using available Hubei province data. Also, a fractional-order model SEIRD model (an SEIR model which includes deceased) is proposed in [37] for COVID-19 pandemic while emphasizing the fractional models possess an inherent memory effect. On the other hand, an epidemic model for COVID-19 which takes into account undetected infective cases and the different sanitary and infectiousness conditions of the hospitalized individuals is discussed in [38] while an extended SEIR model is considered in [39] which incorporates as a new subpopulation, or compartment, the concentration of the coronavirus pathogen in the environment reservoir. Each susceptible individual can become infected through the contaminated environment through either an asymptomatic infectious individual or through a symptomatic one. Also, the dynamics of such a concentration is driven by the exposed and infectious subpopulations. Ageing population layers for control interventions and re-susceptibility and time delay are considered in [40]. This paper is organized as follows.