## 1 Introduction  
For the last two decades, an important effort has been made to propose and analyze new mathematical epidemic models being based on integro-differential equations and/or difference equations. Such models are claimed to describe the evolution through time of the various subpopulationsintegrated in the epidemic model under study whose respective dynamics are coupled. A classical, so-called, SEIR (susceptible-exposed-infectious-recovered) epidemic model splits the total infectious population into two subpopulations (or compartments), namely, the so-called \"infected\" or \"exposed\" (E) (those having the disease but do not present yet external symptoms) and the \"Infectious\" or \"Infective\" (I) (those having external symptoms). The mentioned basic SEIR model has nowadays multiple variants with different degrees of complexity. See, for instance, [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14] and some of the references therein. In particular, one can refer to those ones which admit controls like constant and feedback vaccination and treatment controls and/or impulsive controls (exerted on very short periods of time) or those involving several interacting patches associated to different towns ort regions. For instance, an optimization approach is given to get the vaccination controls in [7]. In [9; 10; 11], some pulse vaccination strategies are proposed. In [12; 11] several strategies are discussed including feedback. In [13], an epidemic model is proposed which is subject to a ratio-dependent saturation incidence rate. On the other hand, a SIR (susceptible-infectious-recovered) epidemic model under impulsive vaccination was investigated in [14] while a non- autonomous SIRVS epidemic model subject to vaccination controls has been proposed and discussed in [15]. Also, an epidemic delayed model with diffusion has been characterized in [16]. It has to be mentioned that a relevant attention has been paid to the investigation of the stability and positivity properties of epidemic models in both the vaccination-free and vaccination control situations. See, for instance, [17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29]. The positivity properties of de solutions and equilibrium points of some epidemic models are studied in detail in [30; 31; 32; 33], while model discretization concerns are discussed in [33; 34] and relationships to thermodynamics concepts are discussed in [35]. On the other hand, epidemics evolution and concerns regarding COVID-19 are being described in more recent works. See, for instance, [36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53; 54; 55; 56; 57] and references in their model. It can be pointed out that the approaches of [19; 20] and [58] are formulated in the stochastic framework context. In particular, the epidemic model proposed in [58] can be interpreted as an interpolation between a SI model and an SIR model. On the other hand, it is well- known that there may be some individuals who are infective so that they can produce contagions to susceptible individuals but do not have significant external symptoms. Those ones are the members of the so-called \"Asymptomatic\" (A) subpopulation, [4; 31; 32; 33]. This subpopulation appears even in the common known influenza disease. If such an asymptomatic subpopulation is incorporated to the model, then it turns out that the exposed might have distinct transitions to the symptomatic infectious and to the asymptomatic ones. In this way, a proportion of the exposed become asymptomatic after a certain time period while another proportion becomes infectious after a certain period of time. In general, both respective proportions might be distinct. In the Ebola disease, the lying dead corpses are also infective [4; 5; 6; 31; 32; 33] which can cause very serious sanitary problems in third world tropical countries with low or scarce sanitary means.