SS symmetry Article On Confinement and Quarantine Concerns on an SEIAR Epidemic Model with Simulated Parameterizations for the COVID-19 Pandemic Manuel De la Sen 1, * , Asier Ibeas 2 and Ravi P. Agarwal 3 1 Campus of Leioa, Institute of Research and Development of Processes IIDP, University of the Basque Country, 48940 Leioa (Bizkaia), Spain 2 Department of Telecommunications and Systems Engineering, Universitat Autònoma de Barcelona, UAB, 08193 Barcelona, Spain; Asier.Ibeas@uab.cat 3 Department of Mathematics, Texas A & M University, 700 Univ Blvd, Kingsville, TX 78363, USA; Ravi.Agarwal@tamuk.edu * Correspondence: manuel.delasen@ehu.eus  Received: 7 September 2020; Accepted: 30 September 2020; Published: 7 October 2020  Abstract: This paper firstly studies an SIR (susceptible-infectious-recovered) epidemic model without demography and with no disease mortality under both total and under partial quarantine of the susceptible subpopulation or of both the susceptible and the infectious ones in order to satisfy the hospital availability requirements on bed disposal and other necessary treatment means for the seriously infectious subpopulations. The seriously infectious individuals are assumed to be a part of the total infectious being described by a time-varying proportional function. A time-varying upper-bound of those seriously infected individuals has to be satisfied as objective by either a total confinement or partial quarantine intervention of the susceptible subpopulation. Afterwards, a new extended SEIR (susceptible-exposed-infectious-recovered) epidemic model, which is referred to as an SEIAR (susceptible-exposed-symptomatic infectious-asymptomatic infectious-recovered) epidemic model with demography and disease mortality is given and focused on so as to extend the above developed ideas on the SIR model. A proportionally gain in the model parameterization is assumed to distribute the transition from the exposed to the infectious into the two infectious individuals (namely, symptomatic and asymptomatic individuals). Such a model is evaluated under total or partial quarantines of all or of some of the subpopulations which have the effect of decreasing the number of contagions. Simulated numerical examples are also discussed related to model parameterizations of usefulness related to the current COVID-19 pandemic outbreaks. Keywords: SEIAR epidemic model; confinement intervention; quarantine intervention; COVID-19; mathematical modelling; numerical modelling 1. Introduction Epidemic mathematical models in different formal frameworks are of crucial interest in recent decades [1,2]. Such mentioned formal frameworks include, for instance, differential, difference and differential/difference hybrid equations, dynamic systems, control theory [3–20], computation [7], and information theory [21–28] as well as combined mathematical analysis of epidemic spreading and control design tools [21,22,27–36]. One of the objectives of such models is to investigate and predict the evolution of epidemic infectious diseases in humans, animals, or plants as well as to elucidate how the interventions, like quarantine actions or regular and impulsive vaccination and treatment controls, can avoid or mitigate their contagious propagation (See References [1,2,11,13,15,17–20,25–36]). It can be pointed out that the quarantine and the confinement interventions are qualitatively similar to Symmetry 2020, 12, 1646; doi:10.3390/sym12101646 www.mdpi.com/journal/symmetry Symmetry 2020, 12, 1646 2 of 33 using impulsive vaccination and treatment controls since all or a part of a targeted subpopulation (typically, the susceptible one or the infectious one or fractions of them) is removed from its associate compartment and the viewable effect is a re-initialization of the corresponding trajectory solution under fewer contagious contacts. Another beneficial effect is that the transmission rate decreases to lower levels because of the mentioned decrease in the number of contacts. The previously mentioned control intervention are very relevant concerning the disease temporary evolution and its steady-state reachable numbers of the various subpopulations since they can increase the values of the transmission rate being compatible with a basic reproduction number being less than unity. This property implies that the disease-free equilibrium point is a global attractor and the disease becomes asymptotically removed as a result. Those above issues have been important in the last months with regard to the evolution of the COVID-19 pandemic around the world (See References [27,29,37–51]). In particular, an “ad hoc” SEIR model parameterization related to the COVID-19 pandemic is investigated in Reference [37] including delayed re-susceptibility caused by the infection. Additionally, a kind of autoregressive model average model (ARMA), known as an ARIMA model for prediction of COVID-19, is presented and simulated in Reference [38] for the data of several countries. The effects of different phases of quarantine actions in the values of the transmission rate are studied in Reference [39]. See Reference [42] for a discussion of related simulated numerical results on the COVID-19 outbreak in Italy. Some results have been recently reported on reaction-diffusion epidemic models of usefulness fore dengue [52], on delayed diffusive epidemic models [53], and on multi-group epidemic discrete models with time delays [54]. (See also related references therein). It can be pointed out that there are links in the descriptions of epidemic models to symmetry and asymmetry concepts with different interpretations. For instance, in Reference [13], the problem of patchy epidemic environments in the multi-node case where there are in-flux and out-flux of populations between different nodes (for instance, with each node describing distinct towns or regions) is focused on. The most general assumption involved is that the matrices of population travelling interchange are asymmetric in the most general case, but they can be symmetric in particular. In Reference [55], a close problem is focused on for epidemic models describing a network with more than one node and the concepts of symmetry and asymmetry play a role in the mathematical description of the proposed model. On the other hand, in Reference [56], the case of reported versus unreported infective cases is calculated. In the case of time-invariant model parameterizations, such a ratio becomes invariant for both the time-instantaneous tests and the time- accumulated tests interpreted as a certain symmetry property. Generally speaking, symmetry is equivalent to an invariance of a certain quantity under transformations. In that way, a related property associated with a number of common epidemic models is their invariance under certain transformations of variables, which is a property that applies even for models of a single node. Assume, for instance, a true-mass action epidemic model [43] with a typical bilinear incidence of the form βSI/N where β is the transmission rate and S and I are the susceptible and infectious instantaneous numbers and N is the total population assumed constant through time. Furthermore, assume that the above model is changed to a normalized one for all the subpopulations leading to the transformed variables s = S/N, i = I/N etc. It can be easily seen that the form of the equations remains invariant with the redefined transmission rate βn = βN. Such an idea is applicable even to epidemic models of a single node. The related existing bibliography on epidemic modelling is abundant and very rich including a variety of epidemic models with several coupled subpopulations including the susceptible, the infectious, and the recovered ones as an elementary starting basis in the simpler SIR epidemic models. Further generalizations lead to the so-called SEIR models, which include the exposed subpopulation (those who do not have external symptoms yet) as a new subpopulation. More complex models, such as SEIADR-type models, which include the asymptomatic infectious and the infective dead subpopulations, have been designed and discussed, for instance, for Ebola disease [15,16]. On the other hand, Covid-19 is an infectious disease caused by a newly discovered coronavirus at the end of 2019. The first outbreak was reported in Wuhan (China) and then it extended worldwide very Symmetry 2020, 12, 1646 3 of 33 fast [57]. At this time, there is no vaccine nor effective treatment available while social distancing, isolation, and quarantine have been revealed to be the most important measurements to counteract its spread. Despite the current death rate of the disease being around 3–4%, acute symptoms may lead to intensive care unities (ICU’s) admission to many seriously infected patients, which provokes the overflow of these units and a situation of stress in the health system of many countries where the situation of their health systems is becoming really difficult over recent months because of the lack of beds, material, and staff for the abundant, seriously infected people. The main objective of this paper is to propose and to investigate a new SEIAR epidemic model with demography, which is a generalized SEIR (susceptible-exposed-infectious-recovered). Such a model includes asymptomatic infectious (A) and eventual illness mortality. All the formulations are developed in a deterministic framework of differential equations with finite jumps in their solutions. A fraction of the exposed subpopulation has a transition to the symptomatic infectious subpopulation while its complementary to unity fraction has a transition to the asymptomatic one. The main results concern the study of the model under, in general, partial quarantines. Total quarantine actions and confinement interventions are considered as particular cases being governed, in each particular situation, by the choices of the gains, which describe the decrease of subpopulation levels subject to eventual reception or transmission of new contagions. Such gains are designed in such a way that the foreseen fraction of the infectious needing hospital care are upper-bounded by a function, which describes the hospital availability on beds and other means, like staff, available amounts and qualification, number of respirators, etc. Before the presentation and discussion of the new proposed SEIAR model, two simple illustrative examples on partial or total quarantines are given on a simpler SIR model without demography and with no mortality. The paper is organized as follows. Section 2 states and briefly describes the simple mentioned SIR model under total or partial quarantine of the susceptible and infectious in order to satisfy the objective that a weighted proportion of the infectious subpopulation is less than a certain upper-bound defined by hospital treatment considerations. Section 3 presents the proposed SEIAR model, which includes the exposed and the asymptomatic subpopulations subject to demography and eventual mortality so that it generalizes the previously discussed SIR model. The properties of positivity and boundedness of the state trajectory solution are stated and proved. Section 4 studies the model under partial or total quarantines of some or all the subpopulations so that a hospital management objective on temporary availability of technical means and bed disposal can be fulfilled. Section 5 presents and discusses some numerically worked examples under model parameterizations of the COVID-19 pandemic to test the efficiency of several quarantine interventions. Lastly, conclusions end the paper. Appendices A and B with details of some auxiliary calculations are also given. 2. A Quarantined SIR Epidemic Model The effects of total or partial quarantine interventions imply that the transmission rates decrease since the number of contagious contacts decreases. See, for instance, References [38,39]. The quarantines on the susceptible have the qualitative effect of an impulsive vaccination since part of the susceptible population is kept away of infective contacts. This is equivalent to reduce, in practice, along a short time interval (instantaneously in an ideal impulsive model), the susceptible subpopulation. Similarly, quarantines on the infectious are similar to impulsive treatment actions on the infectious subpopulation, which is equivalent to reduce its numbers in short time intervals or instantaneously in the ideal case (See References [11,15,18,19]). Quarantine of infectious agents is also referred to as their isolation. However, in order to unify the nomenclature along the paper, the term quarantine will be used for all populations. Example 1 below on an SIR model without mortality is given as an introductory one to fix some basic ideas about quarantine decisions. Such decisions might be made on both susceptible and infectious decisions based on the hospital management regarding availability of beds and other hospital technical means necessary for infection treatment on seriously infected individuals. The solution Symmetry 2020, 12, 1646 4 of 33 for this model is obtained directly from the differential equations and it is then extended to more general SEIR models including quarantine actions and demography with the presence of exposed and asymptomatic individuals. It can be pointed out that the solution for an SIR model with demography has been obtained in Reference [5] by its reduction to an Abel-type equation by using a power-series perturbation approach. 2.1. Example 1 (An SIR Epidemic Model without Demography Subject to Monitored Quarantine) Consider the subsequent SIR (Susceptible “S”- Infectious “I”- Recovered “R”) without demography and disease mortality: . S(t) = −βS(t)I (t) . I (t) = βS(t)I (t) − γI (t) (1) . R(t) = γI (t) with S(0) = S0 , I (0) = I0 ≥ 0 and R(0) = R0 ≥ 0 with min(S0 , I0 , R0 ) ≥ 0, where β and γ are the transmission rate and the recovery rate, respectively. The total population is N (t) = S(t) + I (t) + R(t); ∀t ∈ R0+ with N (t) = N (0) = N0 being constant ∀t ∈ R0+ in view of (1). The solution of (1) is: Rt I (τ) dτ S(t) = e−β 0 S0 Rτ 0 I (σ) dσ Rt Rt β 0 S(τ) dτ−γt e−β I (t) = e I0 = e−γt eβS0 0 dτ I0 (2) Rt R(t) = eγ 0 I (τ)dτ R0 . Let Ih (t) = ρ (t)I (t) the estimated fraction of infectious which require hospital care at time t (referred to as the “seriously ill infectious subpopulation”) and Ih (t) the upper-bound of hospital admission availability at time t. Note the following simple result which is concerned with an eventual quarantine intervention of all the susceptible individuals: Proposition 1. Assume that, for a given ρ(t) ∈ (0 , 1] and an availability upper-bound Ih (t), it is possible to hospitalize all the seriously ill infectious subpopulations at time t via susceptible quarantine of all the susceptible γt subpopulations at time t0 = 0, if I0 ≤ ρe(t) Ih (t) ; ∀t ∈ R0+ . Proof. Assume that S0 = 0, that is, a total quarantine of the existing susceptible subpopulation at time t0 = 0, or absence of an initial susceptibility for an eventual contagion. Then ρ(t)I (t) = ρ(t)e−γt I0 ≤ Ih (t) γt from Equation (2) for any given t ∈ R0+ if I0 ≤ ρe(t) Ih (t) with initial immune amounts given by   γt R0 = N0 − I0 ≥ max 0, N0 − ρe(t) Ih (t) .  t∈R0+ In general, it might not be necessarily a total quarantine on the susceptible to accomplish with the hospital management requirements provided that the number of initial infections is small enough according to Proposition 1. In this sense, Proposition 1 might be easily generalized to the case of partial quarantine of the susceptible. It can be necessary to keep the hospital bed availability at time t along a time interval of a length of at least γ−1 , which is the recovery average period or up until some time exceeding t along the lasting of the most serious period of the illness force. The subsequent Proposition 2 is based on a partial withdrawal of the initial susceptible amounts from the contagion scenario via a quantifying parameter λ ∈ [0, 1] of the fraction population to withdraw from the contagion environment at the time instant where a quarantine intervention is decided.   For instance,   if a quarantine   is decided on  the susceptible subpopulation at time t1 , then S t1+ = (1 − λ)S t− 1 , where S t + 1 = lim S ( t ) and S t − = S(t ) = lim S(t) and the instantaneous net 1 1 t→t1+ t→t−   1    change of the susceptible subpopulation at t1 is ∆ S(t1 ) = S t1+ − S t− 1 = −λS t−1 . In the following, we show an alternative and confluent interpretation to the finite jumps in the solution decreases the Symmetry 2020, 12, 1646 5 of 33 susceptible candidate   individuals available  for contagion at the time instant t = t1 . A Dirac impulse of amplitude −λS t− 1 , namely −λδ ( t − t1 ) S t− 1 , with δ(.) denoting the Dirac delta distribution is applied to the time–derivative of S(t) at t = t1 , which is physically equivalent to the injection of such an instantaneous impulsive vaccination on the susceptibility [43]. Proposition 2 is concerned with a quarantine intervention at t = 0 on the susceptible subpopulation, which leads to the eventual existence bounded discontinuity of the solution at t = 0 and an impulsive corresponding time-derivative as a result. The piecewise continuous solution of the differential system is still unique since the finite jump at the discontinuity is calculated explicitly with its left and right limits at the discontinuity point. Proposition 2. Assume that S(0− ) = S− 0 = S0 and S(0+ ) = S0+ = (1 − λ)S0 for some λ ∈ [0 , 1], where λS0 is the amount of initial susceptible submitted to quarantine at the initial time instant t0 = 0. Then, the following properties hold. (i) The constraint ρ(t)I (t) I (t) is fulfilled for some given t ∈ R+ if λ ≥ λm = ≤ ! h ln Ih (t)−lnρ(t)+γt γt max 1 − Rτ , 0 provided that I0 ≤ ρe(t) Ih (t). βS0 0 e−β 0 I(σ) dσ dτ Rt h i h i (ii) The constraint ρ(θ)I (θ) ≤ Ih (θ) is fulfilled for θ ∈ t, t + γ−1 + η for any given η ∈ η0 − γ−1 , η1 and ! ln Ih (ξ)−lnρ(ξ)+γξ−lnI0 some prefixed η0 , η1 ∈ R0+ if and only if λ ≥ λmθ = max 1 − inf R ξ −βR τ I(σ) dσ , 0 . t≤ξ≤t+γ−1 +η βS0 0 e 0 dτ h i If ρ(θ) = ρ(t) and Ih (θ) = Ih (t); ∀θ ∈ t, t + γ−1 + η then the constraint ρ(θ)I (θ) ≤ Ih (θ) is fulfilled   ln I ( t ) −lnρ ( t )+ γt−lnI h i for θ ∈ t, t + γ + η if λ ≥ max1 − R t+γ−1 +η R τ −1 0  h  , 0. −β 0 I (σ) dσ βS0 0 e dτ Proof. The hospital management availability objective is fulfilled at time t if and only if: Rτ 0 I (σ) dσ Rt e−β ρ(t)e−γt eβ(1−λ)S0 0 dτ I0 ≤ I h ( t ) (3) or, equivalently, if and only if Z t Rτ I (σ) dσ lnρ(t) − γt + β(1 − λ)S0 e−β 0 dτ + lnI0 ≤ ln Ih (t) (4) 0 which is satisfied for some λ ∈ [0 , 1] if and only if λ ≥ λm . Property (i) has been proved. Property (ii) is a direct extension of Property (ii).    Related to Proposition 2, note that S0+ /S− 0 = 1 − λ and ∆S0 /S−0 = S0+ − S−0 /S−0 = −λ. Then, 1 − λ is the instantaneous relative variation of the susceptible subpopulation at the impulsive time instant t0 = 0 where a quarantine intervention is applied on the susceptible. In the same way, −λ is the relative negative decrement of the population at t0 = 0 because of the quarantine action. Similar interpretations apply for any quarantined subpopulation as it will be discussed later on in this section and in the remaining sections. Note that Proposition 1 describes the total initial quarantine of the susceptible as a particular case in Proposition 2 with λ = 1. Proposition 2 establishes a guaranteed fraction of the susceptible submitted to quarantine for achieving the hospital availability of serious cases at a certain time instant or time interval. Example 1 is now extended with two combined quarantine interventions on the susceptible and infectious subpopulations, respectively. For exposition simplicity, it is assumed in example 1 and in example 2 below that the transmission rate becomes constant. In practice, such a parameter decreases as the number of infectious-susceptible contacts decreases and increases since such a number increases. This fact would be taken into account along Sections 4 and 5 where a more general epidemic model will be discussed. Symmetry 2020, 12, 1646 6 of 33 2.2. Example 2 (A Partial Quarantine on the Susceptible Followed by a Later Partial Quarantine on the Infectious) Example 1 can be generalized by programming a first partial quarantine on the susceptible and a later partial quarantine on the infectious so that certain susceptible amounts can be protected from the infection and some infectious are then taken apart from the successive propagation of the infection. This makes sense if there are relatively few susceptible and relevant infectious numbers in the time where the action starts and it is the considered case. Later on, it is considered the case when there are initially relatively few infectious cases, which is a fraction of known cases submitted to quarantine, while, later on, as the infection progresses, there are relevant numbers of susceptible cases, which need to be protected as the infectious cases are increasing. This has been recently the case in many countries around the world concerning the COVID19 has out-broken from epidemic to pandemic [38,46–50]. Thus, assume that, at a time instant t1 , a fraction λ(t1 )S(t1 ) is submitted to quarantine on the interval [t1 , t3 ] and that, at a time instant t2 ∈ [t1 , t3 ), a fraction ρ(t2 )I (t2 ) of the infectious cases is submitted to quarantine on the time interval [t2 , t3 ]. Assume, with no loss in generality, that the initial conditions are taken at t0 = 0. The question arises is how to choose reasonably h λ(t1 ) i∈ (0 , 1), δ(t2 ) ∈ (0, 1), T0 = t1 – t0 , and T1 = t2 − t1 such that ρ(t)I (t) ≤ Ih (t), ∀t ∈ t3 , t3 + γ + η for any −1 h i η ∈ η0 − γ−1 , η1 and some prefixed η0 , η1 ∈ R0+ . The solution trajectory obeys from (1) the following set of equations of which there is at least one per subpopulation, which depends on the infectious evolution only from the initial conditions, in view of Equation (2). Rt I (τ) dτ S(t) = e−β 0 S0 ; ∀t ∈ [0, t1 ) (5) Rt −β 0 1 I (τ) dτ     S t1+ = (1 − λ(t1 ))S t− 1 = ( 1 − λ ( t 1 )) e S0 (6) Rt Rt I (τ) dτ −β I (τ) dτ Rt −β (1 − λ(t1 ))S t−1 = e−β 0 I(τ) dτ (1 − λ(t1 ))S0 ; ∀t ∈ [t1 , ∞)     S(t) = e t1 S t1+ = e t1 (7) Rτ 0 I (σ) dσ Rt Rt e−β I (t) = eβ 0 S(τ) dτ−γt I0 = e−γt eβS0 0 I0 ; ∀t ∈ [0 dτ , t1 ] (8) Rt β S(τ) dτ  +  I (t) = e−γ(t−t1 ) e t1 It R t −βR τ I(σ+t ) dσ R t −βR τ I(σ1) dσ (9) βS0 [(1−λ(t1 )) t e 0 1 1 dτ+ 0 e 0 dτ] = e−γt e 1 I0 ; ∀t ∈ [t1 , t2 ) Rτ Rτ 2 e−β 0 I (σ+t1 ) dσ 1 e−β 0 I (σ) dσ Rt Rt     βS [(1−λ(t1 )) dτ+ dτ] I t2+ = (1 − δ(t2 ))I t−2 = (1 − δ(t2 ))e−γt2 e 0 t1 0 I0 (10) Rt −γ ( t−t ) β t S(τ) dτ  +  I (t) =e 2 e 2 I t2 Rτ Rτ βS0 [(1−λ(t1 )) t 2 e−β 0 I(σ+t1 ) dσ dτ+ 0 1 e−β 0 I(σ) Rt Rt Rt dσ β S(τ) dτ dτ] (11) = e−γt e t2 (1 − δR(t2 ))e 1 I0 τ Rτ −γt+βS0 ((1−λ(t1 )) t e−β 0 I(σ+t1 ) dσ dτ+ 0 1 e−β 0 I(σ) dσ Rt Rt dτ) =e 1 (1 − δ(t2 ))I0 ; ∀t ∈ [t2 , ∞) R τ −β ν I (σ) dσ R γ 0 e−γτ eβS0 0 e Rt Rt 0 dν R ( t ) = eγ 0 I (τ)dτ R0 = e dτ R0 ; ∀t ∈ [0 , t1 ) (12)       R t1+ = R t− 1 + λ(t1 )S t−1 R τ −β ν I (σ) dσ (13) R 1 e−γτ eβS0 0 e Rt 0 dν Rt γ dτ 1 I (τ) dτ =e 0 R0 + λ(t1 )e−β 0 S0 R τ −β ξ I (σ+t ) dσ R Rξ Rt −β I (σ) dσ Rt Rt βS ((1−λ(t1 )) t e 0 1 dξ+ 1 e 0 dξ) γ t I (τ)dτ  +  γ t e−γτ e 0 1 0 I0 dτ R(t) = e 1 R t1 = e 1 Rν (14)  t1 −γτ+βS0 0τ e−β 0 I(σ) dσ dν dτ  R R Rt  ×eγ 0 e R0 + λ(t1 )e−β 0 I(τ) dτ S0 ; ∀t ∈ [t1 , t2 ) 1  Symmetry 2020, 12, 1646 7 of 33 R τ −β ξ I (σ+t ) dσ R Rξ Rt −β I (σ) dσ R t −γτ+βS0 ((1−λ(t1 )) e 0 t1 1 dξ+ 1 e 0 dξ) − = eγ t1 e       2 0 I0 dτ R t2+ = R t− + δ ( t2 ) I t  2R 2 R ν (15)  γ t1 e−γτ eβS0 0τ e−β 0 I(σ) dσ dν dτ R Rt  −β 1 I (τ) dτ   R0 + λ(t1 )e 0 S0 +δ(t2 )I t−  ×e 0  2  Rξ Rξ  R t −γτ+βS0 ((1−λ(t1 )) τ e−β 0 I(σ+t1 ) 1 e−β 0 I (σ) dσ dξ) R dσ Rt dξ+ R   t t γ t I (τ)+ t e 2 0 I0 dτ  Rt  1 γ I (τ)dτ    2 1 R t2+ = e  t2 R(t) = e R τ −β ν I (σ) dσ  R R   γ 0t1 e−γτ+βS0 0 e 0 dν dτ Rt −β 0 1 I (τ) dτ (16) R0 + λ(t1 )e  ×e S0  Rτ Rτ 2 e−β 0 I (σ+t1 ) dσ 1 e−β 0 I (σ) dσ Rt Rt −γt2 +βS0 [(1−λ(t1 )) dτ+ dτ] +δ(t2 )e t1 0 I0 ; ∀t ∈ [t1 , t2 ) We still keep the notation and term of “recovered” for all the subpopulation R(t) while noting the ones which have been transferred as a result of the susceptible quarantine as well as those that do not have permanent or temporary immunity. 2.3. Hospital Management Objective of Example 2 on a Temporary Time Interval hThe hospital imanagementh availabilityi objective is the fulfilment of ρ(t)I (t) ≤ Ih (t); ∀t ∈ t3 , t3 + γ−1 + η for any η ∈ η0 − γ−1 , η1 and some prefixed η0 , η1 ∈ R0+ with time intervals to take the quarantine actions defined by T1 = t1 − t0 and T2 = t2 − T1 with t0 = 0, that is, one gets from (11) that the objective is fulfilled if: Rτ Rτ 3 e−β 0 I (σ+t1 ) dσ 1 e−β 0 I (σ) dσ Rt Rt −γt3 +βS0 ((1−λ(t1 )) dτ+ dτ) ρ(t )e t1 h i 0 (1 − δ(t2 ))I0 ≤ Ih (t); (17) ∀t ∈ t3 , t3 + γ−1 + η , or equivalently,  Rt Rτ Rt Rτ  ln ρ(t) − γt3 + βS0 (1 − λ(t1 )) t e−β 0 I(σ+t1 ) dσ dτ + 0 1 e−β 0 I(σ) dσ dτ + ln(1 − δ(t2 )) + lnI0 ≤ lnIh ; (18) 1 h i ∀t ∈ t3 , t3 + γ−1 + η which holds if  R t +γ−1 +η −βR τ I(σ+t ) dσ Rt Rτ  βS0 (1 − λ(t1 )) t 3 e 0 1 dτ + 0 1 e−β 0 I(σ) dσ dτ + ln(1 − δ(t2 )) + lnI0 1 (19) ≤ min lnIh (t) − ln ρ(t) + γt3 t3 ≤t≤t3 +γ−1 +η h i which still holds under the sufficient condition to be fulfilled on t, t3 + γ−1 + η : " 0 ≤ 1 − λ ( t1 ) ≤ ! # R t1 Rτ −β 0 I (σ) dσ R t +γ−1 +η 1R τ 1 βS0 min lnIh (t) − ln ρ(t) + γt3 − ln(1 − δ(t2 )) − lnI0 − 0 e dτ 3 e−β 0 I (σ+t1 ) dσ dτ t3 ≤t≤t3 +γ−1 +η t1 or, equivalently, if   min I h (t) Rτ Rτ R t +γ−1 +η  t3 ≤t≤t3 +γ−1 +η   R −β 0 I (σ+t1 ) dσ +γt3 + 0 1 e−β 0 I(σ) dσ t 3 dτ−β−1 S−1  e ln dτ t1 0   (1−δ(t2 ))ρ(t) I0  1 ≥ λ ( t1 ) ≥ R t +γ−1 +η Rτ ≥0 (20) 3 e−β 0 I (σ+t1 ) dσ dτ t1 Symmetry 2020, 12, 1646 8 of 33 under the necessary condition that:   min I h (t)   t ≤t≤t + γ −1 + η  R t3 +γ−1 +η −βR τ I(σ+t ) dσ R t1 −βR τ I(σ) dσ 1 + β−1 S−1 0  γt3 + ln 3 3 (1−δ(t2 ))ρ(t) I0  ≥ e 0 1 dτ + e 0 dτ   t1 0  min  I h (t)  (21)  t ≤t≤t + γ −1 +η ≥ β−1 S−1 γt3 + ln 3 3   , 0   (1−δ(t2 ))ρ(t) I0   equivalently, if for a given min Ih (t), t3 satisfies the constraints: t3 ≤t≤t3 +γ−1 +η    R t +γ−1 +η Rτ Rτ  min I h (t)  t γ −1 η e−β 0 I(σ+t1 ) dσ dτ + 0 1 e−β 0 I(σ) dσ dτ − 1 − ln 3(1−δ3(t ))ρ(t) I  R t ≤t≤t + + γ−1 βS0 t 3    1 2 0   min I h (t)   (22)  R t +γ−1 +η Rτ Rτ  −1 +η t γ e−β 0 I(σ+t1 ) dσ dτ + 0 e−β 0 I(σ) dσ dτ − ln 3(1−δ3(t ))ρ(t) I , R t ≤t≤t + ≤ t3 ≤ γ−1 βS0 t 3 1    1 2 0  equivalently, if for a given t3 , Ih (t) satisfies the constraints: R t +γ−1 +η Rτ Rτ 0 I (σ+t1 ) dσ 1 e−β 0 I (σ) dσ Rt βS0 ( 3 e−β dτ+ dτ−1)−γt3 (1 − δ(t2 ))ρ(t) I0 e t1 0 R t +γ−1 +η −βR τ I(σ+t ) dσ Rτ (23) 1 e−β 0 I (σ) dσ Rt βS0 ( t 3 e 0 1 dτ+ dτ)−γt3 ≤ min Ih (t) ≤ (1 − δ(t2 ))ρ(t) I0 e 1 0 t3 ≤t≤t3 +γ−1 +η It becomes clear, as intuitively expected, that the fraction of susceptible allocated in quarantine since time has to be increased as the fraction of infectious in quarantine since time decreases and also as the initial susceptible and infectious subpopulations become larger. Rτ 0 I (σ) dσ Rt e−β I (t) = e−γt eβS0 0 dτ I0 ; ∀t ∈ [0, t1 ] (24) Rτ −γt1 +βS0 0 1 e−β 0 I(σ) dσ     Rt dτ I t1+ = (1 − δ(t1 ))I t− 1 = ( 1 − δ ( t 1 )) e I0 (25) Rτ 0 I (σ) dσ Rt βS0 e−β dτ   I (t) = e−γ(t−t1 ) e t1 I t1+ (26) Rτ 0 I (σ+t1 ) dσ Rt e−β = e−γt eβS0 0 dτ (1 − δ(t1 ))I0 ; ∀t ∈ [t1 , t2 ] (27) Rt −β 0 I (τ) dτ S(t) = e S0 ; ∀t ∈ [0, t1 ] (28) Rt Rt −β( 0 1 I (τ) dτ+ t1 2 I (τ) dτ) S(t) = e S0 (29) R t −γτ+βS R τ e−β 0ξ I(σ) dσ dξ R t −γτ+βS R τ e−β 0ξ I(σ+t1 ) dσ dξ R R 1 −βI0 ( 0 e 0 0 dτ+(1−δ(t1 )) t e 0 0 dτ) =e 1 S0 ; ∀t ∈ [t1 , t2 ) (30) R τ −β ξ I (σ) dσ R τ −β ξ I (σ+t ) dσ R R 1 −βI0 ( 0 1 e−γτ+βS0 0 e dτ+(1−δ(t1 )) t 2 e−γτ+βS0 0 e Rt 0 dξ Rt 0 dξ   dτ) S t2+ = (1 − λ(t2 ))S(t2 ) = (1 − λ(t2 ))e 1 S0 (31) In this case, the quarantine of a fraction of the infectious precedes thath of a fraction i of the susceptible. The objective is now the fulfilment of the ρ(t)I (t) ≤ Ih (t); ∀t ∈ t3 , t3 + γ−1 + η for any h i η ∈ η0 − γ−1 , η1 and some prefixed η0 , η1 ∈ R0+ with time intervals to take the quarantine actions defined by T1 = t1 − t0 and T2 = t2 − T1 with t0 = 0, Rτ 0 I (σ+t2 ) dσ Rt βS(t+ ) e−β dτ h i ρ(t)e−γ(t−t2 ) e 2 t2 I (t2 ) ≤ Ih (t); ∀t ∈ t3 , t3 + γ−1 + η (32) Symmetry 2020, 12, 1646 9 of 33 h i for any η ∈ η0 − γ−1 , η1 and some prefixed η0 , η1 ∈ R0+ with time intervals to take the quarantine actions defined by T1 = t1 − t0 and T2 = t2 − T1 with t0 = 0. Then,  Z t Rτ −β 0 I (σ+t2 ) dσ β(1 − λ(t2 ))S t− 2 e dτ + lnρ(t) − γ(t − t2 ) + lnI (t2 ) ≤ lnIh (t) (33) t2  R t Rτ R t −βR τ I(σ) dσ ! −β 0 I (σ+t2 ) dσ −γt2 eβS0 0 2 e 0 dτ β(1 − λ(t2 ))S t− 2 t e dτ + lnρ ( t ) − γ ( t − t 2 ) + ln e ( 1 − δ ( t 1 )) I0 ≤ lnIh (t) (34) 2 or,  R t Rτ R t2 −βR τ I(σ) dσ −β 0 I (σ+t2 ) dσ β(1 − λ(t2 ))S t− 2 t e dτ + lnρ ( t ) − γt + βS0 0 e 0 dτ + ln(1 − δ(t1 )) + lnI0 ≤ lnIh (t) 2 h i guaranteed for all t ∈ t3 , t3 + γ−1 + η , that is for the largest value of Formula (33) if  R t3 +γ−1 +η Rτ Rt Rτ β(1 − λ(t2 ))S t− 2 t e−β 0 I(σ+t2 ) dσ dτ + lnρ(t) − γt3 + βS0 0 2 e−β 0 I(σ) dσ dτ + ln(1 − δ(t1 )) + lnI0 (35) 2 ≤ min lnIh (t), t3 ≤t≤t3 +γ−1 +η which can be always guaranteed if δ(t1 ) = 1 for the objective min lnIh (t) > 0, that is for the t3 ≤t≤t3 +γ−1 +η removal of all the infectious cases h via quarantine i at t = t1 if the objective is to keep any number of infectious on the time interval t3 , t3 + γ + η since −1 Z t2 Rτ I (σ) dσ − ∞ = lnρ(t) − γt3 + βS0 e−β 0 dτ − ∞ + lnI0 ≤ lnIh (t) (36) 0  Z t3 +γ−1 +η Rτ I (σ+t2 ) dσ β(1 − λ(t2 ))S t− 2 e−β 0 dτ t2 Z t2 Rτ I (σ) dσ + lnρ(t) − γt + βS0 e−β 0 dτ + ln(1 − δ(t1 )) + lnI0 ≤ lnIh (t) (37) 0 Assume that the information about the numbers of infections and those susceptible are not precise. Therefore, the number of known infectious cases is at most µ(t)I (t) and that of susceptible cases is at most ν(t)S(t) with known µ(t) , ν(t) ∈ [0, 1]. Therefore, δ(t1 ) ∈ [0 , µ(t1 )] and λ(t2 ) ∈ [0 , ν(t2 )]. We proceed as follows: 2.4. Algorithm 1 It consists of the following main steps: Symmetry 2020, 12, 1646 10 of 33 Algorithm 1 1 Step 0. Given are t3 > 0, µ , ν , ρ : R0h+ → [0, 1] with i ρ ( t ) ≤ µ ( t ) ; t ∈ R0 + 2 Fix t1 ≥ 0 , t2 ∈ [t1 , t3 ), η0 , η1 ≥ 0η ∈ η0 − γ−1 , η1 3 Define  R t3 +γ−1 +η Rτ 4 L(t1 , t2 , t3 , µ̂(t1 ), ν̂(t2 )) = β(1 − ν̂(t2 )λ(t2 ))S t−2 t e−β 0 I(σ+t2 ) dσ dτ Rτ 2 R t2 5 +lnρ(t) − γt3 + βS0 0 e−β 0 I(σ) dσ dτ + ln(1 − µ̂(t1 )δ(t1 )) + lnI0 6 Step 1. If L(t1 , t2 , t3 , 0 , 0) ≤ min lnIh (t), then the hospital availability objective is t3 ≤t≤t3 +γ−1 +η 7 fulfilled without quarantine actions neither on the infectious patients nor 8 on the susceptible individuals. 9 Go to End. 10 Step 2. If L(t1 , t2 , t3 , µ(t1 ) , 0) ≤ min lnIh (t), then the hospital availability objective is t3 ≤t≤t3 +γ−1 +η 11 ( quarantine action on the infectious at time t = t1 . fulfilled with some ) 12 Defineδ̂(t1 ) = inf µ̂(t1 ) ∈ (0, µ(t1 )] : L(t1 , t2 , t3 , µ̂(t1 ) , 0) ≤ min lnIh (t) . t3 ≤t≤t3 +γ−1 +η 13 Then,the hospital availability h objective i is fulfilled with any eventually partial or total quarantine action 14 with fraction δ(t1 ) ∈ δ̂(t1 ) , µ(t1 ) at time t = t1 on the infectious (which is necessarily total if 15 δ̂(t1 ) = µ(t1 )). Go to End. 16 Step 3. If L(t1 , t2 , t3 , µ(t1 ) , λ(t2 )) ≤ min lnIh (t), then the objective is fulfilled with total t3 ≤t≤t3 +γ−1 +η 17 quarantine action on the infectious cases at time t = t1 and some quarantine action on the 18 susceptible ones( at the time instant t2 . ) 19 Defineλ̂(t2 ) = inf ν̂(t2 ) ∈ (0, ν(t2 )] : L(t1 , t2 , t3 , µ(t1 ) , ν̂(t2 )) ≤ min lnIh (t) . t3 ≤t≤t3 +γ−1 +η 20 Then, the hospital availability h objective i is fulfilled with any eventually partial or total quarantine action 21 with fraction λ(t2 ) ∈ λ̂(t2 ) , ν(t2 ) at time t = t2 on the infectious (which is necessarily total if 22 λ̂(t2 ) = ν(t2 )). Go to End. 23 Step 4. If L(t1 , t2 , t3 , µ(t1 ) , λ(t2 )) > min lnIh (t), then the hospital availability objective t3 ≤t≤t3 +γ−1 +η 24 cannot be solved with the time- scheduling specifications of Step 0. 25 Step 5. If possible, decrease one or both of the 26 quarantine time instants to anticipate the quarantine actions. Go to Step 1. Otherwise, go to End. The following result holds concerning the feasibility of Step 5 of Algorithm 1. Proposition 3. The following properties hold: (i) If S01 = S1 (0) ≤ S02 = S2 (0) and I01 = I1 (0) ≤ I2 (0) = I02 then I1 (t) ≤ I2 (t), ∀t ∈ R0+ . (ii) The “if” part of Step 5 of algorithm 1 can always be performed by fixing any t0 > 0 and t0 ← t1 , t2 in Step 1 unless Step 0 has been performed for quarantine time actions t1 = t2 = 0. Proof. Assume that there exists some t > 0 such that I1 (t) > I2 (t) and I1 (τ) ≤ I2 (τ) for τ ∈ [0 , t) and proceed by contradiction arguments. One has from Equation (2) that: Rτ Rτ 0 I1 (σ) dσ 0 I2 (σ) dσ Rt Rt e−β e−β I01 (t) = e−γt eβS01 0 dτ I01 > I02 (t) = e−γt eβS02 0 dτ I02 what implies that Z t Rτ Z t Rτ I1 (σ) dσ I2 (σ) dσ 1 S01 e−β 0 dτ − S02 e−β 0 dτ > (ln I02 − ln I01 ) ≥ 0 0 0 β so that Z t Rτ Z t Rτ Z t Rτ −β 0 I2 (σ) dσ S02 −β 0 I2 (σ) dσ e dτ ≤ e dτ < e−β 0 I1 (σ) dσ dτ 0 S01 0 0 Symmetry 2020, 12, 1646 11 of 33 Rt Rτ Rt Rτ and then 0 e−β 0 I2 (σ) dσ dτ < 0 e−β 0 I1 (σ) dσ dτ so that there exists some ε ∈ (0 , t) such that Rt Rt t−ε I2 ( σ ) dσ > I (σ) dσ. Thus, there exists some 0 < t0 < t such that I2 (t0) > I1 (t0), which contradicts t−ε 1 the assumption that I1 (t) > I2 (t) and I1 (τ) ≤ I2 (τ) for τ ∈ [0 , t). As a result, I1 (t) ≤ I2 (t); ∀t ∈ R0+ and the Property (i) is proven. Property (ii) is a direct consequence of Property (i) and Step 0, Step 1, and Step 5 of algorithm 1 and the fact that the initial conditions may be taken at any initial time t0 > 0.  Remark 1. It turns out that the average disease transmission rate depends on the number of contagion contacts between susceptible and infectious cases and it increases with the number of average contacts. Therefore, the above example can be directly generalized without difficulty to a piecewise constant disease transmission rate function defined by β(t) = β0 for t ∈ [0, t1 ) (Equations (5)–(8) and (12)–(13)), β = β1 , t ∈ [t1 , t2 ) (Equations (9)–(11) and (14)–(16)) and β = β2 , t ∈ [t2 , t3 ). Similar considerations apply to example 1. 3. An SEIAR Epidemic Model Subject to Confinement or to Partial Quarantines 3.1. The Model The ideas of the examples of Section 2 are extended to a more general model. The following SEIAR (susceptible “S”, exposed “E”, symptomatically infectious “I”, asymptomatically infectious “A” and recovered “R”) is proposed: .    S(t) = b1 − b2 + β I (t) + e βA A ( t ) S ( t ) + η R ( t ) (38) .   E ( t ) = − ( b2 + γ ) E ( t ) + β I ( t ) + e βA A(t) S(t) (39) . I (t) = −(b2 + α + τ0 )I (t) + γpE(t) (40) . A(t) = −(b2 + τ0 )A(t) + γ(1 − p)E(t) (41) . R(t) = −(b2 + η)R(t) + τ0 (I (t) + A(t)) (42) with initial conditions S(0) = S0 ≥ 0, E(0) = E0 ≥ 0, I (0) = I0 ≥ 0, A(0) = A0 ≥ 0 and R(0) = R0 ≥ 0 with max(S0 , E0 , I0 , A0 , R0 ) > 0. The parameters have the subsequent interpretation. b1 is the recruitment rate, b2 is the natural  average death rate, β an e βA β eβA ≤ 1 are the transmission rates of the symptomatic and asymptomatic infectious cases, respectively. η is the average immunity rate, γ is the average transition rate from the exposed to all the infectious, i.e., A(t) + I (t), α is the average extra mortality associated with the disease, τ0 is the average natural immune response rate for the whole infectious subpopulation A(t) + I (t), p and 1 − p are the fractions of the exposed that become symptomatic and asymptomatic infectious cases, respectively. The total population N (t) = S(t) + E(t) + I (t) + R(t) obeys the differential equation: . N (t) = b1 − b2 N (t) − αI (t) (43) With the initial condition N0 = S0 + E0 + I0 + A0 + R0 . The above model has two separate transitions from three exposed to the asymptomatic and symptomatic infectious cases so that the two corresponding transition fractions sum-up unity. At the same time, the model considers that the disease transmission rates of the symptomatic and asymptomatic infectious cases to the susceptible cases are eventually distinct. A reason for that is that the symptomatic case can transmit the illness with stronger force such as by means of a stronger and persistent cough. Symmetry 2020, 12, 1646 12 of 33 3.2. Hospital Management Objective on a Temporary Time Interval The hospital management objective within the time interval [t∗ , t∗ + T ] is the fulfilment of the following constraint. ρ(t)I (t) ≤ IH (t); ∀t ∈ [t∗ , t∗ + T ] for a given ρ : [t∗ , t∗ + T ] → [0, 1] . IH : bt∗ , t∗ + Tc → R0+ defines the hospital maximum admissible individuals according to its availability on bed disposal and sanitary necessary means for the seriously ill symptomatic infectious individuals ρ(t)I (t) who need medical care in the hospital. The quarantine time instants are defined by the strictly increasing sequence {ti }∞ 0 ⊂ R0+ or by a finite subset of this sequence. Any fraction of a subpopulation x(t) submitted to quarantine   at time  ti translates     + into the reduction of its numbers for possible contagions, according to x ti = 1 − δx t− i x t− i   h  i where δx t− i ∈ 0 , µx t − ⊂ [ 0, 1 ] is the fraction of that subpopulation, which is quarantined. i  Usually, its maximum µx t− i is less than one or even small enough than one due to precise knowledge of the current numbers belonging to that population. Since the susceptible, exposed, and asymptomatic infectious cases are not mutually identifiable as separable from the others in most of the cases because, due to the lack of efficient tests or availability for their application, they are typically considered together to establish their numbers in quarantine. The problem is more tractable by considering that the asymptomatic infectious cases are a fraction of the symptomatic ones. The proportionality function is obtained from (40)–(41) in the sequel. First, one gets from (40)–(41), the following solutions: Z t −(b2 +α+τ0 )t I (t) = e I0 + γp e−(α+b2 +τ0 )(t−τ) E(τ)dτ (44) 0 Z t A(t) = e−(b2 +τ0 )t A0 + γ(1 − p) e−(b2 +τ0 )(t−τ) E(τ)dτ (45) 0 so that   αt ϑ I + γ(1 − p) t e(b2 +τ0 )τ E(τ)dτ R A(t) e 0 0 0 ϑ(t) = = Rt (46) I (t) I0 + γp 0 eατ e(b2 +τ0 )τ E(τ)dτ A which is also coherent with a proportionality of initial conditions ϑ(0) = ϑ0 = I00 and which can be calculated for any other previous values of the exposed subpopulation as follows:  Rh  A(t) eαh ϑ t−h I t−h +γ 1−p 0 e ( ) ( ) ( ) ( b 2 + τ0 )( t−h + τ ) E t − h + τ dτ ( ) ϑ(t) = = Rh (47) I (t) I (t − h) + γp 0 e(α+b2 +τ0 )(t−h+τ) E(t − h + τ)dτ and Equations (38)–(39) can then be rewritten via Equation (46) as: .    S ( t ) = b1 − b2 + β 1 + e βA ϑ(t) I (t)S(t) + η R(t) (48) .   E(t) = −(b2 + γ)E(t) + β 1 + e βA ϑ ( t ) I ( t ) S ( t ) (49) The SEIADR model can be alternatively interpreted as a special case of SEIR by removing the explicit evolution of the asymptomatic infectious cases while considering its influence in the susceptible, exposed, and recovered subpopulations via the proportionality function (47). Therefore, the trajectory solution of Equations (38)–(42) in view of Equations (46), (48) and (49), can be expressed by Equations (44) and (45), subject to Equation (46), together with the subsequent equations: Rt Z t Rt (I (τ)+e βA A(τ))dτ S(t) = e−b2 t−β 0 S0 + e−(b2 (t−τ)+β τ (I (ξ)+e βA A(ξ))dξ) (b1 + ηR(τ))dτ (50a) 0 Symmetry 2020, 12, 1646 13 of 33 Rt Z t Rt βA ϑ(τ))I (τ)dτ =e −b2 t−β 0 (1+e S0 + e−(b2 (t−τ)+β τ βA ϑ(ξ))I (ξ)dξ) ( 1 +e (b1 + ηR(τ))dτ (50b) 0 Z t −(b2 +γ)t e−(b2 +γ)(t−τ) I (τ) + e   E(t) = e E0 + β βA A(τ) S(τ)dτ (51a) 0 Z t = e−(b2 +γ)t E0 + β e−(b2 +γ)(t−τ) 1 + e   βA ϑ(τ) I (τ)S(τ)dτ (51b) 0 Z t − ( b2 + η ) t R(t) = e R0 +τ0 e−(b2 +η)(t−τ) (I (τ) + A(τ))dτ (52a) 0 Z t − ( b2 + η ) t =e R0 +τ0 e−(b2 +η)(t−τ) (1 + ϑ(τ))I (τ)dτ (52b) 0 Proposition 4. The following properties hold under finite non-negativity of the initial conditions: (i) All the subpopulations, and then the total population, are non-negative for all time. (ii) All the subpopulations, and then the total population, are bounded for all time. Proof. The proof of property (i) is immediate by inspection of Equations (44), (45), (50), (51), and (56). The proof of property (ii) follows by contradiction. Assume that N : R0+ → R0+ is unbounded. . Then, there is a strictly increasing sequence { ti }∞ i=0 such that N (ti ) → ∞ as ti → ∞ and N (ti ) > b −αI (t ) b 0. Then from Equation (43), N (ti ) < 1 b i ≤ b1 , which contradicts that N (ti ) → ∞ as ti → ∞ . 2 2 Then, N : R0+ → R0+ is bounded. Since all the subpopulations are non-negative for all time from Property (i), then they are bounded as well for all time and property (ii) has been proven.  The following result proves that ϑ(t) = A(t)/I (t) is always bounded for all time. Therefore, it cannot happen that A(t) converges asymptotically to a positive value or that it oscillates provided that I (t) converges asymptotically to zero. This fact is already known as a consequence of fixing α = 0 and p = 1/2 since, in this case, the asymptomatic subpopulation cannot evolve by exceeding the numbers of the infectious subpopulation. Proposition 5. ϑ : R0+ → R0+ is bounded for any finite initial conditions satisfying I0 > 0. Proof. First note that if, I0 > 0, then ϑ(0) = ϑ0 > 0 and finite since A0 ≥ 0 is finite since the initial conditions are non-negative and finite. Note also from (44)–(45) that, since A0 and I0 , are nonzero, then A(t) and I (t) cannot be zero at any finite time even though they can potentially converge asymptotically to zero. Now, assume that ϑ : R0+ → R0+ is not bounded and proceed by contradiction arguments. Note the following:  Claim 1. If ϑ : R0+ → R0+ is unbounded, then there exists a strictly increasing sequence {ti }∞i=0 ⊂ R0+ with ∞ ti + 1 − ti ∞ ϑ ϑ   i=0 being bounded such that ( t i ) − ( ti − h ) i=0 is strictly increasing. Proof of Calim 1. Assume that this is not the case so that ϑ(ti ) − ϑ(ti − h) ∞  i=0 is not strictly increasing for any given strictly increasing {ti }∞ i=0 ⊂ R 0+ and ϑ : R 0+ → R 0+ is unbounded. ∞ However, if ϑ(tk ) ≤ ϑ(tk − h); ∀tk ∈ {ti }i=0 , and since ϑ : R0+ → R0+ is continuous, then ϑ(τ) is bounded for τ ∈ [tk − h , tk ] since this interval is bounded ∀tk ∈ {ti }∞ ∞ i=0 and any sequence {ti }i=0 . Then, ϑ : R0+ → R0+ is bounded, which contradicts its unboundedness. Then, if ϑ : R0+ → R0+ is ∞ unbounded, then there exists a strictly increasing sequence {ti }∞ i=0 ⊂ R0+ such that ϑ(ti ) − ϑ(ti − h) i=0  is strictly increasing. The proof follows by proving that there is no such sequence {ti }∞ i=0 ⊂ R0+ such that ϑ(ti ) − ϑ(ti − h) ∞ ϑ  i=0 is strictly increasing so that : R 0+ → R 0+ is unbounded.  Symmetry 2020, 12, 1646 14 of 33     Claim 2. From claim 1, one has that ϑ t j > ϑ t j − h for any t j ∈ {ti }∞ ∞ i=0 with the sequence {ti }i=0 ⊂ R0+ existing from claim 1. Then, one has from Equation (47) that: ! γ(1−p) R h (b2 +τ0 )(t j −h+τ) eαh     I tj − h + e − h + τ dτ E tj ϑ(t j −h) 0  Rh   >1 (53) I t j − h + γp 0 e(α+b2 +τ0 )(t j −h+τ) E t j − h + τ dτ   so that Rh γ(1−p)eah R h (b2 +τ0 )(t j −h+τ) eαh − 1 I t j − h > γp 0 e(α+b2 +τ0 )(t j −h+τ) E t j − h + τ dτ −        e E tj − h + τ dτ (54) ϑ(t j −h) 0 and then   h γ(1 − p)eah   Z   1 (b2 +τ0 )(t j −h+τ)   α(t j −h+τ)  I tj − h > e γpe −   E t j − h + τ dτ; ∀t j ∈ {ti }∞ (55)  eαh − 1 0  ϑ tj − h  i=0 Now, since ϑ(ti ) − ϑ(ti − h) ∞  i=0 is strictly increasing, then: Case (a) If E : R0+ → R0+ is bounded, and since, ϑ(ti ) ∞  i=0 is unbounded, then γp h Z e(b2 +τ0 )(t j −h+τ) eα(t j −h+τ) E t j − h + τ dτ     I tj − h > (56) eαh −1 0 γp  Z h    (b2 +τ0 )(t j −h+τ) α(t j −h+τ)     lim infI t j − h − αh  e e E t j − h + τ dτ > 0 (57) t j →∞ e −1 0 n o∞ n o∞ ∀ti ∈ tik for some subsequence tik of {ti }∞ i=0 so that either the claimed bounded n o∞ k =0 k =0 → R0+ leads to E tik → 0 as k → ∞ or I (tik ) ∞ ∞     E : tik k =0 k=0 is unbounded and then I (ti ) i=0 is also unbounded. However, E(ti ) ∞  i=0 being bounded implies from Equation (51) and from continuity arguments (since a continuous function cannot be unbounded in a finite interval, it is bounded on its boundary) that I : R0+ → R0+ is bounded for the sequence {ti }∞ i=0 . Since t j+1 − t j ≤ T < ∞; ∀t ∈ {t }∞ ,  j  i i=0 it follows that E, I : R0+ → R0+ are bounded. Thus, if E : R0+ → R0+ is bounded, then E t j → 0 as   t j → ∞ and, from Equation (44), I : R0+ → R0+ is bounded and, also, from Equation (40), I t j → 0 as t j → ∞ . However, as a result, Equation (57) is untrue so that if ϑ : R0+ → R0+ is unbounded then E : R0+ → R0+ and I : R0+ → R0+ are unbounded and Case a is not possible, which leads to consider Case b discussed below. Case (b) If E : R0+ → R0+ is unbounded, then I : R0+ → R0+ is unbounded too. Thus, one gets from (39)–(40) that     Z tj Z tj A t j + I t j = (1 + ϑ(0))I0 − α I (τ)dτ − (b2 + τ0 ) (1 + ϑ(τ))I (τ)dτ (58) 0 0 and the following contradiction arises for any given finite initial conditions since:      R tj R tj  A0 + I0 = (1 + ϑ(0))I0 = lim A t j + I t j + α 0 I (τ)dτ + (b2 + τ0 ) 0 (1 + ϑ(τ))I (τ)dτ = ∞ (59) t j →∞ As a result, E : R0+ → R0+ is bounded if ϑ(ti ) − ϑ(ti − h) ∞  i=0 is strictly increasing and Case b is not possible. Since neither case a nor case b are possible, one concludes that ϑ(ti ) − ϑ(ti − h) ∞  i=0 is ∞ not strictly increasing for any sequence {ti }i=0 and any given h > 0 with a positive finite separation between consecutive elements. As a result, ϑ : R0+ → R0+ is bounded as a result of a contradiction to the initial assumption of the proof. Symmetry 2020, 12, 1646 15 of 33 4. Total Confinement or Partial Quarantines to Achieve the Hospital Availability Objective We consider a sequence (or a finite denumerable set) of isolated time instants Sc = {ti }N c i=0 at which the quarantine intervention actions will be applied in order to satisfy some defined management hospital objective of intensive care of the seriously ill patients. If Sc is a finite set, then Nc = cardSc < ∞. If Sc is a sequence, its cardinal is infinity, denoted by Nc = χ0 , since it is a denumerable set of infinity cardinal. Note the following facts: h i Fact 1. δx (ti ) ∈ 0 , δx (ti ) with δx (ti ) ≤ 1 for x = S, E, A, I, ∀ti ∈ Sc . Usually, δx (.) ≤ δx (.) < 1 for x = S, E, A since the suitable universal confinement of the fraction of population including the non-symptomatic infectious cases cannot be performed because of the fact that basic services in industry, health, transportation, etc. have to be kept and, furthermore, it is not possible to control precisely all the programmed quarantined individuals. Fact 2. Usually in practice, µI (ti ) < 1, ∀ti ∈ Sc because all the infectious individuals are not known because of the limitation of appropriate testing and the difficulty of identifying those with a low level of symptoms. Fact 3. For quasi-universal quarantines, except for the maintenance of the basic services, it can be programmed for the use of identical fractional values δS (ti ) = δE (ti ) = δA (ti ) = µ(ti ), ∀ti ∈ Sc to get the same proportionality of quarantined numbers for all the non-symptomatic infectious cases because of the practical lack of technical means for identifying them separately from the symptomatic infectious. Fact 4. The transmission rate is proportional to the susceptible–infectious contacts so that quarantines or total confinements translate into a decrease in the number of contacts and then into smaller values of the disease transmission rate β. For exposition and simplicity in the equations’ presentation, it is assumed that the transmission rate is a piecewise constant function β(t) = β(ti ), ∀t ∈ [ti , ti+1 ), where ti and ti+1 are any consecutive time instants in Sc . The quarantine fractions for each subpopulation can be programmed at a set of sampling instants belonging to Sc subject to minimum and maximum interval constraints, namely, 0 < T0i ≤ ti+1 − ti = Ti ≤ T < ∞; ∀ti ∈ Sc (60) Assume that the subpopulation R(t) is an “extended like-recovered” subpopulation, which contains the recovered subpopulation while also incorporating the fractions of susceptible, exposed,  and infectious cases, which are quarantined. Let the non-negative real sequences SM (ti ) ti∈Sc ,     EM (ti ) ti∈Sc , IM (ti ) ti∈Sc , AM (ti ) ti∈Sc and RM (ti ) ti∈Sc be the maximum suitable values on [ti , ti+1 ) for         ti ∈ Sc for given values S t− i ,E t− i , I t− i and R t− i , namely, SM (ti ) ≥ sup S(t), EM (ti ) ≥ sup S(t), ti ≤t 0, δ t j ∈ [0, 1]; ∀t j ∈ Timp , Timp is a finite or (denumerable) infinite set or real numbers and x(0− ) ∈ R is given. Then,     i i Y  i h  iZ tk R t  i R t Z t R t τ a(ξ) dξ e τ a(ξ) dξ u(τ)dτ Y   X  h a ( ) dτ + x(t) =  1 − δ t j e 0 x0 +  1 − δ t j   e τ u ( τ ) dτ +      t+  t∗ j=1 k =1 j=k k−1 i     i h i Y  i h  iZ tk R t  i R t a(τ) dτ   Z t R t a(ξ) dξ e τ a(ξ) dξ u(τ)dτ;  Y X + =  1 − δ t j e ` t x t` + 1 − δ t e τ u ( τ ) dτ +  j          t+ t j=` +1 k =` +1 j=k k−1 i ∀t ∈ [ti , ti+1 );∀t` , t`+1 , ti (≥ t`+1 ) ∈ Timp is the unique solution of the above impulsive differential equation for t ∈ [ti , ti+1 ); ∀ti , ti+1 ∈ Timp , and, for all, t ≥ tM if tM is the maximum element of Timp if such a set has a finite cardinal. Proof. It is organized by induction. Take a time instant t ∈ [ti , ti+1 ); ∀ti , ti+1 ∈ Timp . Then,   Z t x(t) = (1 − δ(ti )) Φ(t, ti )x t− i + Φ(t, τ)u(τ)dτ ti+ Rt a(ξ)dξ where Φ(t, τ) = e τ ; ∀τ, t(≥ τ), t0(≥ t) ∈ R0+ . Note that for τ ∈ [t, t0]: R t0 Rτ R t0 a(ξ)dξ a(ξ)dξ+ a(ξ)dξ Φ(t, t0) = e t =et τ = Φ(t, τ)Φ(τ, t0) Symmetry 2020, 12, 1646 29 of 33   By using recursion to compute x t− i from the solution x(ti−1 ) at the preceding ti−1 ∈ Timp and so on until reaching t0 = 0 yields:    R ti  R t x(t) = (1 − δ(ti )) Φ(t, ti ) (1 − δ(ti−1 )) Φ(ti , ti−1 )x t− i−1 + + ti−1 Φ ( ti , τ ) u ( τ ) dτ + t+ Φ(t, τ)u(τ)dτ   Rt Rt i = (1 − δ(ti )) (1 − δ(ti−1 )) Φ(t, ti−1 )x t−i−1 + (1 − δ(ti )) t+ Φ(t, τ)u(τ)dτ + t+ Φ(t, τ)u(τ)dτ i−1 R i Q i  Pi Qi h  i t Rt k=0 [1 − δ(tk )] Φ (t, 0)x(0 ) + k=1 j=k 1 − δ t j Φ(t, tk ) t+k Φ(tk , τ) u(τ)dτ + t+ Φ(t, τ)u(τ)dτ = − Q i  Pi Qi h  iR t k−1 Rt i k=0 [1 − δ(tk )] Φ (t, 0)x(0 ) + k=1 j=k 1 − δ t j + Φ (t, τ) u(τ)dτ + t+ Φ (t, τ)u(τ)dτ = − k tk−1 Q  P Q i h  iR t −t  i   i −) + i + + = k =0 [ 1 − δ ( t k )] Φ ( t, 0 ) x ( 0 k =1 j=k 1 − δ t j 0 k k−1 Φ t, tk−1 + τ u tk−1 + τ dτ R t−t + 0 i Φ(t, ti + τ)u(ti + τ)dτ; ∀t ∈ [ti , ti+1 ) Q  Pi Qi h  iR t −t     i + + k=1 [1 − δ(tk )] Φ (t, 0)x(0 ) + k=1 j=k 1 − δ t j Φ t, tk−1 + τ u tk−1 + τ dτ = + k k−1 R t−t 0 + 0 i Φ(t, ti + τ)u(ti + τ)dτ; ∀t ∈ [ti , ti+1 ); ∀t` , t`+1 , ti (≥ t`+1 ) ∈ Timp and the proof is complete.  In general, for a sequence of potential confinement time instants, Sc = {ti }N c i=0 by taking t0 = 0 with no loss in generality, one obtains the following set of equations for the various subpopulations by proceeding recursively from the above Equations (A1)–(A11) and Theorem A1. Rt (I (τ)+e βA A(τ))dτ Rt −b2 (t−ti )−β(ti )   Rt S(t) =e ti S ti+ + t e−(b2 (t−τ)+β(ti ) τ (I(ξ)+βA A(ξ))dξ) (b1 + ηR(τ))dτ; e i (A12) ∀t ∈ [ti , ti+1 ) Rt βA ϑ(τ))I (τ)dτ Rt −b2 (t−ti )−β(ti ) ( 1 +e   Rt =e ti S ti+ + t e−(b2 (t−τ)+β(ti ) τ (1+βA ϑ(ξ))I(ξ)dξ) (b1 + ηR(τ))dτ; e i (A13) ∀t ∈ [ti , ti+1 ) Q j  −b2 (t−t` )−β(t` )R t (I(τ)+e βA A(τ))dτ  −  = i=` [ 1 − δ S ( t i )] e t` S t` Rt Rt + + Pj k −tk−1 −(b2 (t−tk−1 −τ)+β(tk−1 ) τ (I (tk−1 +ξ)+e βA A(tk−1 +ξ))dξ) Q j    + +k =` +1 i=k [1 − δS (ti )]0R e b1 + ηR tk−1 + τ dτ R t−t −(b (t−t −τ)+β(t j ) t (I(t+ +ξ)+e (A14) βA A(t+ +ξ))dξ)    + 0 je 2 j τ j j b1 + ηR t+ j + τ dτ; ∀t` , t j (≥ t` ), t j+1 ∈ Sc , h  ∀t ∈ t j , t j+1 R t−t −b2 (t−t` )−β(t` ) 0 ` (1+e    βA ϑ(t`+ +τ))I (t`+ +τ)dτ Q j  = i=` [ 1 − δ ( S i t )] e S t−`  R t −t R t + + Pj k k−1 −(b2 (t−tk−1 −τ)+β(tk−1 ) β ϑ ξ ξ Q j    ( 1 + ( t + )) I ( t + ) dξ ) + + k =` +1 [1 − δS (ti )] 0 e τ A b1 + ηR tk−1 + τ dτ e k−1 k−1 i=k Rt (A15) R t−t −(b (t−t −τ)+β(t j ) (1+e βA ϑ ( t + +ξ))I (t+ +ξ)dξ)    + 0 je 2 j τ j j b1 + ηR t+ j + τ dτ; ∀t` , t j (≥ t` ), t j+1 ∈ Sc , h  ∀t ∈ t j , t j+1 Q j [1 − δE (ti )] e−(b2 +γ)(t−t` ) E t−`    E(t) = i=` Pj Qj t −t + k=`+1 i=k [1 − δE (ti )] 0 k k−1 e−(b2 +γ)(t−tk−1 −τ) β(tk−1 ) I tk−1  R        + + + +τ +e βA I tk−1 + τ S tk−1 + τ dτ (A16) R t−t         + 0 j e−(b2 +γ)(t−t j −τ) β t j I t+ h  j +τ +e βA I t + j + τ S t+ j + τ dτ; ∀ti , t j ∈ Sc , ∀t ∈ t j , t j+1 Qj I (t) = i=` [1 − δI (ti )]e−(b2 +α+τ0 )(t−t` ) I t−   R t −t ` Pj Q j +γp k=`+1 i=k [1 − δI (ti )] 0 k k−1 e−(b2 +α+τ0 )(t−tk−1 −τ) E tk−1   + + τ dτ (A17) R t−t   +γp 0 j e−(b2 +α+τ0 )(t−t j −τ) E t+ h  j + τ dτ; ∀ti , t j ∈ Sc , ∀t ∈ t j , t j+1 Q j −(b2 +τ0 )(t−t` ) A t− dτ    A(t) = [ 1 − δA ( t i )] e `  Pj Qi=j `  R t −t k k−1 −(b2 +τ0 )τ  + + γ (1 − p ) k =` +1 [ 1 − δ A ( t i )] 0 e E t − τ dτ (A18) R t−t j i=k h k−1  ( b + τ ) τ +γ(1 − p) 0 e 2 0 E(t − τ)dτ; ∀ti , t j ∈ Sc , ∀t ∈ t j , t j+1 − Symmetry 2020, 12, 1646 30 of 33                 R(t) = e−(b2 +η)(t−t j ) R t−j + δS t j S t−j + δE t j E t−j + δI t j I t−j + δA A t−j R t−t +τ0 0 j e−(b2 +η)τ (I (t − τ) + A(ht − τ))dτ  (A19) = N (t) − S(t) − I (t) − A(t) − R(t); ∀t ∈ t j , t j+1 , ∀t j ∈ Sc Appendix B. Discussion of the Quarantined Choices According to (61)–(66) One gets from (A1)–(A9) that   R Ti RT  −b2 (t−τ)−β(ti ) τ i (I (ξ+ti+ )+e βA A(ξ+ti+ ))dξ sup S(t) ≤ (1 − δS (ti ))S t− i + 0+ e (b1 + η RM (ti )) ≤ SM (ti ) (A20) ti ≤t