entropy Article On the Entropy of Events under Eventually Global Inflated or Deflated Probability Constraints. Application to the Supervision of Epidemic Models under Vaccination Controls Manuel De la Sen 1, * , Asier Ibeas 2 and Raul Nistal 1 1 Institute of Research and Development of Processes IIDP, University of the Basque Country, Campus of Leioa, 48940 Leioa, Spain; raul.nistal@gmail.com 2 Department of Telecommunications and Systems Engineering, Universitat Autònoma de Barcelona (UAB), 08193 Barcelona, Spain; Asier.Ibeas@uab.cat * Correspondence: manuel.delasen@ehu.eus  Received: 27 January 2020; Accepted: 27 February 2020; Published: 29 February 2020  Abstract: This paper extends the formulation of the Shannon entropy under probabilistic uncertainties which are basically established in terms or relative errors related to the theoretical nominal set of events. Those uncertainties can eventually translate into globally inflated or deflated probabilistic constraints. In the first case, the global probability of all the events exceeds unity while in the second one lies below unity. A simple interpretation is that the whole set of events losses completeness and that some events of negative probability might be incorporated to keep the completeness of an extended set of events. The proposed formalism is flexible enough to evaluate the need to introduce compensatory probability events or not depending on each particular application. In particular, such a design flexibility is emphasized through an application which is given related to epidemic models under vaccination and treatment controls. Switching rules are proposed to choose through time the active model, among a predefined set of models organized in a parallel structure, which better describes the registered epidemic evolution data. The supervisory monitoring is performed in the sense that the tested accumulated entropy of the absolute error of the model versus the observed data is minimized at each supervision time-interval occurring in-between each two consecutive switching time instants. The active model generates the (vaccination/treatment) controls to be injected to the monitored population. In this application, it is not proposed to introduce a compensatory event to complete the global probability to unity but instead, the estimated probabilities are re-adjusted to design the control gains. Keywords: Shannon entropy; complete/incomplete systems of events; probabilistic uncertainty; probabilistic inflated/deflated probability constraints; epidemic model; vaccination controls; treatment controls 1. Introduction Classical entropy is a state function in Thermodynamics. The originator of the concept of entropy was the celebrated Rudolf Clausius in the mid-nineteenth century. The property that reversible processes have a zero variation of entropy among equilibrium states while irreversible processes have an increase of entropy is well-known. Reversible processes only occur under ideal theoretical modelling of isolated processes without energy losses such as, for instance, the Carnot cycle. Real processes are irreversible because the above ideal conditions are impossible to fulfil. In classical Thermodynamics, the Clausius equality establishes that the entropy variation in reversible cycle processes is zero. That is, Entropy 2020, 22, 284; doi:10.3390/e22030284 www.mdpi.com/journal/entropy Entropy 2020, 22, 284 2 of 35 the entropy variation is path-independent since it is a state function which takes an identical value at a final state being coincident with an initial one state. However, in irreversible ones the entropy variation is positive. This is the motivating reason which associates a positive variation of entropy to a “disorder increase”. It has to be pointed out that, in order to interpret correctly the non-negative variations of entropy, the system under consideration has to be an isolated one. In other words, some of the mutually interacting subsystems of an isolated system can exhibit negative variations of entropy. Later on Boltzmann, Gibbs and Maxwell have defined entropy under a statistical framework. On the other hand, Shannon entropy (named after Claude E. Shannon, 1916–2001, “the father of information theory”) is a very important tool to measure the amount of uncertainty in processes characterized in the probabilistic framework by sets of events, [1–5]. In some extended studies in the frameworks of physics, economy or fractional calculus, it is admitted the existence of events with negative probabilities, [4–6]. Entropy may be interpreted as information loss [1–3,7–9] and is useful, in particular, to characterize dynamic systems from this point of view [8]. On the other hand, entropy tools have also being used either to model or for complementary modelling support to evaluate certain epidemic models, [10–17]. In some of the studies, the investigation of epidemic models, which are ruled by a differential system of coupled equations involving the various subpopulations, has been proposed in the framework of a patchy environment [13]. The model uncertainty amount is evaluated in such a way that the time-derivative of the entropy is shown to be non-negative at a set of testing sampling instants. In [9], a technique to develop a formal Shannon entropy in the complex framework is proposed. In this way, the components of the overall entropy are calculated so as to determine the real and the imaginary parts of the state complex Shannon entropy as a natural quantum-amplitude generalization of the classical Shannon entropy. Epidemic evolution has been typically studied through models based either on differential equations, difference equations or mixed hybrid models. Such models, because of their structure, become very appropriate to study the equilibrium points, the oscillatory behaviors, the illness permanence and the vaccination and treatment controls. See, for instance, [18–24] and some of the references therein. More recently, entropy-based models have been proposed for epidemic models. See, for instance, [14–16,25] and some of the references therein. In particular, entropy tools for analysis are mixed with differential-type models in [16,25,26]. Also, the control techniques are appropriate for studies of alternative biological problems, [27–29] and, in particular, for implementation of decentralized control techniques in patchy environments where several nodes are interlaced, [27,29]. Under that generic framework, it can be described, for instance, the situation of several towns with different own health centers, where the controls are implemented, and whose susceptible and infectious populations interact through in-coming and out-coming population fluxes. This paper proposes a supervisory design tool to decrease the uncertainty between the observed data related to infectious disease, or those given by a complex model related to a disease, via the use of a set of dynamic integrated simplest models together with a higher-level supervisory switching algorithm. Such a hierarchical structure selects on line the most appropriate model as the one which has a smaller error uncertainty according to an entropy description (the so-called “active model”). Such an active model is re-updated through time in the sense that another active model can enter into operation. In that way, the active model is used to generate the correcting sanitary actions to control the epidemics such as, for instance, the gains of the vaccination and antiviral or antibiotic treatment controls and the corresponding control interventions. The paper is organized as follows: Section 2 recalls some basic concepts of Shannon entropy. Also, those concepts are extended to the case of eventual presence of relative probabilistic uncertainties in the defined set of events compared to the nominal set of probabilities. Section 3 states and proves mathematical results for the Shannon entropy for the case when a nominal complete finite and discrete set of events is eventually subject to relative errors of their associate probabilities in some of all of their various integrating events. The error-free nominal system of events is assumed to be complete. The current system of events under probabilistic errors may be either deflated or inflated in the sense that the total probability for the whole sets might be either below or beyond unity, respectively, so that it Entropy 2020, 22, 284 3 of 35 might be non-complete. In the case of globally inflated of deflated probability of the whole set of events, new compensatory events can be used to accomplish with a unity global probability. The entropy of the current system is compared via quantified worst-case results to that of the nominal system. Section 4 applies partially the results of the former sections to epidemics control in the case that either the disease transmission coefficient rate is not well-known or it varies through time due, for instance, to seasonality. The controls are typically of vaccination or treatment type or appropriate mixed combinations of both of them. A finite predefined set of running models described by a system of coupled differential equations is set. Such a whole discrete set covers a range of variation of such a coefficient transmission rate within known lower-bound and upper-bound limits. Each model is driven by a constant disease transmission rate and the whole set of models covers the whole range of foreseen variation of such a parameter in the real system. Since the set of models is finite, the various values of the disease transmission rate are integrated in a discrete set within the whole range of admissible variation of the true coefficient rate. A supervisory technique of control monitoring is proposed which chooses the so-called active model which minimizes the accumulated entropy of the absolute error data/model within each supervision interval. A switching rule allows to choose another active model as soon as it is detected that the current active model becomes more uncertain than other(s) related to the observed data. The active model supplies the (vaccination and/or treatment) controls to be injected to the real epidemic process. Due to the particular nature of the problem, compensatory events are not introduced for equalizing the global probability to unity. Instead, a re-adjustment of the error probabilities related to the true available data is performed to calculate the control gains provided by the active model. It can be pointed out that some parameters of the epidemic description evolutions, typically the coefficient transmission rate, can vary according to seasonality [30]. This justifies the use of simpler active invariant models to describe the epidemics evolution along time subject to appropriate model switching. On the other hand, the existing medical tests which evaluate the proportions of healthy and infected individuals within the total population are not always subject to confluent worst-case estimation errors. See, for instance [31,32]. In the case that the estimated global probability of the various subpopulations is not unity, the estimated worst-case probabilities need to be appropriately amended before an intervention. This design work is performed and discussed in Section 5 through numerical simulations in confluence with the above mentioned supervision monitoring technique. Finally, conclusions end the paper. 2. Basic Entropy Preliminaries  Assume a finite system of events A = Ai : i ∈ n of respective probabilities pi for Ai ; i ∈ n =  Pn {1, 2, · · · , n} with pi : i ∈ n ⊂ [0, 1], with i=1 pi = 1 which is complete, that is, only one event occurs at each trial (e.g., the appearance of 1 to 6 points in rolling a die). The Shannon entropy is defined as follows: n X H (p1 , p2 , . . . ..pn ) = − pi ln pi (1) i=1 which serves as a suitable measure of the uncertainty of the above finite scheme. The name “entropy” pursues a physical analogy with parallel problems, for instance, in Thermodynamics or Statistics Physics which does not have a similar sense here so that there is no need to go into in the current context. The above entropy has been defined with neperian logarithms but any logarithm with a fixed base could be used instead with no loss in generality. Note that H (p1 , p2 , . . . ..pn ) ≥ 0 with H (p1 , p2 , . . . ..pn ) = 0 if and only if p j = 1 for some arbitrary j ∈ n and pi = 0; ∀i(, j) ∈ n and Hmax = H (p, p, . . . ., p) = H (1/n, 1/n, . . . ., 1/n) = max Pn = −nplnp = −ln(1/n) = lnn ≥ 0 pi ∈[0, 1], i=1 pi =1 Hmax = H (p, p, . . . ., p) = H (1/n, 1/n, . . . ., 1/n) = max Pn = −nplnp = −ln(1/n) = lnn ≥ 0 pi ∈[0, 1], i=1 pi =1 Entropy 2020, 22, 284 4 of 35 with Hmax = 0 if and only if n = p = 1. Assume that the probabilities are uncertain, given by pi (1 + εi ); ∀i ∈ n, subject to the constraints pi (1 − εi0 ) ≤ pi (1 + εi ) ≤ pi (1 + εi1 ); ∀i ∈ n, where εi0 and εi1 are known so that the current entropy (or the entropy of the current system) is uncertain and given by Hε = H (p1 (1 + ε1 ), p2 (1 + ε2 ), . . . ..pn (1 + εn )); ∀i ∈ n. The reference entropy H∗ = H (p1 , p2 , . . . ..pn ) for the case when the probabilities of all the events are precisely known being equal to pi ; ∀i ∈ n is said to be the nominal entropy (or the entropy of the nominal system). Some constraints have to be fulfilled in order for the formulation to be coherent related to the entropy bounds under probabilistic constraints in the set of involved events. The following related result follows:  Lemma 1. Assume a nonempty finite complete nominal system of events A∗ = Ai∗ : i ∈ n of respective nominal probabilities pi ∈ [0, 1] for Ai∗ ; ∀i ∈ n = {1, 2, · · · , n} and a current (or uncertain) version of the system of events A = Ai : i ∈ n of uncertain probabilities pi (1 + εi ) ∈ [0, 1]; ∀i ∈ n, where εi ∈ R are relative probability  errors due to probabilistic uncertainties. Define the following disjoint subsets of A: A+ = {Ai ∈ A : εi > 0}, A− = {Ai ∈ A : εi < 0}, A0 = {Ai ∈ A : εi = 0}. Then, the following constraints hold: (i) A is complete if and only if ni=1 εi pi = 0 (global probabilistic uncertainty mutual compensation) P (ii) max (cardA+ , cardA− ) < n, and min (cardA+ , cardA− ) > 0 or cardA0 = n (and then A+ = A− = ∅) Proof: The proof of Property (i) follows since A∗ being complete implies that A is complete if and only if ni=1 pi (1 + εi ) = 1. P Pn Pn i=1 pi (1 + εi ) = 1 = pi + ni=1 pi εi = 1 + ni=1 pi εi ⇔ ni=1 pi εi = 0 P P P i=1P and A is complete since ni=1 pi (1 + εi ) = 1. Property (i) has been proved, The proof of Property (ii) follows from Property (i). To prove the first constraint, assume, on the contrary, that max (cardA+ , cardA− ) = n. Then, either cardA+ = n and A− = A0 = ∅ or cardA− = n and A+ = A0 = ∅. Assume that cardA+ = n and A− = A0 = ∅. Then, ni=1 εi pi > 0 which contradicts P Pn Property (i). Similarly, if cardA− = n and A+ = A0 = ∅ then i=1 εi pi < 0 which again contradicts Property (i). Thus, max (cardA+ , cardA− ) < n and the first constraint of Property (i) has been proved. Now, assume that min (cardA+ , cardA− ) = 0 and 0 ≤ cardA0 < n. First, assume that cardA− = 0 and cardA+ = cardA − cardA0 = max (cardA+ , cardA− ) ≤ n (from the already proved above first constraint of this property). If cardA+ > 0 then, ni=1 εi pi > 0 which contradicts Property (i). If cardA+ = 0 P then A = A0 and cardA0 = n. So, if cardA− = 0 then cardA+ = 0 and cardA0 = n (then A = A0 ). In the same way, interchanging the roles of A+ and A− , it follows that, if cardA+ = 0 then cardA− = 0 and cardA0 = n. One concludes that either min (cardA+ , cardA− ) > 0 or cardA0 = n and Property (ii) is proved. Note that, in Lemma 1, A0 ⊂ A is the subset of probabilistically certain events of A (in the sent that is elements have a known probability) and A+ ∪ A− ⊂ A is the subset of probabilistically uncertain events of A. For A being nonempty, any of the sets A0 , A+ and A− or pair combinations may be empty. Note also that the probabilistic uncertainties have been considered with fixed values εi ∈ [0, 1]; ∀i ∈ n.  3. Entropy Versus Global Inflated and Deflated Probability Constraints in Incomplete Systems of Events under Probabilistic Uncertainties An important issue to be addressed is how to deal with the case when, due to incomplete knowledge of the probabilistic uncertainties, the sum of probabilities of the current complete system of events, i.e., that related to the uncertain values of individual probability values exceeds unity, that is ni=1 pi (1 + εi ) > 1, a constraint referred to as “global inflated probability”. If both the nominal P individual and nominal global probability constraints, pi ≥ 0; ∀i ∈ n and ni=1 pi = 1 hold then there P is a global inflated probability if and only if the probabilistic disturbances fulfill ni=1 pi εi > 0. It is P Entropy 2020, 22, 284 5 of 35 always possible to include the case of “global deflated probability” if ni=1 pi (1 + εi ) < 1 implying that P Pn Pn i=1 pi εi < 0 provided that i=1 pi = 1. Three possible solutions to cope with this drawback, associate to an exceeding amount of modeled probabilistic uncertainty leading to global inflated probability, are: (a) To incorporate a new (non empty) event An+1 with negative probability which reduces the probabilistic uncertainty so that the global probability constraint of the extended complete system of events Ae = A ∪ An+1 holds. Obviously, A losses is characteristic property of being a complete system of events and Ae is said to be partially complete since it fulfills the global probability constraint but it has one negative probability. (b) To modify all or some of the probabilistic uncertainty relative amounts εi ; i ∈ n so that the global probability constraint of the complete system of events holds for the new amended fixed set εi : i ∈ n .  p (1+εi ) (c) To consider uncertain normalized probabilities pi = Pn i ; i ∈ n. In this case, there i=1 pi (1+εi ) is no need to modify the individual uncertainties of the events but the initial uncertainties are kept. Furthermore, the events would keep its ordination according to uncertainties, namely, if pi (1 + εi ) ≤ p j 1 + ε j for i, j(, i) ∈ n then pi ≤ p j , the modified set of events is complete and keeps the same number of events as the initial one. Two parallel solutions to cope with global deflated probability are: (d) To incorporate a new (non empty) event An+1 with positive probability which decreases the probabilistic uncertainty so that the global probability constraint of the extended complete system of events Ae = A ∪ An+1 holds. As before, the current system of events A losses is completeness and Ae is complete. (e) To modify all or some of the probabilistic uncertainty relative amounts εi ; i ∈ n so that new amended fixed set εi : i ∈ n agrees with the global probability constraint.  Firstly, we note that the introduction of negative probabilities invoked in the first proposed solution has a sense in certain problems. In this context, Dirac commented in a speech in 1942 that negative probabilities can have a sense in certain problems as negative money has in some financial situations. Later on, Feynman has used also this concept to describe some problems of physics [4] and also said that negative trees have nonsense but negative probability can have sense as negative money has in some financial situations. More recently, Tenreiro-Machado has used this concept in the context of fractional calculus [6]. It is now proved in the next result that the incorporation of a new event of negative probability to define from the system of events A an extended partial complete system of events Ae does not alterate the property of the non-negativity of the Shannon entropy in the sense that that of the extended system of events is still non-negative. At the same time, it is proved that (1) If 1 < ni=1 pi (1 + εi ) ≤ 2 then the real part of the entropy of Ae (which is complex because of the P negative probability of the added event for completeness) is smaller that that of A. The interpretation is that the excessive disorder in A, due to its global inflated probability, is reduced (equivalently, “the order amount” Ae is increased in with respect to A) when building its extended version Ae by the contribution of the added event of negative probability. Also, both A and Ae are not complete while Ae is partially complete since it fulfills the global probability constraint although not the partial ones in all the events due to the incorporated one of negative probability. (2) If ni=1 pi (1 + εi ) > 2 then the above qualitative considerations on increase/ decrease of “order” P or” disorder” are reversed with respect to Case a.  Theorem 1. Assume a nonempty finite complete nominal system of events A∗ = Ai∗ : i ∈ n of respective nominal probabilities pi ∈ [0, 1] for Ai∗ ; ∀i ∈ n = {1, 2, · · · , n} and a current (or uncertain) version of the system of events A = Ai : i ∈ n of uncertain probabilities pi (1 + εi ) ∈ [0, 1]; ∀i ∈ n. Assume that 2 ≥ ni=1 pi (1 + εi ) > 1  P (that is, there is a global inflated probability of the current system of events). Define the extended system of events Ae = A ∪ An+1 such that An+1 has a probability pn+1 = pn+1 (ε1 , ε2 , · · · , εn ) = 1 − ni=1 pi (1 + εi ). Then, P Entropy 2020, 22, 284 6 of 35 If 2 ≥ ni=1 pi (1 + εi ) > 1 then ReHε1 ≤ Hε with 0 ≤ ReHε1 = Hε if and only if ni=1 pi (1 + εi ) = 2. P P (i) If ni=1 pi (1 + εi ) > 2 then Hε1 > Hε ≥ 0, P (ii) where: n X Hε = H (p1 (1 + ε1 ), p2 (1 + ε2 ), . . . .., pn (1 + εn )) = pi (1 + εi )(lnpi + ln(1 + εi )) (2) i=1  Hε1 = H (p1 (1 + ε1 ), p2 (1 + ε2 ), . . . .., pn (1 + εn ), pn ) Pn  (3) = H p1 (1 + ε1 ), p2 (1 + ε2 ), . . . .., pn (1 + εn ), 1 − i=1 pi (1 + εi ) are the respective entropies of A and Ae , the second one being complex. Proof: Since ni=1 pi (1 + εi ) > 1 then A = Ai1 : i ∈ n is not complete. Since the nominal system of P  events is complete then ni=1 pi = 1 with pi ≥ 0; ∀i ∈ n and pn+1 = 1 − ni=1 pi (1 + εi ) < 0. The entropy P P of A is and that of the extended system of events Ae is Hε1 . Consider two cases: Case (a) 0 < ni=1 pi (1 + εi ) − 1 ≤ 1 P Since the neperian logarithm of a negative real number exists in the complex field and it is √ real such that its real part is the neperian logarithm of its modulus and the imaginary part is iπ (i = −1 being the imaginary unit), so that it becomes a negative real number, one gets the following set of relations: Pn Pn Pn   −Hε1 = i=1 pi (1 + εi )ln(pi (1 + εi )) + 1 −  i=1 pi (1 + εi ) ln 1 −  i=1 pi (1 + εi ) = −Hε + 1 − ni=1 pi (1 + εi ) ln 1 − ni=1 pi (1 + εi ) P P P    = −Hε − ni=1 pi (1 + εi ) − 1 ln 1 − ni=1 pi (1 + εi ) P (4) P   = −Hε − ni=1 pi (1 + εi ) − 1 ln 1 − ni=1 pi (1 + εi ) + iπ P   = −Hε + ni=1 pi (1 + εi ) − 1 ln 1 − ni=1 pi (1 + εi ) + iπ P P Then:  n n  X  X Re (−Hε1 )= −Hε +  pi (1 + εi ) − 1ln pi (1 + εi ) − 1 (5) i=1 i=1 Pn since 0 < i=1 pi (1 + εi ) − 1 ≤ 1 leads to n n n   X  X  X ln pi (1 + εi ) − 1 ≤ 0, ln1 −  pi (1 + εi ) = −ln 1 − pi (1 + εi ) > 0 (6) i=1 i=1 i=1 One concludes that that ReHε1 ≤ Hε with 0 ≤ ReHε1 = Hε if and only if ni=1 pi (1 + εi ) = 2. P It has to be proved now that ReHε1 ≥ 0. Assume, on the contrary, that ReHε1 < 0. Since ReHε1 < Hε , this happens if and only if the following contradiction holds pi (1+εi ) pi (1+εi ) Pn Pn h i hQ i n i=1 pi ( 1 + ε i )( lnpi + ln ( 1 + εi )) = ln ( p i ( 1 + εi )) = ln ( pi ( 1 + εi )) P  i=P 1 P i=1 = Hε < ni=1 pi (1 + εi1 ) − 1 ln ni=1 pi (1 + εi1 ) − 1 = ni=1 pi εi1 ln ni=1 pi εi1 P (7) P  P  P  n pi εi1 P  = ni=1 pi εi1 ln ni=1 pi εi1 = ln n i=1 pi εi1 i=1 since ni=1 pi (1 + εi ) > 1 in the assumption of global inflated probability. Then, 0 ≤ ReHε1 ≤ Hε . P Case (b) 1 < ni=1 pi (1 + εi ) − 1 ≤ z for some z(> 1) ∈ R+ . Then ln ni=1 pi (1 + εi ) − 1 > 0 and P P then the last above identity changes the second term right-hand-side term resulting in: Entropy 2020, 22, 284 7 of 35 Pn P  n −Hε1 = Re (−Hε1 ) = −Hε − i=1 pi (1 + εi ) − 1 ln i=1 pi (1 + εi ) − 1  n  n X  X ReHε1 = Hε +  pi (1 + εi ) − 1ln     pi (1 + εi ) − 1 > Hε ≥ 0 (8) i=1 i=1  It is now proved in the next result for the case of deflated probability that the introduction of an additional event of positive probability to define from the system of events A an extended partial complete system of events Ae does not modify the property of the non-negativity of the Shannon entropy in the sense that that of the extended system of events is still non-negative. At the same time, it is proved that the entropy of Ae is larger that that of A. The interpretation is that the excessively low disorder in A, due to its global deflated probability, is increased (equivalently, “the order amount” is decreased) when building its extended version Ae by the contribution of the added event An+1 of positive probability. Also, A is not complete while Ae is complete.  Theorem 2. Assume a nonempty finite complete nominal system of events A∗ = Ai∗ : i ∈ n of respective nominal probabilities pi ∈ [0, 1] for Ai∗ ; ∀i ∈ n = {1, 2, · · · , n} and a current (or uncertain) version of the system of events A = Ai : i ∈ n of uncertain probabilities pi (1 + εi ) ∈ [0, 1]; ∀i ∈ n. Assume that ni=1 pi (1 + εi ) < 1  P (that is, there is a global deflated probability of the current system of events). Define the extended system of events Ae0 = A ∪ An+1 such that An+1 has a probability pn+1 = pn+1 (ε1 , ε2 , · · · , εn ) = 1 − ni=1 pi (1 + εi ). Then, P Hε0 > Hε ≥ 0, where: n X Hε = H (p1 (1 + ε1 ), p2 (1 + ε2 ), . . . .., pn (1 + εn )) = pi (1 + εi )(lnpi + ln(1 + εi )) (9) i=1 Hε0 = H (p1 (1 + ε1 ), p2 (1 + ε2 ), . . . .., pn (P 1 + εn ), pn+1 )  (10) = H p1 (1 + ε1 ), p2 (1 + ε2 ), . . . .., pn (1 + εn ), ni=1 pi (1 + εi ) − 1 are the respective entropies of A and Ae0 . Proof: Since ni=1 pi (1 + εi ) < 1 then A = Ai : i ∈ n is not complete. Since the nominal system of P  events is complete then ni=1 pi = 1 with pi ≥ 0; ∀i ∈ n and pn+1 = 1 − ni=1 pi (1 + εi ) > 0. The entropy P P of A is Hε and that of Ae0 is: Hε0 = H (p1 (1 + ε1 ), p2 (1 + ε2 ), . . . .., pn (P 1 + εn ) , p n + 1 )  (11) = H p1 (1 + ε1 ), p2 (1 + ε2 ), . . . .., pn (1 + εn ), ni=1 pi (1 + εi ) − 1 Pn Since 0 < 1 − i=1 pi (1 + εi ) < 1, one gets the following set of relations: Pn  Pn  Pn  −Hε0 = i=1 pi (1 + εi )ln(pi (1 + εi )) + 1 −  i=1 pi (1 + εi ) ln 1 −  i=1 pi (1 + εi ) = −Hε + 1 − ni=1 pi (1 + εi ) ln 1 − ni=1 pi (1 + εi ) P P (12)     = −Hε − 1 − ni=1 pi (1 + εi ) ln 1 − ni=1 pi (1 + εi ) < −Hε P P One concludes that that Hε0 > Hε ≥ 0. Note that since we entropy is defined with a sum of weighted logarithms of nonnegative real numbers bounded by unity in the usual cases (which exclude negative probabilities) then the Shannon entropy of the worst case of the probability uncertainty is not necessarily larger than or equal to the nominal one.  Entropy 2020, 22, 284 8 of 35  Example 1. Assume that the nominal system of two events is A = A1 (p1 ), A2 (p2 ) with p1 = p2 = 0.5, then H∗ = −2 × 0.5 × ln0.5 = −ln0.5 = 0.6931. Assume that it probabilistic uncertainty is given by ε = 0.1, p1 (1 + ε) = 0.5 = 0.55, p2 (1 − ε) = 0.45 = 0.55. Then, the entropy under the uncertain probability of Aε = A1 (p1 (1 + ε)), A2 (p2 (1 − ε)) is Hε = −0.55ln(0.55) − 0.45ln(0.45) = 0.68814 < H∗ .  Example 2. Consider the complete system of events A = Ai : i ∈ n such that pi = λi p; ∀i ∈ n, for some p > 0,  is the probability of Ai , ∀i ∈ n and constants λi ∈ R0+ = R+ ∪ {0}; ∀i ∈ n satisfying that ni=1 λi = 1/p. Note P Pn that i=1 pi = 1 so that there is no negative probability and neither global inflated or global deflated probabilities. The Shannon entropy is the everywhere continuous real concave function: n X Ha = Ha (p) = H (λ1 p, λ2 p, . . . ., λn p) = − λi pln(λi p) (13) i=1 whose maximum value is reached at a real constant p∗ if dH/dp]p=p∗ = 0. Direct calculation yields, since Pn i=1 λi = 1/p, dHa (p) = − ni=1 (λi ln(λi p) + λi ) = ni=1 λi (1 + lnp + lnλi ) P P dp (14) − ni=1 λi (1 + lnp) + ln λi λi = − p − ni=1 ln λi λi = P  1+lnp   P    = 0 Such a p∗ is unique for the given set λi : i ∈ n , since the function Ha (p) is concave in its whole definition  domain, and admissible provided that p∗ ∈ [0, 1]. The unique solution for each given set λi : i ∈ n is  p∗ = p∗ (λ1 , λ2 , . . . , λn ) satisfying the constraint: n   1 + lnp∗ X ln λi λi < 0  f (p∗) = = Kλ = − (15) p∗ i=1 Since f : [0 , 1] → R is continuous and bounded from above with f (0) = −∞ and f (1) = 1, and K < 0 such that a real p∗ exists in (0, 1] and it is unique, since there is a unique p∗ such that f (p∗) = Kλ , for each given set λi : i ∈ n and the maximum entropy becomes  n X Hmax = Ha (p∗) = H (λ1 p∗, λ2 p∗, . . . ., λn p∗) = − λi p ∗ ln(λi p∗) (16) i=1 In particular, if n = 2 with p1 = λ1 p and p2 = λ2 p = 1 − λ1 p then Ha (p) = −λ1 pln(λ1 p) − (1 − λ1 p)ln(1 − λ1 p) and: = −λ1 ln(λ1 p) − λ1 p λλ1p + λ1 ln(1 − λ1 p) − (1 − λ1 p) 1−λ dHa (p) −λ1 dp 1 p 1 (17) = −λ1 (ln(λ1 p) − ln(1 − λ1 p)) = 0 leading to λ2 p = 1 − λ1 p = 1/2 so that λ1 p = λ2 p = 1/2 and p∗ = 1/2λ1 . Note that this is equivalent to λ p∗ λ p∗ ln 1−λ1 p∗ = 0, that is 1−λ1 p∗ = 1. In particular, if λ1 = λ2 = 1 then p∗ = 1/2. Assume now that one uses any 1 1 basis b of logarithms to define the entropy so that:   Hab (p) = −λ1 plogb λ1p − (1 − λ1 p)logb (1 − λ1 p) (18) leading to: dHab (p)   = λ1 logb (λ1 p) − logb (1 − λ1 p) = 0 (19) dp which is satisfied for the same conditions as above, that is, if λ1 p∗ = 1 − λ1 p∗ = 1/2 so that p∗ = 1/2λ1 again. However, Hab (p∗) , Ha (p∗) if b , e (e being the basis of the neperian algorithms). Entropy 2020, 22, 284 9 of 35 Example 3. Consider the particular case of Example 2 for n = 2 with global deflated probability, that is, p1 = λ1 p and p2 = λ2 p < 1 − λ1 p. The system of two events {A1 , A2 } is not complete and we extend it with a new event A3 of positive probability p3 = 1 − (λ1 + λ2 )p so that the extended system of events is complete. Now, one has for such an extended system: Ĥa (p) = −λ1 pln(λ1 p) − λ2 pln(λ2 p) − (1 − (λ1 + λ2 )p)ln(1 − (λ1 + λ2 )p) (20) = −λ1 ln(λ1 p) − λ1 p λλ11p − λ2 ln(λ2 p) − λ2 p λλ12p dĤa (p) dp −(λ +λ ) +(λ1 + λ2 )ln(1 − (λ1 + λ2 )p) − (1 − (λ1 + λ2 )p) 1−(λ1 +λ2 )p 1 2 = −λ1 ln(λ1 p) − λ1 − λ2 ln(λ2 p) − λ2 + (λ1 + λ2 )ln(1 − (λ1 + λ2 )p) + λ1 + λ2 (21) = −λ1 ln(λ1 p) − λ2 ln(λ2 p) + (λ1 + λ2 )ln(1 − (λ1 + λ2 )p) (1−(λ1 +λ2 )p)λ1 +λ2 = ln =0 (λ1 p)λ1 p (λ2 p)λ2 p (1−(λ1 +λ2 )p∗)λ1 +λ2 which is satisfied for p = p∗ such that = 1. Note that Ĥa (p) > Ha (p) with Ha (p) being (λ1 p∗)λ1 p∗ (λ2 p∗)λ2 p∗ obtained in Example 2 for the non-deflated probability case, as expected from Theorem 2. Note also that the value of p∗ giving the maximum entropy is less than the one producing the maximum entropy in the non-deflated case of Example 2 since that one p∗ = 1/(λ1 + λ2 ) with give the contradiction 0=1 in the above formula for p = p∗ and p∗ < 1/(λ1 + λ2 ) would give the contradiction 1 < 0 in such a mentioned formula. The next result relies on the case where the probabilities are uncertain but each nominal probability is assumed to be eventually subject to the whole set of probabilistic uncertainties so as to cover a wider class of potential probabilistic uncertainties. It is also assumed (contrarily to the assumptions of Theorem 1 and Theorem 2) that the probability disturbances never increase each particular probability of the nominal system of event since they are all constrained to the real interval [0, 1]. As a result, the cardinalities of the extended current and nominal systems of events are, in general, distinct. It is found that the entropy of the current system of events is non-smaller than the nominal entropy. Theorem 3. Let Hεδ = H (p1 δ1 , p1 δ2 , . . . .., p1 δm , . . . ., pn δ1 , pn δ2 , . . . .., pn δm )n and H∗ = H (p1 , po2 , . . . .., pn ) the extended perturbed and current entropies of the sets of nonempty events Aα = Aij : (i, j) ∈ n × m and A∗ = Ai : i ∈ n , respectively, of respective cardinalities n × m and n, with pi , δi ∈ [0, 1] and ni=1 pi = m i=1 δi = 1.  P P Then, the following properties hold: (i) Hεδ = H∗ + H eδ = H (δ1 , δ2 , . . . .., δm ) is the incremental entropy due to the uncertainties eδ ≥ H∗ , where H and Hεδ = H∗ if and only if δ j = 1 and δi = 0; ∀i(, j) ∈ m and some arbitrary j ∈ m. If m = n and δi = 1 + εi with εi ∈ [0, 1]; ∀i ∈ n and ni=1 εi = 1 − n then the above relations hold with Hεδ = H∗ if P and only if ε j = 0 and εi = −1; ∀i(, j) ∈ n and some arbitrary j ∈ n. Assume that m = n and δi = 1 − εi with εi ∈ [−1, 0]; ∀i ∈ n and ni=1 εi = n − 1. Then, P (ii)   e1−ε = H n − k + Pn−k εi , 1 + εk+1 , . . . ., 1 + εn + H i=1 ! Pn−k P j  Pn−k−1 εi +n−2 εn−k +1 (22) j=1 ε i=1 i + j H i=1 n−k , n−k i=1 εi +n−1 i=1 εi +n−1 P P Pn−1 for any given integer n ≥ 3 and any k ∈ n − 2 such that 0 < n − k + i=1 εi < 1. (iii) The following identity holds under the constraints of Property (ii): Pn−k−1 ! Pn−k P j  εi +n−2 εn−k +1 j=1 +jH ε i=1 i , i=1 Pn−k Pn−k ε i=1 i +n−1 i=1 εi +n−1 (23)   eδ − H n − k + n−k εi , 1 + εk+1 , . . . ., 1 + εn ≥ 0 =H P i=1 Pn−1 for any given integer n ≥ 3 and any k ∈ n − 2 such that 0 < n − k + i=1 εi < 1. Entropy 2020, 22, 284 10 of 35 Proof: Note from the additive property of the entropy that Hεδ = H∗ + H eδ and pi , δ j ∈ [0, 1]; ∀i ∈ n, Pm Pn ∀j ∈ m, i=1 δi = 1, equivalently, εi (= δi − 1) ∈ [−1, 0] and i=1 εi = n − 1 if m = n. Then, Property (i) follows from the non-negativity property of the entropy and the fact that H eδ = 0 if and only if δ j = 1 and δi = 0; ∀i(, j) ∈ m and some arbitrary j ∈ m what implies that H (δ1 , δ2 , . . . .., δm ) =   H 0, 0, . . . , δ j (= 1), 0, . . . .., 0 = 0. The property for m = n and εi = δi − 1; ∀i ∈ n is a particular case of the above one. Property (i) has been proved. On the other hand, one gets from the recursive property of the entropy if m = n and εi = δi − 1; ∀i ∈ n with pi ∈ [0, 1], εi ∈ [−1, 0]; ∀i ∈ n, ni=1 pi = 1 and ni=1 εi = 1 − n that: P P e1−ε = H (2 + ε1 + ε2 , 1 + ε3 , . . . ., 1 + εn ) + (2 + ε1 + ε2 )H 1+ε1 , 1+ε2   H 2+ε1 +ε2 2+ε1 +ε2 = H (3 + ε1 + ε2 + ε3 , 1 + ε4 , . . . ., 1 + εn )  +(3 + ε1 + ε2 + ε3 )H 3+2ε+1ε+1ε+2ε+2ε3 , 3+ε11 + ε3 +ε2 +ε3  1+ε 1 + ε +(2 + ε1 + ε2 )H 2+ε1 +1ε2 , 2+ε1 +2ε2   P2 ε + 2  = H (3 + ε1 + ε2 + ε3 , 1 + ε4 , . . . ., 1 + εn ) + 3j=1 i=1 εi + 3 H Pi3=1 i , P3ε3 +1 P P j εi +3 i=1 εi +3  P3 P j   P2i=1 εi +2 ε +1 (24) = H (3 + ε1 + ε2 + ε3 , 1 + ε4 , . . . ., 1 + εn ) + j=1 i=1 εi + 3 H P3 i=1 , P3 3   i = 1 εi +3 i = 1 εi +3 Pn−k = H n − k + i=1 εi , 1 + εk+1 , . . . ., 1 + εn + ! Pn−k P j  Pn−k−1 εi +n−2 εn−k +1 j=1 i=1 i ε + j H Pn−k i=1 , Pn−k i=1 εi +n−1 i=1 εi +n−1  Pn−1  Pn−1 P j    Pn−2 εi +n−2 ε +1  = H n − 1 + i=1 εi , 1 + εn + j=1 i=1 εi + j H Pn−1 i=1 , Pn−1n−1 i=1 εi +n−1 i=1 εi +n−1 for any given integer n ≥ 3 each equality for a right-hand-side H (n − k + (. . .), . . . ., , 1 + εn ) is valid for 1−p h i each k ∈ n − 2 such that pi ∈ [0, 1], εi ∈ −1, p i ; ∀i ∈ n, 0 < n − k + n−1 i=1 εi < 1 implying that the sum P i of the two first argument of the entropy is positive, that is, ε1 + ε2 > −2. Property (ii) has been proved. Property (iii) is a consequence of Property (ii).  The next result relies on the case where the probabilities are uncertain but belonging to a known admissibility real interval, rather than fixed as it has been assumed in Lemma 1. Contrarily to Theorem 3, it is not assumed that the probabilistic disturbances are within [0, 1] so that they can increase the corresponding nominal probabilities.  Theorem 4. Assume a finite complete system of events Ai : i ∈ n such that the nominal and current probabilities of the event Ai are pi and pi (1 + εi ), respectively, with εi ∈ [−εi0 , εi1 ], such that εi1 + εi0 ≥ 0 (so that the constraint 1 + εi1 ≥ 1 − εi0 holds, εi0 ∈ [(pi − 1)/pi , 1] and −1 ≤ εi1 ≤ (1 − pi )/pi ; ∀i ∈ n and Pn i=1 pi (1 + εi ) = 1. Then, the following properties hold: (i) Hε ≥ max [(1 + ε0 )H∗ − (1 + ε1 ), 0] with ε1 = max |εi | and ε0 = min |εi | where Hε = 1≤i≤n 1≤i≤n H (p1 (1 + ε1 ), p2 (1 + ε2 ), . . . .., pn (1 + εn )) and H∗ = H (p1 , p2 , . . . .., pn ) (ii) (1 − m0 )H∗ ≥ Hε ≥ (1 + m1 )H∗ (25) where:   n ∞ ε ! X X 1  m0 = 1 − pi lnpi   + i0 (−εi0 )2 j+1 (1 + εi1 ) (26)  2 j + 1 2( j + 1)  i=1 j=0 ∞ εi1 ! X 1 2 j+1 m1 = − ε (1 + εi1 ) − 1 (27) 2 j + 1 2( j + 1) i1 j=0 Entropy 2020, 22, 284 11 of 35 (iii) If εi0 ∈ [(pi − 1)/pi , 1) and εi1 ∈ (−1, (1 − pi )/pi ] then: n n X εi0 X pi lnpi (1 − εi0 ) ≥ Hε ≥ − pi lnpi (1 + εi1 )εi1 (28) |1 − εi0 | i=1 i=1 (iv) If εi0 ∈ [(pi − 1)/pi , 1) and εi1 ∈ (−1, (1 − pi )/pi ] then: n  n 1/2 X X  − pi lnpi (1 + εi1 )εi1 = H0 ≤ Hε ≤ H1 =  ε2i0  H (p1 , p2 , . . . ..pn ) (29) i=1 i=1 such that an upper-bound of the lower-bound H0 of Hε is:  n 1/2 X 2  H0 = H∗  (1 + εi1 ) ε2i1  (30) i=1 Proof: Note that, since εi > −1; ∀i ∈ n then ln(1 + εi ) ≤ εi ; ∀i ∈ n so that: −Hε = ni=1 pi (1 + εi )(lnpi + ln(1 + εi )) P ≤ ni=1 pi (1 + εi1 )(εi1 + lnpi ) P ≤ −H∗ + ni=1 pi lnpi εi + ni=1 pi (1 + εi )εi P P (31) ≤ −H∗ − ε0 H∗ + (1 + ε1 ) ni=1 pi εi P = −H∗ − ε0 H∗ + (1 + ε1 ) = − ( 1 + ε 0 ) H∗ + ( 1 + ε 1 ) and Property (i) follows directly. On the other hand, note that: Pn pi (1 − εi0 )(lnpi + ln(1 − εi0 )) i=1 P (32) ≤ −H∗ ≤ ni=1 pi (1 + εi )(lnpi + ln(1 + εi )) Note that, since max(εi0 , εi1 ) ≤ 1; ∀i ∈ n, we can use Taylor’s expansion series around 1 for the above neperian logarithms to get: j+1 (−1)n εi1 P∞ ε2i1j+1 P∞ εi12( j+1) ln(1 + εi1 ) = ∞ P j=0 j+1 = j=0 2 j+1 − j=0 2( j+1) 2 j+1 εi1 2( j+1) εi1 !   (33) εi1 2 j+1 = ∞ P∞ j=0 2 j+1 − 2( j+1) εi1 P 1 j=0 2 j+1 − 2( j+1) = ∞ ∞ (−1) j (−εi0 ) j+1 X εi0 ! X 1 ln(1 − εi0 ) = = + (−εi0 )2 j+1 (34) j+1 2j + 1 2( j + 1) j=0 j=0 so that: Pn i=1 pi lnpi (1 − m0 ) = −(1 − m0 )H∗ = −H1 ≤ −Hε (35) ≤ −H0 = −(1 + m1 )H∗ = ni=1 pi lnpi (1 + m1 ) P ∞ (−1)n ε j+1 X ∞ ε2 j + 1 X ∞ ε2 ( j + 1 ) X i1 i1 i1 ln(1 + εi1 ) = = − (36) j+1 2j + 1 2( j + 1) j=0 j=0 j=0 Entropy 2020, 22, 284 12 of 35 2 j+2 2 j+2 so that H1 ≥ H∗ ≥ H0 . Since εi1 min 2 j+1 = lim = 1 then m1 ≥ −1 so that H0 ≥ 0 and H1 ≥ H0 ≥ 0. j≥0 j→∞ 2 j+1 Furthermore, (26) and (27) imply that m1 ∈ [−1, 0] and m0 ∈ [0, 1]. The proof of Property (i) has been completed. Now, since εi0 < 1 and εi1 > −1; ∀i ∈ n then (25) follows since: ln(1 + εi ) ≤ ln(1 + εi1 ) ≤ εi1 (37) since εi1 > −1; ∀i ∈ n: εi0 εi0 ln(1 + εi ) ≥ ln(1 − εi0 ) ≥ − = (38) 1 − εi0 εi0 − 1 since −εi0 > −1; ∀i ∈ n, equivalently, εi0 < 1; ∀i ∈ n so that: Pn εi0 Pn − i=1 pi lnpi (1 − εi0 ) |1−εi0 | = i=1 pi lnpi sgn(1 − εi0 )εi0 Pn ≤ i=1 pi lnpi (1 − εi0 )ln(1 − εi0 ) −H (p1 (1 + ε1 ), p2 (1 + ε2 ), . . . ..pn (1 + εn )) = ni=1 pi lnpi (1 + εi )ln(1 + εi ) P ≤ (39) Pn ≤ i=1 pi lnpi (1 + εi1 )ln(1 + εi1 ) ≤ ni=1 pi lnpi (1 + εi1 )εi1 P and Property (ii) has been proved. On the other hand, Property (iii) is proved as follows: Pn Pn (1−εi0 )εi0 Hε = − i=1 pi lnpi (1 + εi )ln(1 + εi ) = i=1 pi lnpi |1−εi0 | Pn εi0 n i=1 pi lnpi (1 − εi0 ) |1−εi0 | = i=1 pi lnpi sign(1 − εi0 )εi0 P P n 2  1/2 P n 2 1/2  (40) ≤ i=1 ( sign ( 1 − εi0 ) ε i0 ) ( pi lnpi ) P 1/2 i=1 n ≤ i=1 i0ε 2 |H∗ | After using of Cauchy-Schwartz inequality for summable sequences and the constraint 2 1/2 P  n i=1 (pi lnpi ) ≤ H∗ , one gets, since εi1 > −1 and εi0 < 1, that Property (iii) holds since: Pn Pn (1+εi1 )εi1 − i=1 pi lnpi (1 + εi1 )ln(1 + εi ) =− i=1 pi lnpi (1 + εi )ln(1 + εi ) (1+εi )ln(1+εi ) !1/2 Pn (1+εi1 )2 ε2i1 ≤ i=1 (1+ε )2 ln2 (1+ε ) H (p1 , p2 , . . . ..pn ) (41) i i !1/2 Pn (1+εi1 )2 ε2i1 ≤ i=1 H (p1 , p2 , . . . ..pn ) (1−εi0 )2 ln2 (1−εi0 ) and Property (iv) follows from Property (iii) and the fact that: H0 = − ni=1 pi lnpi (1 + εi1 )εi1 P P  n 2 2 1/2 P n 2 2 1/2  P n 2 2 1/2  (42) ≤ i=1 pi ln pi i=1 (1 + εi1 ) εi1 ≤ H0 = i=1 (1 + εi1 ) εi1 H∗  4. Applications to Epidemic Systems This section is devoted to the application of entropy tools to the modelling of epidemics. It should be pointed first out that some parameters of the epidemic description evolutions, typically the coefficient transmission rate, can vary according to seasonality [30]. It can be also pointed out that the existing medical tests to evaluate the proportions of healthy and infected individuals are not always subject to confluent estimation of the errors. That is the estimated worst-case errors for reach of the populations can have different running ranges due to the fact that they are performed with different techniques. See, for instance, [31,32] where the mentioned question is justified taking as a basis medical tests. Entropy 2020, 22, 284 13 of 35 -To solve the first problem, one considers the usefulness of designing a parallel scheme of alternative time-invariant models, being ran by a supervisory switching law, to choose the most appropriate active time-invariant model through time. The whole set of models covers the whole range of expected variations of the model parameters through time. The objective of the parallel structure is to select the one which describes the registered data more tightly along a certain period of time. -On the other hand, the fact that the existing tests of errors on the subpopulation integrating the model not always give similar worst-case allowed estimated errors for all the subpopulations, justifies and adjustment of the probabilities in the case when the sum of all of them does not equalize unity. Two potential actions to overcome this drawback are: (a) the introduction of events of positive (respectively, negative) probabilities in the case of deflated (respectively, inflated) global probability; (b) to readjust all the individuals estimated probabilities via normalization by the current sum which is distinct of unity. In the first case, the re-adjustment is made by an algebraic sum manipulation. In the second one, be readjusting via normalization all the individual probabilities. In both cases, it is achieved that the amended global probability is unity. Assume h ian epidemic disease with unknown time-varying bounded coefficient transmission rate β(t) ∈ β , β , where β0 and β1 are known, which is defined by the following differential system of n 0 1 first-order differential equations: . x(t) = Ψ(x(t), β(t) , p)x(t) + Γu(t); x(0) = x0 (43) where x(t) ∈ Rn is the state-vector and p is the vector of parameters containing all other parameters that the coefficient transmission rate, like recovery rate mortality rate, average survival rate, average expectation of life irrespective of the illness etc. In practice, Equation (43) can be replaced by a non-parameterized description, based on the state measurements through time, where x(t) is given by provided experimental on-line data on the subpopulation which in this case, should be discretized with a small sampling period. In this way, (43) can be either a more sophisticated mathematical model, than those simplified ones provided later on, which provides data x(t) on the illness close to the real measurements or the listed real data themselves got from the disease evolution. The state vector contains the subpopulations integrated in the model which depend on the type of model itself such as susceptible (E), infectious (I) and recovered (or immune) (R) in the so-called SIR models to which it is added, in the so-called SEIR models the exposed subpopulation (E) which are those in the first infection stages with no external symptoms. The models can also contain a vaccinated subpopulation (V) and can have also several nodes or patches, describing, for instance, different environments, in general coupled, each having their own set of coupled subpopulations which interact with the remaining ones though population fluxes. The vector u(t) ∈ Rm is the control vector. There are typically either one control, namely, vaccination on the susceptible, or two controls, namely, vaccination on the susceptible and (either antiviral or antibiotic) treatment on the infectious in the case when there is only one node. Those controls might be applied to each subsystem associated to one patch if there are several patches integrated in the model. The matrix function of dynamics Ψ(x(t), β(t) , p) is a real n × n-matrix for each t ∈ R0+ . The control matrix Γ has as many columns as controls are applied and it typically consists of entries being “o” (i.e., no control applied on the corresponding state component associated to one subpopulation), “−1” if the control leads to a decrease of the rate of growing of a subpopulation, for instance, vaccination effort on the susceptible) and “+1” if it leads to a compensatory increase rate of a subpopulation due to a corresponding decrease of another one, for instance, the increase in the recovered in the vaccination case (when the susceptible are decreased via vaccination) or again the recovered in the treatment case (when the infectious are decreased via treatment). For simplicity, it is assumed that p is constant and there are no delays in the dynamics. The control architecture which is proposed consists of a scheme of approximated models located in a parallel disposal, one of them being chosen by a higher- level supervisory switching scheme as the active model to select the controller gains along each current time interval. Entropy 2020, 22, 284 14 of 35 In particular, it is proposed to run a set of Q + 1 approximated models of the same dimension as Equation (43) with a constant coefficient transmission rate βi ; ∀i ∈ Q + 1 being chosen as: .   xi (t) = Ψi xi (t), βi , pi x(t) + Γui (t); xi (0) = x0 , ∀i ∈ Q + 1 (44) with xi (t) ∈ Rn ,where: β1 − β0 βi+1 = βi + ; ∀i(> 1) ∈ Q, β1 = β0 (45) Q in such way that β1 = β0 and βQ+1 = β1 . Note that the approximated models (44) and (45) are parameterized by a constant coefficient transmission rate contrarily to the real model (43) whose coefficient transmission rate is time-varying. The remaining parameters are constant and eventually time- varying, respectively. Note also that then models (44)o are initialized to the initial conditions. We consider a set of (Q + 1) event errors E = Ei : i ∈ Q + 1 of the states of the models (44) and (45) with respect to (43), that is, ei (t) = xi (t) − x(t); ∀i ∈ Q + 1. Each event Ei is integrated by a set of events Eij which are the errors of each of its integrating subpopulations with respect to the real system, that is, n o Ei = Eij : j ∈ n ; ∀i ∈ Q + 1. Define the instantaneous error entropies of each error event by summing up all the component-wise contributions, that is, H (Ei , t) = − nj=1 pij (t)lnpij (t); ∀i ∈ Q + 1, while the corresponding accumulated P continuous-time and discrete-time entropies on the time interval [t, t + T ) are defined in a natural way from the instantaneous ones, respectively, as follows: Z t+T Z n t+T X Hca (Ei , [t, t + T ]) = − H (Ei , t + jτ)dτ = − pij (σ)lnpij (σ)dσ (46) t t j=1 and: α X α X X n Hda (Ei , [t, t + T ]) = − H (Ei , t + jτ) = − pij (t + kτ)lnpij (t + kτ) (47) j=0 k =0 j=1 provided that T = ατ so that α = T/τ is the set of sampling intervals on T of period τ which is a submultiple of T with T and τ being design parameters satisfying these constraints. The control effort is calculated by applying on a time interval [t, ti+1 ) the control which has made the accumulated entropy of the error on an error event to be smallest one among all the error events on a tested previous time interval [t − ti , t) which defines the so-called active model on [t, ti+1 ). To simplify the exposition, and with no loss in generality, the accumulated discrete - time entropy is the particular one used for testing in the sequel. Then, the following switching Algorithm 1 is proposed: Entropy 2020, 22, 284 15 of 35 Algorithm 1 (all the subpopulations are ran by the same active model within each inter-switching interval) Step 0- Auxiliary design parameters: Define the prefixed minimum inter-sample period threshold Tmin > 0 and σ being an auxiliary time interval, 0 < σ << Tmin , to measure the possible degradation of the current active model in operation what foresees a new coming switching. Step 1-Initial control: u(t) = ui (t) for t = t0 , with t0 = 0, for some arbitrary model i ∈ Q + 1 and make k(∈ Z0+ ) = 0, the initial active control being a(0) = i ∈ Q + 1 an the initial running integer for switching time instants is k = 0. Step 2- Eventually switched control: ( )   u(t) = ui (t) : Hda (Ei , [tk , t)) = min Hda E j , [tk , t) ; ∀t ∈ [tk , tk + Tk ) 1≤ j≤Q+1   !) (48) ∧ Hda (Ei , [tk , tk + τ)) > min Hda E j , [tk , tk + τ) 1≤ j(,i)≤Q+1 such that: -current active model: a(t) = i ∈ Q + 1; ∀t ∈ [tk , tk+1 ) is the active model in the set of (Q + 1) models which generates the control on [tk , tk+1 ), - next active model:  a(t) = j(, i) ∈ Q + 1, such that Hda E j , [tk+1 , tk+1 − σ] < min Hda (E` , [tk+1 , tk+1 − σ]) is the next active model to be in operation 1≤`≤Q+1 at the next switching time instant t = tk+1 . - The switching time instants tk+1 = tk+1 (tk ) = tk + Tk and the inter-switching time periods Tk = Tk (tk ) ≥ Tmin for j ∈ Z0+ depend on the set of preceding sampling instants t0 = 0, t1 (t0 ), t2 (t1 ), · · · , tk (tk−1 ), . . . .  - The control law (48) is calculated by computing the accumulated discrete- time entropies (47) with the following probabilistic rule: ki j (t) ei j (t + kτ) pi j (t + kτ) = 1 − Pn ; ∀i ∈ Q + 1, ∀ j ∈ n, ∀k ∈ T/τ ∪ {0}, ∀t ∈ R0+ (49) j=1 ei j (t + kτ)   with kij (t) ∈ 0, ki j (t) ; ∀i ∈ Q + 1, ∀j ∈ n, and eij (t) = xij (t) − x j (t); ∀i ∈ Q + 1, ∀ j ∈ n, ∀t ∈ R0+ Step 3-Updating the activation of the next active control and inter-switching time interval: Do k ← k + 1 with tk+1 ← tk + Tk being the next controller switching time instant to the next active model a(tk+1 ) ∈ Q + 1, re-initialize all the models to the measured data, that is, xi (tk+1 ) = x(tk+1 ); ∀i ∈ Q + 1 and Go to Step 2. Remarks: (1) Note that the initial control run on a time interval lasting at least the designed time interval length Tmin . In the case of availability of some “a priori” knowledge about the adequacy of the various models to the epidemic process in the initial stage of the disease, this knowledge can be used to overcome the arbitrariness in the selection of the initial controller. Pn j=1 eij (t+ jτ) (2) Note also that if kij ≤ ; i ∈ Q + 1, ∀j ∈ n, ∀t ∈ R0+ then the global probability of the eij (t+ jτ) error event Ei cannot be inflated but it can be deflated. If kij (t) = n − 1; i ∈ Q + 1, ∀j ∈ n, ∀t ∈ R0+ then such a global probability is neither inflated nor deflated ! for all time. Pn Pn eij (t+ jτ) j=1 pij (t + jτ) = j = 1 1 − ( n − 1 ) Pn   = n − (n − 1) = 1; ∀i ∈ Q + 1, ∀t ∈ R0+ . Note j=1 eij (t+ jτ) also that the fact that kij (t) can be time-varying so that the probabilities have a margin for experimental design adjustment relies with the problem statement of probabilities subject to possible errors in the theoretical statements of the above sections. (3) Note that in the above algorithm there are zero-probability (although non impossible) events of leading a unique solution like, for instance, the first accumulated entropy equality in (48) of the strict inequality leading the next active controller. If the event a lack of uniqueness be detected one can fix any valid solution (from the set Entropy 2020, 22, 284 16 of 35 of valid ones) to run or simply to give an uniqueness rule, as for instance, to get as active the nearest indexed model to the last active one active among the set of valid ones. (4) Note also that the events Eij for j ∈ n and each given i ∈ Q + 1 are not mutually independent since they consist of the solutions of all the subpopulations, each given by a single-order differential equation, between the n-th differential equation of the i-th model. The reason is that first-order equations of the differential system are coupled. (5) It can be observed that the switching time instants are also resetting times of initial conditions with the real (or more tightly) data provided by the real model (43). (6) The use of Equation (49) to calculate the probabilities together with their companion saturation rules for the control gains, is sufficient to evaluate the entropies of Equation (48) in order to implement the algorithm. So, it is no need of introduction of a compensatory event of negative probability to equalize to unity the global probability. An alternative to the above switching algorithm consists of defining the error events, one- per subpopulation of the epidemic description (44) versus their real one counterparts in (43). For instance, we can think of choosing the “susceptible event error” with the Q + 1 first error components of all the solutions of (44) compared to the first component of (43) and one proceeds so on for the various remaining subpopulations. So, we can chose online the best simplified model (that is, the closest one to the true complex model) for each subpopulation. Note that this can be reasonable since the approximated models are designed by choosing parameterization of the coefficient transmission rate which sweep a region where the true one is point-wise allocated through time. In parallel, each whole set of error events, associated to each of the subpopulations, consists of mutually independent events since, in this case, the data of all the set of models, including those of the active one, are generated by different differential equations n on nonzeroo measure intervals. In this case, the set of events is {E1 , E2 , · · · , En }, such that E j = Eij : i ∈ Q + 1 ; n o n o ∀j ∈ n is associated to the j-th subpopulation, instead of E1 , E2 , · · · , EQ+1 with Ei = Eij : j ∈ n ; ∀i ∈ Q + 1 as associated to one model as it was sated in the former design. Then, we have the following Algorithm 2. Algorithm 2 (each subpopulation can be ran by a different active model within each inter-switching interval) Step 0- Auxiliary design parameters: it is similar to that of Algorithm 1. Step 1-Initial control: it is similar to that of Algorithm 1. Step 2- Eventually switched control: ( )   u j (t) = u ji (t) : Hda E j , [tk , t) = min Hda (Ei , [tk , t)) ; ∀t ∈ [tk , tk + Tk ) 1≤i≤n     !) (50) ∧ Hda E j , [tk , tk + τ) > min Hda E j , [tk , tk + τ) 1≤i(, j)≤n such that u j (t) is the j-th controller component, and one can distinguish: - the current active model for the j-th subpopulation: a(t) = i ∈ n; ∀t ∈ [tk , tk+1 ) is the active model in the set of n models which generates the control on [tk , tk+1 ), - the next active model for the j-th subpopulation:  a(t) = j(, i) ∈ n, such that Hda E j , [tk+1 , tk+1 − σ] < min Hda (E` , [tk+1 , tk+1 − σ]) is the next active model to be in operation at 1≤`≤n the next switching time instant t = tk+1 . - The switching time instants are obtained in a similar way as in Algorithm 1. - The control law (33) is calculated by computing the accumulated discrete- time entropies (47) with a similar probabilistic rule as that of Algorithm1 with errors ei j (t) = xi j (t) − x j (t); ∀i ∈ Q + 1, ∀ j ∈ n, , ∀t ∈ R0+ Step 3- Updating the activation of the next active control and inter-switching time interval: it is similar to that of Algorithm 1. Remark 6. Related to (50) in Algorithm 2 versus (48) in Algorithm 1, note that u j (t) is the j-th controller component. Assume that the epidemic model is a SEIR one such that x = (x1 , x2 , x3 , x4 )T = Entropy 2020, 22, 284 17 of 35  T S−susceptible , E−exposed , I−in f ectious , R−immune . The precise meaning of the sentence is that the active vaccination control is got, Entropy 2020, 22, for 284 instance, from the active controller 1 ∈ n if feedback vaccination control is used as19being of 38 proportional to the susceptible subpopulation. And the active treatment control is got, for instance, from the 5. Simulation active controllerExamples 3 ∈ n if feedback treatment control is used being proportional to the infectious subpopulation. If there are no more controllers the remaining controllers would be zeroed for all time. The models for the This section contains some simulation examples illustrating the application of the proposed other components could be omitted from the whole scheme or simply used for information of the estimation of Entropy paradigm to the multi-model epidemic system discussed in Section 4. Thus, the behavior of subpopulations since no controls are got from them to be injected to the real population. Algorithms 1 and 2 will be shown in this section through numerical examples in open and closed- loop. The accurate model considered as the one generating the actual or true data is the SEIR one described 5. Simulation in [27] with vaccination: Examples This section contains some dS simulation examples illustrating the application of the proposed = μ − β ( t )S ( t )I ( t ) − μS ( t ) + δR( t ) − V ( t ) Entropy paradigm to the multi-model dt epidemic system discussed in Section 4. Thus, the behavior of Algorithms 1 and 2 will be shown dE in this section through numerical examples in open and closed-loop. = β ( t )S ( t ) I ( t ) − ( μ + ε ) E ( t ) The accurate model considereddtas the one generating the actual or true data is the SEIR one described (51) in [27] with vaccination: dI = εE ( t ) − ( μ + γ ) I ( t ) dt dt ==µγI− β(t)S(t)I (t) − µS(t) + δR(t) − V (t) dS dR dE( t ) − ( μ + δ )R( t ) + V ( t ) dt dt = β(t)S(t)I (t) − (µ + ε)E(t) (51) dt = εE(t) − (µ + γ)I (t) dI −1 −1 where μ = 2. 0 years is the growth and death rate of the population, ε = 1.0 years , δ = 0.1 dt = γI (t) − (µ + δ)R(t) + V (t) dR −1 −1 days , γ = 0. 02 days are the instantaneous per capita rates of leaving the exposed, infected and where recovered µ =stages, 2.0 years −1 is the growth and death rate of the population, ε = 1.0 years−1 , δ = 0.1 respectively, and V( t ) denotes the vaccination. This model fits in the structure days , γ = 0.02 days−1 are the instantaneous per capita rates of leaving the exposed, infected and −1 given by (26) recovered where stages, u( t ) = V( t ) V (tas respectively, andacts the control ) denotes the command. vaccination. The initial This conditions model are structure fits in the given by S( 0 ) =byE((26) given 0 ) =where I( 0 ) = uR((t0) )== 0V.1(t.) All actsthe parameters as the are assumed control command. Thetoinitial be constant areβgiven except conditions ( t ) , the by (0) = Etransmission Sdisease (0) = I (0) =coefficient, R(0) = 0.1. whichAll describes the parameters are assumed the seasonality in thetoinfection be constantrateexcept β(t), the and is given by transmission coefficient, which describes the ( the widely accepted Dietz’s model, [30], β( t ) = β0 1+ bcos(2πt ) with β0 = 6.2 and b = 0.6 . The disease seasonality in the )infection rate and is given by the widely accepted Dietz’s model, [30], β(t) = β0 (1 + bcos(2πt)) with β0 = 6.2 and b = 0.6. The function βfunction (t) describesβ (t )annual describes annual in seasonality seasonality this example. in this example. Figure 1 shows Figure 1 shows of the behavior thethis behavior systemof in this the system in absence ofthe anyabsence external of action any external (i.e., inaction open (i.e., loop). in open loop). As it can beAs it can beinobserved observed Figure 1,intheFigure disease1, the is disease is persistent since the infectious do not converge to a zero steady-state persistent since the infectious do not converge to a zero steady-state value asymptotically. This situation value asymptotically. Thisbe will situation tackled in will be tackled Section 5.2 byin Subsection means 5.2 by means of the vaccination of theinvaccination function function order to generate in order to a closed-loop generate a closed-loop system system whose infectious tend to zero. whose infectious tend to zero. 0.7 0.6 0.5 S E I R 0.4 Population 0.3 0.2 0.1 0 0 50 100 150 200 250 300 350 400 450 time (days) Figure 1. Dynamics of the accurate seasonal epidemic model. Figure 1. Dynamics of the accurate seasonal epidemic model. This accurate time-varying model is described by a number of simplified time-invariant models Thisinaccurate running parallel time-varying with the samemodel is described constant parameterbyvalues a number of simplified and fixed β obtained according values oftime-invariant models running in parallel with the same constant parameter values and fixed values of β obtained according to (44). In this example Q = 9 , so that 10 models will be running in parallel. The Figures 2– 5 show the trajectory followed by these fixed models for the different considered constant values of β . Entropy 2020, 22, 284 20 of 38 Entropy 2020, 22, 284 18 of 35 Entropy 2020, 22, 284 20 of 38 to (44). In2020, Entropy this22, example 284 Q = 9, so that 10 models will be running in parallel. The Figures 2–5 show the 20 of 38 trajectory followed by these fixed models for the different considered constant values of β. Figure 2. Dynamics of susceptible (S) for different values of constant β. Figure 2. Dynamics of susceptible (S) for different values of constant β. Figure 2. Dynamics of susceptible (S) for different values of constant β. Figure 2. Dynamics of susceptible (S) for different values of constant β . Figure 3. Dynamics of exposed (E) for different values of constant β . Figure 3. Dynamics of exposed (E) for different values of constant β. Figure 3. Dynamics of exposed (E) for different values of constant β . Figure 3. Dynamics of exposed (E) for different values of constant β . Figure 4. Dynamics of infectious (I) for different values of constant β. Figure 4. Dynamics of infectious (I) for different values of constant β . It is worth to mention that none of the models whose trajectory is depicted in Figures 2–5 can solely describe the dynamics of the whole accurate model, since none of them is able to reproduce Figure 4. Dynamics of infectious (I) for different values of constant β . the complex behavior generated by the time-varying seasonal incidence rate, β(t). In this way, the switching mechanisms given by Algorithms 1 and 2 inspired in the entropy paradigm are employed to 4. Dynamics of infectious (I) for different values of constant β . generate a switchedFigure piecewise constant model able to describe the time-varying system by means of time-invariant ones. Section 5.1 shows the capabilities of the switched model to reproduce the behavior of the accurate model in open loop, i.e., in the absence of any external action. Entropy Entropy 2020, 2020, 22, 284 22, 284 21 19 of of 38 35 Figure 5. Dynamics of immune (R) for different values of constant β. Figure 5. Dynamics of immune (R) for different values of constant β . 5.1. Open-Loop Switched Model It is worth In this to mention subsection that none of the vaccination the models function is setwhose trajectory to zero so that isalldepicted the models in Figures 2–5 canin are running solely describe the dynamics of the whole accurate model, since none of them open loop. The simulations are completed for different values of the switching time T (in days) is able to reproduce andthethecomplex behavior probability gainsgenerated kij of (49)byinthe time-varying order to observe seasonal incidenceon their influence the β rate, ( t ) . In thistrajectories. open-loop way, the Theswitching values ofmechanisms the probability given gains include 1the by Algorithms and case of inflation 2 inspired in the as well. paradigm entropy Thus, theareFigures employed6–10 to generate display a switched the trajectories ofpiecewise the switchedconstant model system able to describe for Algorithm the time-varying 1, different values of the system by means switching time of time-invariant T and a constant gain ones. of Subsection kij = 15.8 5.1 shows while the capabilities Figures 11–15 depictof the theswitched model trajectories fortodifferent reproduce the values behavior of the of the accurate probability gains kijmodel and ainconstant open loop, i.e., of value in the T =absence 5 days. of any external In all action. the subsequent figures, the blue line denotes the output of the time-varying system while the output of the switched system is 5.1. Open-Loop Switched Model displayed in red; legends have been then omitted from figures for the sake of clarity. On the22other Entropy 2020, 22, 284 of 38 hand, Figures In this 16–24 display subsection thethe trajectories vaccination underisthe function setsame conditions to zero butthe so that all formodels Algorithm 2. are running in open loop. The simulations are completed for different values of the switching time T (in days) and the probability gains kij of T(49) =3 in order to observe their influence on theT open-loop =5 trajectories. The 0.6 0.6 values of the probability gains include the case of inflation as well. Thus, the Figures 6–10 display the 0.5of the switched system for Algorithm 1, different trajectories 0.5 values of the switching time T and a Population S Population S gain of kij = 15.8 while Figures 11–15 depict constant 0.4 0.4 the trajectories for different values of the probability gains kij and a constant value of T = 5 days. In all the subsequent figures, the blue line 0.3 0.3 denotes the output of the time-varying system while the output of the switched system is displayed 0.2 in red; legends have been then omitted from figures0.2for the sake of clarity. On the other hand, Figures 16–24 display the trajectories under the same conditions but for Algorithm 2. 0.1 0.1 0 100 200 300 400 500 0 100 200 300 400 500 time (days) time (days) T =7 T =10 0.6 0.6 0.5 0.5 Population S Population S 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 100 200 300 400 500 0 100 200 300 400 500 time (days) time (days) Figure 6. Trajectory of the susceptible for the switched system (Algorithm 1) and different values of T. Figure 6. Trajectory of the susceptible for the switched system (Algorithm 1) and different values of T . 0 100 200 300 400 500 0 100 200 300 400 500 time (days) time (days) Figure 6. Trajectory of the susceptible for the switched system (Algorithm 1) and different values of T . Entropy 2020, 22, 284 20 of 35 Entropy 2020, 22, 284 23 of 38 Figure 7. Trajectory of the exposed for the switched system (Algorithm 1) and different values of T. Figure 7. Trajectory of the exposed for the switched system (Algorithm 1) and different values of T =3 T =5 0.1 T . 0.1 0.08 0.08 Population I Population I 0.06 0.06 0.04 0.04 0.02 0.02 0 100 200 300 400 500 0 100 200 300 400 500 time (days) time (days) T =7 T = 10 0.1 0.1 0.08 0.08 Population I Population I 0.06 0.06 0.04 0.04 0.02 0.02 0 100 200 300 400 500 0 100 200 300 400 500 time (days) time (days) Figure 8. Trajectory of the infectious for the switched system (Algorithm 1) and different values of T. Figure 8. Trajectory of the infectious for the switched system (Algorithm 1) and different values of T . T =3 T =5 0.1 0.1 0.08 0.08 pulation I pulation I 0.06 0.06 0.04 0.04 0 100 200 300 400 500 0 100 200 300 400 500 time (days) time (days) Figure 8. Trajectory of the infectious for the switched system (Algorithm 1) and different values of T . Entropy 2020, 22, 284 21 of 35 T =3 T =5 0.1 0.1 Population I 0.08 0.08 Population I 0.06 0.06 0.04 0.04 0.02 0.02 0 0 0 100 200 300 400 500 0 100 200 300 400 500 time (days) time (days) T =7 T = 10 0.1 0.1 0.08 0.08 Population I Population I 0.06 0.06 0.04 0.04 0.02 0.02 0 0 0 100 200 300 400 500 0 100 200 300 400 500 Entropy 2020, 22, 284 time (days) time (days) 24 of 38 Figure 9. Trajectory of the immune for the switched system (Algorithm 1) and different values of T. Figure 9. Trajectory of the immune for the switched system (Algorithm 1) and different values of T. T=3 T=5 10 10 8 8 Population I Population I 6 6 4 4 2 2 0 0 0 100 200 300 400 500 0 100 200 300 400 500 time (days) time (days) T=7 T = 10 10 10 8 8 Population I Population I 6 6 4 4 2 2 0 0 0 100 200 300 400 500 0 100 200 300 400 500 time (days) time (days) Figure 10. Active model within time intervals for the different values of T (Algorithm 1). Figure 10. Active model within time intervals for the different values of T (Algorithm 1). k ij = 0.8 k ij = 5.8 0.6 0.6 0.5 0.5 Population S Population S 0.4 0.4 0.3 0.3 0 0 0 100 200 300 400 500 0 100 200 300 400 500 time (days) time (days) Figure 10. Active model within time intervals for the different values of T (Algorithm 1). Entropy 2020, 22, 284 22 of 35 k ij = 0.8 k ij = 5.8 0.6 0.6 Population S 0.5 0.5 Population S 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 100 200 300 400 500 0 100 200 300 400 500 time (days) time (days) k ij = 15.8 k ij = 150.8 0.6 0.6 0.5 0.5 Population S Population S 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 100 200 300 400 500 0 100 200 300 400 500 time (days) time (days) Entropy 2020, 11. Figure 22, 284 25 of 38 Trajectory of the susceptible for the switched system (Algorithm 1) and different values of kij . 11. Trajectory of the susceptible for the switched system (Algorithm 1) and different values of Figure kij . k ij = 0.8 k ij = 5.8 0.7 0.7 0.6 0.6 Population E Population E 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 100 200 300 400 500 0 100 200 300 400 500 time (days) time (days) k ij = 15.8 k ij = 150.8 0.7 0.7 0.6 0.6 Population E Population E 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 100 200 300 400 500 0 100 200 300 400 500 time (days) time (days) Figure 12. Trajectory of the exposed for the switched system (Algorithm 1) and different values of ki j . Figure 12. Trajectory of the exposed for the switched system (Algorithm 1) and different values of kij . k ij = 0.8 k ij = 5.8 0.1 0.1 0.08 0.08 opulation I opulation I 0.06 0.06 time (days) time (days) Figure 12. Trajectory of the exposed for the switched system (Algorithm 1) and different values of kij Entropy .2020, 22, 284 23 of 35 k ij = 0.8 k ij = 5.8 0.1 0.1 0.08 0.08 Population I Population I 0.06 0.06 0.04 0.04 0.02 0.02 0 100 200 300 400 500 0 100 200 300 400 500 time (days) time (days) k ij = 15.8 k ij = 150.8 0.1 0.1 0.08 0.08 Population I Population I 0.06 0.06 0.04 0.04 0.02 0.02 0 100 200 300 400 500 0 100 200 300 400 500 Entropy 2020, 22, 284 time (days) time (days) 26 of 38 Figure 13. Trajectory of the infectious for the switched system (Algorithm 1) and different values of ki j . Figure 13. Trajectory of the infectious for the switched system (Algorithm 1) and different values of k ij = 0.8 k ij = 5.8 kij . 0.1 0.1 0.08 0.08 Population R Population R 0.06 0.06 0.04 0.04 0.02 0.02 0 0 0 100 200 300 400 500 0 100 200 300 400 500 time (days) time (days) k ij = 15.8 k ij = 150.8 0.1 0.1 0.08 0.08 Population R Population R 0.06 0.06 0.04 0.04 0.02 0.02 0 0 0 100 200 300 400 500 0 100 200 300 400 500 time (days) time (days) Figure 14. Trajectory of the immune for the switched system (Algorithm 1) and different values of ki j . Figure 14. Trajectory of the immune for the switched system (Algorithm 1) and different values of kij . k ij = 0.8 k ij = 5.8 10 10 8 8 pulation R pulation R 6 6 4 4 time (days) time (days) Figure 14. Trajectory of the immune for the switched system (Algorithm 1) and different values of kij . Entropy 2020, 22, 284 24 of 35 k ij = 0.8 k ij = 5.8 10 10 Population R 8 8 Population R 6 6 4 4 2 2 0 0 0 100 200 300 400 500 0 100 200 300 400 500 time (days) time (days) k ij = 15.8 k ij = 150.8 10 10 8 8 Population R Population R 6 6 4 4 2 2 0 0 0 100 200 300 400 500 0 100 200 300 400 500 Entropy 2020, 22, 284 time (days) time (days) 27 of 38 Figure 15. Active model within each time interval for different values of ki j (Algorithm 1). Figure 15. Active model within each time interval for different values of T=3 k (Algorithm 1). T = 5 ij 0.6 0.6 0.5 0.5 Population S Population S 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 100 200 300 400 0 100 200 300 400 time (days) time (days) T=7 T = 10 0.6 0.6 0.5 0.5 Population S Population S 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 100 200 300 400 0 100 200 300 400 time (days) time (days) Figure 16. Trajectory of the susceptible for the switched system (Algorithm 2) and different values of T. Figure 16. Trajectory of the susceptible for the switched system (Algorithm 2) and different values of T . T =3 T =5 0.6 0.6 0.5 0.5 Population E Population E 0.4 0.4 0.3 0.3 0 100 200 300 400 0 100 200 300 400 time (days) time (days) Figure 16. Trajectory of the susceptible for the switched system (Algorithm 2) and different values of T . Entropy 2020, 22, 284 25 of 35 T =3 T =5 0.6 0.6 Population E 0.5 0.5 Population E 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 100 200 300 400 0 100 200 300 400 time (days) time (days) T =7 T = 10 0.6 0.6 0.5 0.5 Population E Population E 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 100 200 300 400 0 100 200 300 400 Entropy 2020, 22, 284 time (days) time (days) 28 of 38 Figure 17. Trajectory of the exposed for the switched system (Algorithm 2) and different values of T. Figure 17. Trajectory of theTexposed =3 for the switched system (Algorithm 2) Tand =5 different values of T . 0.1 0.1 0.08 0.08 Population I Population I 0.06 0.06 0.04 0.04 0.02 0.02 0 100 200 300 400 0 100 200 300 400 time (days) time (days) T=7 T = 10 0.1 0.1 0.08 0.08 Population I Population I 0.06 0.06 0.04 0.04 0.02 0.02 0 100 200 300 400 0 100 200 300 400 time (days) time (days) Figure 18. Trajectory of the infectious for the switched system (Algorithm 2) and different values of T. Figure 18. Trajectory of the infectious for the switched system (Algorithm 2) and different values of T . T =3 T =5 0.1 0.1 0.08 0.08 pulation R pulation R 0.06 0.06 0.04 0.04 time (days) time (days) Figure 18. Trajectory of the infectious for the switched system (Algorithm 2) and different values of T . Entropy 2020, 22, 284 26 of 35 T =3 T =5 0.1 0.1 Population R 0.08 0.08 Population R 0.06 0.06 0.04 0.04 0.02 0.02 0 0 0 100 200 300 400 0 100 200 300 400 time (days) time (days) T =7 T = 10 0.1 0.1 0.08 0.08 Population R Population R 0.06 0.06 0.04 0.04 0.02 0.02 0 0 0 100 200 300 400 0 100 200 300 400 Entropy 2020, 22, 284 29 of 38 time (days) time (days) Figure 19. Trajectory of the immune for the switched system (Algorithm 2) and different values of T. Figure 19.10Trajectory of the immune for the switched10system (Algorithm 2) and different values of T . 8 8 Active model Active model 6 6 4 4 2 2 0 0 0 100 200 300 400 0 100 200 300 400 time (days) time (days) 10 10 8 8 Active model Active model 6 6 4 4 2 2 0 0 0 100 200 300 400 0 100 200 300 400 time (days) time (days) Figure 20. Active model within each time interval for Algorithm 2 and values T = 5 days and ki j = 15.8. Figure 20. Active model within each time interval for Algorithm 2 and values T = 5 days and kij = 15.8 . k ij = 0.8 k ij = 5.8 0.6 0.6 0.5 0.5 Population S Population S 0.4 0.4 0.3 0.3 0.2 0.2 time (days) time (days) Figure 20. Active model within each time interval for Algorithm 2 and values T = 5 days and kij = 15.8 . Entropy 2020, 22, 284 27 of 35 k ij = 0.8 k ij = 5.8 0.6 0.6 Population S 0.5 0.5 Population S 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 100 200 300 400 0 100 200 300 400 time (days) time (days) k ij = 15.8 k ij = 150.8 0.6 0.6 0.5 0.5 Population S Population S 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 100 200 300 400 0 100 200 300 400 time (days) time (days) Entropy 2020, 22, 284 30 of 38 Figure 21. Trajectory of the susceptible for the switched system (Algorithm 2) and different values of kij . 21. Trajectory of the susceptible for the switched system (Algorithm 2) and different values of Figure kij . k ij = 0.8 k ij = 5.8 0.6 0.7 0.5 0.6 Population E Population E 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 100 200 300 400 0 100 200 300 400 time (days) time (days) k ij = 15.8 k ij = 150.8 0.6 0.6 0.5 0.5 Population E Population E 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 100 200 300 400 0 100 200 300 400 time (days) time (days) Figure 22. Trajectory of the exposed for the switched system (Algorithm 2) and different values of ki j . Figure 22. Trajectory of the exposed for the switched system (Algorithm 2) and different values of kij . k ij = 0.8 k ij = 5.8 0.1 0.1 0.08 0.08 Population I Population I 0.06 0.06 time (days) time (days) Figure 22. Trajectory of the exposed for the switched system (Algorithm 2) and different values of kij . Entropy 2020, 22, 284 28 of 35 k ij = 0.8 k ij = 5.8 0.1 0.1 0.08 0.08 Population I Population I 0.06 0.06 0.04 0.04 0.02 0.02 0 100 200 300 400 0 100 200 300 400 time (days) time (days) k ij = 15.8 k ij = 150.8 0.1 0.1 0.08 0.08 Population I Population I 0.06 0.06 0.04 0.04 0.02 0.02 0 100 200 300 400 0 100 200 300 400 Entropy 2020, 22, 284 time (days) time (days) 31 of 38 Figure 23. Trajectory of the infectious for the switched system (Algorithm 2) and different values of ki j . Figure 23. Trajectory of the k =infectious 0.8 for the switched system (Algorithm k =2)5. and 8 different values of ij ij 0.1 0.1 kij . 0.08 0.08 Population R Population R 0.06 0.06 0.04 0.04 0.02 0.02 0 0 0 100 200 300 400 0 100 200 300 400 time (days) time (days) k ij = 15.8 k ij = 150.8 0.1 0.1 0.08 0.08 Population R Population R 0.06 0.06 0.04 0.04 0.02 0.02 0 0 0 100 200 300 400 0 100 200 300 400 time (days) time (days) Figure 24. Trajectory of the immune for the switched system (Algorithm 2) and different values of ki j . Figure 24. Trajectory of the immune for the switched system (Algorithm 2) and different values of From kij . the above Figures we can conclude that the switched system, either with Algorithm 1 or with Algorithm 2, is able to reproduce the shape of the trajectories of the accurate-time-varying system. Consequently, the presented approach is useful to describe the overall dynamics of a complex set From the above Figures we can conclude that the switched system, either with Algorithm 1 or of data, coming either from real measurements or from a complex model, by means of a simpler with Algorithm 2, is able to reproduce the shape of the trajectories of the accurate-time-varying piece-wise time-invariant model. The switching rules based on the entropy of the errors between each system. Consequently, the presented approach is useful to describe the overall dynamics of a complex set of data, coming either from real measurements or from a complex model, by means of a simpler piece-wise time-invariant model. The switching rules based on the entropy of the errors between each model and the data have revealed to be an adequate frame for such a task. Figures 10 and 15 show the active model selected within each time interval by Algorithm 1 to parameterize the switched Entropy 2020, 22, 284 29 of 35 model and the data have revealed to be an adequate frame for such a task. Figures 10 and 15 show the active model selected within each time interval by Algorithm 1 to parameterize the switched system. Moreover, Figure 20 shows the active model when Algorithm 2 is employed. It is seen in Figure 20 how the algorithm selects a different model for each one of the state components of the system, which is the particularity of the algorithm. From the simulation results we can also observe that the probability value has a slight influence on the approximation performance and the selection of an appropriate value is not that critical. However, the switching time plays a more important role in the obtained performance as it can be deduced from the above examples. Moreover, there are no specific criteria for the selection of the switching time while a trial-error work can be performed in order to obtain an appropriate value for it. Finally, it can also be observed that Algorithm 1 is able to attain closer curves to the actual system than Algorithm 2, especially when it comes to the Susceptible population. The above figures show that the disease is persistent in the sense that the infectious do not vanish with time. Therefore, in the next Section 5.2 vaccination is used as external control to make the infectious converge to zero asymptotically. 5.2. Closed-Loop Control In this subsection, vaccination is employed to avoid the persistency of the disease. To this end, the following state-feedback type vaccination law is used [29]: V (t) = KS Sactive (t) + KI Iactive (t) with KS = 0.1 and KI = 0.01 being the state-feedback gains and Sactive (t), Iactive (t) the state components of the corresponding active model according to Algorithms 1 and 2. Along this section, T = 5 days and kij = 15.8. Figure 25 displays the evolution of the closed-loop system when Algorithm 1 is employed while Figure 26 shows the corresponding active model selected to parameterize the control command within each time interval. Figure 27 shows the trajectories for Algorithm 2 and Figure 28 shows the corresponding active model selected to parameterize the control command within each time interval. From Figures 25 and 27 we can conclude that the output of the real system and the output of the active model are practically the same when a feedback control action is included in the system. Thus, the approximation errors appearing in open loop vanish in closed-loop because of the control action. Moreover, there are no differences in the closed-loop trajectories generated by Algorithms 1 and 2. It can also be observed in Figures 25 and 27 that the infectious tend to zero eradicating the disease from the population, as desired. Therefore, the control objective is achieved. Figures 29 and 30 display the vaccination function calculated by using both algorithms. It can be seen that both control commands are very similar with only some peaks associated with the switching process making the difference between one and another. Overall, the proposed approach has been showed to be a powerful tool to model the complex time-varying system. to zero eradicating the disease from the population, as desired. Therefore, the control objective is achieved. Figures 29 and 30 display the vaccination function calculated by using both algorithms. It can be seen that both control commands are very similar with only some peaks associated with the switching process making the difference between one and another. Overall, the proposed approach has been Entropy 2020,showed 22, 284 to be a powerful tool to model the complex time-varying system. 30 of 35 0.1 0.12 0.08 0.1 Population S Population E 0.08 0.06 0.06 0.04 0.04 0.02 0.02 0 0 0 100 200 300 400 500 0 100 200 300 400 500 time (days) time (days) 0.1 1 0.08 0.8 Population R Population I 0.06 0.6 0.04 0.4 0.02 0.2 0 0 0 100 200 300 400 500 0 100 200 300 400 500 time (days) time (days) Entropy 2020, 22, 284 33 of 38 Figure 25. Evolution of the closed-loop system when vaccination is applied and Algorithm 1 is used. Figure 25. Evolution of the closed-loop 9 system when vaccination is applied and Algorithm 1 is used. 8 7 6 Active model 5 4 3 2 1 0 50 100 150 200 250 300 350 400 450 time (days) Figure 26. Active model when Algorithm 1 is employed. Figure 26. Active model when Algorithm 1 is employed. 0.1 0.12 0.08 0.1 Population S Population E 0.08 0.06 0.06 0.04 0.04 0.02 0.02 0 0 0 100 200 300 400 500 0 100 200 300 400 500 time (days) time (days) 0.1 1 0.08 0.8 pulation R pulation I 0.06 0.6 0.04 0.4 2 1 0 50 100 150 200 250 300 350 400 450 time (days) Figure 26. Active model when Algorithm 1 is employed. Entropy 2020, 22, 284 31 of 35 0.1 0.12 0.08 0.1 Population S Population E 0.08 0.06 0.06 0.04 0.04 0.02 0.02 0 0 0 100 200 300 400 500 0 100 200 300 400 500 time (days) time (days) 0.1 1 0.08 0.8 Population R Population I 0.06 0.6 0.04 0.4 0.02 0.2 0 0 0 100 200 300 400 500 0 100 200 300 400 500 Entropy 2020, 22, 284 34 of 38 time (days) time (days) Figure 27. Evolution of the closed-loop system when vaccination is applied and Algorithm 2 is used. Figure 27. Evolution of the closed-loop system when vaccination is applied and Algorithm 2 is used. 10 10 8 8 Active model (S) Active model (E) 6 6 4 4 2 2 0 0 0 100 200 300 400 500 0 100 200 300 400 500 time (days) time (days) 10 10 8 8 Active model (R) Active model (I) 6 6 4 4 2 2 0 0 0 100 200 300 400 500 0 100 200 300 400 500 time (days) time (days) Figure 28. Active model when Algorithm 2 is employed. Figure 28. Active model when Algorithm 2 is employed. 0.012 0.01 0.008 V(t) 0.006 0.004 0.002 0 0 0 100 200 300 400 500 0 100 200 300 400 500 0 0 0 100 time 200 (days) 300 400 500 0 100 time 200 (days) 300 400 500 time (days) time (days) Figure 28. Active model when Algorithm 2 is employed. Entropy 2020, 22, 284 Figure 28. Active model when Algorithm 2 is employed. 32 of 35 0.012 0.012 0.01 0.01 0.008 0.008 V(t) 0.006 V(t) 0.006 0.004 0.004 0.002 0.002 0 0 50 100 150 200 250 300 350 400 450 time (days) 0 0 50 100 150 200 250 300 350 400 450 time (days) Figure 29. Vaccination function for Algorithms 1 and 2. Figure 29. Vaccination function for Algorithms 1 and 2. Figure 29. Vaccination function for Algorithms 1 and 2. -3 10 3.6 10 -3 3.53.6 3.43.5 3.33.4 3.23.3 3.13.2 33.1 2.9 3 2.82.9 120 140 160 180 200 220 240 time (days) 2.8 120 140 160 180 200 220 240 time (days) Figure 30. Zoom on the vaccination function for Algorithms 1 and 2. Figure 30. Zoom on the vaccination function for Algorithms 1 and 2. Figure 5.3. Example with Actual 30. Zoom on the vaccination function for Algorithms 1 and 2. Data In this subsection the proposed entropy approach is applied to the real case of measles in the city of New York. In this way, it was stated in [33] that measles outbreaks in NYC during the period 1930–1970 can be appropriately described by a SIR model with a time-varying contact rate. Moreover, [33] estimates a monthly contact rate for the model while [34] proposes a Dietz-type contact rate for this problem. Thus, this real situation fits in with the formulation treated in the application problem. The authors of [34] gathered the weekly amount of reported measles cases for 93 years corresponding to the period 1891–1984 and made them publicly available as a supplementary material of [34]. In this simulation, the Algorithm 2 from Section 4 will be used to generate a switched model describing the data set of the year 1960. The SIR model describing the problem is given by: dt = ν − β(t)SI − µS dS dt = β(t)SI − (γ + µ)I dI (52) dt = γI − µR dR The parameters of the model are estimated in [34] to be γ−1 = 13 days, µ = 0.02yr−1 , βmin = 1.79 · 10−10 , βmax = 5.3831 · 10−10 and the yearly number of new born is approximately 105 . The initial values of the populations are S(0) = 7782000, the population of New York in 1960, I (0) = 225 and R(0) = 0 while T = 20 days. There are 40 time-invariant models linearly spaced between βmin and βmax . Actual data for the infectious are used in (49) in order to calculate the error corresponding to each one of models running in parallel. The Figure 31 shows the number infectious predicted by the switched model compared to the actual data during 1960. The x axis of the figure represents the 52 weeks of the year. and R (0) = 0 while T = 20 days. There are 40 time-invariant models linearly spaced between β min and β max . Actual data for the infectious are used in (49) in order to calculate the error corresponding to each one of models running in parallel. The Figure 31 shows the number infectious predicted by the switched Entropy model compared to the actual data during 1960. The x axis of the figure represents 2020, 22, 284 33 of the 35 52 weeks of the year. 3500 I switched Actual data 3000 2500 Population I 2000 1500 1000 500 0 0 10 20 30 40 50 60 time (weeks) Figure 31. Comparison of the output of the switched model with the actual data set for 1960. Figure 31. Comparison of the output of the switched model with the actual data set for 1960. As it can be observed from Figure 31 the proposed approach succeeds at reproducing the trend As itincan contained thebe observed data from Figure and predicting 31 the the time proposed when approach the outbreak succeeds reaches at reproducing the peak. the trend Finally, the Figure 32 contained Entropy 2020, in 22, the 284 data and predicting the time when the outbreak reaches the peak. Finally, shows the active model at each time for the infectious population. Since the data are only available forthe Figure 36 of 38 32 shows the active model at each time for the infectious population. the infectious, only this population has been displayed in the figures. Since the data are only available for the infectious, only this population has been displayed in the figures. 40 38 36 Active model for the Infectious 34 32 30 28 26 24 22 0 10 20 30 40 50 60 time (weeks) Figure 32. Active model selected at each time interval for the infectious population. Figure 32. Active model selected at each time interval for the infectious population. 6. Conclusions 6. Conclusions This paper has presented results for the Shannon entropy when a complete the events of a complete finite This and discrete set are paper has eventually presented subject results fortothe relative errorsentropy Shannon of their when associate probabilities. a complete As a result, the events of a the currentfinite complete system andofdiscrete events,seteventually undersubject are eventually probabilistic errors, to relative canof errors lose itsassociate their completeness since it probabilities. may As aberesult, either deflated the currentor inflated systeminoftheevents, sense that the totalunder eventually probability for the whole probabilistic errors,sets canmight lose be its either below orsince completeness beyond unity. it may be Later eitheron the previous deflated technical or inflated in theresults havethe sense that been applied total to control probability of for the epidemics whole setsevolution might bein the case either that below oreither beyond theunity. disease transmission Later coefficient on the previous rate isresults technical not well-known have been or it varies applied to through control oftime due, forevolution epidemics instance, to in seasonality. the case thatFor such either thea purpose, a finite predefined disease transmission set coefficient of running rate models, described is not well-known by coupled or it varies throughdifferential time due, for equations, instance,istochosen whichFor seasonality. covers such aa range of purpose, variation of such a coefficient a finite predefined transmission set of running models,rate within known described lower-bound by coupled and equations, differential upper-bound limits. is chosen Each which one of such covers models a range of is driven by variation of asuch constant disease transmission rate a coefficient rate and the known within whole set of models lower-bound covers the whole range and upper-bound limits.ofEach foreseen one of variation such modelsof such a parameter is driven in the real by a constant system. disease In a general transmission rate context, and thedifferent whole setuncertain of models parameters, covers theorwhole groups of parameters, range of foreseenother that the variation of coefficient transmission such a parameter in the rate realcould system.be In checked by the a general proposed context, minimum-error different entropy supervisory uncertain parameters, or groupsscheme. Two monitoring of parameters, other that algorithms have been the coefficient proposedrate transmission to select could thebeactive one which checked by theminimizes proposed theminimum-error accumulated entropy of entropy supervisory scheme. Two monitoring algorithms have been proposed to select the active one which minimizes the accumulated entropy of the absolute error data/model within each supervision time interval. A switching rule allows to choose another active model as soon as it is detected that the current active model becomes more uncertain than other (s) related to the observed data. Some numerical results have been also performed and discussed including a discussion on a real registered Entropy 2020, 22, 284 34 of 35 the absolute error data/model within each supervision time interval. A switching rule allows to choose another active model as soon as it is detected that the current active model becomes more uncertain than other (s) related to the observed data. Some numerical results have been also performed and discussed including a discussion on a real registered case. The active model, which is currently in operation, generates the vaccination and/or treatment controls to be injected to the real epidemic process. Author Contributions: Data curation, A.I.; Formal analysis, M.D.l.S.; Funding acquisition, M.D.l.S.; Investigation, M.D.l.S.; Methodology, M.D.l.S.; Project administration, M.D.l.S.; Resources, M.D.l.S. and A.I.; Software, A.I.; Supervision, M.D.l.S.; Validation, A.I. and R.N.; Visualization, A.I.; Writing—review and editing, R.N. All authors have read and agreed to the published version of the manuscript. Funding: The authors are grateful to the Spanish Government for Grant RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE) and to the Basque Government for Grant IT1207-19. Conflicts of Interest: The authors declare no conflict of interest. References 1. Khinchin, A.I. Mathematical Foundations of Information Theory; Dover Publications Inc.: New York, NY, USA, 1957. 2. Aczel, J.D.; Daroczy, Z. On Measures of Information and Their Generalizations; Academic Press: New York, NY, USA, 1975. 3. Ash, R.B. Information Theory; Interscience Publishers: New York, NY, USA; John Wiley and Sons: Hoboken, NJ, USA, 1965. 4. Feynman, R.P. Simulating Physics and Computers. Int. J. Theor. Phys. 1982, 21, 467–488. [CrossRef] 5. Burgin, M.; Meissner, G. Larger than one probabilities in mathematical and practical finance. Rev. Econ. Financ. 2012, 2, 1–13. 6. Tenreiro-Machado, J. Fractional derivatives and negative probabilities. Commun. Nonlinear Sci. Numer. Simul. 2019, 79, 104913. [CrossRef] 7. Baez, J.C.; Fritz, T.; Leinster, T. A characterization of entropy in terms of information loss. Entropy 2011, 2011, 1945–1957. [CrossRef] 8. Delyon, F.; Foulon, P. Complex entropy for dynamic systems. Ann. Inst. Henry Poincarè Phys. Théor. 1991, 55, 891–902. 9. Nalewajski, R.F. Complex entropy, resultant information measures. J. Math. Chem. 2016, 54, 1777–2782. [CrossRef] 10. Goh, S.; Choi, J.; Choi, M.Y.; Yoon, B.G. Time evolution of entropy in a growth model: Dependence of the description. J. Korean Phys. Soc. 2017, 70, 12–21. [CrossRef] 11. Wang, W.B.; Wu, Z.N.; Wang, C.F.; Hu, R.F. Modelling the spreading rate of controlled communicable epidemics through and entropy-based thermodynamic model. Sci. China Phys. Mech. Astron. 2013, 56, 2143–2150. [CrossRef] 12. Tiwary, S. The evolution of entropy in various scenarios. Eur. J. Phys. 2019, 41, 1–11. [CrossRef] 13. Koivu-Jolma, M.; Annila, A. Epidemic as a natural process. Math. Biosci. 2018, 299, 97–102. [CrossRef] 14. Artalejo, J.R.; Lopez-Herrero, M.J. The SIR and SIS epidemic models. A maximum entropy approach. Theor. Popul. Biol. 2011, 80, 256–264. [CrossRef] [PubMed] 15. Erten, E.Y.; Lizier, J.T.; Piraveenan, M.; Prokopenko, M. Criticality and information dynamics in epidemiological models. Entropy 2017, 19, 194. [CrossRef] 16. De la Sen, M. On the approximated reachability of a class of time-varying systems based on their linearized behaviour about the equilibria: Applications to epidemic models. Entropy 2019, 2019, 1–31. 17. Meyers, L. Contact network epidemiology: Bond percolation applied to infectious disease prediction and control. Bull. Am. Math. Soc. 2007, 44, 63–86. [CrossRef] 18. Cui, Q.; Qiu, Z.; Liu, W.; Hu, Z. Complex dynamics of an SIR epidemic model with nonlinear saturated incidence and recovery rate. Entropy 2017, 2017, 1–16. 19. Nistal, R.; de la Sen, M.; Alonso-Quesada, S.; Ibeas, A. Supervising the vaccinations and treatment control gains in a discrete SEIADR epidemic model. Int. J. Innov. Comput. Inf. Control 2019, 15, 2053–2067. Entropy 2020, 22, 284 35 of 35 20. Verma, R.; Sehgal, V.K.; Nitin, V. Computational stochastic modelling to handle the crisis occurred during community epidemic. Ann. Data Sci. 2016, 3, 119–133. [CrossRef] 21. Iggidr, A.; Souza, M.O. State estimators for some epidemiological systems. Math. Biol. 2019, 78, 225–256. [CrossRef] 22. Yang, H.M.; Ribas-Freitas, A.R. Biological view of vaccination described by mathematical modellings: From rubella to dengue vaccines. Math. Biosci. Eng. 2018, 16, 3185–3214. 23. De la Sen, M. On the design of hyperstable feedback controllers for a class of parameterized nonlinearities. Two application examples for controlling epidemic models. Int. J. Environ. Res. Public Health 2019, 16, 2689. [CrossRef] 24. De la Sen, M. Parametrical non-complex tests to evaluate partial decentralized linear-output feedback control stabilization conditions for their centralized stabilization counterparts. Appl. Sci. 2019, 9, 1739. [CrossRef] 25. Abdullahi, A.; Shohaimi, S.; Kilicman, A.; Ibrahim, M.H.; Salari, N. Stochastic SIS modelling: Coinfection of two pathogens in two-host communities. Entropy 2019, 22, 54. [CrossRef] 26. Li, Y.; Cai, W.; Li, Y.; Du, X. Key node ranking in complex networks: A novel entropy and mutual information-based approach. Entropy 2019, 22, 52. [CrossRef] 27. Nakata, Y.; Kuniya, T. Global dynamics of a class of SEIRS epidemic models in a periodic environment. J. Math. Anal. Appl. 2010, 363, 230–237. [CrossRef] 28. De la Sen, M.; Alonso-Quesada, S. Control issues for the Beverton-Holt equation in ecology by locally monitoring the environment carrying capacity: Non-adaptive and adaptive cases. Appl. Math. Comput. 2009, 215, 2616–2633. [CrossRef] 29. De la Sen, M.; Ibeas, A.; Alonso-Quesada, S.; Nistal, R. On a SIR model in a patchy environment under constant and feedback decentralized controls with asymmetric parameterizations. Symmetry 2019, 11, 430. [CrossRef] 30. Buonomo, B.; Chitnis, N.; d’Onofrio, A. Seasonality in epidemic models: A literature review. Ric. Mat. 2018, 67, 7–25. [CrossRef] 31. Pewsner, D.; Battaglia, M.; Minder, C.; Markx, A.; Bucher, H.C. Ruling a diagnosis in or with “SpPin” and “SnOut”: A note of caution. Br. Med. J. 2004, 329, 209–213. [CrossRef] 32. Facente, S.N.; Dowling, T.; Vittinghoff, E.; Sykes, D.L.; Colfax, G.N. False positive rate of rapid oral fluid VIH test increases as kits near expiration date. PLoS ONE 2009, 4, e8217. [CrossRef] 33. London, W.P.; Yorke, J. Recurrent outbreaks of measles, chickenpox and mumps. I Seasonal variations in contact rates. Am. J. Epidemiol. 1973, 98, 453–468. [CrossRef] 34. Hempel, K.; Earn, D.J.D. A century of transitions in New York City’s measles dynamics. J. R. Soc. Interface 2015, 12, 20150024. [CrossRef] [PubMed] © 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).