On the Use of Entropy Issues to Evaluate and Control the Transients in Some Epidemic Models

This paper studies the representation of a general epidemic model by means of a first-order differential equation with a time-varying log-normal type coefficient. Then the generalization of the first-order differential system to epidemic models with more subpopulations is focused on by introducing the inter-subpopulations dynamics couplings and the control interventions information through the mentioned time-varying coefficient which drives the basic differential equation model. It is considered a relevant tool the control intervention of the infection along its transient to fight more efficiently against a potential initial exploding transmission. The study is based on the fact that the disease-free and endemic equilibrium points and their stability properties depend on the concrete parameterization while they admit a certain design monitoring by the choice of the control and treatment gains and the use of feedback information in the corresponding control interventions. Therefore, special attention is paid to the evolution transients of the infection curve, rather than to the equilibrium points, in terms of the time instants of its first relative maximum towards its previous inflection time instant. Such relevant time instants are evaluated via the calculation of an “ad hoc” Shannon’s entropy. Analytical and numerical examples are included in the study in order to evaluate the study and its conclusions.


Introduction
Some classical works by Boltzmann, Gibbs and Maxwell have defined entropy under a statistical framework. A useful entropy concept is the Shannon entropy since it is a basic tool to quantify the amount of uncertainty in many kinds of physical or biological processes [1][2][3][4][5][6]. It may be interpreted as a quantification of information loss [1][2][3][7][8][9]. On the other hand, entropy-based tools have been also proposed to evaluate the propagation of epidemics and related public control interventions (see, for instance, [10][11][12][13][14][15][16][17] and some of the references therein). There are also models whose basic framework relies on the use of entropy tools, as for instance [13][14][15][16]. It can be also pointed out that the control designs might be incorporated to some epidemic propagation and other biological problems, see, for instance, [18][19][20][21][22][23][24][25][26][27], and, in particular, for the synthesis of decentralized control in patchy (or network node-based) interlaced environments [24,27]. A typical situation is that of several towns each with its own health center, whose susceptible and infectious populations, apart from their coupled self-dynamics among their integrating subpopulations, might also mutually interact with the subpopulations of the neighboring nodes through in-coming and out-coming fluxes.

The Basic Model Description and Some Related Technical Results
Since disease propagation can be interpreted as a thermodynamic system, it can be assumed that the rate of increase or decrease is proportional to the infection at the previous day following the approach of modelling the rate of chemical reactions, [11]. Thus, assume that the infection evolution obeys the following time-varying differential equation: . I(t) = α(t)I(t); I(0) = I 0 > 0 (1) where α : R 0+ → R 0+ is continuous and time differentiable on (0, +∞). The particular structure of the varying coefficient α(t) depends on the balances between the spreading mechanism and the exerted controls during the public intervention. Such a coefficient contains the available information related to the incorporation of all the control mechanisms and the coupling dynamics between the infectious populations and the remaining interacting ones such as the susceptible, immune or vaccinated ones. By taking time-derivatives with respect to time in (1), one gets: ..
α(t) + α 2 (t) I(t); . I(0) = . I 0 = α(0)I 0 (2) It is proposed in [11] to consider two relevant time instants in the disease evolution, namely: (1) The inflection time instant of I(t) which is the date in the infection evolution at which the controlling actions take effect on the evolution. Typically, this time instant is the undulation point date in the evolution of I(t), that is the zero of .. It turns out that, along the whole disease evolution, several successive inflection points and relative maxima can happen. The subsequent result which is concerned with the non-negativity, boundedness and asymptotic vanishing property of the infection as time tends to infinity and its two first-time derivatives is immediate from the above expressions (1) and (2) . I(t) = α(t)e t 0 α(τ)dτ I 0 ;

I(t), provided that the first non-zero derivative of
..

I(t))
is still bounded under the conditions of Theorem 1 (v) (respectively, Theorem 1 (vii)) on R + .
Note also that the vanishing infection condition of Theorem 1 typically occurs under convergence of the solution to the disease-free equilibrium point if the disease reproduction number is less than one [19,[22][23][24]27,29,30,36]. However, it can happen that the infection oscillates around some stable equilibrium or that it converges to a nonzero positive constant defining the corresponding component of the endemic equilibrium steady-state as it is discussed in the next result.
Let us introduce the following definitions and lemma of usefulness for the proof of the subsequent theorem [36]: Definition 1. Let f : R → R be everywhere continuous and twice differentiable at t 0 ∈ R. Then, t 0 is an undulation point (or pre-inflection point) of f if ..
Inflection points of the continuous and twice-differentiable f : R → R are the undulation points of the function where the curvature changes its sign, that is, points of change of local convexity to local concavity or vice-versa. They are also the isolated extrema of . f : R → R . A well-known technical definition and a related result on inflection points follow: Definition 2. Let f : R → R be everywhere continuous and twice differentiable at t 0 ∈ R which is an isolated extremum of f (that is, a local maximum or minimum, and also an undulation point of, f as a result). (i) Let f : R → R be everywhere continuous and twice differentiable at t 0 ∈ R. Then, t 0 is an inflection point (ii) Let f : R → R be everywhere continuous and an odd number k(≥ 3) -times differentiable, within a neighborhood of t 0 ∈ R which is an undulation point of f satisfying f ( j) (t 0 ) = 0 for j = 2, 3, . . . k − 1 and f (k) (t 0 ) 0. Then, t 0 is an inflection point of f .
The subsequent result has a very technical proof leading to the basic result that the zeros at finite time instants of . I(t) and .. I(t) alternate if I(t) is sufficiently smooth and α(t) is sufficiently smooth. In order to simplify the result proof, it is assumed, with no loss in generality, that the disease dynamics (1)- (2) has no equilibrium points such that the zeros under study are isolated.
, where c , E ∈ R + and g, h : R 0+ → R 0+ are everywhere continuous and time-differentiable such that g(0) = 0 with lim t→0 ln(g(t)/E) h(t) ≤ −ε for some ε ∈ R 0+ , and furthermore, α : R 0+ → R 0+ fulfills the constraints: for any given positive real number L, with D i ∈ D S and L i ∈ L S , where D S = D ∈ R + : α(D) = g(D) = E ⊂ R 0+ and L S = L ∈ R + : . α(L) + α 2 (L) = 0 ⊂ R 0+ are assumed to be nonempty and of zero Lebesgue measure. Then, the following properties hold: , and (a) if card(D S ) = card(L S ) ≤ ℵ 0 (with ℵ 0 denoting the infinite cardinality of denumerable sets) then L i < D i < L i+1 ; ∀i ∈ Z 0+ for any pairs D i , D i+1 ∈ D S and L i , .
Proof. First, note that . I(D) = .. I(L) = 0; ∀D ∈ D S , ∀L ∈ L S since α(D) = 0 even if I(D) 0. On the other hand, L S is the set of undulation points of I : R 0+ → R 0+ and it is clear that D S is contained in the set of relative maximum and minimum points of I(t). The properties (i)-(iii) are now proved:

Proof of Property (i).
It is now proved that D S is the set of extreme points of I(t) which is disjoint to its set of undulation points L S . Assume, on the contrary, that there is some D D S such that . I(D) = 0. Then, I(D) = 0 since α(D) 0, and then the disease-free equilibrium point is reached in finite time contradicting the fact that is only zero at finite time for a discrete set of time instants satisfying g(t) = E so that I(D) = 0 is a disease-free equilibrium point which is reached in finite time which contradicts the given hypothesis. So, it is easy to see that L S and D S are discrete sets of non-negative real time instants which can be strictly ordered. Note also from (1)-(2) that: .
and, if L i ∈ L S and since h(L i ) > 0, one has: g(L i ) 0 from (9) since c 0 and ln(g(L i )/E) = 0 and, furthermore, one gets from (8) from the first identity of (8). Then, 0 . α(L i ) = 0 is a contradiction so that L i L S ∩ D S . Equivalently, D S ∩ L S = ∅. Property (i) has been proved.

Proof of Property
Since the zeros of α(t) and those of its first time-derivative do not coincide since D S ∩ L S = ∅ (from Property (i)), it turns out that the two sets of respective zeros alternate if there are not two zeros of α(t) within any open time interval of two consecutive zeros of . α(t) or vice-versa. One proceeds by contradiction arguments by assuming two cases which are both rebutted.
Case 1: Assume that there are two consecutive zeros of . I(t) between two consecutive zeros of ..

I(t)
, then satisfying the constraint 0 ≤ L i < D i < D i+1 < L i+1 for some two consecutive time instants D i , D i+1 in D S and two consecutive time instants L i , L i+1 in L S so that α(D i ) = α(D i+1 ) = ..
Assume that I(t) = 0 for some t ∈ (D i , D i+1 ) then . I(t) = α(t)I(t) = 0 so that t ∈ D S and then D i , D i+1 are not consecutive time instants in D S and this case has to be excluded from further reasoning. Now, assume that I(t) 0 for all t ∈ (D i , D i+1 ) and .
Case 2: Assume now that there are two consecutive zeros of ..
0 for all t ∈ (L i , L i+1 ) since, otherwise, there exists some t ∈ (L i , L i+1 ) such that t ∈ D S , and then the previously claimed constraint 0 ≤ D i < L i < L i+1 < D i+1 does not hold, and also . α(t) −α 2 (t) < 0 for all t ∈ (L i , L i+1 ) since, otherwise, there exists some t ∈ (L i , L i+1 ) such that t ∈ L S and then L i and L i+1 are not two consecutive time instants in L S as claimed. Also, note that.
α(L i ) + α 2 (L i ) = 0 with α(L i ) 0 and α(L i+1 ) 0 since L i , L i+1 D S . But then, by continuity arguments on . α(t) + α 2 (t), there is a change of sign point t ∈ (L i , L i+1 ) which zeroes this function which contradicts . α(t) −α 2 (t) < 0 for all t ∈ (L i , L i+1 ). Then, Case 2 is impossible so that 0 ≤ D i < L i < L i+1 < D i+1 cannot happen and Property (ii) has been proved. I(L 1 ) = 0 and then it exists some L 2 ∈ (L 1 , D 1 ) such that .. I(L 2 ) = 0 and L 2 ∈ L S . As a result, there is D 1 > L 2 > L 1 and then there are two consecutive undulation time instants what contradicts Property (ii). As a result, D 1 > L 1 as claimed.

Remark 1.
In Theorem 2, note that the sets D S and L S have the following properties: They are nonempty so that there is at least one D ∈ D S such that α(D) = 0 implying that . I(D) = 0 and at least one L ∈ L S such that . α(L) = −α 2 (L) implying that .. I(L) = 0. Otherwise, the infection could converge asymptotically to zero as time goes to infinity but it would not have finite zeros, They are sets of zero Lebesgue measure so that they are denumerable discrete sets of strictly ordered isolated real points, for any real numbers, They fulfill that cardL S = cardD S + ϑ with ϑ = {0, 1} so that they are of either identical finite or infinite cardinal or the cardinal of L S is finite and exceeds that of D S by one, If ϑ = 0 then card(D S ) = card(L S ) ≤ ℵ 0 , that is, if both sets have infinity cardinal or identical finite one then any ordered points of L S and D S alternate.
On the other hand, note that: Equation (4) establishes that D S is the set of zeros of α(t). At those zeros, the first-time derivative of the infection function is zeroed from (1) without such a function being necessarily zero while on the other hand, Equation (5) is a nonzero real constant for any finite undulation time instant L i ≤ L of I : R 0+ → R 0+ zeroing the second derivative of the infection function according to (2) which holds if c = K from (5). The fact that (5) is constant follows easily under periodicity conditions of the same or integer multiple/submultiple periods of g(t) and h(t).
Since α : R 0+ → R 0+ has no finite zero coincident with a zero of its first time-derivative, by hypothesis, then g(L i ) = E ⇔ h(L i ) . g(L i ) 0 since c 0 from inspection of (8)- (9). This is equivalent to D S ∩ L S = ∅, that is, the finite zeros which make zero . I(t) and which do not make zero I(t) do not make zero either .. I(t) = 0 from (2), provided that α : R 0+ → R 0+ is twice everywhere continuously differentiable in [0, +∞) but this can only happen as time tends to infinity for certain structures of g(t) and h(t). Note that the constraint (5) also implies that the auxiliary functions g, h : R 0+ → R 0+ used to define the function α : R 0+ → R 0+ in (1) fulfill the constraint h(L i ) . g(L i ) ln(g(L i )/E) . h(L i )g(L i ); ∀L i ∈ L S . By examining Definitions 1 and 3 and Lemma 1, it turns out that the set L S of undulation points of I(t) includes but, maybe non-properly, the set of its inflection points. However, it suffices to give some further weak conditions on α : R 0+ → R 0+ , that is, on g, h : R 0+ → R 0+ to guarantee that every undulation point of I(t) is also an inflection point. Some such conditions are discussed in the next corollary.

Corollary 2.
The following properties hold: where: . h(t)ln(g(t)/E); ∀t ∈ R 0+ Then, the set L S of undulation points of I(t) is the set of its inflection points. (ii) Assume that f , g : R 0+ → R 0+ are twice continuously differentiable at each undulation point L i ∈ L S . Then, the sets of undulation points and that of the inflection points of I(t) coincide if Proof. Note that .. ..
On the other hand, if f , g : R 0+ → R 0+ are twice continuously differentiable at each undulation ; ∀t ∈ R 0+ yields: .

Remark 2.
Note that Theorem 2 applies, in particular, to the case when there are equilibrium points with the initial conditions being distinct from such points. It can be also extended by including the above case by redefining finite discrete sets of the zeros of .

I(t) and
.. I(t) alternate although an equilibrium points has not still been reached provided that it exists.
Inspired in Theorem 2, some conditions are discussed in the next result which imply that the first undulation point of the infection evolution function (i.e., the first zero of its second-time derivative) precedes the first zero of its first time-derivative. It is not required that the infection has necessarily a disease-free equilibrium point or that it might be oscillatory leading to successive zeros of its timederivative along time.
, where c , E ∈ R + and g, h : R 0+ → R 0+ are everywhere continuous and time-differentiable and satisfy the constraints: Assume also that I 0 > 0. Then, min I(t), .

I(t).
Assume that this is not the case so that there is some (2) and the infection extinguishes in a finite time D 1 < D. This leads to a contradiction since The following example describes the basic model proposed in [11] under a first-order differential equation for the infection evolution without any entropy considerations at this stage: [11] satisfies all the conditions of Theorem 3 with h(t) = g(t) = t and E = D. It satisfies, in addition, that α(0) = +∞. This function satisfies also the given further conditions of Theorem 2 Note that the condition α(0) > 0 of Theorem 3 avoids that It can be argued that the proposed basic model (1) is a very simple time-varying differential equation of first-order which describes the infective population time-evolution. Note that the use of appropriate particular structures in the definition of the time-varying coefficient α(t) can take care of the eventual incorporation of the necessary supplementary environment information to make such an equation well-posed to practically describe a concrete disease evolution through time. The incorporation which can be incorporated is the eventual couplings of the infectious subpopulation with another ones (such as the susceptible, recovered or vaccinated subpopulations and their associated dynamics) or the information about the feedback information controls in more elaborated models. The next section develops some work in this direction.

Further Examples of Linking the Basic Model to Some Existing Epidemic Models Incorporating Other Subpopulations
The infection description via (1) assumes implicitly that it has a first-order dynamics. It has been argued that α(t) in (1) contains the information about the controls and other coupled subpopulations influencing the disease evolution through time. It can be of interest to discuss its application to infection descriptions described by differential equations of orders higher than one which is a very common situation in disease transmission mathematical models.
It is now seen how a well-known epidemic model can be also discussed under the point of view of Theorem 3. In the subsequent example, the above characterization, based on the first zero of infection evolution time-derivative and on the undulation point of the infection evolution, is used for a model with three subpopulations via an appropriate choice of g(t) and h(t) in the definition of α(t).
where S(t), I(t) and R(t) are, respectively, the susceptible, infectious and recovered (or immune) subpopulations, under nonzero initial conditions being subject to min(S(0), I(0), R(0)) ≥ 0, where β is the coefficient transmission rate and γ is the removal or recovery rate (its inverse γ −1 being the average infectious period).
The mathematical study of this model and their variants is not easy as seen in [30,40]. First, note that the total population I 0 > 0. The solution of (10) becomes in closed form: Note that by combining the above equations that: Note from (11) that S : R 0+ → R 0+ is non-increasing so that there exists a susceptible equilibrium subpopulation S e = lim t→∞ S(t) ≤ S 0 for any given non-negative initial conditions. Note also from for some λ a > 0 from (10) and (11) is non-increasing and then it converges to S e satisfying 0 ≤ S e ≤ S 0 . By inspection of the second equation of (11), it also follows that I(t) → I e and R(t) → R e as t → ∞ satisfying I e ≥ 0 and R e ≥ 0. Assume that I e > 0 then S e = 0 from the first equation of (11). But if S e = 0 then I e = 0 since then I : R 0+ → R 0+ is strictly decreasing on [t a , ∞) for some finite t a > 0 from the second equation of (11). Hence, a contradiction to I e > 0 follows implying that I e = 0 if S e = 0. Now, assume that γ/β > S e > 0. Then, from the second equation of (11), I(t) → I e = 0 as t → ∞ . But then S e > 0, from the first equation of (12), since γ/β > S e if I 0 > 0 and then R e = N 0 − S e . From the second equation of (12) and, under a similar reasoning as that of Case a, I : The fact that I : R 0+ → R 0+ is strictly increasing on some initial time interval is of interest from the point of view of hospital management of availability of beds and other sanitary specific means in the event that the disease might have a relevant number of seriously infected individuals. Since S : R 0+ → R 0+ is non-increasing then either . Then, from the first equation of (12), S(t) → 0 as t → ∞ . Then, S e = 0 which contradicts that S e > γ/β , As a result, 0 ≤ S e ≤ γ/β . Now, assume that S e = 0. Then, from (11), I(t) → I e = 0 and I : R 0+ → R 0+ being square-integrable, and following a similar argument as that of Cases a-b, one again concludes that S e > 0 so that S e ∈ (0, γ/β ] and R e = N 0 − S e , as a result. But, since S e ≤ γ/β then I e = 0 from (11) since I : R 0+ → R 0+ is strictly decreasing after some finite time instant t 0 and integrable on [0, ∞) and a following again the reasoning of Cases a-b, one concludes that S e > 0. As a result, if S 0 > R −1 * and I 0 > 0, then I e = 0, S e > 0 and R e = N 0 − S e . Thus, the relevant conclusions on the disease-free equilibrium point which is a disease-free one are similar for the three above cases.
On the other hand, since S : .
I 0 > 0 under the reasonable assumption that I 0 is sufficiently small (the initial numbers of infectious is usually very small in practice) satisfying I 0 < As a result, we find that if the basic reproduction number exceeds unity then the infection curve corresponding to the endemic solution has a minimum at a larger time instant that the one defining its undulation point. That situation corresponds to the situation of small initial infection force with reproduction number greater than one. On the other hand, if ..
Comparing the infectious subpopulation evolution to (1) and the structure of the function in Theorem 3 yields: . ∀t ∈ R 0+ . If one defines g(t) = t; ∀t ∈ R 0+ and h(t) = It is easy to verify that these functions satisfy the conditions of Theorem 3.
In the case when the reproduction number is less than unity and it is an upper-bound of the normalized susceptible population, each primary infection generates, in average, less than one secondary one so that the infection extinguishes asymptotically. According to this particular model, also the susceptible subpopulation extinguishes asymptotically. See Case a referred to (11). Thus, ..
I(t) → 0 as t → ∞ but there are no finite time instants of minimum and undulation of the infectious curve to the light of Theorem 3. However, we can have a practical visualization of the disease removal by defining a design quadruple (k 1 , k 2 , k 3 , ε) ∈ R 4 + and the following cut associate time instants: Note that t I2 (k 2 , ε) and t I3 (k 3 , ε) generalize the roles of the time instants D and L, that is, the finite minimum infection and undulation time instants, respectively, within prescribed margins when those time instants do not exist.
The solution trajectory converges to the disease-free equilibrium point at exponential rate. Then, one gets by combining (10)-(12) and (18) that: implying that: which leads to: and: − k 3 ε ≤ ..
Consider the following SIS model with vaccination and antiviral or antibiotic controls: . .
subject to S(0) = S 0 , I(0) = I 0 with min(S 0 , I 0 ) ≥ 0 where the vaccination and treatment feedback controls on the susceptible and infectious are, respectively, V(t) = k V S(t) and T(t) = k T I(t) with min(k V , k T ) ≥ 0.
If it is assumed that the total population N(t) = N 0 = S 0 + I 0 ; ∀t ∈ R 0+ is constant through time then there is a complementary recovered (or immune) subpopulation present which obeys the differential equation The solution is: The following result links the above SIS model with a complementary recovered subpopulation to the generic one (1) under a minimum number of initial susceptible and sufficiently large number of initial infectious with initial growing rate.
is strictly increasing on [0, t Imax ], and I max = I(t max ) = max(I(t) : t ∈ [0 , t Imax ], t Imax = min(t ∈ R 0+ : S(t) = (γ + k T )/β)) with t Imax ≥ t Smin , (iv) There is t und < t Imax which is an undulation and, furthermore, strict inflection time instant of I(t), (v) Assume, in addition, that I 0 is large enough to satisfy I 0 > Then, the epidemic model (26) can be written in the form (1) on [0, t Imax ] with the following function α : [0, t Imax ] → R 0+ : which is of the form α(t) = . Property (i) has been proved.
and I(t) has a relative maximum I max at t = t = t Imax which is also the absolute maximum on [0, t max ]. Property (iii) has been proved. Note also that since ..

I(t) is continuous and
..
I 0 > 0, there exists some t und < t such that t und is an undulation point of I(t). Note furthermore that .. ..

I(t und − ε)
.. I(t und + ε) < 0; ∀ε ∈ B(0, r) and some r ∈ R + then t und is in fact an inflection time instant of I(t). Assume that there is some arbitrarily small ε ∈ R + such that ..

I(t und + τ)dτ
It is known that 0 < ε I ≤ . I(t und ) < . I 0 so that, for some arbitrarily small ε ∈ R + such that ..
such that the following joint constraints hold: . ..
Example 4 is tested numerically in the sequel with the following data β = 30, γ = 50 years −1 , implying that the average infectious period is T γ = 365/50 = 7.3 days, k V = 1 and k T = 50. The time scale of the figures is in a scale of years accordingly with the above numerical values. In Figure 1, the solution trajectories of all the subpopulation are shown with the constraints of Theorem 4 being fulfilled by the initial conditions, in particular S 0 > γ+k T β , I 0 = 1 − S 0 and R 0 = 0 so that N 0 is normalized to unity. It is seen that the infectious subpopulation trajectory has a maximum at a finite time and that the state trajectory solution converges asymptotically to an endemic equilibrium point. In Figure 2, the state trajectory solution is shown with N 0 = 1 when S 0 = (γ + k T )/β which violates the conditions of Theorem 4 with . I 0 = 0. In this case, there is no relative maximum of the infectious subpopulation at finite time. In both situations, it has been observed by extending the overall simulation time that the susceptible and the infectious subpopulations converge asymptotically to zero while the recovered subpopulation converges to unity as time tends to infinity. The controls are suppressed in Figure 3 with N 0 = 1. In this case, the recovered subpopulation may be deleted from the model since it is unnecessary while being identically zero. The infectious and susceptible subpopulations are in an endemic equilibrium point for all time so that the infection results to be permanent in the sense that it cannot be asymptotically removed. See Theorem 4(vi) for the case k V = 0. Figure 4 exhibits a trajectory solution which agrees with Theorem 4 while there is no normalization of the initial conditions to unity. In this case, the maximum of the infectious subpopulation at a finite time becomes very apparent. 0 infectious subpopulation at finite time. In both situations, it has been observed by extending the overall simulation time that the susceptible and the infectious subpopulations converge asymptotically to zero while the recovered subpopulation converges to unity as time tends to infinity. The controls are suppressed in Figure 3 with N0 = 1. In this case, the recovered subpopulation may be deleted from the model since it is unnecessary while being identically zero. The infectious and susceptible subpopulations are in an endemic equilibrium point for all time so that the infection results to be permanent in the sense that it cannot be asymptotically removed. See Theorem 4(vi) for the case kV= 0. Figure 4 exhibits a trajectory solution which agrees with Theorem 4 while there is no normalization of the initial conditions to unity. In this case, the maximum of the infectious subpopulation at a finite time becomes very apparent.

Comparison of the Epidemic Model and Reference Model Information Entropies
Since (1)

Comparison of the Epidemic Model and Reference Model Information Entropies
Since (1) is a scalar equation, a valid solution for the particular model-dependent time-varying coefficient α(t) = −cln(g(t)/E)/h(t) of Theorem 2 and Theorem 3 is, according to Theorem 1: Under the particular constraints E = D, c = (1 − ln(L/D))/ln 2 (L/D) and g(t) = h(t) = t, it is got in [11] that α(t) = (ln(L/D) − 1)/ln 2 (L/D) t −1 ln(t/D) and (32), namely: approaches the log-normal distribution: for reference values D = D r and L = L r of the maximum and inflection reference time instants where µ r = lnD r + σ 2 r and σ r is given by the principle of extreme entropy production rate, typically σ r ≈ 0.408 gives the width of the distribution function for the maximum dissipation rate for the usual definition of the Shannon entropy. The main reason for the limitation of such a width is that the medical and social interventions are a dissipation mechanism which controls and limits the disease propagation. Comparing (33) and (34), one gets that k = √ 2π σ 3 r I 0 after solving the indetermination 0/0 at t = 0 via L´Hôpital rule leading to the "infection reference evolution" I r (t) = I p (t), that is by equalizing (23) and (24), under the above set of particular constraints, where: Now, equalize I(t) = I r (t) + I(t); ∀t ∈ R + for some perturbation function I : R + → R 0+ resulting to be from (32) and (35) for I 0 > 0: The Shannon entropy of the infection S I (η) results to be given by the following Riemann-Stieljes integral which quantifies the entropy error S I (η) of that associated with any given model related to the entropy of the "infection reference evolution" given by the log-normal function S I r (η) = S I r (η, σ r ) for the given reference width value σ r = 1/2η: Entropy 2020, 22, 534 19 of 31 after using I(t) = I r (t) 1 + I −1 r (t) I(t) and its equivalent expression I(t) = −I r (t) 1 − I −1 r (t)I(t) , where the reference entropy based on the identification of the log-normal function (34)with the solution of (1), that is, (33), yields for σ r = 1/2η: after converting the Riemann-Stieljes integral (39) in a Riemann integral via differentiation of dt η by using (35). Note that it is assumed that both current and reference entropies are evaluated for the same parameter η which is typically chosen as η = 3. At the same time, it is assumed that the maximum dissipation rate proportional to the maximum rate of entropy production is governed by the width of the distribution function σ. So the current model can potentially have a value σ σ r . See [11] for the normalized case obtained for I 0 = 1, and, also one gets the following entropy error: It turns out obvious that the integrand of (39) is identically zero if I(t) ≡ 0, so that I(t) ≡ I r (t), leading to S I (η) ≡ 0. The expression (37), subject to (38)-(39), parameterizes the incremental entropy with the same parameter η which parameterizes the reference entropy S I r ( η r ). Now, define the error: so that S I (η) ≡ 0 if δ(t) ≡ 0 and, expanding ln I(t) I(t) I r (t) Ir(t) via the Newton-Mercator series for the logarithm, leads to: and such a series converges to ln(1 + δ(t)) for all t ∈ R 0+ provided that δ(t) ∈ (−1, 1], equivalently, I(t) ∈ (−I r (t), I r (t)]; ∀t ∈ R 0+ ; ∀t ∈ R 0+ . Thus, the following description in linear and higher-order additive terms of the entropy error follows from (40)-(41) into (39): where: The subsequent results hold related to the case when the error between the infectious functions of the model and the reference one associated to the log-normal function converges asymptotically to zero as time tends to infinity. The first result, stated separately by convenience concerned its proof, discusses the simplest case for η = 1.  I r (τ) Irτ t (η−1)(Ir(ς)−I(τ)) dτ < +∞.
Proof. Note that S I (1) < +∞ and that, from the uniform continuity of everywhere in R 0+ , the boundedness of its integral on [0, ∞) and Barbalat´s lemma, it follows that I(t) I(t) I r (t) Ir(t) t (η−1)(Ir(t)−I(t)) → 1 as t → +∞ what implies that: If η > 1 and lnt → ∞ as t → ∞ then there exists some strictly increasing real sequence can hold only if lim k→+∞ I(t k )lnI(t k ) − I r (t k )lnI r (t k ) = +∞. But, since I r : R 0+ → R 0+ is bounded for all time, this implies that I(t k ) → +∞ as t k ∈ {t i } ∞ i=0 → +∞ and I : R 0+ → R 0+ is unbounded. But then and a contradiction follows to the above limit to be zero. As a result, lim t→+∞ (I(t) − I r (t)) = 0 if η > 1.
It is now briefly discussed the fact that the boundedness hypothesis of Proposition 2 is not very restrictive for some of the given examples, like for instance, Examples 2,3, where the infectious subpopulation converges asymptotically to zero. For such a purpose, note from (35) that I r (t) → 0 exponentially fast as t → ∞ . In example 2, I(t) → 0 exponentially as t → ∞ so their difference function also converges to zero exponentially as t → ∞ . The integral boundedness invoked in the assumption of Proposition 2 is of the form F = I r (t) Ir(t) t (η−1)(Irt−I(t)) is everywhere differentiable with respect to time. In order to convert the elevant Riemann-Stieljes integral into a standard Riemann one, take dx = .
x(t)dt and, later on, perform the change of variable x → u defined by u = lnx, du = dx/x to yield: x(t) x(t) = 0 for any t ≥ 0 if and only the following constraint

z(t)
∅ is an event of zero probability. Thus, the boundedness hypothesis of Proposition 2 happens almost surely in the event that the infectious subpopulation converges asymptotically to zero as time tends to infinity.
Concerning Proposition 3, note that the boundedness of S I ( η) does not guarantee that the linear part and the remaining part of higher-order terms in the decomposition of (42), subject to (43) and (44), are both finite. It could "a priori" happen that they both tend to infinity with opposite signs. But if any of them is bounded, the other one should be bounded as well according to Proposition 3. Fortunately, this does not happen under weak extra assumptions. In particular, the following result holds: Proposition 4. Assume that η ∈ R 0+ , I, I r : R 0+ → R 0+ are bounded, and Then, S IL (η) < +∞; t ∈ R 0+ .

Proof.
It is direct to see that S IL (η) < +∞. Also, and again from Barbalat´s lemma,  I r (t) Ir(t) t (η−1)(Ir(t)−I(t)) dt < +∞ then, again from Proposition 3, S I (η) < +∞ and Note that the above results agree with the asymptotic results of Examples 1-4, where I(t) → 0 as t → ∞ , and with Theorem 1, since the reference I r (t) → 0 , jointly implying (I(t) − I r (t)) → 0 as t → ∞ . (1) is to reduce the higher-order epidemic model with two or more states to a single-order differential equation based on the assumption that the log-normal distribution is a sufficiently accurate model for the infectious evolution. It is apparent that the profile of the log-normal distribution remembers the behavior of the strong infections in their blowing-up evolution phase along time. However, it is obvious that the epidemic models have the concourse of several coupled subpopulations so that it the model is reduced to a first-order dynamics the influence of the remaining dynamics should be accounted for through a time-varying parameterization and dynamics uncertainty in (1) since the model order is reduced to unity. The accuracy of the modeling procedure is evaluated by means of the entropy through (37). Hence if the actual infectious population curve is close to the reference one, then we have S I (η) = S I r (η) which generates the dissipation rate of the model. On the other hand, if the current system differs from the reference model, then the entropy becomes corrected with the additional term S I (η). Therefore, the contributing terms in (37) provide an estimation of the modeling uncertainty based on the assumed log-normal reference distribution. As a result, the best approximation of the current model to the reference one is that which minimizes the error entropy S I (η), i.e., the one which reduces as much as possible the uncertainty introduced by the approximation.
That is, the inflection point of the truncated entropy occurs at the relative extreme values of I(t). In particular, if the infection is in its first expanding phase, this occurs at its maximum t = D.

Estimation of Errors of the Distribution Widths between the Log-Normal Reference and Current Model Information Entropies
One gets from (38) for the usual reference entropy definition based on the log-normal distribution of width σ r = 1 √ 2η , [11,33,37], that: and the particular value: is the width distribution maximum value which makes the reference entropy to cease to increase while giving the maximum dissipation rate which leads to: Note that the above reference description is easily associated to an epidemic model given by a first-order differential equation involving only the infection evolution. Note also, in particular, that the infection curve solution is of exponential order as it is the log-normal function. Such an order is negative if the disease-free equilibrium point is globally asymptotically stable (that is, the reproduction number is less than one) so that the infection converges exponentially to zero. In other words, the curves (43) and (44) can be reasonably identified with each other as it has been made in the above subsection by considering the influence of the initial conditions. In more sophisticated models involving the concourse of more subpopulations (say susceptible, immune, etc.), like those discussed in the above section, the differential equation is of higher-order than one so that the α(t) -function describing the time evolution of I(t) depends on the remaining subpopulations. This translates into the following facts: (1) Fact 1: It is known that, for η = 3, σ r = 1 6 ≈ 0.408; D r L r = 1.649; f (L r ) = 2.120, [11]. (2) Fact 2: A modification of the relevant time instants D and L of maximum infection and previous inflection point with respect to D r and L r , and the corresponding entropies as it has been discussed analytically in Section 4.1. Those parameters depend on each particular model. This also will translate, as a result, into a change of the distribution width σ related to the reference width σ r for the maximum dissipation concerns. I(t), respectively which also lead directly to their corresponding rates. (4) Fact 4: The entropy of the current model might be interpreted in terms of the maximum dissipation rate by assuming a description via a log-normal distribution. However, it is easy to verify that the log-normal function is zeroed as its argument is either zero or +∞, although its profile is close, but not identical, to the solution of a first-order differential equation describing a decaying exponential infection evolution towards a disease-free equilibrium point. For this reason, and having in mind the comparison of the solution of models with more than one subpopulation (with associated differential system of order larger than one) to the log-normal distribution f (t) which is zero at zero and at infinity and which satisfies ∞ 0 f (t)d(t) = 1, we first normalize the infectious subpopulation of the current model in order to get a comparable entropy to the reference one associated with the log-normal function, that is, we define:

Some Numerical Tests on Reference and Current Model Entropies
Now Example 2 and Example 4 are compared to the infection study of [11], by introducing the appropriate tools of normalized infection entropy (48) associated with the maximum dissipation rate for the choice η = 1. Recall the basic notation D r , L r , D and L being the first time instants such that 1/2 that the parameterized reference entropy is: and one gets for Example 2 that its associated normalized entropy for η = 1 being un-parameterized in (D, σ) becomes from (48): Numerical experimentation with Example 2: Note that D is the first time instant such that . I(D) = 0 and I(D) is a relative maximum, which in practice, gives the maximum expected infectious numbers. Also, L is the first time instant such that .. I(L) = 0. Note also that the basic model, of response being close to a log-normal function, has only an infectious subpopulation while the examples of Section 3 have more subpopulations integrated in the models. Therefore, the reasonable condition that the initial conditions of the infectious subpopulation are the one percent of the total population, we consider a total population of N 0 = S 0 + I 0 + R 0 = 1 for Example 2 in order to get a feasible comparison.
Thus, we perform several alternative experiments as follows: (a) We get the values of the time instants D and L and the corresponding infection numbers I(D) and I(L), from the solution trajectory of Example 2 and its first two-time derivatives trajectories through time, as well as the normalized entropy S I n (1) from (50). Later on, by equalizing (50) to (49), one then gets the value of D rm which specifies the time instant given a maximum infectious subpopulation with a maximum dissipation rate in a log normal distribution. This equalization yields: We equalize again (49) by fixing D r = D in (50). Then, we get the necessary value σ rm for such an equality to hold. (c) We define the variance with distribution function I n (t) and log-normal distribution resulting to be: where ω(t) = I n (t) or ω(t) = x(t, D r , σ r ), the log-normal distribution. Then, we obtain the necessary σ rmv = σ rmv (var(I n ), D) got from var(I n ) = var(x(D r = D, σ rmv )) One observes that, in general, σ rmv σ r = 1  (51) is also plotted. The corresponding infectious subpopulations are displayed in Figure 6. Figure 7 gives the entropies of (50) and (49). On the other hand, Figure 8 displays σ rm , σ rmv and the variance of the normalized infectious I n (t) of (52). It is basically concluded that for the model of example 2 which has three subpopulations, the results are distinct from to those obtained from the log-normal distribution which we can recall that behave closely to the solution of a first-order differential equation involving the infectious only for initial infection being close to zero and small susceptible amounts. The above discrepancy increases as the quotient S 0 /I 0 increases. The reason of the approximation discrepancy is that the couplings of the infectious subpopulations with the remaining ones becomes increasingly relevant to the transient responses evolution as the proportion of susceptible to infectious increases.  (51) is also plotted. The corresponding infectious subpopulations are displayed in Figure 6. Figure 7 gives the entropies of (50) and (49). On the other hand, Figure 8 displays rm  , rmv  and the variance of the normalized infectious   t I n of (52). It is basically concluded that for the model of example 2 which has three subpopulations, the results are distinct from to those obtained from the log-normal distribution which we can recall that behave closely to the solution of a first-order differential equation involving the infectious only for initial infection being close to zero and small susceptible amounts. The above discrepancy increases as the quotient increases. The reason of the approximation discrepancy is that the couplings of the infectious subpopulations with the remaining ones becomes increasingly relevant to the transient responses evolution as the proportion of susceptible to infectious increases.            Numerical experimentation with Example 4: The initial values satisfy a normalization constraint N 0 = S 0 + R 0 = 1 with subpopulations S 0 = 0.99, i o = 0.01 (that, is the initial infectious subpopulation is 1% of the total one) and R 0 = 0 since the recovered populations is compensatory in the model in order to take into account the effects of the intervention controls. The parameters β and γ are fixed as in Example 2. In particular, Figures 9 and 10 show the maximum infection and its previous value at the inflection time instant and the corresponding time instants without vaccination and with a vaccination effort rate of k T = 290 for different values of the vaccination control gain. It is basically seen that the maximum and inflection amounts decrease as the treatment control gain gives a skip from zero to an important effort as that, in parallel, the above values also decrease as the vaccination control gain increases. Figures 10 and 11 describe parallel experiments where the roles of the vaccination and treatment control gains are reversed with respect to the data of Figures 9 and 10. The obtained conclusions are similar. The time instants of maximum infection and the inflection value are reached without and with vaccination control as the treatment control effort increases for Example 4 are plotted in Figure 12. The corresponding entropies for those to experiments compared to the reference entropy are displayed in Figures 13 and 14. Note that the entropies (48) and (50) reach negative values because of the normalization of the infection by the total infection integral contribution (48) used to evaluate the normalized entropy (50). Note that the vaccination control does not affect to the entropy as significantly as the treatment control gains since it influences less significantly to the model dynamics.
are reached without and with vaccination control as the treatment control effort increases for Example 4 are plotted in Figure 12. The corresponding entropies for those to experiments compared to the reference entropy are displayed in Figures 13 and 14. Note that the entropies (48) and (50) reach negative values because of the normalization of the infection by the total infection integral contribution (48) used to evaluate the normalized entropy (50). Note that the vaccination control does not affect to the entropy as significantly as the treatment control gains since it influences less significantly to the model dynamics.   Example 4 are plotted in Figure 12. The corresponding entropies for those to experiments compared to the reference entropy are displayed in Figures 13 and 14. Note that the entropies (48) and (50) reach negative values because of the normalization of the infection by the total infection integral contribution (48) used to evaluate the normalized entropy (50). Note that the vaccination control does not affect to the entropy as significantly as the treatment control gains since it influences less significantly to the model dynamics.

Conclusions
This paper has investigated the extensions of a first-order differential system describing the infection propagation through time to epidemic models integrating more than one subpopulation. The main involved tool has been the consideration of the coupling of inter-populations dynamics and the control intervention information through the structure of the time-varying coefficient which drives the basic differential equation model of first-order. The control of the infection along its transient to fight more efficiently against a potential initial exploding transmission from a high initial growth rate is considered relevant. Special attention has been paid throughout the manuscript to the discussion of the profiles of the transients of the infection curve in terms of the time instants of its first relative maximum towards its previous inflection time instant, so the study is mainly focused on the transient behavior characterization rather than on the steady-state equilibrium points. The

Conclusions
This paper has investigated the extensions of a first-order differential system describing the infection propagation through time to epidemic models integrating more than one subpopulation. The main involved tool has been the consideration of the coupling of inter-populations dynamics and the control intervention information through the structure of the time-varying coefficient which drives the basic differential equation model of first-order. The control of the infection along its transient to fight more efficiently against a potential initial exploding transmission from a high initial growth rate is Entropy 2020, 22, 534 29 of 31 considered relevant. Special attention has been paid throughout the manuscript to the discussion of the profiles of the transients of the infection curve in terms of the time instants of its first relative maximum towards its previous inflection time instant, so the study is mainly focused on the transient behavior characterization rather than on the steady-state equilibrium points. The time instants leading to the maximum infection and inflection numbers have been investigated via the Shannon´s information entropy for the maximum dissipation rate linked to a previous background study for a first-order differential equation describing the infection propagation. Since it is relevant to know the time instants of maximum infection and inflection as well as its numbers in order to monitor the availability of hospitalization resources, some examples related to existing epidemic models integrated by more than a subpopulation have been studied. The obtained results have been compared, both via theoretical work and also by numerical experimentation, to the background results obtained from a reference model, just involving a single infectious population, which is based on a description via a log-normal distribution which has a close profile to the solution response of a first-order differential equation. In those examples, special attention is paid to the comparisons of the maximum infection and inflection time dates for different values of initial conditions and to the entropy discrepancies related to the reference one. It can be concluded that the influence of the couplings of the dynamics of other subpopulations in the model to the infectious one is relevant to the infection evolution, especially, in the cases when the initial amounts of the susceptible are significantly large compared to the initial amounts of the infectious.