**1. 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 [1–6]. It may be interpreted as a quantification of information loss [1–3,7–9]. 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, [10–17] and some of the references therein). There are also models whose basic framework relies on the use of entropy tools, as for instance [13–16]. It can be also pointed out that the control designs might be incorporated to some epidemic propagation and other biological problems, see, for instance, [18–27], 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.

It can be pointed out that the knowledge or estimation of the transient behavior of the infection is very relevant for the hospital management of the disease since it is necessary to manage the availability of beds and other sanitary utensils and sanitary means, in general. The work by Wang et al. in [11] pays mainly attention to the description of the transient behavior of the evolution of epidemics rather than to the equilibrium states. The main purpose in that paper was to formulate the time interval occurring between the time instant of the maximum of the infection, which gives a relative maximum of the infection evolution through time (and which zeroes the first time-derivative of the infection function), and the time instant giving its previous inflection time instant. It turns out that the knowledge of the first part of the transient evolution is very relevant to fight against the initial exploding of the illness since any eventual control intervention is typically much more efficient as far as it is taken as quickly as possible. The model proposed in [11] is a time-varying differential equation of first-order describing the infectious population which is the unique explicit one in the model. It is also pointed out in that paper that the time-varying coefficient might potentially contain the supplementary environment information to make such an equation well-posed to practically describe a concrete disease evolution. An interesting point of that work is that the infection evolution is identified with a log-normal distribution whose parameterization is selected in such a way that the entropy production rate is maximized. The above proposed theoretical first-order model has been proved to be very efficient to describe the data of SARS 2003. Alternative interpretations of the entropy in terms of maximum entropy or maximum entropy rate are given, for instance, in [12–14] and some references therein.

This paper studies how to link the extension of the first-order differential system proposed in [11] for the study of infection propagations to epidemic models with more integrated coupled subpopulations (such as susceptible, immune, vaccinated etc.) by introducing the coupling and control information through the time-varying coefficient which drives the basic differential equation model. It is considered relevant the control of the infection along its transient to fight more efficiently against a potential initial exploding transmission. Note 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 controls. See, for instance [19,27]. Therefore, special attention is paid to the transients of the infection curve evolution in terms of the time instants of its first relative maximum towards its previous inflection time instant since there is a certain gap in the background literature concerning the study of such transients. The ratio of such time instants is later on considered subject to some worst-case uncertainty relations via the calculation and analysis of an "ad hoc" Shannon's entropy. Note that entropy issues have been considered in the study of biological, evolution and epidemic models by incorporating techniques of information theory. See, for instance [11–13,28–32]. It is well-known that the entropy production theorems might be classified according to a generalized sequence of stable thermodynamic states. Also, the thermodynamic equilibrium, which is characterized by the absence of gradients of state or kinematic variables, is in a state of maximum entropy and zero entropy production [33,34]. Furthermore, linear non-equilibrium processes are associated with entropy production so that the entropy concept may be also invoked in transient processes [35]. On the other hand, it may be pointed out that uncertainties can appear in the characterization of the infection evolution through time, even in deterministic models, due to parameterization uncertainties, fluxes of populations or existing uncertainties in the initial conditions. Other mathematical techniques of interest which combine analytical and numerical issues have been also been applied to the analysis and discussion of epidemic models with eventual support of mathematical techniques on homotopy analysis and distribution functions as, forinstance, the log-normal distribution [36,37]. For instance, in [38], the SIR and SIS epidemic models are solved through the homotopy analysis method. A one-parameter family of series solutions is obtained which gives a method to ensure convergent series solutions for those kinds of models. On the other hand, in [39], the analytic solutions of an SIR epidemic model are investigated in parametric form. It is also found that the generalization of a SIR model

including births and mortality with vital dynamics might be reduced to an Abel-type which greatly simplify the analysis.

The paper is organized as follows: Section 2 gives an extension of the basic model of [11] to be then compared in subsequent sections with some existing models with several subpopulations. Such a model only considers the infection evolution through time and it is based on the action of two auxiliary non-negative functions which define appropriately the time-varying coefficient which defines the first-order differential equation of the infection evolution. The model includes, as particular case, that of the abovementioned reference where both such auxiliary functions are identical to the time argument. Particular choices of those functions make it possible to consider alternative effects linked to the basic model like, for instance, the influence on the infectious subpopulation of other coupled subpopulations in more general models like, for instance, the susceptible, exposed, recovered or vaccinated ones. It is also possible to include the control effects through such a varying coefficient, if any, like for instance, the vaccination and treatment controls. Some basic formal results are stated and proved mainly concerning with the first relative maxima and inflection time instants of the infection curve through time. The above two time instants are relevant to take appropriate control interventions to fight against an initially exploding infectious disease.

Section 3 links the basic model of Section 2 with some known epidemic models which integrate more subpopulations than just the infectious one, like for instance, the susceptible and recovered subpopulations, The time-varying coefficient driving the infection evolution is defined explicitly for each of the discussed epidemic models. Basically, it is taken in mind that some relevant information of higher-order differential epidemic models concerning the transient trajectory solution can be captured by a parameter-dependent and time-varying coefficient which drives a first-order differential equation to the light of the basic model of Section 2. So, the time-varying coefficient describing the infection evolution depends in those cases of the remaining subpopulations integrated in the model. The maximum and inflection time instants are characterized for some given examples involving epidemic models of several subpopulations. In particular, the last one of the discussed theoretical examples includes the effects of vaccination and treatment intervention controls generated by linear feedback of the susceptible and infectious subpopulations, respectively. Later on, Section 4 investigates the entropy associated with the infection accordingly to the generalizations of Section 2 concerning the specific structure of the time-varying coefficient describing the infection dynamics and its links with the theoretical examples discussed in Section 3. The error of the entropy related to the reference one associated with the log-normal distribution is estimated. In practice, that property can be interpreted in terms of public medical and social interventions which control the disease propagation when introducing the controls of the last example discussed in Section 3. The second part of Section 4 is devoted to linking the entropy and inflection and maximum infection time instants and their reached values of the discussed multi-population structures to their counterparts of the maximum dissipation rate being associated to the formulation of a simpler model based on the log-normal distribution and one-dimensional infection dynamics. Some numerical tests are performed for comparisons of the entropies and its width of the basic model with two of the discussed examples in the previous sections which involve the presence of more than one integrated subpopulations. Finally, conclusions end the paper.

*Notation*

$$\mathcal{R}\_{+} = \{ r \in \mathcal{R} \, : \, r > 0 \}; \; \mathcal{R}\_{0+} = \{ r \in \mathcal{R} \, : \, r \ge 0 \} = \mathcal{R}\_{+} \cup \{ 0 \}$$

$$\mathcal{Z}\_{+} = \{ r \in \mathcal{Z} \, : \, r > 0 \}; \; \mathcal{Z}\_{0+} = \{ r \in \mathcal{Z} \, : \, r \ge 0 \} = \mathcal{Z}\_{+} \cup \{ 0 \}$$

$$\overline{n} = \{ 0, 1, \dots, n \}$$

#### **2. 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:

$$\dot{I}(t) = \alpha(t)I(t);\ I(0) = I\_0 > 0\tag{1}$$

where α : *R*0<sup>+</sup> → *R*0<sup>+</sup> 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:

$$\begin{aligned} \ddot{I}(t) &= \dot{a}(t)I(t) + a(t)\dot{I}(t) \\ &= \left(\dot{a}(t)/a(t) + a(t)\right)\dot{I}(t) \\ &= \left(\dot{a}(t) + a^2(t)\right)l(t); \dot{I}(0) = \dot{\dot{I}}\_0 = a(0)I\_0 \end{aligned} \tag{2}$$

It is proposed in [11] to consider two relevant time instants in the disease evolution, namely:


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):

#### **Theorem 1.** *The following properties hold:*

(i) *The infection population and its two first-time derivatives obey the following time evolution equations:*

$$I(t) = e^{\int\_0^t a(\tau)d\tau} I\_0; \dot{I}(t) = a(t)e^{\int\_0^t a(\tau)d\tau} I\_0; \ddot{I}(t) = \left(\dot{a}(t) + a^2(t)\right)e^{\int\_0^t a(\tau)d\tau} I\_0; \forall t \in \mathbb{R}\_{0+} \tag{3}$$



Note that <sup>α</sup>(*t*) (respectively, <sup>α</sup>(*t*) <sup>+</sup> . α(*t*)) is infinity at *t* = 0 while it is bounded for *t* > 0, as it happens for instance with the <sup>α</sup>—function proposed in [11], then . *<sup>I</sup>*(*t*) (respectively, .. *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 [19,22–24,27,29,30,36]. 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.

#### **Corollary 1.** *The following properties hold:*


**Proof of Property (i).** Follows directly from (1)–(3). On the other hand, since α : *R*0<sup>+</sup> → *R*0<sup>+</sup> is uniformly continuous and the limit *lim t*→+∞ *t* <sup>0</sup> <sup>α</sup>(τ)*d*<sup>τ</sup> = *<sup>C</sup>* exists and it is finite then <sup>α</sup>(*t*) <sup>→</sup> <sup>0</sup> as *<sup>t</sup>* <sup>→</sup> +<sup>∞</sup> (Barbalat´s Lemma) and *<sup>I</sup>*(*t*) <sup>→</sup> *eCI*<sup>0</sup> as *<sup>t</sup>* <sup>→</sup> <sup>+</sup><sup>∞</sup> from (3), . *I* : *R*<sup>+</sup> → *R*0<sup>+</sup> is bounded, since being continuous, it cannot diverge in finite time, and . *<sup>I</sup>*(*t*) <sup>→</sup> <sup>0</sup> as *<sup>t</sup>* <sup>→</sup> <sup>+</sup><sup>∞</sup> from (1). If, furthermore, . α : *R*0<sup>+</sup> → *R*0<sup>+</sup> is uniformly continuous and, since *lim t*→+∞ *t* 0 . α(τ)*d*τ = *lim <sup>t</sup>*→+∞α(*t*) <sup>−</sup> <sup>α</sup>(0) <sup>=</sup> <sup>−</sup>α(0) then . <sup>α</sup>(*t*) <sup>→</sup> <sup>0</sup> as *<sup>t</sup>* <sup>→</sup> <sup>+</sup><sup>∞</sup> (again from Barbalat´s Lemma). Since <sup>α</sup>(*t*), . <sup>α</sup>(*t*) <sup>→</sup> <sup>0</sup> as *<sup>t</sup>* <sup>→</sup> <sup>+</sup><sup>∞</sup> then .. *I*(*t*) → 0 as *t* → +∞ from (2).

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 di*ff*erentiable at t*<sup>0</sup> ∈ *R. Then, t*<sup>0</sup> *is an undulation point (or pre-inflection point) of f if* .. *f*(*t*0) = 0.

*Inflection points of the continuous and twice-di*ff*erentiable 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 di*ff*erentiable at t*<sup>0</sup> ∈ *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).*

**Lemma 1.** *The following properties hold:*


The subsequent result has a very technical proof leading to the basic result that the zeros at finite time instants of . *<sup>I</sup>*(*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.

**Theorem 2.** *Assume that the function* <sup>α</sup> : *<sup>R</sup>*0<sup>+</sup> <sup>→</sup> *<sup>R</sup>*0<sup>+</sup> *defined by* <sup>α</sup>(*t*) <sup>=</sup> <sup>−</sup>*cln*(*g*(*t*)/*E*) *<sup>h</sup>*(*t*) *, where c* , *E* ∈ *R*<sup>+</sup> *and g*, *h* : *R*0<sup>+</sup> → *R*0<sup>+</sup> *are everywhere continuous and time-di*ff*erentiable such that g*(0) = 0 *with lim t*→0 *ln*(*g*(*t*)/*E*) *<sup>h</sup>*(*t*) ≤ −ε *for some* ε ∈ *R*0+*, and furthermore,* α : *R*0<sup>+</sup> → *R*0<sup>+</sup> *fulfills the constraints:*

$$
\alpha(D\_i) = 0; \; \dot{\alpha}(L\_i) = -\alpha^2(L\_i) \tag{4}
$$

$$\frac{\ln(L\_i)\dot{\mathbf{g}}(L\_i) - \ln(\mathbf{g}(L\_i)/E)\dot{\mathbf{h}}(L\_i)\mathbf{g}(L\_i)}{\mathbf{g}(L\_i)\ln^2(\mathbf{g}(L\_i)/E)} = K > 0; \; \forall L\_i \in L\_S \cap \left[0, \overline{L}\right] \tag{5}$$

*for any given positive real number L, with Di* <sup>∈</sup> *DS and Li* <sup>∈</sup> *LS, where DS* = *D* ∈ *R*<sup>+</sup> : α(*D*) = *g*(*D*) = *E* ⊂ *<sup>R</sup>*0<sup>+</sup> *and LS* <sup>=</sup> *<sup>L</sup>* <sup>∈</sup> *<sup>R</sup>*<sup>+</sup> : . α(*L*) + α2(*L*) = 0 ⊂ *R*0<sup>+</sup> *are assumed to be nonempty and of zero Lebesgue measure. Then, the following properties hold:*


**Proof.** First, note that . *<sup>I</sup>*(*D*) <sup>=</sup> .. *I*(*L*) = 0; ∀*D* ∈ *DS*, ∀*L* ∈ *LS* since α(*D*) = 0 even if *I*(*D*) - 0. On the other hand, *LS* is the set of undulation points of *I* : *R*0<sup>+</sup> → *R*0<sup>+</sup> and it is clear that *DS* 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 *DS* is the set of extreme points of *I*(*t*) which is disjoint to its set of undulation points *LS*. Assume, on the contrary, that there is some *<sup>D</sup> DS* 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 <sup>α</sup>(*t*) <sup>=</sup> <sup>−</sup>*cln*(*g*(*t*)/*E*) *<sup>h</sup>*(*t*) is only zero at finite time for a discrete set of time instants satisfying *<sup>g</sup>*(*t*) <sup>=</sup> *<sup>E</sup>* so that . *<sup>I</sup>*(*D*) <sup>=</sup> 0 if and only if *<sup>D</sup>* <sup>∈</sup> *DS*. Then, *<sup>I</sup>*(*D*) <sup>=</sup> . *<sup>I</sup>*(*D*) <sup>=</sup> .. *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 *LS* and *DS* are discrete sets of non-negative real time instants which can be strictly ordered. Note also from (1)–(2) that:

$$\ln(D\_i) = -\frac{\text{cln}(\text{g}(D\_i)/E)}{h(D\_i)} = 0; \; \forall D\_i \in D\_S \tag{6}$$

$$
\dot{\alpha}(L\_i) = -\alpha^2(L\_i);\ \forall L\_i \in L\_S \tag{7}
$$

If *Di* ≤ *D* < +∞ then *g*(*Di*) = *E* since *c* - 0. Also, <sup>α</sup>(*t*) <sup>=</sup> <sup>−</sup>*cln*(*g*(*t*)/*E*) *<sup>h</sup>*(*t*) and, if *Li* ∈ *LS* and since *h*(*Li*) > 0, one has:

$$\begin{split} a^2(L\_i) &= -\dot{a}(L\_i) = \frac{d}{dt} \Big[ \frac{\text{cln}(\mathcal{G}(t)/E)}{h(t)} \Big]\_{t=L\_i} \\ &= c \frac{h(L\_i)\dot{\mathcal{G}}(L\_i)/\mathcal{G}(L\_i) - \ln(\mathcal{G}(L\_i)/E)\dot{h}(L\_i)}{h^2(L\_i)} = c^2 \frac{\ln^2(\mathcal{G}(L\_i)/E)}{h^2(L\_i)} \end{split} \tag{8}$$

$$\varepsilon = \frac{\ln(L\_i)\dot{\mathbf{g}}(L\_i) - \ln(\mathcal{g}(L\_i)/E)\dot{\mathbf{h}}(L\_i)\mathbf{g}(L\_i)}{\mathbf{g}(L\_i)\ln^2(\mathcal{g}(L\_i)/E)} > 0 \tag{9}$$

Now, if there is some *Li* ∈ *LS* ∩ *DS*, equivalently, *LS* ∩ *DS* - <sup>∅</sup>, then *<sup>g</sup>*(*Li*) <sup>=</sup> *<sup>E</sup>* <sup>⇔</sup> *<sup>h</sup>*(*Li*) . *g*(*Li*) - 0 from (9) since *c* - 0 and *ln*(*g*(*Li*)/*E*) <sup>=</sup> 0 and, furthermore, one gets from (8) that . α(*Li*) - 0 since *<sup>g</sup>*(*Li*) <sup>=</sup> *<sup>E</sup>*. But one also has that . <sup>α</sup>(*Li*) <sup>=</sup> <sup>α</sup>(*Li*) <sup>=</sup> 0, since . <sup>α</sup>(*Li*) = <sup>−</sup>α2(*Li*); <sup>∀</sup>*Li* <sup>∈</sup> *LS* from the first identity of (8). Then, 0 - . <sup>α</sup>(*Li*) <sup>=</sup> 0 is a contradiction so that *Li LS* <sup>∩</sup> *DS*. Equivalently, *DS* <sup>∩</sup> *LS* <sup>=</sup> <sup>∅</sup>. Property (i) has been proved.

**Proof of Property (ii).** Since . <sup>α</sup>(*Li*) <sup>=</sup> <sup>−</sup>α2(*Li*) then .. α(*Li*) = −2α(*Li*) . α(*Li*) so that:

$$\ddot{I}(L\_i) = \left(\dot{\alpha}(L\_i) + \alpha^2(L\_i)\right) \dot{I}(L\_i) = 0$$

$$\dddot{I}(L\_i) = \left(\ddot{\alpha}(L\_i) + 2\alpha(L\_i)\dot{\alpha}(L\_i)\right) \dot{I}(L\_i) + \left(\dot{\alpha}(L\_i) + \alpha^2(L\_i)\right) \dot{I}(L\_i)$$

$$= \left(\ddot{\alpha}(L\_i) - 2\alpha^3(L\_i)\right) \dot{I}(L\_i)$$

Since the zeros of <sup>α</sup>(*t*) and those of its first time- derivative do not coincide since *DS* <sup>∩</sup> *LS* <sup>=</sup> <sup>∅</sup> (from Property (i)), it turns out that the two sets of respective zeros alternate if there are not two zeros of <sup>α</sup>(*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 . *<sup>I</sup>*(*t*) between two consecutive zeros of .. *I*(*t*), then satisfying the constraint 0 ≤ *Li* < *Di* < *Di*+<sup>1</sup> < *Li*+<sup>1</sup> for some two consecutive time instants *Di* , *Di*+<sup>1</sup> in *DS* and two consecutive time instants *Li* , *Li*+<sup>1</sup> in *LS* so that <sup>α</sup>(*Di*) <sup>=</sup> <sup>α</sup>(*Di*+1) <sup>=</sup> .. *<sup>I</sup>*(*Li*) <sup>=</sup> .. *I*(*Li*+1) = 0. Assume that *<sup>I</sup>*(*t*) <sup>=</sup> 0 for some *<sup>t</sup>* <sup>∈</sup> (*Di*, *Di*+1) then . *I*(*t*) = α(*t*)*I*(*t*) = 0 so that *t* ∈ *DS* and then *Di* , *Di*+<sup>1</sup> are not consecutive time instants in *DS* and this case has to be excluded from further reasoning. Now, assume that *I*(*t*) - 0 for all *<sup>t</sup>* <sup>∈</sup> (*Di*, *Di*+1) and . α(*t*) - 0, otherwise, if . α(*t*) = 0 then *t* ∈ *Ds* and *Di* , *Di*+<sup>1</sup> are not consecutive time instants in *DS*. Thus, <sup>α</sup>(*t*) <sup>=</sup> <sup>α</sup>(*Di*) <sup>+</sup> *<sup>t</sup> Di* . <sup>α</sup>(τ)*d*<sup>τ</sup> = *<sup>t</sup> Di* . α(τ)*d*τ for all *<sup>t</sup>* <sup>∈</sup> (*Di*, *Di*+1). Since . α(*t*) - 0 for all *t* ∈ (*Di*, *Di*+1), it has no sign change in (*Di*, *Di*+1) so that *lim t*→*D*<sup>−</sup> *i*+1 α(*t*) - 0 and since α : *R*0<sup>+</sup> → *R*0<sup>+</sup> is continuous then α(*Di*+1) -0 which contradicts that

*Di*+<sup>1</sup> ∈ *DS*. It has been proved that Case 1 is impossible 0 ≤ *Li* < *Di* < *Di*+<sup>1</sup> < *Li*+<sup>1</sup> cannot happen.

*Case 2*: Assume now that there are two consecutive zeros of .. *I*(*t*) between two consecutive zeros of . *I*(*t*), that is 0 ≤ *Di* < *Li* < *Li*+<sup>1</sup> < *Di*+<sup>1</sup> for some consecutive time instants *Di* , *Di*+<sup>1</sup> in *DS* and some two consecutive time instants *Li* , *Li*+<sup>1</sup> in *LS*. Then, α(*t*) - 0 for all *t* ∈ (*Li*, *Li*+1) since, otherwise, there exists some *t* ∈ (*Li*, *Li*+1) such that *t* ∈ *DS*, and then the previously claimed constraint <sup>0</sup> <sup>≤</sup> *Di* <sup>&</sup>lt; *Li* <sup>&</sup>lt; *Li*+<sup>1</sup> <sup>&</sup>lt; *Di*+<sup>1</sup> does not hold, and also . α(*t*) - <sup>−</sup>α2(*t*) <sup>&</sup>lt; 0 for all *<sup>t</sup>* <sup>∈</sup> (*Li*, *Li*+1) since, otherwise, there exists some *t* ∈ (*Li*, *Li*+1) such that *t* ∈ *LS* and then *Li* and *Li*+<sup>1</sup> are not two consecutive time instants in *LS* as claimed. Also, note that. .

<sup>α</sup>(*Li*) <sup>+</sup> <sup>α</sup>2(*Li*) <sup>=</sup> . α(*Li*) + α2(*Li*) = 0 with α(*Li*) - 0 and α(*Li*+1) - 0 since *Li* , *Li*+<sup>1</sup> *DS*. But then, by continuity arguments on . <sup>α</sup>(*t*) + <sup>α</sup>2(*t*), there is a change of sign point *<sup>t</sup>* <sup>∈</sup> (*Li*, *Li*+1) which zeroes this function which contradicts . α(*t*) - <sup>−</sup>α2(*t*) <sup>&</sup>lt; 0 for all *<sup>t</sup>* <sup>∈</sup> (*Li*, *Li*+1). Then, Case 2 is impossible so that 0 ≤ *Di* < *Li* < *Li*+<sup>1</sup> < *Di*+<sup>1</sup> cannot happen and Property (ii) has been proved.

**Proof of Property (iii).** Assume that, contrarily to the statement, *<sup>D</sup>*<sup>1</sup> <sup>≤</sup> *<sup>L</sup>*1. If *<sup>L</sup>*<sup>1</sup> <sup>=</sup> *<sup>D</sup>*<sup>1</sup> then . *<sup>I</sup>*(*L*1) <sup>=</sup> .. *I*(*L*1) and the equilibrium point is reached in finite time what is impossible, since *I*<sup>0</sup> > 0, for a non-trivial

solution of a continuous-time first-order differential equation with continuous-time parameterization. Then, *<sup>L</sup>*<sup>1</sup> <sup>=</sup> *<sup>D</sup>*<sup>1</sup> is impossible. Now, assume that *<sup>L</sup>*<sup>1</sup> <sup>&</sup>gt; *<sup>D</sup>*<sup>1</sup> and 0 <sup>=</sup> . *<sup>I</sup>*(*L*1) <sup>=</sup> . *<sup>I</sup>*(*D*1) <sup>+</sup> *<sup>D</sup>*<sup>1</sup> *L*1 .. *I*(τ)*d*τ = *D*<sup>1</sup> *L*1 .. *<sup>I</sup>*(τ)*d*<sup>τ</sup> with .. *<sup>I</sup>*(*L*1) <sup>=</sup> 0 and then it exists some *<sup>L</sup>*<sup>2</sup> <sup>∈</sup> (*L*1, *<sup>D</sup>*1) such that .. *I*(*L*2) = 0 and *L*<sup>2</sup> ∈ *LS*. As a result, there is *D*<sup>1</sup> > *L*<sup>2</sup> > *L*<sup>1</sup> and then there are two consecutive undulation time instants what contradicts Property (ii). As a result, *D*<sup>1</sup> > *L*<sup>1</sup> as claimed.
