Research Article on an Sir Epidemic Model for the COVID-19 Pandemic and the Logistic Equation

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Hindawi Discrete Dynamics in Nature and Society Volume 2020, Article ID 1382870, 17 pages https://doi.org/10.1155/2020/1382870

Research Article On an Sir Epidemic Model for the COVID-19 Pandemic and the Logistic Equation

Manuel De la Sen 1 and Asier Ibeas2

1Institute of Research and Development of Processes IIDP, Department of Electricity and Electronics, University of the Basque Country, Campus of Leioa, Leioa (Bizkaia), P.O. Box 48940, Spain 2Department of Telecommunications and Systems Engineering, Universitat Auto`noma de Barcelona, UAB, 08193-Barcelona, Spain

Correspondence should be addressed to Manuel De la Sen; [email protected]

Received 24 July 2020; Revised 10 September 2020; Accepted 18 November 2020; Published 9 December 2020

Academic Editor: Leonid Shaikhet

Copyright © 2020 Manuel De la Sen and Asier Ibeas. *is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*e main objective of this paper is to describe and interpret an SIR (Susceptible-Infectious-Recovered) epidemic model though a logistic equation, which is parameterized by a Malthusian parameter and a carrying capacity parameter, both being time-varying, in general, and then to apply the model to the COVID-19 pandemic by using some recorded data. In particular, the Malthusian parameter is related to the growth rate of the infection solution while the carrying capacity is related to its maximum reachable value. *e quotient of the absolute value of the Malthusian parameter and the carrying capacity ﬁxes the transmission rate of the disease in the simplest version of the epidemic model. *erefore, the logistic version of the epidemics’ description is attractive since it oﬀers an easy interpretation of the data evolution especially when the pandemic outbreaks. *e SIR model includes recruitment, demography, and mortality parameters, and the total population minus the recovered population is not constant though time. *is makes the current logistic equation to be time-varying. An estimation algorithm, which estimates the transmission rate through time from the discrete-time estimation of the parameters of the logistic equation, is proposed. *e data are picked up at a set of samples which are either selected by the adaptive sampling law or allocated at constant intervals between consecutive samples. Numerical simulated examples are also discussed.

1. Introduction available which consider the spatial-temporal spread of the disease. For instance, several neighboring domains with Epidemic mathematical models under different formal diversity of populations are considered in [23] concerning frameworks are of major interest along the last years [1, 2]. the HIV/AIDS spread. It is considered that different levels of Some of such models rely on the dynamic aspects of the awareness are present in different groups which might affect, disease evolution and are stated in terms of differential, for instance, to the hardness of the actions to be taken for difference, and differential/difference hybrid equations, transmission prevention. In [24], a Latin hypercube sam- dynamic systems, and/or control theory (so as to deal with pling method is discussed to calculate relevant reproduction vaccination and treatment intervention strategies) [3–18], numbers from distributions of some relevant model-related computation tools [7], information theory, [19–22], etc. On parameters. the contrary, some of the above invoked modelling tech- It can be pointed out that continuous-time models are niques combine several analysis and design tools. See, for widely used since they are more tractable mathematically instance, [19, 20] and some references therein are related to than the discrete models while having a direct physical designs, the optimality of the controls, and the use of interpretation [25]. In particular, the positivity of the so- fractional- order models. *ere are also epidemic models lution under nonnegative initial conditions can be proved 2 Discrete Dynamics in Nature and Society from the differential system describing the model without instance, [47]. *e main objective of this paper is to propose the use of extradiscretization adjusting parameters. Fur- and to link a SIR (susceptible-infectious-recovered) with thermore, it is a direct task to pick up values from the recruitment and demography to a proposed time-varying solution at appropriate sampling points (for instance, daily logistic equation parameterized by a time-varying carrying or weekly) of the continuous-time solution of the model, capacity and a time-varying Malthusian parameter and provided that such an information is needed for storing whose solution is attractive and of simple interpretation. *e statistic data or in order to decide the implementation of methodological analysis key point is to compare the current some vaccination or other alternative public intervention SIR epidemic model with eventual recruitment, demogra- controls such as partial quarantines or confinements. On the phy, and disease-related mortality parameters to the so- contrary, the use of discrete models has been shown to be called nominal (or reference) case, where those parameters specifically useful in the context of dissemination of disease- are zeroed and, in addition, the recovered are considered related information such as its current state or evolution constant being equal to their initial values along with the tendency, for instance, through social networks. *at as- time interval of interest. *e total reference population is sociated information can help to prevent or to reduce also constant due to the absence of recruitment, demog- possible foreseen outbreaks [26]. raphy, and disease-related mortality. It turns out that the A basic goal is to investigate and predict the evolution of above nominal logistic-type model can work efficiently along infectious diseases as well as to focus on how the public certain transient periods of time (of the order of weeks) interventions, for example, quarantines or vaccination and when the infection is growing up very fast, but there are no treatment controls can mitigate their outbreaks and un- still recovered subpopulation, and the total population does controlled propagations. See, for instance, [1, 2] and not vary significantly. *e nominal case might be described, [11, 13, 15, 17, 18, 27, 28]. A beneficial effect of quarantine equivalently, by a time-invariant logistic equation which has interventions on a part of a targeted subpopulation, in a unique solution for the infectious depending on the initial particular, either fractions or the susceptible subpopulation conditions and on two constant (reference Malthusian and or fractions of the infectious one is their “removal” from its reference carrying capacity) parameters which depend on associate compartment. In this way, the most apparent effect the transmission rate, the recovery rate, and the initial levels in the disease evolution through time is a reinitialization of of total population and immune subpopulation. In partic- the corresponding trajectory solution under fewer numbers ular, the Malthusian parameter quantifies the rate of either of contagious contacts. A second beneficial effect is that the growth or decrease of the infection and the (infection) disease transmission rate decreases to more moderate levels carrying capacity fixes the maximum level of such an in- as a result of the reduction in the contagion contacts. *ose fection along its growing time period (say, the maximum of concerns have been seen to be relevant concerning the the infectious subpopulation). evolution of the current COVID-19 pandemic. See, for *en, a most complex current SIR model is described instance, [29–46] and some of the references therein. In through the abovementioned time-varying logistic equation particular, an SEIR model is proposed in [41] for the whose solution is characterized as that of the nominal one COVID-19 pandemic. Such a model includes delayed plus an error function which depends on the new added resusceptibility caused by the infection. Also, a kind of parameters (recruitment rate, demography rate, and disease- autoregressive model average model (ARMA), so-called an related mortality). A practical way of monitoring the time- ARIMA model, for prediction of COVID-19 is presented varying logistic equation in a simple way is to reinitialize it and simulated in [42] for the data of several countries. *e by simple resetting of initial conditions, when necessary, and effects of different phases of quarantine actions in the values to keep its parameterization in operation along time in- of the transmission rate are studied in [43], see also [46], for tervals of appropriate moderate adaptive lengths before the a comparative and exhaustive discussion of related simu- next resetting. For each updated parameterization, the lo- lated numerical results on the COVID-19 outbreak in Italy. gistic equation is run as time-invariant until its next re- *e related existing bibliography is abundant and very setting. In this way, the current Malthusian parameter and rich including a variety of epidemic models with several carrying capacity are modelled by piecewise constant coupled subpopulations included as an elementary starting functions whose values are updated, in general, at a sequence basis of the susceptible, the infectious, and the recovered of sampling instants subject, in general, to nonuniform ones in the simpler SIR epidemic models. Further gener- intersample periods. Such periods are adapted to the in- alizations lead to the so-called SEIR models which include fection variation so that the larger the infection variation, the the exposed subpopulation, that is, those who do not have smaller the intersample period. external symptoms yet, as a new subpopulation. More *e paper is organized as follows. Section 2 states the SIR complex models, such as SEIADR-type models, which in- (susceptible subpopulation S-infectious subpopulation clude the asymptomatic infectious and the infective dead I-recovered subpopulation R) epidemic model with pop- subpopulations, have been designed and analyzed, for in- ulation recruitment, demography, and disease mortality and stance, Ebola disease [15, 16]. establishes and proves its positivity and boundedness On the contrary, it is well known that the celebrated properties. Section 3 is split into several sections. *e first logistic equation of Velhulst is a model of population growth one rewrites the model under the equivalent form of a time- which is parameterized by two constant parameters, namely, varying logistic equation parameterized by a the Malthusian the Malthusian parameter and the carrying capacity, see, for parameter, which defines the exponential order of the Discrete Dynamics in Nature and Society 3 solution form, and a carrying capacity which is related to the 2. The SIR Epidemic Model and Its Positivity maximum attainable infection level along its growing pe- and Boundedness Properties riod. Both parameters are, in general, time-varying functions which are defined based on the primary model parameters Consider the following SIR (susceptible S -infectious I-recovered and the values of the subpopulations through time. Section R) epidemic model with demography: 3.2 discusses the description of a simplified version of the S_(t) � ] − ( + I(t))S(t),S( ) � S , ( ) above epidemic model to the light of a time-invariant logistic μ β 0 0 1 equation parameterized by constant values of the Malthu- _ sian parameter and the carrying capacity. It also assumes I(t) �(βS(t) − c − μ)I(t),I(0) � I0, (2) that the increment of the total population over the recovered _ subpopulation remains constant. Such a time-invariant lo- R(t) �(1 − ρ)cI(t) − μR(t),R(0) � R0, (3) gistic equation is referred to as the nominal (or reference) logistic equation. *e simplified version of the SIR model, subject to arbitrary nonnegative finite initial conditions, rewritten equivalently as the nominal logistic equation, is i.e., min(S0,I0,R0) ≥ 0, where β is the disease transmission recruitment, demography, and mortality free. *ose sim- rate μ and ρ ∈ [0, 1] are, respectively, the natural and plifications are reasonable along periods lasting some disease-related mortality rates, and ]( ≥ μ) is the months since the disease-associated mortality is usually population recruitment rate. *e total population is small compared to the total population, while the recruit- N(t) � S(t) + I(t) + R(t) whose derivative with respect to ment and demography rates are also usually small related to time is obtained by summing-up (1)–(3): the total population amounts along short periods of time N_ (t) � ] − μN(t) − ρcI(t), compared with the species life expectation. An important (4) motivating result of the problem approach is that the carrying N(0) � N0 � S0 + I0 + R0. capacity is the quotient between the absolute Malthusian Denote in the following R � {}z ∈ R: z > 0 , parameter and the carrying capacity. In this way, the nominal + R � {}z ∈ R: z ≥ 0 � R ∪ {}0 . *e subsequent result relies logistic equation characterizes a constant disease trans- 0+ + on the positivity and boundedness of all the solution of mission rate for the simplified SIR model of no public in- (1)–(3) under any given finite nonnegative initial conditions. terventions of confinement or quarantine type is performed. In the same way, the current time-varying logistic equation establishes a time-varying disease transmission rate for the Theorem 1. Model (1)–(3) is positive, in the sense that more complex model which also depends on such public ∀t ∈ R0+ and globally stable, that is, the solutions of (1)–(3) interventions, in the case where they are performed. Section are bounded for all time for any given finite nonnegative 3.3 obtains in a closed form the solution of the time-varying initial conditions. As a result, it holds that logistic equation related to the nominal one and computes lim supt⟶∞(sup0≤ξ≤tI(ξ)) ≤ ]/c. also the worst-case of the error through time among both of t β � S(ξ)dξ− (c+μ)t them. It also discusses a technique to evaluate the current Proof. *e solution of (2) is I(t) � e 0 I0 ≥ 0; solution along with a set of samples which are updated ∀t ∈ R0+, since I0 ≥ 0. *erefore, I: R0+ ⟶ R0+, for any through adaptive sampling laws which decrease the inter- given nonnegative initial conditions. On the contrary, since sampling period as the infection variation between S0 ≥ 0 and S: R0+ ⟶ R0+ are everywhere continuous (since consecutive samples increases and vice versa. Finally, it is everywhere time-differentiable), it is nonnegative until Section 3.4 gives a pseudocode algorithm to implement the S(t1) � 0, for some t1 ≥ 0, in the event that such a t1 exists. _ above ideas to estimate the Malthusian and carrying capacity However, by inspecting (1), one gets S(t1) � ] ≥ 0 (even if the time-varying parameters from registered infection data on + model is recruitment-free, i.e., if ] � 0). *en, S(t1 ) ≥ 0. sequences of time instants. With those parameters, the time- *erefore, no t ∈ R0+ can exist such that S(t) < 0. *erefore, carrying capacity is also sample-dependent calculated and its S: R0+ ⟶ R0+ for any given nonnegative initial conditions. averaged value along the time period of each public inter- On the contrary, the solution of (3) is nonnegative in any vention (quarantines, confinements, de-escalating post- interval [0 , t2) under any nonnegative initial conditions _ quarantine phases, or public intervention void) is also such that R(t2) � 0, but then R(t2) � (1 − ρ)cI(t2) ≥ 0 so estimated. Section 4 is devoted to perform some numerical that no t ∈ R0+ exists such that R(t) < 0 and R: R0+ ⟶ R0+. examples as well as to discuss the obtained results with It has been proved that the solutions of (1)–(3) are always special emphasis on the influences on the infection prop- nonnegative for any given nonnegative initial conditions. agation of partial or total quarantines of some or all the *e total population has, from (4), the following solution subpopulations or de-escalating phases after quarantines. equation: Model parameterizations with available recorded data related to the COVID-19 pandemic are used for the testing N(t) � S(t) + I(t) + R(t) t process and the related transmission rate obtained. Finally, − μt − μ(t− ξ) � e R0 + � e (] − cI(ξ))dξ ≥ 0, ∀t ∈ R0+, Section 5 ends the paper. Some auxiliary calculations related 0 to the estimations of the Malthusian parameter and the (5) carrying capacity concerned with a relevant step of the estimation algorithm of Section 3 4 are given in Appendix A. so that one has, for any t0, t( ≥ t0) ∈ R0+, that 4 Discrete Dynamics in Nature and Society

t t − (t− ) − (t− ) 3. The Epidemic Model in a Logistic � e μ ξ I(ξ)dξ ≤ � e μ ξ I(ξ)dξ t0 0 Equation Form (6) 1 ] 3.1. “Ad hoc” Time-Varying Logistic Equation. Verhulst´s or ≤ �e− μtR + �1 − e− μt �� c 0 μ logistic equation is a celebrated model of population growth which was proposed in 1838 by Pierre Verhulst (1804–1849). so that, for any given finite θ ∈ R+, one has by using the It can also be considered in an “ad-hoc” time-varying mean value theorem that, for any strictly increasing se- version to describe the epidemic model of the above section ∞ ∞ under a certain modification of the basic logistic equation quence �tn �n�0 ⊂ R0+ and some sequence �ςn �n�0 ⊂ (tn+1, tn), after replacing S(t) � N(t) − I(t) − R(t) in (2). *us, one − μθn gets the following infectious solution with the contribution tn+θ θ�1 − e � ] − μ()tn− ξ ( ) � e I(ξ)dξ � Iςn � ≤ , 7 to its dynamics of the susceptible and recovered t μ cμ n subpopulations: so that ] I_(t) � β(N(t) − I(t) − R(t))I(t) − (c + μ)I(t) lim sup Iςn � ≤ , ∀t ∈ R0+. (8) n⟶∞ cθ � − βI2(t) +[β(N(t) − R(t)) − (c + μ)]I(t) (13) Since I: R0+ ⟶ R0+ is continuous and θ finite, one 2 concludes that lim supn⟶∞I(tn) ≤ ]/cθ, and under the same I (t) I(t) I(t) ] c � − + , ∀t ∈ R +, reasoning, one gets lim supt⟶∞ ≤ / θ. Since a(t)c(t) a(t) 0 I: R0+ ⟶ R0+ is continuous, it cannot be unbounded on any finite time interval [0, t1) so that the above ultimate where a(t)c(t) � 1/β, and boundedness property as time tends to infinity also implies I [ , ] 1 the boundedness of : R0+ ⟶ R0+ in 0 ∞ . a(t) � , For any nonnegative initial conditions and then β(N(t) − R(t)) − c − μ I: R0+ ⟶ R0+ is bounded. *is implies from (5) that � � (14) � � 1 1 � c + μ� N(t) N + + c I(t) + . ( ) c(t) � ��N(t) − R(t) − �, ∀t ∈ R +. ≤ 0 �] sup � < ∞ 9 |a(t)| � � 0 μ t≥0 β β On the contrary, one has from (2) that the (empty or *e subsequent result will be then used related to the nonempty, connected, or nonconnected) real interval interpretation of the epidemic models (1)–(3) as a time- varying logistic equation related to a nominal time-invariant c + μ one, in the case that S(t) + I(t) ≥ S0 + I0 for all t ∈ R0+. US � �t ∈ R0+: S(t) > � � � ti, ti+1 � (10) β i ∈Z0+ Proposition 1. Ae following results hold for a normalized model (1)–(3) at the initial time, that is, if N0 � 1 has a Lebesgue measure 0 ≤ Lb(US) < + ∞ (zero if it is empty) and finite since otherwise, limt⟶∞I(t) � +∞ which (i) Ae inequality contradicts the boundedness of I: R0+ ⟶ R0+ according to (8). *en, the complementary set US in R0+ is S(t) + I(t) � N(t) − R(t) ≥ S0 + I0 � N0 − R0 (15) c + μ R \U ��t ∈ R : S(t) ≤ � 0+ S 0+ β is guaranteed for a given t ∈ R+ if (11)

� � ti, ti+1 � � � ]ti, ti+1[ , t i∈Z0+ i∈Z0+ − μ(t− ξ) − μt � e (] − cI(ξ))dξ ≥ �1 − e �N0 − R0 �. (16) 0 which has 0 ≤ Lb(R0+\US) � +∞ and, therefore, S: R0+ ⟶ R0+\US is bounded while S: R0+ ⟶ US cannot and the inequality is reversed, namely, for a given t ∈ R+, be unbounded since S: R0+ ⟶ R0+ is continuous and US has a ﬁnite Lebesgue measure. As a result, S: R0+ ⟶ R0+ is bounded for any nonnegative initial conditions. Since S(t) + I(t) � N(t) − R(t) ≤ S0 + I0 � N0 − R0 (17) S, I, R, N: R0+ ⟶ R0+ and S, I, N: R0+ ⟶ R0+ are bounded, then if

0 ≤ R(t) � N(t) − S(t) − I(t) < + ∞, ∀t ∈ R0+, (12) t and R: R0+ ⟶ R0+ is bounded. *e theorem has been fully − μ(t− ξ) − μt � e (] − cI(ξ))dξ ≤ �1 − e �N0 − R0 �. (18) proved. □ 0 Discrete Dynamics in Nature and Society 5

(ii) Ae above inequality is guaranteed in [0, t] for any 3.2. Nominal Logistic Equation. Assume that the epidemic given t ∈ R+ if models (1)–(3) are considered without recruitment, demography, and mortality, that is, ] � μ � ρ � 0. *us, one has from (4) and (13) and (14) that N_ (t) ≡ 0 so that ] − μ N(t) � N ; ∀t ∈ R . It is also assumed that the total sup I(ξ) ≤ , (19) 0 0+ 0≤ξ≤t c population excess over the recovered subpopulation remains constant. *is assumption keeps the logistic equation time- so that it is guaranteed in [0, ∞) as well. Ae above inequality invariant, for instance, identical to their initial condition also guarantees that liminf (S(t) + I(t)) ≥ S + I is t⟶∞ 0 0 excess N0 − R0. *e above hypotheses is reasonable along fulfilled since periods of several months over which the above population/ recovered subpopulation excess does not vary substantially ] − μ lim sup⎛⎝ sup I(ξ)⎞⎠ ≤ . (20) in practice. *us, the parameterization of the (so-called c t⟶∞ 0≤ξ≤t nominal or reference) logistic equation is time invariant accordingly to 1 Proof. *e solution to (3) is a(t) ≡ ar � , βN0 − R0 � − c t ( ) − μt − μ(t− ξ) � � 26 R(t) � e R0 + c(1 − ρ) � e I(ξ)dξ, ∀t ∈ R0+, 1 � c� 0 c(t) ≡ c � � � ��N − R − �, r � � � 0 0 � (21) β�ar� β which together with (5) yields and (13) has the following nominal expression parameter- t ized by (26): − μt − μ(t− ξ) N(t) − R(t) � e N0 − R0 � + � e (] − cI(ξ))dξ, ∀t ∈ R0+. 2 0 I (t) I (t) rI (t)K − I (t) � I_ (t) � r − r � r r r ,I ( ) � I , ( ) r r 0 0 22 ar arcr Kr (27) Note that S(t) + I(t) � N(t) − R(t) ≥ S0 + I0 � N0 − R0 t for some ∈ R0+ if and only if where t − μ(t− ξ) − μt ( ) 1 � e (] − cI(ξ))dξ ≥ �1 − e �N0 − R0 �, 23 r � � N − R � − c, 0 β 0 0 (28) ar and the first part of Property (i) is proved. *e reversed is the Malthusian parameter, namely, the rate of maximum inequality and associated condition are proved “mutatis- infection growth which can be positive, negative, or zero, mutandis.” On the contrary, the above inequality holds on and [0, t] if � � � c� − μt − μt � � c�1 − e � ]�1 − e � Kr � rarcr � cr ��N − R − � (29) − μt � 0 0 β� sup I(ξ) ≤ − �1 − e �S0 + I0 �, μ 0≤ξ≤t μ is the carrying capacity, namely, the maximum sustainable (24) infectious subpopulation. *e normalized by Kr differential for a normalized model at the initial conditions, that is, for equation (27) becomes for S + I ≤ N � 1 if sup (ξ) ≤ ] − μ/c. Also, 0 0 0 0≤ξ≤t _ lim inf (S(t) + I(t)) ≥ S + I if Irn � rIrn(t) 1 − KrInr(t)�, t⟶∞ t 0 0 lim sup � e− μ(t− ξ)I(ξ)dξ ≤ 1/c(]/μ − tS n − qI ) t⟶∞ 0 0 0 (30) which is guaranteed if Ir0 t I (0) � I � , � e− μ(t− ξ)I( ) ] − c rn rn0 K lim supt⟶∞ 0 ξ dξ ≤ μ/ μ which is guaran- r teed if lim sup (sup I(ξ)) ≤ ] − μ/c. Property (ii) has t⟶∞ 0≤ξ≤t I (t) � I (t) K been proved. □ where rn r / r and whose solution is the sigmoid function:

From the nonnecessary normalized model at the initial rt conditions, one has the following result which follows di- 1 Irnoe Irn(t) � − 1 − rt � rt rectly from (24) in the proof of Proposition 1: 1 +�Irn0 − 1 �e Irnoe +1 − Irno � (31) rt Ir e Ir Proposition 2. S(t) + I(t) ≥ S + I holds for all t ∈ [0, ta], 0 0 0 0 � rt � − rt − rt. provided that S0 + I0 ≤ ]/μ if Ir0�e − 1 � + Kr Ir0�1 − e � + Kre 1 sup I(t) ≤ ] − μS0 + I0 ��. (25) *en, the unnormalized infectious evolution is 0≤t≤ta c 6 Discrete Dynamics in Nature and Society

KrIr0 parameterization of the nominal one. *en, note that the Ir(t) � KrIrn(t) � I �− e− rt � + K e− rt combination of (28) and (29) yields the constraint: r0 1 r � � � � �βN0 − R0 � − c� |r| c I Kr � � , (36) � r r0 ( ) β β − t/ar − t/ar 32 Ir0�1 − e � + cre which implies that the transmission rate satisﬁes β � |r|/Kr. c I *e remaining results of Property (ii) follow directly from � r r0 , − t/ar (28) and (29). On the contrary, note, from (28) and (29) that Ir0 +cr − Ir0 �e rβ2 � − rβ1 � � N0 − R0 �β2 − β1 �, and, for Ir0 ≠ 0, (37) Kr Kr cβ2 − β1 � Ir(t) � − rt � (33) Krβ2 � − Krβ1 � � , r() tM d− t 1 +Kr/Ir0 − t1 �e 1 + e β1β2

tM dr if e � Kr/Ir0 − 1, namely, if and also one has from (34) that − 2 1 Kr − Ir cr − Ir t c I t � ln 0 � a ln 0 d M d 1 β r0 M d r � × − r Ir Ir N − R � − c 1 0 0 dβ β 0 0 N0 − R0 − cβ − Ir0 (34) � − � � 1 � − 1 1 �N0 − R0 � − β c� N − R N − R − cβ � ln� − 1�. − 0 0 × ln� 0 0 − 1� βN0 − R0 � − c Ir0 2 I βN0 − R0 � − c� r0 _ It also holds that the maximum value of Ir(t), which is (38) the zero of I€ (t) (or the growth inversion point), takes place r so that dt /dβ � o(β− 3). Property (iii) has been at t � t , equation (34), from (33). *is is easily seen from M d M d proved. the nominal version of the logistic equation (13) with □ I(t) ≡ Ir(t), a(t) ≡ ar, and c(t) ≡ cr, which leads to _ Remark 1. Note from (27) that if r > 0(that is if the trans- € Ir(t) 2Ir(t) Ir (t) � �1 − � � 0, (35) mission rate is large enough to satisfy β > c/(N0 − R0)) and ar cr Ir0 < Kr, then Ir(t) is strictly increasing until it reaches its _ maximum value Kr. If r turns to a negative value, that is if which implies that Ir(t) � cr/2 � Kr/2, if Ir(t) ≠ 0, which β < c/(N0 − R0), then the Ir(t) reverses its growth turning to happens at t � tM d from (33). *erefore, tM d is the growth inversion point time instant. On the contrary, from (27), a strictly decreasing profile. *e effective way of achieving this behaviour is to decrease the number of effective in- there is no relative finite maximum value of Ir(t) since _ fectious contacts to the susceptible either by social discipline Ir(t) � 0 holds for t � ∞ if r > 0. We have, as basic relations for the key parameters of the nominal logistic equation or eventually via quarantines or confinements. associated with (1)–(3), the Malthusian parameter (28), the carrying capacity (29), and the growth inversion point (34). 3.3. Comparison Results of the Logistic Equation with Its *e following result of interest holds. Nominal Version. Note that the time-varying logistic equation (13) may be compared to the nominal one (27) by Proposition 3. Ae nominal logistic equation has the fol- expressing the errors between both current and nominal lowing properties: Malthusian parameters and carrying capacities. For that (i) It satisfies the positivity and boundedness conditions purpose, define parametrical errors: of Aeorem 1. 1 1 a�(t) � a(t) − a � − , (ii) Ae disease transmission rate satisfies the constraint r β(N(t) − R(t)) − c − μ βN − R � − c β � |r|/K so that either β > 0 and K is finite or β � 0 0 0 r r � � � � r � − c K � c N − R � � � � with and r ∞. If β > / 0 0 (respec- � c + μ� � c� �c(t) � c(t) − cr � �N(t) − R(t) − � − �N0 − R0 − � tively, β < c/N0 − R0), then r > 0 (respectively, r < 0). � β � � β� � c N − R r � If β / 0 0, then 0. (39) (iii) Ae Malthusian parameter and the carrying capacity increase (respectively, decrease) as the disease so that transmission rate increases (respectively, decreases). 1 1 1 Ae same property holds with respect to β for the r(t) � � � + �r(t) � r + �r(t), growth inversion point time instant if β < 1. a(t) a + a�(t) a r r (40) � K(t) � c(t) � cr + �c(t) � Kr + K(t), Proof. Property (i) is a corollary of *eorem 1 for the nominal logistic equation which is a particular where Discrete Dynamics in Nature and Society 7

a�(t) β N(t) − R(t) − N + R � − μ which can be rewritten for each time instant t from a � 0 0 r(t) � r(t) − rr � − � , t − T(t) ar + a�(t) β(N(t) − R(t)) − c − μ previous time instant as follows: � � � � � � � � � � c + μ� � c� crI(t − T(t)) K(t) � �c(t) � c(t) − cr � �N(t) − R(t) − � − �N0 − R0 − �, I(t) � � β � � β� − T(t)/ar I(t − T(t)) +cr − I0 �e

1 1 ar�c(t) + cra�(t) + �c(t)a�(t) t � b�(t) � − � − . rK(ξ) 2 ⎜ � I (ξ)dξ ⎟ a(t)c(t) arcr arcr + a�(t)cr + �c(t)ar + �c(t)a�(t) ⎜⎛ � ⎟⎞ ⎜ t− T(t) Kr + K(ξ) ⎟ ⎜ ⎟ (41) ⎜ ⎟ +⎜ ⎟. ⎜ ⎟ ⎝⎜ t 1 K� (ξ) ⎠⎟ *us, (13) can be rewritten as follows: + � �r( )I( ) − − I( ) ξ ξ �1 � � � ξ �dξ t− T(t) Kr K + K(ξ) I(t) I(t) I(t) r I_(t) � �− � �(r + �r(t))I(t)� − � ( ) 1 1 � 46 a(t) c(t) Kr + K(t) Assume that, for a given t0 ∈ R+ and each t( ≥ t0) ∈ R+, 1 K� (t) T(t) is selected so that K� (t) and �r(t) are almost constant �(r + �r(t))I(t)� − � − �I(t)� 1 � with absolute errors less than positive real constants ε and Kr Kr + K(t) �r |K� (t)| K supt≥0 ≤ εK� < r (since the current carrying capacity I(t) K(t) � Kr + ε� is positive) in [t − T(t), t]. *us, one has � rI(t)�1 − � + f(t), K K r rT(t) t � ε� 2 rK(ξ) 2 (42) − K sup I (ξ) ≤ � I (ξ)dξ K − t− T(t) � r εK� t− T(t)≤ξ≤t Kr + K(ξ) where rT(t) � � εK� 2 rK(t) 2 1 K(t) I ( ), f(t) � I (t) + �r(t)I(t)�1 − � − �I(t)�. ≤ sup ξ � � Kr t− T(t) t Kr + K(t) Kr Kr + K(t) ≤ξ≤ (43) (47)

*en, one has that K �1 + ε � + ε ⎛⎝ r K� K� ⎞⎠ t ε rε T(t) inf I(ξ) sup I(ξ) I(ξ) � K� t− T(t) t ≤ξ≤ Kr�Kr − ε � t− T(t)≤ξ≤t I(t) ��I0 + r � I(ξ)�1 − �dξ� K� 0 K r t K� (ξ) 1 ≤ � �r(ξ)I(ξ)�1 +� − �I(ξ)�dξ t � rK� (ξ) t− T(t) Kr + K(ξ) Kr � I2(ξ)dξ ⎜⎛ � ⎟⎞ ⎜ 0 Kr + K(ξ) ⎟ ⎜ ⎟ ⎛⎝ 1 ⎞⎠ ⎜ ⎟ ≤ ε�rε�T(t) sup I(ξ) 1 + sup I(ξ) , +⎜ ⎟. K K ⎜ ⎟ t− T(t)≤ξ≤t r t− T(t)≤ξ≤t ⎝⎜ t 1 K� (ξ) ⎠⎟ + � �r( )I( )� − � − �I( )� ξ ξ 1 � ξ dξ (48) 0 Kr Kr + K(ξ) |K� (t)| (K , ) (44) provided that supt≥0 ≤ εK� < min r 1 . On the contrary, if the incremental carrying capacity related to the *e ﬁrst right-hand side term of (45) is the solution of nominal one is large enough to satisfy the nominal logistic equation (i.e., (32) or (33)). *e second � inf t≥0|K(t)| ≥ Kr/|Kr − 1|, which implies also that ε� > 1 one is the integrated disturbance function f(ς) on [0 , t], K t while it is also subject to Kr > 1, then � f( ) equation (44), namely, 0 ξ dξ due to the error between the current and nominal solutions. In view of the nominal � � K − � − K K(t) 1 Kr − 1 �K(t) − Kr εK� r 1 r solution expression (32), one has that (45) can be expressed − ≤ ≤ K + K� (t) K K K + K� (t) � K2 equivalently as follows: r r r r r (49) c I t c I I(t) � r 0 + � f(ξ)dξ � r 0 − t/ar − t/ar so that the upper bound in (47) becomes modiﬁed as follows: I0 +cr − I0 �e 0 I0 +cr − I0 �e

t rK� (ξ) ε�Kr − 1 � − Kr +�� I2(ξ)dξ ε ε T(t) sup I(ξ)⎛⎝1 + K sup I(ξ)⎞⎠, � �r K� 2 0 Kr + K(ξ) t− T(t)≤ξ≤t Kr t− T(t)≤ξ≤t (50) t 1 K� (ξ) + � �r( )I( )� − � − �I( )� �, ξ ξ 1 � ξ dξ 0 Kr Kr + K(ξ) *en, one has under the upper bound of (47), for the case when sup |K� (t)| ≤ ε < K if ε � min(ε , ε ), that (45) t≥0 K� r �r K� 8 Discrete Dynamics in Nature and Society

(1) Step 0: Choose t0 ≥ 0 and the initial intervention Phase Ph0 to start to run the procedure. (2) Smooth the recorded infection curve I(t), if the recorded data are discrete, to make the amended one defined everywhere since the logistic equation used for modelling is continuous through time. (3) Define Ci for i � 1, 2, 3 and Tm ,TM to then run (53)–(55) to generate the sequences IS and ISP of sampling instants and intersampling intervals (or sampling periods) with TM − Tm being small enough for the local time-invariance parameterization of the logistic equation to work efficiently. (4) Step 1: Given the current sampling instant ti ∈ IS, locate the intervention phase Phj to which it belongs. Calculate the intersampling interval Ti ∈ ISP via (54) and some of the equations (55)–(57) and then calculate the next sampling instant ti+1(∈ IS) � ti + Ti (5) Step 2: Read I(ti+1) from recorded data (6) Step 3: *e Malthusian parameter and the carrying capacity at time ti are calculated “a posteriori” at time ti+2 � ti+2Ti from the registered I(ti), I(ti+1) � I(ti + Ti) and I(ti + 2Ti) evaluated at the current and next time instants ti and ti+1 � ti + Ti and an “a � priori” estimation ti+2 � ti+1 + Ti � ti + 2Ti of ti+2 from equations (A.12)–(A.14) and (A.5) in Appendix A (Remark A.1). (7) Step 4: Calculate the β(ti, ti+1) � |r(ti, ti+1)/K(ti, ti+1)| (see Proposition 3 (ii)). (8) Step 5: If ti+1 ∈ Phj then make i⟵i + 1 and GoTo Step 1. Else GoTo Step 6 (9) Step 6: Calculate the average disease transmission rate of the Phase Ph β(Ph ) � 1/l � β(t , t ), make j⟵j + 1 and if j j J ti,ti+1∈Phj i i+1 the whole number of checked phases is unfinished GoTo Step 1. Else GoTo Step 7 (10) Step 7: End

ALGORITHM 1: Evaluation of the Malthusian parameter and the carrying capacity.

K I(t − T(t)) 1 I(t) � r + ε T(t) sup − T(t)r K� 0.9 I(t − T(t)) + Kr − I(t − T(t))�e t− T(t)≤ξ≤t 0.8 K (r − 1) + ε K − 1 � ⎛⎝ r K� r ⎞⎠ I(ξ) ε�r + 2 sup I(ξ) + o(εT(t)), 0.7 Kr t− T(t)≤ξ≤t 0.6 (51) 0.5 and one also has, for the case when � 0.4 |K(t)|inf t≥0 ≥ Kr/|Kr − 1|, according to (49), that 0.3 KrI(t − T(t)) 0.2 I(t) � − T(t)r I(t − T(t)) + Kr − I(t − T(t))�e 0.1 0 r + ε�r 0 20 40 60 80 100 120 140 160 180 200 + ε T(t) sup I(ξ)⎛⎝ε + sup I(ξ)⎞⎠ K �r Time (days) t− T(t)≤ξ≤t Kr t− T(t)≤ξ≤t S + o(εT(t)). I (52) R t Figure 1: Evolution of the SIR model describing the spread of On the contrary, note that � �r(ξ)I(ξ)(1 − (1/Kr � � t− T(t) COVID-19 in Italy. − tK(ξ)/Kr + K(ξ))I(ξ))dξ is slowly time varying if εT(t) is sufficiently small which is a reasonable approximation for a sufficiently small interval T(t) from the time instant where demography-related parameters μ and ] are also as small as the initial infection values are picked up until its current suited. *ese considerations suggest that, by choosing the calculation. *e reason is that this implies in addition that ε central interval point for such an evaluation of the incre- is also sufficiently small which leads to a small variation of mental solution, one gets as an alternative expression to the N(ξ) − R(ξ) along [t − T(t), t] for a sufficiently small in- infectious subpopulation evolution through time that terval T(t) with that additional advantage that the

K(t − nT(t))I(t − nT(t)) I(t) � + δ(ε(nT(t)), nT(t)), (53) I(t − nT(t)) +(K(t − nT(t)) − I(t − nT(t)))e− nT(t)r(t− nT(t)) Discrete Dynamics in Nature and Society 9

0.5 ×10–3 6 0.45 5 0.4 0.35 4

0.3 3 0.25

I ( t ) 2 0.2 1 0.15 0.1 0

0.05 –1 0 0 20 40 60 80 100 120 140 160 180 200 –2 0 20 40 60 80 100 120 140 160 180 200 Time (days) Time (days) SIR model, I (t) Reported data r (t) Figure Figure 2: Number of active cases per day obtained from the SIR 4: Evolution of the Malthusian function, r(t), parameter- model and from reported data. izing the time-varying logistic equation.

0.8 0.5 0.7 0.45 0.4 0.6 0.35 0.5 0.3 0.4 0.25 I ( t ) 0.2 0.3 0.15 0.2 0.1 0.1 0.05

0 0 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 Time (days) Time (days)

I c (t) TV logistic equation Figure 5: Evolution of the carrying capacity, c(t), parameterizing Figure 3: Number of active cases per day obtained from the the time-varying logistic equation. complete SIR model and from the time-varying logistic equation. T , T T , ⎪⎧ M if ia > M for given n ∈ Z+ and T(t) ∈ R+ leading to a performed ⎨ [t − nT(t), t] Ti �⎪ Tia, if Tia ∈ �Tm,TM �, (54) calculation along a time interval . A further ⎩⎪ improvement might be performed without diﬃculty over a T , if T < T , n m ia m sequence, or eventually a ﬁnite set IS ≡ �ti �i�0 of sampling instants by using recorded data with a monitored adaptation where there are a number of possibilities to generate the of the intersampling interval to the rate of variation of the auxiliary tentative sampling period Tia from the classical adaptive sampling background literature (see [48–51] and infection curve in such a way that t0 � 0, ti+1 � ti + Ti, and n related references therein) as, for instance, the three ones the intersample period sequence ISP ≡ �Ti �i�0 are being updated as follows: which follow: 10 Discrete Dynamics in Nature and Society

1.2 ×10–3 7

1 6

0.8 Nominal logistic 5 equation holds

0.6 4

3 0.4

2 0.2

1 20 40 60 80 100 120 140 160 180 Time (days) 0 0 5 1015202530354045 N(t) – R(t) Time (days) N0 – R0 r (t) Figure 6: Evolution of (N(t) − R(t)) with time. r (t)

Figure 8: Estimation of the Malthusian parameter from the SIR 0.8 Agreement with model. 0.7 nominal logistic 0.6 considerations. *e adaptive sampling laws (54) and (55) evaluate explicitly the absolute value of time derivative, while 0.5 the adaptive sampling law (54), which subjects to (56), evaluates it by ﬁnite increments. *e sampling period is 0.4 constant (thus, it is nonupdated through time) if C � 0, I ( t ) 2 T � T � C C 0.3 ia m 1/ 3. *us, one gets from (53) the following relation in-between two consecutive sampling instants: 0.2 K ti �I ti � I(t) � , ∀t ∈ [ ti, ti+1 ), ∀ti ∈ IS. − Tir() ti 0.1 I ti � + K ti � − I ti ��e (58) 0 0 20 40 60 80 100 120 140 160 180 200 Time (days)

SIR model, TV logistic 3.4. Estimation Algorithm of the Malthusian Parameter and Nominal logistic equation the Carrying Capacity along Time. *e parameters which Reported data characterize the current logistic equation are time varying, but it proposed the following simple algorithm (Algo- Figure 7: Comparison between the SIR model, the time-varying rithm 1) expressed in general pseudocode style. It is assumed logistic equation, and the nominal one. that the pandemic is being focused on under different public intervention phases (Phi, i ∈ Z+), such as free mobility, confinements, partial quarantines, or different phases of de- C T � 1� �, escalation of quarantines. According to the evaluation of the ia � _ � (55) C3 + C2�I ti �� pandemic growth force, the algorithm identifies from recorded data of the infection curve the Malthusian pa-

C1 rameters and the carrying capacity, and then it estimates the Tia � � � (56) averaged disease transmission rate per intervention phase. C +�C �I t � − I t ��/T ), 3 2 i i− 1 i− 1 Note that the calculation process is repeated for all the pairs of samples (t , t ) and (t , t ). *e transmission C i i+1 i+1 i+2 � 1 � rate is calculated for each such a pair from Proposition 3 (ii). Tia � � �. (57) C3 + C2�I ti � − I ti− 1 ��

For some prescribed design real constants C1 ,C2 > 0, Remark 2. For the pairs (ti , ti+1) with sampling instants C3 ≥ 0, and Tm(TM > Tm) with TM − Tm being suﬃciently belonging to consecutive phases, we can include the option small in accordance with the above solution approximation of considering their associated transmission rates for both Discrete Dynamics in Nature and Society 11

0.8 4

0.7 3.5 3 0.6 2.5 0.5 2 0.4 1.5

0.3 1

0.2 0.5

0 0.1 –0.5 0 0 5 1015202530354045 0 5 1015202530354045 Time (days) Time (days) c (t) c (t) c (t) Figure 11: Estimation of the carrying capacity from reported data (Italy). Figure 9: Estimation of the carrying capacity from the SIR model.

0.7

1.5 0.6 0.5

1 0.4

0.3 0.5 0.2

0 0.1 0

–0.5 –0.1

–0.2 –1 –0.3 0 5 1015202530354045 –1.5 Time (days) 0 5 1015202530354045 Time (days) r (t)

r (t) Figure 12: Estimation of the Malthusian parameter from filtered reported data (Italy). Figure 10: Estimation of the Malthusian parameter from reported data (Italy). predictions can be replaced alternatively by the use for a set of previously recorded data corresponding phases in Algorithm 1 or for none of them as in the above to the current time instant under checking by the given version form of Step 6. algorithm. Note that both methods can involve the selection of time instants being an integer multiple of Remark 3. Note that, in Appendix A, two ways of imple- the minimum period of supplied data, typically, one menting Step 3 of Algorithm 1 to estimate the carrying day. In this case, the adaptive sampling law just capacity and the Malthusian parameter are given: selects a time-varying sampling period which is an integer multiple integer of the mentioned minimum (a) *e first one involves the use of infection recorded sampling period reported in the recorded data. A data after the current evaluation time instant. particular case is simply to use the minimum sam- (b) *e second one is summarized in Remark A.1, and it pling period as a constant running sampling period involves the only use of past data. Concerning the while removing the adaptive sampling law design first method, one has to point out that the use of from Algorithm 1. 12 Discrete Dynamics in Nature and Society

2 0.4

0.35 1 0.3 0 0.25

–1 0.2

0.15 –2 0.1 –3 0.05

–4 0 0 5 1015202530354045 0 5 10 15 20 25 30 35 40 45 Time (days) Time (days) r (t) c (t) Figure Figure 13: Estimation of the carrying capacity from ﬁltered re- 15: Estimation of the Malthusian parameter from reported ported data (Italy). data (Italy) and lsqcurveﬁt Matlab function.

1200 40

1000 20

800 0

600 –20

400 –40

200 –60

0 –80

–200 0 5 1015202530354045 –100 0 5 1015202530354045 Time (days) Time (days) β (t) c (t) Figure 14: Estimation of β from ﬁltered reported data (Italy). Figure 16: Estimation of the carrying capacity from reported data (Italy) and lsqcurveﬁt Matlab function. 4. Numerical Examples *e simulation starts on February 26, 2020, and spans for 200 *is section is devoted to present some numerical simulation days. Moreover, Figure 2 displays the number of active cases per examples illustrating the results discussed in the previous day obtained from the SIR model along with the reported sections. To this end, we consider the case of COVID-19. It is number of cases obtained from [53]. As it is proved in *eorem shown in [52] that the spread of COVID-19 in various regions, 1, the solution of the SIR model remains nonnegative and such as China, South Korea, and Italy (among others) can be bounded for all time given nonnegative initial conditions. *e appropriately described by a SIR model. For Italy, the SIR model evolution of the number of active cases per day (the state I) can describing the spread of coronavirus is given by β � 0.18, be equivalently described by the time-varying (TV) logistic − 1 c � 0.037, and ρ � μ � ] � 0, all of them in units of day . *e equation (13). *us, Figure 3 displays the evolution of the state I initial values for the states are given by S(0) � 240, 000, I(0) � obtained from the SIR model and the solution to the time- 240, and R(0) � 0. As it is performed in [52], the model is scaled varying equation (13). It is concluded from Figure 3 that both by a factor f � 2.4 · 105 in such a way that the scaled initial systems are equivalent so that the ad hoc time-varying logistic − conditions read S(0) � 1 and I(0) � 10 3. *e evolution of the equation is a single equation that appropriately describes the scaled SIR model with these parameters is depicted in Figure 1. number of active cases per day. Furthermore, Figures 4 and 5 Discrete Dynamics in Nature and Society 13

16 20

14 10

12 0 10 –10 8 –20 6 –30 4

2 –40

0 –50 0 5 1015202530354045 0 5 1015202530354045 Time (days) Time (days)

β (t) c (t) Figure 17: Estimation of β from reported data (Italy) and Figure 19: Estimation of the carrying capacity from reported lsqcurvefit Matlab function. filtered data (Italy) and lsqcurvefit Matlab function.

0.35 2

0.3 1.8

1.6 0.25 1.4

0.2 1.2

1 0.15 0.8

0.1 0.6

0.4 0.05 0.2 0 0 5 1015202530354045 0 0 5 1015202530354045 Time (days) Time (days) r (t) β (t) Figure 18: Estimation of the Malthusian parameter from reported Figure 20: Estimation of β from reported filtered data (Italy) and filtered data (Italy) and lsqcurvefit Matlab function. lsqcurvefit Matlab function. display the values of the Malthusian function, r(t), and carrying subpopulation of immune. Moreover, Figure 7 supports this capacity, c(t), defined in (14), which parameterize the time- conclusion where the plots of the SIR model and the time- varying logistic equation. varying and the nominal logistic equations are displayed. It Under certain conditions, the time-varying logistic is observed in Figure 7 that the nominal logistic equation is equation can be approximated by a nominal time invariant close to the solution of the SIR model during the first days of one that simplifies the model of infection spreading. *e simulation. *erefore, the nominal logistic equation may be main condition for this to hold is the excess (N(t) − R(t)) used as a simplified model to describe the spread of an being constant and equal to (N(0) − R(0)) along time. Fig- infection during the first stages, while the time-varying one ure 6 shows the evolution of (N(t) − R(t)) with time. It is may be used to describe the spreading for a longer period. It observed in Figure 6 that only during the first days of can also be concluded from Figure 6 that spreading this condition holds so that the nominal logistic N(t) − R(t) ≤ N(0) − R(0) for all time since the condition equation is an appropriate model only while the incubation in Proposition 1 trivially holds in this case, where the in- period has not supplied with individuals yet the fectious are always nonnegative and ρ � μ � ] � 0. 14 Discrete Dynamics in Nature and Society

×104 10 9 8 7 6 5 I ( t ) 4 3 2 1 0 0 5 1015202530354045 Time (days)

Reported data Nominal logistic: least squares Nominal logistic: algorithm 1 with filter

Figure 21: Comparison of reported data with the output of the estimated nominal logistic equation.

As a consequence, it is of interest to have an algorithm to Figures 12 and 13. Moreover, the estimated value of β � |r/c| estimate the values of the Malthusian and carrying capacity is depicted in Figure 14. From these figures, it can be con- from data in order to obtain a description of the infection cluded that the filter smoothens the output of the estimation, spreading during the first stages. *erefore, the algorithm but the filtered data still have too much variability to extract discussed in Section 4 will be applied now to estimate these the value of β as the mean value of the series. *e value of β at parameters during the first days of spreading. Initially, the the last stage of simulation (which is the most stable one) is algorithm is applied to the numerical data obtained by β � 0.412. simulation from the original SIR model. *us, consider a Furthermore, a least squares method is now applied to constant sampling time of TM � Tm � Ti � 1 hour, which is a approximate the reported data to a nominal logistic equa- low sampling time for epidemic systems and the use of tion. To this end, the lsqcurvefit Matlab function will be equations (A.5)–(A.11) from the Appendix to estimate the used. In this way, the fitting procedure is repeated every time values of the carrying capacity and Malthusian parameter. a new data is collected. Figures 15–17 display the evolution Since the approximation to the nominal logistic equation of the estimated parameters with time when the least squares only holds during the first days of simulation, the first 45 method is employed. days will only be considered. Figures 8 and 9 display the *e value at the end of simulation is β � 0.558. A filtered evolution of the actual and estimated Malthusian parameter version of the data is also used to perform the estimation. In and actual and estimated carrying capacity, respectively. It is this way, the same filter as before is used to smoothen the concluded from Figures 8 and 9 that the estimated Mal- data prior to the application of the least squares method. thusian parameter is close to the actual value, while the *us, Figures 18–20 display the evolution of estimated carrying capacity is estimated with an error of 20%. *e parameters when filtered data are used. estimated value of β is therefore 0.2246 days− 1, in com- *e value of β � 0.5204 is obtained at the end of parison to the actual one of 0.18 days− 1. *e algorithm simulation. Finally, Figure 21 depicts the reported data provides an estimation of the transmission rate by mea- along with the plot of a nominal logistic equation pa- suring only the number of active cases in three consecutive rameterized by the values obtained at the end of least time samples. Now, the method will be applied to reported squares estimation and a nominal logistic equations pa- data from Italy, where data are collected every day, i.e., rameterized by the values obtained at the end of Algo- TM � Tm � Ti � 1 day. rithm 1 execution with filtered data. It is observed in Figures 10 and 11 show the estimated values of the Figure 21 that the least-squares estimated nominal logistic Malthusian parameter and carrying capacity when the equation is close to the reported data while the nominal method is applied to reported data from Italy. It can be equation parametrized by Algorithm 1 is slightly further. deduced from these figures that raw data contain too much Overall, the nominal logistic equation is an appropriate variability to perform the estimation directly. *erefore, an model to describe the spread of an infection during its first averaging filter is employed to smoothen the data prior to stages. *e estimation of logistic equation parameters al- applying the estimation procedure. *us, we use the filter lows obtaining an estimation of the transmission rate of the 4 ykf � (1/5) �i�0 yk− i, which is the mean value of the last 5 disease, which provides useful information for decision samples. *e estimation with filtered values provides makers. Algorithm 1 provides an estimation of the Discrete Dynamics in Nature and Society 15 transmission rate with less computational burden than the *en, the above equations are calculated under the least-squares algorithm. However, the treatment of raw constraints: data may require the application of denoising or filtering � � � procedures in order data to be smoothen before processing, K ti � � K ti, ti+1 � � K ti, ti+2 �, which is a future research line It is of interest to extend the (A.4) r t � � �r t , t � � �r t ,�t �, obtained modeling results with use of vaccination controls i i i+1 i i+2 [54] and to evaluate results with future evolution of the pandemic under vaccination data. which lead to

− Tir() ti 5. Conclusions I ti + Ti �I ti ��1 − e � K ti � � − Tir() ti *is paper has compared an SIR epidemic model with I ti � − I ti + Ti �e recruitment, demography, and disease-related mortality (A.5) with a parallel description through an “ad hoc” logistic − Tir() ti equation, which is time varying. *e logistic equation I ti + 2Ti �I ti + Ti ��1 − e � version of the model allows easily to interpret it via the � − Tir() ti Malthusian parameter (related to the exponential rate of I ti + Ti � − I ti + 2Ti �e variation of the solution), the carrying capacity (related to the maximum values which are reached), and the trans- so that mission rate, whose value is related to the absolute value of the quotient of the absolute value of the Malthusian pa- I t + T �I t � I t + T � − I t + 2T �e− Tir() ti i i i × i i i i � 1 rameters and the carrying capacity. To address the process − T r t I t � − I t + T �e i ()i I ti + 2Ti �I ti + Ti � of picking up the data registration in order to estimate the i i i above parameters, an adaptive sampling law is proposed (A.6) which updates the discrete data accordingly to their rate of variation leading to an optional nonuniform disposal of so that samples to calculate the, in general, sample-dependent Malthusian parameter and carrying capacity. *e trans- − 2Tir() ti − Tir() ti aie − bie + ci � 0, (A.7) mission rate is then calculated from the above ones and can be averaged on the various intervention phases of quar- whose roots are antine interventions, interventions absence, and pro- �� grammed de-escalated decisions. Numerical examples 2 bi ± bi − 4aici which illustrate the methodology are given and discussed e− Tir() ti � , (A.8) based on COVID-19 parameterizations. 2ai where Appendix a � I t + T �I t + 2T �, A Calculations Related to Step 3 of Algorithm 1 i i i i i 2 � bi � I ti + Ti � + I ti �I ti + 2Ti �, (A.9) Take two consecutive updated estimates ti+2 of the sampling instant ti+2, once the current sampling instant and sampling ci � I ti �I ti + Ti �, period ti and Ti are known so that ti+1 � ti + Ti is known: leading to �t � t + T � t + 2T . (A.1) i+2 i+1 i i i 2 2 bi − 4aici � I ti + Ti � − I ti �I ti + 2Ti �� ≥ 0 (A.10) � *en, rewrite (58) for ti+1 and ti+2 such that one-step ahead and two-step ahead estimates of K(ti) and r(ti) are so that the roots of (A.6) are real. *e positive root leads to � the Malthusian parameter at t : calculated “a posteriori” after ti+2 has occurred as follows i from the registered measures I(ti), I(ti+1) � I(ti + Ti), and � 1 2ai I(ti+ ) � I(ti+ + Ti) � I(ti + 2Ti): �� 2 1 r ti � � ln , (A.11) Ti 2 bi + bi − 4aici − Ti�r() ti,ti+1 I ti+1 �I ti ��1 − e � � (A.2) and then the carrying capacity is calculated from (A.5). K ti, ti+1 � � , − Ti�r() ti,ti+1 I ti � − I ti+1 �e Remark A.1. If past measures of the infection are used − T r t ,�t � i�i i+2 � Iti+2 �I ti+1 ��1 − e � instead of either a prediction or a selection of provided � � (A.3) K ti, ti+2 � � . registered values after the current time instant ti, one uses − T r t ,�t � i�i i+2 � I ti+1 � − Iti+2 �e the subsequent equations: 16 Discrete Dynamics in Nature and Society

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