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

Authors: Manuel De la Sen; Asier Ibeas
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

The 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. The 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. Therefore, the logistic version of the epidemics’ description is attractive since it offers an easy interpretation of the data evolution especially when the pandemic outbreaks. The SIR model includes recruitment, demography, and mortality parameters, and the total population minus the recovered population is not constant though time. This 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. The 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.


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## Body

## 1. Introduction

Epidemic mathematical models under different formal frameworks are of major interest along the last years [1, 2]. Some of such models rely on the dynamic aspects of the disease evolution and are stated in terms of differential, difference, and differential/difference hybrid equations, dynamic systems, and/or control theory (so as to deal with vaccination and treatment intervention strategies) [3–18], computation tools [7], information theory, [19–22], etc. On the contrary, some of the above invoked modelling techniques combine several analysis and design tools. See, for instance, [19, 20] and some references therein are related to designs, the optimality of the controls, and the use of fractional- order models. There are also epidemic models available which consider the spatial-temporal spread of the disease. For instance, several neighboring domains with diversity of populations are considered in [23] concerning the HIV/AIDS spread. It is considered that different levels of awareness are present in different groups which might affect, for instance, to the hardness of the actions to be taken for transmission prevention. In [24], a Latin hypercube sampling method is discussed to calculate relevant reproduction numbers from distributions of some relevant model-related parameters.It can be pointed out that continuous-time models are widely used since they are more tractable mathematically than the discrete models while having a direct physical interpretation [25]. In particular, the positivity of the solution under nonnegative initial conditions can be proved from the differential system describing the model without the use of extradiscretization adjusting parameters. Furthermore, it is a direct task to pick up values from the solution at appropriate sampling points (for instance, daily or weekly) of the continuous-time solution of the model, provided that such an information is needed for storing statistic data or in order to decide the implementation of some vaccination or other alternative public intervention controls such as partial quarantines or confinements. On the contrary, the use of discrete models has been shown to be specifically useful in the context of dissemination of disease-related information such as its current state or evolution tendency, for instance, through social networks. That associated information can help to prevent or to reduce possible foreseen outbreaks [26].A basic goal is to investigate and predict the evolution of infectious diseases as well as to focus on how the public interventions, for example, quarantines or vaccination and treatment controls can mitigate their outbreaks and uncontrolled propagations. See, for instance, [1, 2] and [11, 13, 15, 17, 18, 27, 28]. A beneficial effect of quarantine interventions on a part of a targeted subpopulation, in particular, either fractions or the susceptible subpopulation or fractions of the infectious one is their “removal” from its associate compartment. In this way, the most apparent effect in the disease evolution through time is a reinitialization of the corresponding trajectory solution under fewer numbers of contagious contacts. A second beneficial effect is that the disease transmission rate decreases to more moderate levels as a result of the reduction in the contagion contacts. Those concerns have been seen to be relevant concerning the evolution of the current COVID-19 pandemic. See, for instance, [29–46] and some of the references therein. In particular, an SEIR model is proposed in [41] for the COVID-19 pandemic. Such a model includes delayed resusceptibility caused by the infection. Also, a kind of autoregressive model average model (ARMA), so-called an ARIMA model, for prediction of COVID-19 is presented and simulated in [42] for the data of several countries. The effects of different phases of quarantine actions in the values of the transmission rate are studied in [43], see also [46], for a comparative and exhaustive discussion of related simulated numerical results on the COVID-19 outbreak in Italy.The related existing bibliography is abundant and very rich including a variety of epidemic models with several coupled subpopulations included as an elementary starting basis of the susceptible, the infectious, and the recovered ones in the simpler SIR epidemic models. Further generalizations lead to the so-called SEIR models which include the exposed subpopulation, that is, those who do not have external symptoms yet, as a new subpopulation. More complex models, such as SEIADR-type models, which include the asymptomatic infectious and the infective dead subpopulations, have been designed and analyzed, for instance, Ebola disease [15, 16].On the contrary, it is well known that the celebrated logistic equation of Velhulst is a model of population growth which is parameterized by two constant parameters, namely, the Malthusian parameter and the carrying capacity, see, for instance, [47]. The main objective of this paper is to propose and to link a SIR (susceptible-infectious-recovered) with recruitment and demography to a proposed time-varying logistic equation parameterized by a time-varying carrying capacity and a time-varying Malthusian parameter and whose solution is attractive and of simple interpretation. The methodological analysis key point is to compare the current SIR epidemic model with eventual recruitment, demography, and disease-related mortality parameters to the so-called nominal (or reference) case, where those parameters are zeroed and, in addition, the recovered are considered constant being equal to their initial values along with the time interval of interest. The total reference population is also constant due to the absence of recruitment, demography, and disease-related mortality. It turns out that the above nominal logistic-type model can work efficiently along certain transient periods of time (of the order of weeks) when the infection is growing up very fast, but there are no still recovered subpopulation, and the total population does not vary significantly. The nominal case might be described, equivalently, by a time-invariant logistic equation which has a unique solution for the infectious depending on the initial conditions and on two constant (reference Malthusian and reference carrying capacity) parameters which depend on the transmission rate, the recovery rate, and the initial levels of total population and immune subpopulation. In particular, the Malthusian parameter quantifies the rate of either growth or decrease of the infection and the (infection) carrying capacity fixes the maximum level of such an infection along its growing time period (say, the maximum of the infectious subpopulation).Then, a most complex current SIR model is described through the abovementioned time-varying logistic equation whose solution is characterized as that of the nominal one plus an error function which depends on the new added parameters (recruitment rate, demography rate, and disease-related mortality). A practical way of monitoring the time-varying logistic equation in a simple way is to reinitialize it by simple resetting of initial conditions, when necessary, and to keep its parameterization in operation along time intervals of appropriate moderate adaptive lengths before the next resetting. For each updated parameterization, the logistic equation is run as time-invariant until its next resetting. In this way, the current Malthusian parameter and carrying capacity are modelled by piecewise constant functions whose values are updated, in general, at a sequence of sampling instants subject, in general, to nonuniform intersample periods. Such periods are adapted to the infection variation so that the larger the infection variation, the smaller the intersample period.The paper is organized as follows. Section2 states the SIR (susceptible subpopulation S-infectious subpopulation I-recovered subpopulation R) epidemic model with population recruitment, demography, and disease mortality and establishes and proves its positivity and boundedness properties. Section 3 is split into several sections. The first one rewrites the model under the equivalent form of a time-varying logistic equation parameterized by a the Malthusian parameter, which defines the exponential order of the solution form, and a carrying capacity which is related to the maximum attainable infection level along its growing period. Both parameters are, in general, time-varying functions which are defined based on the primary model parameters and the values of the subpopulations through time. Section 3.2 discusses the description of a simplified version of the above epidemic model to the light of a time-invariant logistic equation parameterized by constant values of the Malthusian parameter and the carrying capacity. It also assumes that the increment of the total population over the recovered subpopulation remains constant. Such a time-invariant logistic equation is referred to as the nominal (or reference) logistic equation. The simplified version of the SIR model, rewritten equivalently as the nominal logistic equation, is recruitment, demography, and mortality free. Those simplifications are reasonable along periods lasting some months since the disease-associated mortality is usually small compared to the total population, while the recruitment and demography rates are also usually small related to the total population amounts along short periods of time compared with the species life expectation. An important motivating result of the problem approach is that the carrying capacity is the quotient between the absolute Malthusian parameter and the carrying capacity. In this way, the nominal logistic equation characterizes a constant disease transmission rate for the simplified SIR model of no public interventions 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 more complex model which also depends on such public interventions, in the case where they are performed. Section 3.3 obtains in a closed form the solution of the time-varying logistic equation related to the nominal one and computes also the worst-case of the error through time among both of them. It also discusses a technique to evaluate the current solution along with a set of samples which are updated through adaptive sampling laws which decrease the intersampling period as the infection variation between consecutive samples increases and vice versa. Finally, Section 3.4 gives a pseudocode algorithm to implement the above ideas to estimate the Malthusian and carrying capacity time-varying parameters from registered infection data on sequences of time instants. With those parameters, the time-carrying capacity is also sample-dependent calculated and its averaged value along the time period of each public intervention (quarantines, confinements, de-escalating postquarantine phases, or public intervention void) is also estimated. Section 4 is devoted to perform some numerical examples as well as to discuss the obtained results with special emphasis on the influences on the infection propagation of partial or total quarantines of some or all the subpopulations or de-escalating phases after quarantines. Model parameterizations with available recorded data related to the COVID-19 pandemic are used for the testing process and the related transmission rate obtained. Finally, Section 5 ends the paper. Some auxiliary calculations related to the estimations of the Malthusian parameter and the carrying capacity concerned with a relevant step of the estimation algorithm of Section 3 4 are given in Appendix A.


## 2. The SIR Epidemic Model and Its Positivity and Boundedness Properties

Consider the following SIR (susceptibleS -infectious I-recovered R) epidemic model with demography:(1)S˙t=ν−μ+βItSt,S0=S0,(2)I˙t=βSt−γ−μIt,I0=I0,(3)R˙t=1−ργIt−μRt,R0=R0,subject to arbitrary nonnegative finite initial conditions, i.e., minS0,I0,R0≥0, where β is the disease transmission rate μ and ρ∈0,1 are, respectively, the natural and disease-related mortality rates, and ν≥μ is the population recruitment rate. The total population is Nt=St+It+Rt whose derivative with respect to time is obtained by summing-up (1)–(3):(4)N˙t=ν−μNt−ργIt,N0=N0=S0+I0+R0.Denote in the followingR+=z∈R:z>0, R0+=z∈R:z≥0=R+∪0. The subsequent result relies on the positivity and boundedness of all the solution of (1)–(3) under any given finite nonnegative initial conditions.Theorem 1.
Model (1)–(3) is positive, in the sense that ∀t∈R0+ and globally stable, that is, the solutions of (1)–(3) are bounded for all time for any given finite nonnegative initial conditions. As a result, it holds that limsupt⟶∞sup0≤ξ≤tIξ≤ν/γ.Proof.
The solution of (2) is It=eβ∫0tSξdξ−γ+μtI0≥0; ∀t∈R0+, since I0≥0. Therefore, I:R0+⟶R0+, for any given nonnegative initial conditions. On the contrary, since S0≥0 and S:R0+⟶R0+ are everywhere continuous (since it is everywhere time-differentiable), it is nonnegative until St1=0, for some t1≥0, in the event that such a t1 exists. However, by inspecting (1), one gets S˙t1=ν≥0 (even if the model is recruitment-free, i.e., if ν=0). Then, St1+≥0. Therefore, no t∈R0+ can exist such that St<0. Therefore, S:R0+⟶R0+ for any given nonnegative initial conditions. On the contrary, the solution of (3) is nonnegative in any interval 0,t2 under any nonnegative initial conditions such that Rt2=0, but then R˙t2=1−ργIt2≥0 so that no t∈R0+ exists such that Rt<0 and R:R0+⟶R0+. It has been proved that the solutions of (1)–(3) are always nonnegative for any given nonnegative initial conditions. The total population has, from (4), the following solution equation:(5)Nt=St+It+Rt=e−μtR0+∫0te−μt−ξν−γIξdξ≥0,∀t∈R0+,so that one has, for any t0,t≥t0∈R0+, that(6)∫t0te−μt−ξIξdξ≤∫0te−μt−ξIξdξ≤1γe−μtR0+νμ1−e−μtso that, for any given finite θ∈R+, one has by using the mean value theorem that, for any strictly increasing sequence tnn=0∞⊂R0+ and some sequence ςnn=0∞⊂tn+1,tn,(7)∫tntn+θe−μtn−ξIξdξ=θ1−e−μθnμIςn≤νγμ,so that(8)lim supn⟶∞Iςn≤νγθ,∀t∈R0+.
SinceI:R0+⟶R0+ is continuous and θ finite, one concludes that limsupn⟶∞Itn≤ν/γθ, and under the same reasoning, one gets limsupt⟶∞It≤ν/γθ. Since I:R0+⟶R0+ is continuous, it cannot be unbounded on any finite time interval 0,t1 so that the above ultimate boundedness property as time tends to infinity also implies the boundedness of I:R0+⟶R0+ in 0,∞.
For any nonnegative initial conditions and thenI:R0+⟶R0+ is bounded. This implies from (5) that(9)Nt≤N0+1μν+γsupt≥0It<+∞.
On the contrary, one has from (2) that the (empty or nonempty, connected, or nonconnected) real interval(10)US=t∈R0+:St>γ+μβ=∪i∈Z0+ti,ti+1has a Lebesgue measure 0≤LbUS<+∞ (zero if it is empty) and finite since otherwise, limt⟶∞It=+∞ which contradicts the boundedness of I:R0+⟶R0+ according to (8). Then, the complementary set US in R0+ is(11)R0+\US=t∈R0+:St≤γ+μβ=∪i∈Z0+ti,ti+1¯=∩i∈Z0+ti,ti+1,which has 0≤LbR0+\US=+∞ and, therefore, S:R0+⟶R0+\US is bounded while S:R0+⟶US cannot 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,I,R,N:R0+⟶R0+ and S,I,N:R0+⟶R0+ are bounded, then(12)0≤Rt=Nt−St−It<+∞,∀t∈R0+,and R:R0+⟶R0+ is bounded. The theorem has been fully proved.


## 3. The Epidemic Model in a Logistic Equation Form

### 3.1. “Ad hoc” Time-Varying Logistic Equation

Verhulst´s or logistic equation is a celebrated model of population growth which was proposed in 1838 by Pierre Verhulst (1804–1849). It can also be considered in an “ad-hoc” time-varying version to describe the epidemic model of the above section under a certain modification of the basic logistic equation after replacingSt=Nt−It−Rt in (2). Thus, one gets the following infectious solution with the contribution to its dynamics of the susceptible and recovered subpopulations:(13)I˙t=βNt−It−RtIt−γ+μIt=−βI2t+βNt−Rt−γ+μIt=−I2tatct+Itat,∀t∈R0+,where atct=1/β, and(14)at=1βNt−Rt−γ−μ,ct=1βat=Nt−Rt−γ+μβ,∀t∈R0+.The subsequent result will be then used related to the interpretation of the epidemic models (1)–(3) as a time-varying logistic equation related to a nominal time-invariant one, in the case that St+It≥S0+I0 for all t∈R0+.Proposition 1.
The following results hold for a normalized model (1)–(3) at the initial time, that is, if N0=1(i)
The inequality(15)St+It=Nt−Rt≥S0+I0=N0−R0is guaranteed for a givent∈R+ if(16)∫0te−μt−ξν−γIξdξ≥1−e−μtN0−R0.and the inequality is reversed, namely, for a givent∈R+,(17)St+It=Nt−Rt≤S0+I0=N0−R0if(18)∫0te−μt−ξν−γIξdξ≤1−e−μtN0−R0.(ii)
The above inequality is guaranteed in0,t for any given t∈R+ if(19)sup0≤ξ≤tIξ≤ν−μγ,so that it is guaranteed in 0,∞ as well. The above inequality also guarantees that liminft⟶∞St+It≥S0+I0 is fulfilled since(20)lim supt⟶∞sup0≤ξ≤tIξ≤ν−μγ.Proof.
The solution to (3) is(21)Rt=e−μtR0+γ1−ρ∫0te−μt−ξIξdξ,∀t∈R0+,which together with (5) yields(22)Nt−Rt=e−μtN0−R0+∫0te−μt−ξν−γIξdξ,∀t∈R0+.
Note thatSt+It=Nt−Rt≥S0+I0=N0−R0 for some t∈R0+ if and only if(23)∫0te−μt−ξν−γIξdξ≥1−e−μtN0−R0,and the first part of Property (i) is proved. The reversed inequality and associated condition are proved “mutatis-mutandis.” On the contrary, the above inequality holds on 0,t if(24)γ1−e−μtμsup0≤ξ≤tIξ≤ν1−e−μtμ−1−e−μtS0+I0,for a normalized model at the initial conditions, that is, for S0+I0≤N0=1 if sup0≤ξ≤tξ≤ν−μ/γ. Also, liminft⟶∞St+It≥S0+I0 if limsupt⟶∞∫0te−μt−ξIξdξ≤1/γν/μ−S0−I0 which is guaranteed if limsupt⟶∞∫0te−μt−ξIξdξ≤ν−μ/γμ which is guaranteed if limsupt⟶∞sup0≤ξ≤tIξ≤ν−μ/γ. Property (ii) has been proved.From the nonnecessary normalized model at the initial conditions, one has the following result which follows directly from (24) in the proof of Proposition 1:Proposition 2.
St+It≥S0+I0 holds for all t∈0,ta, provided that S0+I0≤ν/μ if(25)sup0≤t≤taIt≤1γν−μS0+I0.


### 3.2. Nominal Logistic Equation

Assume that the epidemic models (1)–(3) are considered without recruitment, demography, and mortality, that is, ν=μ=ρ=0. Thus, one has from (4) and (13) and (14) that N˙t≡0 so that Nt=N0; ∀t∈R0+. It is also assumed that the total population excess over the recovered subpopulation remains constant. This assumption keeps the logistic equation time-invariant, for instance, identical to their initial condition excess N0−R0. The above hypotheses is reasonable along periods of several months over which the above population/recovered subpopulation excess does not vary substantially in practice. Thus, the parameterization of the (so-called nominal or reference) logistic equation is time invariant accordingly to(26)at≡ar=1βN0−R0−γ,ct≡cr=1βar=N0−R0−γβ,and (13) has the following nominal expression parameterized by (26):(27)I˙rt=Irtar−Ir2tarcr=rIrtKr−IrtKr,Ir0=I0,where(28)r=1ar=βN0−R0−γ,is the Malthusian parameter, namely, the rate of maximum infection growth which can be positive, negative, or zero, and(29)Kr=rarcr=cr=N0−R0−γβis the carrying capacity, namely, the maximum sustainable infectious subpopulation. The normalized by Kr differential equation (27) becomes for(30)I˙rn=rIrnt1−KrInrt,Irn0=Irn0=Ir0Kr,where Irnt=Irt/Kr and whose solution is the sigmoid function:(31)Irnt=11+Irn0−1−1e−rt=IrnoertIrnoert+1−Irno=Ir0ertIr0ert−1+Kr=Ir0Ir01−e−rt+Kre−rt.Then, the unnormalized infectious evolution is(32)Irt=KrIrnt=KrIr0Ir01−e−rt+Kre−rt=crIr0Ir01−e−t/ar+cre−t/ar=crIr0Ir0+cr−Ir0e−t/ar,and, for Ir0≠0,(33)Irt=Kr1+Kr/Ir0−1e−rt=Kr1+ertMd−tif etMdr=Kr/Ir0−1, namely, if(34)tMd=1rlnKr−Ir0Ir0=arlncr−Ir0Ir0=1βN0−R0−γlnN0−R0−β−1γIr0−1.It also holds that the maximum value ofI˙rt, which is the zero of I¨rt (or the growth inversion point), takes place at t=tMd, equation (34), from (33). This is easily seen from the nominal version of the logistic equation (13) with It≡Irt, at≡ar, and ct≡cr, which leads to(35)I¨rt=I˙rtar1−2Irtcr=0,which implies that Irt=cr/2=Kr/2, if I˙rt≠0, which happens at t=tMd from (33). Therefore, tMd is the growth inversion point time instant. On the contrary, from (27), there is no relative finite maximum value of Irt since I˙rt=0 holds for t=∞ if r>0. We have, as basic relations for the key parameters of the nominal logistic equation associated with (1)–(3), the Malthusian parameter (28), the carrying capacity (29), and the growth inversion point (34).The following result of interest holds.Proposition 3.
The nominal logistic equation has the following properties:(i)
It satisfies the positivity and boundedness conditions of Theorem1.(ii)
The disease transmission rate satisfies the constraintβ=r/Kr so that either β>0 and Kr is finite or β=0 with r=−γ and Kr=∞. If β>γ/N0−R0 (respectively, β<γ/N0−R0), then r>0 (respectively, r<0). If β=γ/N0−R0, then r=0.(iii)
The Malthusian parameter and the carrying capacity increase (respectively, decrease) as the disease transmission rate increases (respectively, decreases). The same property holds with respect toβ for the growth inversion point time instant if β<1.Proof.
Property (i) is a corollary of Theorem1 for the nominal logistic equation which is a particular parameterization of the nominal one. Then, note that the combination of (28) and (29) yields the constraint:(36)Kr=βN0−R0−γβ=rβ,which implies that the transmission rate satisfies β=r/Kr. The remaining results of Property (ii) follow directly from (28) and (29). On the contrary, note, from (28) and (29) that(37)rβ2−rβ1=N0−R0β2−β1,Krβ2−Krβ1=γβ2−β1β1β2,and also one has from (34) that(38)dtMddβ=1βN0−R0−γ×γβ−2Ir0N0−R0−γβ−1−Ir0−N0−R0βN0−R0−γ2×lnN0−R0−γβ−1Ir0−1so that dtMd/dβ=oβ−3. Property (iii) has been proved.Remark 1.
Note from (27) that if r>0(that is if the transmission rate is large enough to satisfy β>γ/N0−R0) and Ir0<Kr, then Irt is strictly increasing until it reaches its maximum value Kr. If r turns to a negative value, that is if β<γ/N0−R0, then the Irt reverses its growth turning to a strictly decreasing profile. The effective way of achieving this behaviour is to decrease the number of effective infectious contacts to the susceptible either by social discipline or eventually via quarantines or confinements.


### 3.3. Comparison Results of the Logistic Equation with Its Nominal Version

Note that the time-varying logistic equation (13) may be compared to the nominal one (27) by expressing the errors between both current and nominal Malthusian parameters and carrying capacities. For that purpose, define parametrical errors:(39)a˜t=at−ar=1βNt−Rt−γ−μ−1βN0−R0−γ,c˜t=ct−cr=Nt−Rt−γ+μβ−N0−R0−γβso that(40)rt=1at=1ar+a˜t=1ar+r˜t=r+r˜t,Kt=ct=cr+c˜t=Kr+K˜t,where(41)r˜t=rt−rr=−a˜tar+a˜t=βNt−Rt−N0+R0−μβNt−Rt−γ−μ,K˜t=c˜t=ct−cr=Nt−Rt−γ+μβ−N0−R0−γβ,b˜t=1atct−1arcr=−arc˜t+cra˜t+c˜ta˜tarcr+a˜tcr+c˜tar+c˜ta˜t.Thus, (13) can be rewritten as follows:(42)I˙t=Itat1−Itct=r+r˜tIt1−ItKr+K˜t=r+r˜tIt1−1Kr−K˜tKr+K˜tIt=rIt1−ItKr+ft,where(43)ft=rK˜tKr+K˜tI2t+r˜tIt1−1Kr−K˜tKr+K˜tIt.Then, one has that(44)It=I0+r∫0tIξ1−IξKrdξ+∫0trK˜ξKr+K˜ξI2ξdξ+∫0tr˜ξIξ1−1Kr−K˜ξKr+K˜ξIξdξ.The first right-hand side term of (45) is the solution of the nominal logistic equation (i.e., (32) or (33)). The second one is the integrated disturbance function fς on 0,t, equation (44), namely, ∫0tfξdξ due to the error between the current and nominal solutions. In view of the nominal solution expression (32), one has that (45) can be expressed equivalently as follows:(45)It=crI0I0+cr−I0e−t/ar+∫0tfξdξ=crI0I0+cr−I0e−t/ar+∫0trK˜ξKr+K˜ξI2ξdξ+∫0tr˜ξIξ1−1Kr−K˜ξKr+K˜ξIξdξ,which can be rewritten for each time instant t from a previous time instant t−Tt as follows:(46)It=crIt−TtIt−Tt+cr−I0e−Tt/ar+∫t−TttrK˜ξKr+K˜ξI2ξdξ+∫t−Tttr˜ξIξ1−1Kr−K˜ξKr+K˜ξIξdξ.Assume that, for a givent0∈R+ and each t≥t0∈R+, Tt is selected so that K˜t and r˜t are almost constant with absolute errors less than positive real constants εr˜ and supt≥0K˜t≤εK˜<Kr (since the current carrying capacity Kt=Kr+εK˜ is positive) in t−Tt,t. Thus, one has(47)−rTtεK˜Kr−εK˜supt−Tt≤ξ≤tI2ξ≤∫t−TttrK˜ξKr+K˜ξI2ξdξ≤rTtεK˜Krsupt−Tt≤ξ≤tI2ξ,(48)εr˜εK˜Ttinft−Tt≤ξ≤tIξKr1+εK˜+εK˜KrKr−εK˜supt−Tt≤ξ≤tIξ≤∫t−Tttr˜ξIξ1+K˜ξKr+K˜ξ−1KrIξdξ≤εr˜εK˜Ttsupt−Tt≤ξ≤tIξ1+1Krsupt−Tt≤ξ≤tIξ,provided that supt≥0K˜t≤εK˜<minKr,1. On the contrary, if the incremental carrying capacity related to the nominal one is large enough to satisfy inft≥0K˜t≥Kr/Kr−1, which implies also that εK˜>1 while it is also subject to Kr>1, then(49)K˜tKr+K˜t−1Kr≤Kr−1K˜t−KrKrKr+K˜t≤εK˜Kr−1−KrKr2so that the upper bound in (47) becomes modified as follows:(50)εr˜εK˜Ttsupt−Tt≤ξ≤tIξ1+εK˜Kr−1−KrKr2supt−Tt≤ξ≤tIξ,Then, one has under the upper bound of (47), for the case when supt≥0K˜t≤εK˜<Kr if ε=minεr˜,εK˜, that(51)It=KrIt−TtIt−Tt+Kr−It−Tte−Ttr+εK˜Ttsupt−Tt≤ξ≤tIξεr˜+Krr−1+εK˜Kr−1Kr2supt−Tt≤ξ≤tIξ+oεTt,and one also has, for the case when K˜tinft≥0≥Kr/Kr−1, according to (49), that(52)It=KrIt−TtIt−Tt+Kr−It−Tte−Ttr+εK¯Ttsupt−Tt≤ξ≤tIξεr˜+r+εr˜Krsupt−Tt≤ξ≤tIξ+oεTt.On the contrary, note that∫t−Tttr˜ξIξ1−1/Kr−K˜ξ/Kr+K˜ξIξdξ is slowly time varying if εTt is sufficiently small which is a reasonable approximation for a sufficiently small interval Tt from the time instant where the initial infection values are picked up until its current calculation. The reason is that this implies in addition that ε is also sufficiently small which leads to a small variation of Nξ−Rξ along t−Tt,t for a sufficiently small interval Tt with that additional advantage that the demography-related parameters μ and ν are also as small as suited. These considerations suggest that, by choosing the central interval point for such an evaluation of the incremental solution, one gets as an alternative expression to the infectious subpopulation evolution through time that(53)It=Kt−nTtIt−nTtIt−nTt+Kt−nTt−It−nTte−nTtrt−nTt+δεnTt,nTt,for given n∈Z+ and Tt∈R+ leading to a performed calculation along a time interval t−nTt,t. A further improvement might be performed without difficulty over a sequence, or eventually a finite set IS≡tii=0n of sampling instants by using recorded data with a monitored adaptation of the intersampling interval to the rate of variation of the infection curve in such a way that t0=0, ti+1=ti+Ti, and the intersample period sequence ISP≡Tii=0n are being updated as follows:(54)Ti=TM,ifTia>TM,Tia,ifTia∈Tm,TM,Tm,ifTia<Tm,where there are a number of possibilities to generate the auxiliary tentative sampling period Tia from the classical adaptive sampling background literature (see [48–51] and related references therein) as, for instance, the three ones which follow:(55)Tia=C1C3+C2I˙ti,(56)Tia=C1C3+C2Iti−Iti−1/Ti−1,(57)Tia=C1C3+C2Iti−Iti−1.For some prescribed design real constantsC1,C2>0, C3≥0, and TmTM>Tm with TM−Tm being sufficiently small in accordance with the above solution approximation considerations. The adaptive sampling laws (54) and (55) evaluate explicitly the absolute value of time derivative, while the adaptive sampling law (54), which subjects to (56), evaluates it by finite increments. The sampling period is constant (thus, it is nonupdated through time) if C2=0, Tia=Tm=C1/C3. Thus, one gets from (53) the following relation in-between two consecutive sampling instants:(58)It≅KtiItiIti+Kti−Itie−Tirti,∀t∈ti,ti+1,∀ti∈IS.


### 3.4. Estimation Algorithm of the Malthusian Parameter and the Carrying Capacity along Time

The parameters which characterize the current logistic equation are time varying, but it proposed the following simple algorithm (Algorithm1) expressed in general pseudocode style. It is assumed 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-escalation of quarantines. According to the evaluation of the pandemic growth force, the algorithm identifies from recorded data of the infection curve the Malthusian parameters and the carrying capacity, and then it estimates the averaged disease transmission rate per intervention phase.Algorithm 1:Evaluation of the Malthusian parameter and the carrying capacity.
(1)
Step 0: Chooset0≥0 and the initial intervention Phase Ph0 to start to run the procedure.(2)
Smooth the recorded infection curveIt, 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)
DefineCi for i=1,2,3 and Tm,TM to then run (53)–(55) to generate the sequences IS and ISP of sampling instants and inter-sampling 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 instantti∈IS, locate the intervention phase Phj to which it belongs. Calculate the inter-sampling 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: ReadIti+1 from recorded data(6)
Step 3: The Malthusian parameter and the carrying capacity at timeti are calculated “a posteriori” at time ti+2=t+i2Ti from the registered Iti, Iti+1=Iti+Ti and Iti+2Ti evaluated at the current and next time instants ti and ti+1=ti+Ti and an “a priori” estimation t^i+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 theβti,ti+1=rti,ti+1/Kti,ti+1 (see Proposition 3 (ii)).(8)
Step 5: Ifti+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 PhasePhjβPhj=1/lJ∑ti,ti+1∈Phjβti,ti+1, make j⟵j+1 and if the whole number of checked phases is unfinished GoTo Step 1. Else GoTo Step 7(10)
Step 7: EndNote that the calculation process is repeated for all the pairs of samplesti,ti+1 and ti+1,ti+2. The transmission rate is calculated for each such a pair from Proposition 3 (ii).Remark 2.
For the pairsti,ti+1 with sampling instants belonging to consecutive phases, we can include the option of considering their associated transmission rates for both phases in Algorithm 1 or for none of them as in the above given version form of Step 6.Remark 3.
Note that, in Appendix A, two ways of implementing Step 3 of Algorithm1 to estimate the carrying capacity and the Malthusian parameter are given:(a)
The first one involves the use of infection recorded data after the current evaluation time instant.(b)
The second one is summarized in Remark A.1, and it involves the only use of past data. Concerning the first method, one has to point out that the use of predictions can be replaced alternatively by the use for a set of previously recorded data corresponding to the current time instant under checking by the algorithm. Note that both methods can involve the selection of time instants being an integer multiple of the minimum period of supplied data, typically, one day. In this case, the adaptive sampling law just selects a time-varying sampling period which is an integer multiple integer of the mentioned minimum sampling period reported in the recorded data. A particular case is simply to use the minimum sampling period as a constant running sampling period while removing the adaptive sampling law design from Algorithm1.


## 3.1. “Ad hoc” Time-Varying Logistic Equation

Verhulst´s or logistic equation is a celebrated model of population growth which was proposed in 1838 by Pierre Verhulst (1804–1849). It can also be considered in an “ad-hoc” time-varying version to describe the epidemic model of the above section under a certain modification of the basic logistic equation after replacingSt=Nt−It−Rt in (2). Thus, one gets the following infectious solution with the contribution to its dynamics of the susceptible and recovered subpopulations:(13)I˙t=βNt−It−RtIt−γ+μIt=−βI2t+βNt−Rt−γ+μIt=−I2tatct+Itat,∀t∈R0+,where atct=1/β, and(14)at=1βNt−Rt−γ−μ,ct=1βat=Nt−Rt−γ+μβ,∀t∈R0+.The subsequent result will be then used related to the interpretation of the epidemic models (1)–(3) as a time-varying logistic equation related to a nominal time-invariant one, in the case that St+It≥S0+I0 for all t∈R0+.Proposition 1.
The following results hold for a normalized model (1)–(3) at the initial time, that is, if N0=1(i)
The inequality(15)St+It=Nt−Rt≥S0+I0=N0−R0is guaranteed for a givent∈R+ if(16)∫0te−μt−ξν−γIξdξ≥1−e−μtN0−R0.and the inequality is reversed, namely, for a givent∈R+,(17)St+It=Nt−Rt≤S0+I0=N0−R0if(18)∫0te−μt−ξν−γIξdξ≤1−e−μtN0−R0.(ii)
The above inequality is guaranteed in0,t for any given t∈R+ if(19)sup0≤ξ≤tIξ≤ν−μγ,so that it is guaranteed in 0,∞ as well. The above inequality also guarantees that liminft⟶∞St+It≥S0+I0 is fulfilled since(20)lim supt⟶∞sup0≤ξ≤tIξ≤ν−μγ.Proof.
The solution to (3) is(21)Rt=e−μtR0+γ1−ρ∫0te−μt−ξIξdξ,∀t∈R0+,which together with (5) yields(22)Nt−Rt=e−μtN0−R0+∫0te−μt−ξν−γIξdξ,∀t∈R0+.
Note thatSt+It=Nt−Rt≥S0+I0=N0−R0 for some t∈R0+ if and only if(23)∫0te−μt−ξν−γIξdξ≥1−e−μtN0−R0,and the first part of Property (i) is proved. The reversed inequality and associated condition are proved “mutatis-mutandis.” On the contrary, the above inequality holds on 0,t if(24)γ1−e−μtμsup0≤ξ≤tIξ≤ν1−e−μtμ−1−e−μtS0+I0,for a normalized model at the initial conditions, that is, for S0+I0≤N0=1 if sup0≤ξ≤tξ≤ν−μ/γ. Also, liminft⟶∞St+It≥S0+I0 if limsupt⟶∞∫0te−μt−ξIξdξ≤1/γν/μ−S0−I0 which is guaranteed if limsupt⟶∞∫0te−μt−ξIξdξ≤ν−μ/γμ which is guaranteed if limsupt⟶∞sup0≤ξ≤tIξ≤ν−μ/γ. Property (ii) has been proved.From the nonnecessary normalized model at the initial conditions, one has the following result which follows directly from (24) in the proof of Proposition 1:Proposition 2.
St+It≥S0+I0 holds for all t∈0,ta, provided that S0+I0≤ν/μ if(25)sup0≤t≤taIt≤1γν−μS0+I0.


## 3.2. Nominal Logistic Equation

Assume that the epidemic models (1)–(3) are considered without recruitment, demography, and mortality, that is, ν=μ=ρ=0. Thus, one has from (4) and (13) and (14) that N˙t≡0 so that Nt=N0; ∀t∈R0+. It is also assumed that the total population excess over the recovered subpopulation remains constant. This assumption keeps the logistic equation time-invariant, for instance, identical to their initial condition excess N0−R0. The above hypotheses is reasonable along periods of several months over which the above population/recovered subpopulation excess does not vary substantially in practice. Thus, the parameterization of the (so-called nominal or reference) logistic equation is time invariant accordingly to(26)at≡ar=1βN0−R0−γ,ct≡cr=1βar=N0−R0−γβ,and (13) has the following nominal expression parameterized by (26):(27)I˙rt=Irtar−Ir2tarcr=rIrtKr−IrtKr,Ir0=I0,where(28)r=1ar=βN0−R0−γ,is the Malthusian parameter, namely, the rate of maximum infection growth which can be positive, negative, or zero, and(29)Kr=rarcr=cr=N0−R0−γβis the carrying capacity, namely, the maximum sustainable infectious subpopulation. The normalized by Kr differential equation (27) becomes for(30)I˙rn=rIrnt1−KrInrt,Irn0=Irn0=Ir0Kr,where Irnt=Irt/Kr and whose solution is the sigmoid function:(31)Irnt=11+Irn0−1−1e−rt=IrnoertIrnoert+1−Irno=Ir0ertIr0ert−1+Kr=Ir0Ir01−e−rt+Kre−rt.Then, the unnormalized infectious evolution is(32)Irt=KrIrnt=KrIr0Ir01−e−rt+Kre−rt=crIr0Ir01−e−t/ar+cre−t/ar=crIr0Ir0+cr−Ir0e−t/ar,and, for Ir0≠0,(33)Irt=Kr1+Kr/Ir0−1e−rt=Kr1+ertMd−tif etMdr=Kr/Ir0−1, namely, if(34)tMd=1rlnKr−Ir0Ir0=arlncr−Ir0Ir0=1βN0−R0−γlnN0−R0−β−1γIr0−1.It also holds that the maximum value ofI˙rt, which is the zero of I¨rt (or the growth inversion point), takes place at t=tMd, equation (34), from (33). This is easily seen from the nominal version of the logistic equation (13) with It≡Irt, at≡ar, and ct≡cr, which leads to(35)I¨rt=I˙rtar1−2Irtcr=0,which implies that Irt=cr/2=Kr/2, if I˙rt≠0, which happens at t=tMd from (33). Therefore, tMd is the growth inversion point time instant. On the contrary, from (27), there is no relative finite maximum value of Irt since I˙rt=0 holds for t=∞ if r>0. We have, as basic relations for the key parameters of the nominal logistic equation associated with (1)–(3), the Malthusian parameter (28), the carrying capacity (29), and the growth inversion point (34).The following result of interest holds.Proposition 3.
The nominal logistic equation has the following properties:(i)
It satisfies the positivity and boundedness conditions of Theorem1.(ii)
The disease transmission rate satisfies the constraintβ=r/Kr so that either β>0 and Kr is finite or β=0 with r=−γ and Kr=∞. If β>γ/N0−R0 (respectively, β<γ/N0−R0), then r>0 (respectively, r<0). If β=γ/N0−R0, then r=0.(iii)
The Malthusian parameter and the carrying capacity increase (respectively, decrease) as the disease transmission rate increases (respectively, decreases). The same property holds with respect toβ for the growth inversion point time instant if β<1.Proof.
Property (i) is a corollary of Theorem1 for the nominal logistic equation which is a particular parameterization of the nominal one. Then, note that the combination of (28) and (29) yields the constraint:(36)Kr=βN0−R0−γβ=rβ,which implies that the transmission rate satisfies β=r/Kr. The remaining results of Property (ii) follow directly from (28) and (29). On the contrary, note, from (28) and (29) that(37)rβ2−rβ1=N0−R0β2−β1,Krβ2−Krβ1=γβ2−β1β1β2,and also one has from (34) that(38)dtMddβ=1βN0−R0−γ×γβ−2Ir0N0−R0−γβ−1−Ir0−N0−R0βN0−R0−γ2×lnN0−R0−γβ−1Ir0−1so that dtMd/dβ=oβ−3. Property (iii) has been proved.Remark 1.
Note from (27) that if r>0(that is if the transmission rate is large enough to satisfy β>γ/N0−R0) and Ir0<Kr, then Irt is strictly increasing until it reaches its maximum value Kr. If r turns to a negative value, that is if β<γ/N0−R0, then the Irt reverses its growth turning to a strictly decreasing profile. The effective way of achieving this behaviour is to decrease the number of effective infectious contacts to the susceptible either by social discipline or eventually via quarantines or confinements.


## 3.3. Comparison Results of the Logistic Equation with Its Nominal Version

Note that the time-varying logistic equation (13) may be compared to the nominal one (27) by expressing the errors between both current and nominal Malthusian parameters and carrying capacities. For that purpose, define parametrical errors:(39)a˜t=at−ar=1βNt−Rt−γ−μ−1βN0−R0−γ,c˜t=ct−cr=Nt−Rt−γ+μβ−N0−R0−γβso that(40)rt=1at=1ar+a˜t=1ar+r˜t=r+r˜t,Kt=ct=cr+c˜t=Kr+K˜t,where(41)r˜t=rt−rr=−a˜tar+a˜t=βNt−Rt−N0+R0−μβNt−Rt−γ−μ,K˜t=c˜t=ct−cr=Nt−Rt−γ+μβ−N0−R0−γβ,b˜t=1atct−1arcr=−arc˜t+cra˜t+c˜ta˜tarcr+a˜tcr+c˜tar+c˜ta˜t.Thus, (13) can be rewritten as follows:(42)I˙t=Itat1−Itct=r+r˜tIt1−ItKr+K˜t=r+r˜tIt1−1Kr−K˜tKr+K˜tIt=rIt1−ItKr+ft,where(43)ft=rK˜tKr+K˜tI2t+r˜tIt1−1Kr−K˜tKr+K˜tIt.Then, one has that(44)It=I0+r∫0tIξ1−IξKrdξ+∫0trK˜ξKr+K˜ξI2ξdξ+∫0tr˜ξIξ1−1Kr−K˜ξKr+K˜ξIξdξ.The first right-hand side term of (45) is the solution of the nominal logistic equation (i.e., (32) or (33)). The second one is the integrated disturbance function fς on 0,t, equation (44), namely, ∫0tfξdξ due to the error between the current and nominal solutions. In view of the nominal solution expression (32), one has that (45) can be expressed equivalently as follows:(45)It=crI0I0+cr−I0e−t/ar+∫0tfξdξ=crI0I0+cr−I0e−t/ar+∫0trK˜ξKr+K˜ξI2ξdξ+∫0tr˜ξIξ1−1Kr−K˜ξKr+K˜ξIξdξ,which can be rewritten for each time instant t from a previous time instant t−Tt as follows:(46)It=crIt−TtIt−Tt+cr−I0e−Tt/ar+∫t−TttrK˜ξKr+K˜ξI2ξdξ+∫t−Tttr˜ξIξ1−1Kr−K˜ξKr+K˜ξIξdξ.Assume that, for a givent0∈R+ and each t≥t0∈R+, Tt is selected so that K˜t and r˜t are almost constant with absolute errors less than positive real constants εr˜ and supt≥0K˜t≤εK˜<Kr (since the current carrying capacity Kt=Kr+εK˜ is positive) in t−Tt,t. Thus, one has(47)−rTtεK˜Kr−εK˜supt−Tt≤ξ≤tI2ξ≤∫t−TttrK˜ξKr+K˜ξI2ξdξ≤rTtεK˜Krsupt−Tt≤ξ≤tI2ξ,(48)εr˜εK˜Ttinft−Tt≤ξ≤tIξKr1+εK˜+εK˜KrKr−εK˜supt−Tt≤ξ≤tIξ≤∫t−Tttr˜ξIξ1+K˜ξKr+K˜ξ−1KrIξdξ≤εr˜εK˜Ttsupt−Tt≤ξ≤tIξ1+1Krsupt−Tt≤ξ≤tIξ,provided that supt≥0K˜t≤εK˜<minKr,1. On the contrary, if the incremental carrying capacity related to the nominal one is large enough to satisfy inft≥0K˜t≥Kr/Kr−1, which implies also that εK˜>1 while it is also subject to Kr>1, then(49)K˜tKr+K˜t−1Kr≤Kr−1K˜t−KrKrKr+K˜t≤εK˜Kr−1−KrKr2so that the upper bound in (47) becomes modified as follows:(50)εr˜εK˜Ttsupt−Tt≤ξ≤tIξ1+εK˜Kr−1−KrKr2supt−Tt≤ξ≤tIξ,Then, one has under the upper bound of (47), for the case when supt≥0K˜t≤εK˜<Kr if ε=minεr˜,εK˜, that(51)It=KrIt−TtIt−Tt+Kr−It−Tte−Ttr+εK˜Ttsupt−Tt≤ξ≤tIξεr˜+Krr−1+εK˜Kr−1Kr2supt−Tt≤ξ≤tIξ+oεTt,and one also has, for the case when K˜tinft≥0≥Kr/Kr−1, according to (49), that(52)It=KrIt−TtIt−Tt+Kr−It−Tte−Ttr+εK¯Ttsupt−Tt≤ξ≤tIξεr˜+r+εr˜Krsupt−Tt≤ξ≤tIξ+oεTt.On the contrary, note that∫t−Tttr˜ξIξ1−1/Kr−K˜ξ/Kr+K˜ξIξdξ is slowly time varying if εTt is sufficiently small which is a reasonable approximation for a sufficiently small interval Tt from the time instant where the initial infection values are picked up until its current calculation. The reason is that this implies in addition that ε is also sufficiently small which leads to a small variation of Nξ−Rξ along t−Tt,t for a sufficiently small interval Tt with that additional advantage that the demography-related parameters μ and ν are also as small as suited. These considerations suggest that, by choosing the central interval point for such an evaluation of the incremental solution, one gets as an alternative expression to the infectious subpopulation evolution through time that(53)It=Kt−nTtIt−nTtIt−nTt+Kt−nTt−It−nTte−nTtrt−nTt+δεnTt,nTt,for given n∈Z+ and Tt∈R+ leading to a performed calculation along a time interval t−nTt,t. A further improvement might be performed without difficulty over a sequence, or eventually a finite set IS≡tii=0n of sampling instants by using recorded data with a monitored adaptation of the intersampling interval to the rate of variation of the infection curve in such a way that t0=0, ti+1=ti+Ti, and the intersample period sequence ISP≡Tii=0n are being updated as follows:(54)Ti=TM,ifTia>TM,Tia,ifTia∈Tm,TM,Tm,ifTia<Tm,where there are a number of possibilities to generate the auxiliary tentative sampling period Tia from the classical adaptive sampling background literature (see [48–51] and related references therein) as, for instance, the three ones which follow:(55)Tia=C1C3+C2I˙ti,(56)Tia=C1C3+C2Iti−Iti−1/Ti−1,(57)Tia=C1C3+C2Iti−Iti−1.For some prescribed design real constantsC1,C2>0, C3≥0, and TmTM>Tm with TM−Tm being sufficiently small in accordance with the above solution approximation considerations. The adaptive sampling laws (54) and (55) evaluate explicitly the absolute value of time derivative, while the adaptive sampling law (54), which subjects to (56), evaluates it by finite increments. The sampling period is constant (thus, it is nonupdated through time) if C2=0, Tia=Tm=C1/C3. Thus, one gets from (53) the following relation in-between two consecutive sampling instants:(58)It≅KtiItiIti+Kti−Itie−Tirti,∀t∈ti,ti+1,∀ti∈IS.


## 3.4. Estimation Algorithm of the Malthusian Parameter and the Carrying Capacity along Time

The parameters which characterize the current logistic equation are time varying, but it proposed the following simple algorithm (Algorithm1) expressed in general pseudocode style. It is assumed 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-escalation of quarantines. According to the evaluation of the pandemic growth force, the algorithm identifies from recorded data of the infection curve the Malthusian parameters and the carrying capacity, and then it estimates the averaged disease transmission rate per intervention phase.Algorithm 1:Evaluation of the Malthusian parameter and the carrying capacity.
(1)
Step 0: Chooset0≥0 and the initial intervention Phase Ph0 to start to run the procedure.(2)
Smooth the recorded infection curveIt, 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)
DefineCi for i=1,2,3 and Tm,TM to then run (53)–(55) to generate the sequences IS and ISP of sampling instants and inter-sampling 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 instantti∈IS, locate the intervention phase Phj to which it belongs. Calculate the inter-sampling 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: ReadIti+1 from recorded data(6)
Step 3: The Malthusian parameter and the carrying capacity at timeti are calculated “a posteriori” at time ti+2=t+i2Ti from the registered Iti, Iti+1=Iti+Ti and Iti+2Ti evaluated at the current and next time instants ti and ti+1=ti+Ti and an “a priori” estimation t^i+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 theβti,ti+1=rti,ti+1/Kti,ti+1 (see Proposition 3 (ii)).(8)
Step 5: Ifti+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 PhasePhjβPhj=1/lJ∑ti,ti+1∈Phjβti,ti+1, make j⟵j+1 and if the whole number of checked phases is unfinished GoTo Step 1. Else GoTo Step 7(10)
Step 7: EndNote that the calculation process is repeated for all the pairs of samplesti,ti+1 and ti+1,ti+2. The transmission rate is calculated for each such a pair from Proposition 3 (ii).Remark 2.
For the pairsti,ti+1 with sampling instants belonging to consecutive phases, we can include the option of considering their associated transmission rates for both phases in Algorithm 1 or for none of them as in the above given version form of Step 6.Remark 3.
Note that, in Appendix A, two ways of implementing Step 3 of Algorithm1 to estimate the carrying capacity and the Malthusian parameter are given:(a)
The first one involves the use of infection recorded data after the current evaluation time instant.(b)
The second one is summarized in Remark A.1, and it involves the only use of past data. Concerning the first method, one has to point out that the use of predictions can be replaced alternatively by the use for a set of previously recorded data corresponding to the current time instant under checking by the algorithm. Note that both methods can involve the selection of time instants being an integer multiple of the minimum period of supplied data, typically, one day. In this case, the adaptive sampling law just selects a time-varying sampling period which is an integer multiple integer of the mentioned minimum sampling period reported in the recorded data. A particular case is simply to use the minimum sampling period as a constant running sampling period while removing the adaptive sampling law design from Algorithm1.


## 4. Numerical Examples

This section is devoted to present some numerical simulation examples illustrating the results discussed in the previous sections. To this end, we consider the case of COVID-19. It is shown in [52] that the spread of COVID-19 in various regions, such as China, South Korea, and Italy (among others) can be appropriately described by a SIR model. For Italy, the SIR model describing the spread of coronavirus is given by β=0.18, γ=0.037, and ρ=μ=ν=0, all of them in units of day−1. The initial values for the states are given by S0=240,000, I(0) = 240, and R(0) = 0. As it is performed in [52], the model is scaled by a factor f=2.4·105 in such a way that the scaled initial conditions read S(0) = 1 and I(0) =  10−3. The evolution of the scaled SIR model with these parameters is depicted in Figure 1. The simulation starts on February 26, 2020, and spans for 200 days. Moreover, Figure 2 displays the number of active cases per day obtained from the SIR model along with the reported number of cases obtained from [53]. As it is proved in Theorem 1, the solution of the SIR model remains nonnegative and bounded for all time given nonnegative initial conditions. The evolution of the number of active cases per day (the state I) can be equivalently described by the time-varying (TV) logistic equation (13). Thus, Figure 3 displays the evolution of the state I obtained from the SIR model and the solution to the time-varying equation (13). It is concluded from Figure 3 that both systems are equivalent so that the ad hoc time-varying logistic equation is a single equation that appropriately describes the number of active cases per day. Furthermore, Figures 4 and 5 display the values of the Malthusian function, r(t), and carrying capacity, c(t), defined in (14), which parameterize the time-varying logistic equation.Figure 1
Evolution of the SIR model describing the spread of COVID-19 in Italy.Figure 2
Number of active cases per day obtained from the SIR model and from reported data.Figure 3
Number of active cases per day obtained from the complete SIR model and from the time-varying logistic equation.Figure 4
Evolution of the Malthusian function,r(t), parameterizing the time-varying logistic equation.Figure 5
Evolution of the carrying capacity,c(t), parameterizing the time-varying logistic equation.Under certain conditions, the time-varying logistic equation can be approximated by a nominal time invariant one that simplifies the model of infection spreading. The main condition for this to hold is the excess (N(t) − R(t)) being constant and equal to (N(0) − R(0)) along time. Figure 6 shows the evolution of (N(t) − R(t)) with time. It is observed in Figure 6 that only during the first days of spreading this condition holds so that the nominal logistic equation is an appropriate model only while the incubation period has not supplied with individuals yet the subpopulation of immune. Moreover, Figure 7 supports this conclusion where the plots of the SIR model and the time-varying and the nominal logistic equations are displayed. It is observed in Figure 7 that the nominal logistic equation is close to the solution of the SIR model during the first days of simulation. Therefore, the nominal logistic equation may be used as a simplified model to describe the spread of an infection during the first stages, while the time-varying one may be used to describe the spreading for a longer period. It can also be concluded from Figure 6 that Nt−Rt≤N0−R0 for all time since the condition in Proposition 1 trivially holds in this case, where the infectious are always nonnegative and ρ=μ=ν=0.Figure 6
Evolution of(N(t) − R(t)) with time.Figure 7
Comparison between the SIR model, the time-varying logistic equation, and the nominal one.As a consequence, it is of interest to have an algorithm to estimate the values of the Malthusian and carrying capacity from data in order to obtain a description of the infection spreading during the first stages. Therefore, the algorithm discussed in Section4 will be applied now to estimate these parameters during the first days of spreading. Initially, the algorithm is applied to the numerical data obtained by simulation from the original SIR model. Thus, consider a constant sampling time of TM = Tm = Ti = 1 hour, which is a low sampling time for epidemic systems and the use of equations (A.5)–(A.11) from the Appendix to estimate the values of the carrying capacity and Malthusian parameter. Since the approximation to the nominal logistic equation only holds during the first days of simulation, the first 45 days will only be considered. Figures 8 and 9 display the evolution of the actual and estimated Malthusian parameter and actual and estimated carrying capacity, respectively. It is concluded from Figures 8 and 9 that the estimated Malthusian parameter is close to the actual value, while the carrying capacity is estimated with an error of 20%. The estimated value of β is therefore 0.2246 days−1, in comparison to the actual one of 0.18 days−1. The algorithm provides an estimation of the transmission rate by measuring only the number of active cases in three consecutive time samples. Now, the method will be applied to reported data from Italy, where data are collected every day, i.e., TM = Tm = Ti = 1 day.Figure 8
Estimation of the Malthusian parameter from the SIR model.Figure 9
Estimation of the carrying capacity from the SIR model.Figures10 and 11 show the estimated values of the Malthusian parameter and carrying capacity when the method is applied to reported data from Italy. It can be deduced from these figures that raw data contain too much variability to perform the estimation directly. Therefore, an averaging filter is employed to smoothen the data prior to applying the estimation procedure. Thus, we use the filter ykf=1/5∑i=04yk−i, which is the mean value of the last 5 samples. The estimation with filtered values provides Figures 12 and 13. Moreover, the estimated value of β=r/c is depicted in Figure 14. From these figures, it can be concluded that the filter smoothens the output of the estimation, but the filtered data still have too much variability to extract the value of β as the mean value of the series. The value of β at the last stage of simulation (which is the most stable one) is β=0.412.Figure 10
Estimation of the Malthusian parameter from reported data (Italy).Figure 11
Estimation of the carrying capacity from reported data (Italy).Figure 12
Estimation of the Malthusian parameter from filtered reported data (Italy).Figure 13
Estimation of the carrying capacity from filtered reported data (Italy).Figure 14
Estimation ofβ from filtered reported data (Italy).Furthermore, a least squares method is now applied to approximate the reported data to a nominal logistic equation. To this end, the lsqcurvefit Matlab function will be used. In this way, the fitting procedure is repeated every time a new data is collected. Figures15–17 display the evolution of the estimated parameters with time when the least squares method is employed.Figure 15
Estimation of the Malthusian parameter from reported data (Italy) and lsqcurvefit Matlab function.Figure 16
Estimation of the carrying capacity from reported data (Italy) and lsqcurvefit Matlab function.Figure 17
Estimation ofβ from reported data (Italy) and lsqcurvefit Matlab function.The value at the end of simulation isβ=0.558. A filtered version of the data is also used to perform the estimation. In this way, the same filter as before is used to smoothen the data prior to the application of the least squares method. Thus, Figures 18–20 display the evolution of estimated parameters when filtered data are used.Figure 18
Estimation of the Malthusian parameter from reported filtered data (Italy) and lsqcurvefit Matlab function.Figure 19
Estimation of the carrying capacity from reported filtered data (Italy) and lsqcurvefit Matlab function.Figure 20
Estimation ofβ from reported filtered data (Italy) and lsqcurvefit Matlab function.The value ofβ=0.5204 is obtained at the end of simulation. Finally, Figure 21 depicts the reported data along with the plot of a nominal logistic equation parameterized by the values obtained at the end of least squares estimation and a nominal logistic equations parameterized by the values obtained at the end of Algorithm 1 execution with filtered data. It is observed in Figure 21 that the least-squares estimated nominal logistic equation is close to the reported data while the nominal equation parametrized by Algorithm 1 is slightly further. Overall, the nominal logistic equation is an appropriate model to describe the spread of an infection during its first stages. The estimation of logistic equation parameters allows obtaining an estimation of the transmission rate of the disease, which provides useful information for decision makers. Algorithm 1 provides an estimation of the transmission rate with less computational burden than the least-squares algorithm. However, the treatment of raw data may require the application of denoising or filtering procedures in order data to be smoothen before processing, which is a future research line It is of interest to extend the obtained modeling results with use of vaccination controls [54] and to evaluate results with future evolution of the pandemic under vaccination data.Figure 21
Comparison of reported data with the output of the estimated nominal logistic equation.


## 5. Conclusions

This paper has compared an SIR epidemic model with recruitment, demography, and disease-related mortality with a parallel description through an “ad hoc” logistic equation, which is time varying. The logistic equation version of the model allows easily to interpret it via the Malthusian parameter (related to the exponential rate of variation of the solution), the carrying capacity (related to the maximum values which are reached), and the transmission rate, whose value is related to the absolute value of the quotient of the absolute value of the Malthusian parameters and the carrying capacity. To address the process of picking up the data registration in order to estimate the above parameters, an adaptive sampling law is proposed which updates the discrete data accordingly to their rate of variation leading to an optional nonuniform disposal of samples to calculate the, in general, sample-dependent Malthusian parameter and carrying capacity. The transmission rate is then calculated from the above ones and can be averaged on the various intervention phases of quarantine interventions, interventions absence, and programmed de-escalated decisions. Numerical examples which illustrate the methodology are given and discussed based on COVID-19 parameterizations.



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*Source: 1382870-2020-12-09.xml*