An algorithm for the robust estimation of the COVID-19 pandemic’s population by considering undetected individuals.

Abstract

Due to the current COVID-19 pandemic, much effort has been put on studying the spread of infectious diseases to propose more adequate health politics. The most effective surveillance system consists of doing massive tests. Nonetheless, many countries cannot afford this class of health campaigns due to limited resources. Thus, a transmission model is a viable alternative to study the dynamics of the pandemic. The most used are the Susceptible, Infected and Removed type models (SIR). In this study, we tackle the population estimation problem of the A-SIR model, which takes into account asymptomatic or undetected individuals. By means of an algebraic differential approach, we design a model-free (no copy system) reduced-order estimation algorithm (observer) to determine the different non-measured population groups. We study two types of estimation algorithms: Proportional and Proportional-Integral. Both shown fast convergence speed, as well as a minimal estimation error. Additionally, we introduce random fluctuations in our analysis to represent changes in the external conditions and which result in poor measurements. The numerical results reveal that both model-free estimators are robust despite the presence of these fluctuations. As a point of reference, we apply the classical Luenberger type observer to our estimation problem and compare the results. Finally, we consider real data of infected individuals in Mexico City, reported from February 2020 to March 2021, and estimate the non-measured populations. Our work’s main goal is to proportionate a simple and therefore, an accessible methodology to estimate the behavior of the COVID-19 pandemic from the available data, such that the competent authorities can propose more adequate health politics.

1. Introduction

The pandemic caused by the novel severe acute respiratory syndrome SARS-CoV-2 virus in the late December 2019 quickly escalated into a world health crisis and overwhelmed the health care system of every nation. The mortality rate of the SARS-CoV-2 virus has been estimated to be between 3–$7\%$ [1], [2]. Moreover, this virus is highly contagious. The World Health Organization considers a basic reproduction number around 2, this is, each individual infects on average at least two more people [3], [4], [5], [6]. Nonetheless, some researchers have suggested that this global consensus might be low, since values of up to 4 and 6 have been reported [7], [8]. On the other hand, evidence suggests the presence of a considerable amount of asymptomatic individuals, which are as infective as the symptomatic carriers [9], [10], [11]. These individuals are a formidable vehicle of contagion, as they and those who get in contact, have no reason to take special cautions

To face the COVID-19 pandemic, governments around the world have been forced to adopt strict politics, such as massive viral testing campaigns and non-pharmaceutical interventions. However, due to economic, medical and social conditions, not all nations are equally capable to develop adequate surveillance systems [12]. To overcome the absence of medical and public resources, experts suggest the utilization of mathematical models to understand the course of the pandemic and design effective mitigation strategies.

The most simple approach consists in making use of a SIR (Susceptible, Infected and Removed) or SEIR (Susceptible, Exposed, Infected and Removed) type model [13], [14], [15]. These models consider direct transmission due to human-to-human contact, which can be through a sneeze or cough, through skin-skin contact, or through the exchange of body fluids. However, more complex models have been proposed to study the current pandemic. To mention just a few, in [16], the authors took into account hospitalized or in quarantine at home individuals, hospitalized that will die, people that have died and asymptomatic individuals. In [17] were introduced vulnerable groups in a standard SIR model, since evidence indicates large differences in hospitalization and fatality rates between age groups. Some research teams have proposed fractional order models that consider environmental factors and non-pharmaceutical interventions. The most interesting part of these, is the derivative order, which plays the role of an additional curve fitting parameter [18], [19]. Statistical and probabilistic models have also been developed to describe the pandemic and fit the parameters [20], [21], [22]. Also, an interesting review of several models was realized in [23], where a Quadratic Linear Regression model was found to be the best. Additionally, artificial intelligence and machine learning algorithms, also have proved to be a good modelling tool [24], [25].

Although a complex model incorporates more biological and epidemiological information about the epidemic and is more biologically realistic, an increased number of parameters need to be estimated. The fitting of these parameters is difficult and depends on the available data. This task is even harder if we consider that external conditions affect the pandemic. Evidence suggests that weather conditions have an impact on the transmission of the SARs-CoV-2, such as air pollution [26], wind velocity, daily sunlight, air pressure, temperature and humidity [27], [28], [29] and even noise pollution [30]. In particular, cold and dry conditions boost the spread [31]. Based on the above, for some cases it might be better to use a simple model. In fact, in [32], by means of the Akaike Information Criterion, was shown that a simple model is good enough to describe the COVID-19 pandemic, in particular, the SIR type models.

Additionally, some research groups have focused on the development of observers. This is, the estimation of the diverse population groups, the parameters or other factors that influence the behavior of the model. In [33], the authors proposed an Extended State Observer for disturbances in a SIR type model and implemented an observer-based U-model control to design a population restriction policy. In [34], is considered a discrete-time epidemic model and a nonlinear state observer is proposed to compute the non-measured variables from the number of hospitalized patients. In [35], the parameters of a metapopulation SEIR model, this is, a spatially structured population model, are estimate by means of a machine learning algorithm known as sliding window algorithm. Meanwhile, in [36], the authors proposed a High-Gain Observer, whose design assumes the availability of a distributed measurement, this is, parts of the different populations of a SIR model.

In order to contribute to the effort to face the COVID-19 pandemic, we propose a model-free reduced order estimation algorithm (observer). Our main concern is to develop an efficient, simple and accessible estimation methodology. We propose to use differential algebraic tools [37]. These allow us to study the solution of differential equations by means of differential polynomials. Specifically, we consider the algebraic observability condition (AOC) which, shortly speaking, allow us to estimate the system/model unknown variables through differential polynomials of the available data. Moreover, since the presence of asymptomatic individuals is particularly relevant in the current pandemic scenario, and as we look for a simple methodology, here we consider a SIR type model known as A-SIR model (Asymptomatic, Susceptible, Infected and Removed) [38]. Also, we explore the robustness of the propose estimation algorithm by considering additive noise in the output of the A-SIR model. The purpose is to emulate the effect of climatic fluctuations.

The contributions and key points of our work are summarized in the following list:

• •The proposed estimation algorithms is a modified, augmented system (immersion) of the A-SIR model, whose additional dynamical variable (artificial variable) is selected based on the variables of interest and the AOC.
• •The estimation algorithm is non redundant, this is, there is no reconstruction of the measured variables, just the variables of interest are estimated.
• •The observer is model-free. This is, we do not require the full knowledge of the model. This is practical in a scenario like the current pandemic, where many parameters have not been measured or fully accepted by the scientific community.
• •The estimation algorithm is robust against measurement noise. This is important since in the current pandemic, measurements might be poor.
• •When the measured variable is noise-free, the estimation error of the observer is asymptotically stable. When there is noise in the output, then the estimation error is ultimate uniformly bounded, this is, the error remains bounded in a compact set of given radius.
• •Most of observers to date consider observability conditions which are based on complex geometric differential criteria. Ours, uses an algebraic differential approach, which is far simple and therefore, easy to implement.
• •The A-SIR model (see Eqs. (7) to (11)) is a Liouvillian system (see Definition 2), from which we can reconstruct the variables of interest by means of integration processes.
• •There are some indicators that the estimates obtained by our estimation algorithm are correct. In particular, these keep similitude with real data reported by the Mexican Government (see Section 5.2 and Figs. 9 to 12).
• •Whenever the artificial variable depends on time derivatives of the measured variable, we assure that there exist variables related to estimates of those variables of interest (see Corollary 1).

Fig. 9.

COVID-19 pandemic in Mexico City: Infected cases reported from February 22nd, 2020 to March 13th, 2021. In red, the local average of the reported cases obtained with move to average function from MatLab. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 12.

COVID 19 pandemic evolution in Mexico City according to our estimates. February 22nd, 2020 to March 13th, 2021.

The paper is organized as follows: in Section 2, the mathematical A-SIR model is described. In Section 3, we introduce some differential algebra notions. Model-free Proportional and Proportional-Integral reduced order state observers are presented in Section 4. In Section 5, we present the numerical results obtained by the reduced order estimation algorithms, with and without additive noise in the output (number of asymptomatic individuals). As a point of reference, we compare these results with the estimates made by the well known Luenberger observer. Then, we apply our estimation methodology to real data reported in Mexico City from February 2020 to March 2021. Finally, in Section 6 we mention the corresponding concluding remarks of this work.

2. Mathematical model

As has been mentioned, one of the main characteristics of the COVID-19 pandemic is the existence of a considerable amount of individuals with high loads of the virus that present no or mild symptoms. These are as contagious as symptomatic individuals. However, as they do not require medical attention, most remain undetected in the surveillance system. To represent this scenario, we consider the following SIR-type model, known as A-SIR model [38]:

$$\dot{S}\left(t\right)=-\beta S\left(t\right)\left[A\right(t)+I(t\left)\right],$$ (1)

$$\dot{I}\left(t\right)=\rho \beta S\left(t\right)\left[A\right(t)+I(t\left)\right]-\gamma I\left(t\right),$$ (2)

$$\dot{A}\left(t\right)=(1-\rho )\beta S\left(t\right)\left[A\right(t)+I(t\left)\right]-\nu A\left(t\right),$$ (3)

$$\dot{R}\left(t\right)=\gamma I\left(t\right)+\nu A\left(t\right),$$ (4)

$$Y\left(t\right)=I\left(t\right),$$ (5)

where $S\left(t\right)$ and $R\left(t\right)$ represent the populations of susceptible and removed individuals (recovered or deceased), respectively. The infected population is divided in symptomatic individuals $I\left(t\right)$ and asymptomatic individuals $A\left(t\right)$. The system’s output is $Y\left(t\right)$. We suppose that the available data is only the number of symptomatic individuals, i.e., those that have been reported to authorities. The initial conditions of model (1)-(5) are: $S\left(0\right)=S_0,$ $I\left(0\right)=I_0,$ $A\left(0\right)=A_0$ and $R\left(0\right)=R_0$.

The parameter $\beta >0$ is known as the disease transmission rate and is given by $\beta =\delta \tau /N$. Where $\delta$ is the number of contacts that each infected has with susceptible individuals, $\tau$ is the fraction of contacts that result in a transmission (transmissibility) and $N$ is the total population size. The removal rates for symptomatic and asymptomatic individuals are $\gamma >0$ and $\nu >0,$ respectively. In other words, the average number of days between infection and recovery or dead of symptomatic people is ${\gamma }^{-1}$ and of asymptomatic individuals is ${\nu }^{-1}$. On the other hand, $\rho >0$ represents the fraction of infected individuals that are symptomatic. These parameters depend on the characteristics of the disease and on social behavior, however, are independent of the population size.

Remark 1

One can notice that the removed population includes those asymptomatic individuals that recover and contribute to the population immunity.

The model (1)-(5) assumes permanent immunity of individuals who recovered, a constant population (no natural births or deaths) and that all infected individuals are immediately infective. Since population is constant, we have that:

$$S\left(t\right)+I\left(t\right)+A\left(t\right)+R\left(t\right)=N,\forall t\ge 0$$ (6)

For simplicity, we will consider normalized populations, i.e., $s\left(t\right)=S\left(t\right)/N,$ $i\left(t\right)=I\left(t\right)/N,$ $a\left(t\right)=A\left(t\right)/N$ and $r\left(t\right)=R\left(t\right)/N$. Therefore, these variables take values between 0 and 1. Then, the model (1)-(5) now is:

$$\dot{s}\left(t\right)=-\beta s\left(t\right)\left[a\right(t)+i(t\left)\right],$$ (7)

$$\dot{i}\left(t\right)=\rho \beta s\left(t\right)\left[a\right(t)+i(t\left)\right]-\gamma i\left(t\right),$$ (8)

$$\dot{a}\left(t\right)=(1-\rho )\beta s\left(t\right)\left[a\right(t)+i(t\left)\right]-\nu a\left(t\right),$$ (9)

$$\dot{r}\left(t\right)=\gamma i\left(t\right)+\nu a\left(t\right),$$ (10)

$$y\left(t\right)=i\left(t\right),$$ (11)

with initial conditions: $s\left(0\right)=s_0,$ $i\left(0\right)=i_0,$ $a\left(0\right)=a_0$ and $r\left(0\right)=r_0$. Meanwhile, (6) is:

$$s\left(t\right)+i\left(t\right)+a\left(t\right)+r\left(t\right)=1,\forall t\ge 0$$ (12)

The basic reproduction number ${\mathcal{R}}_0=\delta \tau /(\gamma +\nu )$ and the effective reproductive number ${\mathcal{R}}_e=s\left(0\right)\delta \tau /(\gamma +\nu )$ are two helpful parameters to describe any disease spread. ${\mathcal{R}}_0$ is an estimate of how many new infections are originated from a single infected individual in the initial phase of the pandemic. ${\mathcal{R}}_e$ is the average number of secondary cases per infected individual.

If the entire population is initially susceptible, i.e., $s\left(0\right)\approx 1,$ then ${\mathcal{R}}_e$ is approximately equal to ${\mathcal{R}}_0$. This assumption is reasonable for a new virus such as the SARS-CoV-2, where a vaccine was not initially available and information about innate immunity is still unclear [39].

Theorem 1

[40]If${\mathcal{R}}_e\le 1,$then the infected population decreases monotonically to zero as$t\to \infty$. If${\mathcal{R}}_e>1,$then the infected population starts increasing, reaches its maximum, and then decreases to zero as$t\to \infty$. We call this scenario of increasing numbers of infected individuals an epidemic.

From Theorem 1, it follows that a disease can cause an epidemic if ${\mathcal{R}}_0>1$ or $\delta \tau >\gamma +\nu$. Moreover, both cases reach an endemic equilibrium point, i.e., when $\dot{s}=\dot{i}=\dot{a}=\dot{r}=0$.

In Fig. 1 , the evolution of the A-SIR model with different effective reproduction numbers is shown. In Fig. 1.a, the total population is initially susceptible and therefore, ${\mathcal{R}}_e\approx {\mathcal{R}}_0>1$. This is the case for the COVID-19 pandemic, where researchers have reported values between 2 and 6 for the basic reproduction number [8], [41], [42]. On the other hand, in Fig. 1.b we observe the case when ${\mathcal{R}}_e\le 1$. An example of this scenario is the Nipah virus, whose basic reproduction number is about 0.48 [43].

Fig. 1.

A-SIR model: Normalized population’s evolution with ${\mathcal{R}}_e>1$ (a) and with ${\mathcal{R}}_e\le 1$ (b). In (a) we can observe an epidemic scenario, meanwhile in (b), the infected population decreases monotonically to zero. Both cases reach an endemic equilibrium, each one with different characteristics.

3. Preliminaries

Let us introduce the following definitions [44]. Consider a nonlinear system described by:

$$\dot{x}\left(t\right)=f\left(x\right),x\left(0\right)=x_0$$ (13)

$$y\left(t\right)=h\left(x\right)$$ (14)

Where $f:X\to {\mathbb{R}}^n$ is continuously differentiable, $x\left(t\right)\in X\subset {\mathbb{R}}^n$ is a state vector, $y\left(t\right)\in \mathbb{R}$ is the output of the system and $h:X\to \mathbb{R}$ is an analytic function.

Definition 1 Algebraic observability

The system (13), (14) is said to be algebraically observable if satisfies the differential polynomial

$$x\left(t\right)=P(y,\dot{y},\ddot{y},\cdots ,y^{\left(\mu \right)})\left(t\right),$$

for some $\mu \in \mathbb{N}$ time derivative. We refer to the above condition as the algebraic observability condition (AOC).

Definition 2 Liouvillian system

The dynamical system (13), (14) is said to be Liouvillian if its elements (for example, state variables or parameters) can be obtained by an adjunction of integrals or exponentials of integrals of elements of $\mathbb{R}$.

We can notice that, in the first instance, the system (7), (8), (9), (10), (11) does not satisfy the AOC, since some state variables do not satisfy a differential polynomial of the output. Nevertheless, it is a Liouvillian system, i.e.:

$$s\left(t\right)=-\frac{1}{\rho }{\int }_0^t(\dot{y}\left(u\right)+\gamma y\left(u\right))du=\overline{s},$$ (15)

$$i\left(t\right)=y\left(t\right),$$ (16)

$$a\left(t\right)={\int }_0^t[\frac{1-\rho }{\rho }(\dot{y}\left(u\right)+\gamma y\left(u\right))(1+\frac{\nu }{\beta {\int }_0^t(\dot{y}\left(u\right)+\gamma y\left(u\right))\mathit{du}})+\nu y\left(u\right)]\mathit{du}=\overline{a},\rho \ne 0$$ (17)

$$r\left(t\right)=1-y\left(t\right)-\overline{s}-\overline{a}$$ (18)

Then, it is possible to estimate the different populations of the A-SIR model from the available data.

4. Estimation methodology

In the following, we present a methodology to estimate the susceptible, removed, asymptomatic and symptomatic populations by means of Proportional and Proportional-Integral reduced order estimation algorithms. This methodology is inspired in an algebraic differential approach [37], [45].

Let us define the new state variable $\eta \left(x\right)$ (artificial variable) in function of the A-SIR model populations, i.e.:

$$\eta \left(x\right)=\beta s\left(t\right)\left[a\right(t)+i(t\left)\right]$$ (19)

Then, the A-SIR model (7)-(11) can be expressed as the following augmented system (nonlinear system’s immersion [45]):

$$\dot{s}\left(t\right)=-\eta \left(x\right),$$ (20)

$$\dot{i}\left(t\right)=\rho \eta \left(x\right)-\gamma i\left(t\right),$$ (21)

$$\dot{a}\left(t\right)=(1-\rho )\eta \left(x\right)-\nu a\left(t\right),$$ (22)

$$\dot{\eta }\left(x\right)=\Omega \left(x\right),$$ (23)

$$\dot{r}\left(t\right)=\gamma i\left(t\right)+\nu a\left(t\right),$$ (24)

$$y\left(t\right)=i\left(t\right),$$ (25)

where $x\left(t\right)={[s,i,a,\eta ,r]}^T$ and $\Omega \left(x\right)$ is an unknown bounded dynamics. Notice that the problem of estimating the different populations of the A-SIR model is now an estimation problem of the variable $\eta \left(x\right)$. In general, we need to impose certain conditions on $\eta \left(x\right)$ and $\Omega \left(x\right)$. Let us assume the following assumptions:

• H1.$\eta \left(x\right)$ satisfies the AOC.
• H2.There exist auxiliary variables ${\alpha }_1$ and ${\alpha }_2$ that are real valued functions ${\alpha }_1,{\alpha }_2\in C^1$.
• H3.$\Omega \left(x\right)$ is bounded, i.e., $\exists M_1>0$ such that $\parallel \Omega \left(x\right)\parallel \le M_1,$ $\forall x\in X$.

Remark 2

It is not hard to see that for our particular problem and selection of $\eta \left(x\right),$ assumption H1 is fulfilled, i.e.:

$$\eta =\frac{1}{\rho }(\dot{y}+\gamma y),\rho \ne 0$$ (26)

Now, we propose the following Proportional-Integral reduced order state observer to solve the estimation problem.

Theorem 2

Let$\widehat{\eta }$be the estimate of$\eta$. If the assumptions H1–H3 are satisfied, then the system

$$\dot{\widehat{\eta }}={\kappa }_p(\eta -\widehat{\eta })+{\kappa }_i{\widehat{\eta }}_1,$$ (27)

$${\dot{\widehat{\eta }}}_1=\eta -\widehat{\eta }.$$ (28)

with constant gains such that${\kappa }_p>\frac{1}{2}({\kappa }_i+1),$${\kappa }_i>0,$is a model-free Proportional-Integral reduced order estimation algorithm for the unknown dynamics$\Omega \left(x\right)$of the system(20)–(25), and whose estimation error$e=\eta -\widehat{\eta }$is asymptotically stable.

Proof

As the estimation error is $e=\eta -\widehat{\eta },$ we have that the estimation error dynamics is given by:

$$\dot{e}=\Omega \left(x\right)-{\kappa }_{p}e-{\kappa }_i{\widehat{\eta }}_1,$$ (29)

$${\dot{\widehat{\eta }}}_1=e$$ (30)

or in matrix form, is:

$$\dot{E}=KE+\overline{\Omega }$$ (31)

where

$$\begin{array}{c}\hfill E=\left[\begin{array}{c}e\\ {\widehat{\eta }}_1\end{array}\right],K=\left[\begin{array}{cc}-{\kappa }_p& -{\kappa }_i\\ 1& 0\end{array}\right],\overline{\Omega }=\left(\begin{array}{c}\Omega \left(x\right)\\ 0\end{array}\right)\end{array}$$

Now, let us consider the following Lyapunov candidate function

$$V\left(E\right)=E^{T}PE+{\int }_0^t{\parallel \overline{\Omega }\left(x\left(\tau \right)\right)\parallel }^{2}d\tau$$ (32)

The function above is well defined since the second term is bounded, $V\left(0\right)=0$ and $V\left(E\right)>0$ for all $E\ne 0$. Then, it follows that

$$\begin{array}{cc}\hfill \dot{V}\left(E\right)& ={\dot{E}}^{T}PE+E^{T}P\dot{E}+{\parallel \overline{\Omega }\parallel }^2\hfill \\ & =[E^T{K}^T+{\overline{\Omega }}^T]PE+E^{T}P[KE+\overline{\Omega }]+{\parallel \overline{\Omega }\parallel }^2\hfill \\ & =E^T{K}^{T}PE+E^{T}PKE+{\overline{\Omega }}^{T}PE+E^{T}P\overline{\Omega }+{\overline{\Omega }}^T\overline{\Omega }\hfill \end{array}$$ (33)

which we can express as

$$\begin{array}{c}\hfill \dot{V}\left(E\right)=\left(\begin{array}{cc}{E}^T& {\overline{\Omega }}^T\end{array}\right)\left(\begin{array}{cc}{K}^{T}P+PK& P\\ P& I\end{array}\right)\left(\begin{array}{c}E\\ \overline{\Omega }\end{array}\right)\end{array}$$ (34)

Therefore, if there exists $P=P^T>0$ such that

$$\begin{array}{c}\hfill \left(\begin{array}{cc}{K}^{T}P+PK& P\\ P& I\end{array}\right)<0\end{array}$$ (35)

then the estimation error will be asymptotically stable. In particular, let us consider

$$\begin{array}{c}\hfill P=\left(\begin{array}{cc}{\kappa }_p-\frac{1}{2}& \frac{1}{2}{\kappa }_i\\ \frac{1}{2}{\kappa }_i& \frac{1}{2}{\kappa }_i\end{array}\right)>0,\end{array}$$ (36)

then, from the Sylvester criterion, we have that ${\kappa }_p>({\kappa }_i+1)/2$ and ${\kappa }_i>0$. Thus, condition (35) fulfills and the proof is completed □

Corollary 1

If$\eta \left(x\right)$satisfies H1 and can be expressed as

$$\eta =c\dot{y}+d\left(y\right),$$ (37)

where$c$is constant and$d\left(y\right)$is a bounded function, then there exist functions${\alpha }_1$and${\alpha }_2\in C^1$such that the model-free Proportional-Integral reduced order estimation algorithm(27)–(28) is equivalent to

$$\begin{array}{ccc}\hfill {\dot{\alpha }}_1& =& -{\kappa }_p{\alpha }_1+{\alpha }_2+{\kappa }_p[d\left(y\right)-{\kappa }_p\mathit{cy}]+{\kappa }_i\mathit{cy},{\alpha }_1\left(0\right)={\alpha }_{1_0},\hfill \\ \hfill {\dot{\alpha }}_2& =& -{\kappa }_i{\alpha }_1+{\kappa }_i[d\left(y\right)-{\kappa }_p\mathit{cy}],{\alpha }_2\left(0\right)={\alpha }_{2_0},\hfill \\ \hfill \widehat{\eta }& =& {\alpha }_1+{\kappa }_p\mathit{cy},\hfill \\ \hfill {\widehat{\eta }}_1& =& {\alpha }_2+{\kappa }_i\mathit{cy}\hfill \end{array}$$

Proof

By substituting (37) in (27), (28)

$$\begin{array}{cc}\hfill \dot{\widehat{\eta }}& ={\kappa }_p(c\dot{y}+d\left(y\right)-\widehat{\eta })+k_i{\widehat{\eta }}_1,\hfill \\ \hfill {\dot{\widehat{\eta }}}_1& =c\dot{y}+d\left(y\right)-\widehat{\eta }\hfill \end{array}$$ (38)

Let us define the auxiliary variables ${\alpha }_1$ and ${\alpha }_2\in C^1$ as

$$\begin{array}{cc}\hfill {\alpha }_1& =\widehat{\eta }-{\kappa }_{p}cy,\hfill \\ \hfill {\alpha }_2& ={\widehat{\eta }}_1-{\kappa }_{i}cy\hfill \end{array}$$ (39)

whose time derivatives are given by

$$\begin{array}{cc}\hfill {\dot{\alpha }}_1& =\dot{\widehat{\eta }}-{\kappa }_{p}c\dot{y},\hfill \\ \hfill {\dot{\alpha }}_2& ={\dot{\widehat{\eta }}}_1-{\kappa }_{i}c\dot{y}\hfill \end{array}$$ (40)

Then, after some algebraic manipulation, from (38), (39) and (40) we get

$$\begin{array}{ccc}\hfill {\dot{\alpha }}_1& =& -{\kappa }_p{\alpha }_1+{\alpha }_2+{\kappa }_p[d\left(y\right)-{\kappa }_p\mathit{cy}]+{\kappa }_i\mathit{cy},{\alpha }_1\left(0\right)={\alpha }_{1_0},\hfill \\ \hfill {\dot{\alpha }}_2& =& -{\kappa }_i{\alpha }_1+{\kappa }_i[d\left(y\right)-{\kappa }_p\mathit{cy}],{\alpha }_2\left(0\right)={\alpha }_{2_0},\hfill \end{array}$$

 □

4.1. Model-free Proportional reduced order state observer construction

Let us discard the Integral part of observer (27)–(28), i.e., ${\widehat{\eta }}_1=0$. Then, we obtain a model-free Proportional reduced order estimation algorithm for the unknown dynamics (23):

$$\dot{\widehat{\eta }}=\kappa (\eta -\widehat{\eta }),$$ (41)

where ${\kappa }_p=\kappa$. Now, substituting (26) in (41), we have

$$\dot{\widehat{\eta }}=\kappa [\frac{1}{\rho }(\dot{y}+\gamma y)-\widehat{\eta }]$$ (42)

As $\dot{y}$ is not available, we propose the following auxiliary variable:

$$\alpha =\widehat{\eta }-\frac{\kappa }{\rho }y,$$ (43)

whose derivative is

$$\dot{\alpha }=\dot{\widehat{\eta }}-\frac{\kappa }{\rho }\dot{y}$$ (44)

By using (44) and (42), we get

$$\dot{\alpha }=\frac{\kappa \gamma }{\rho }y-\kappa \widehat{\eta }$$ (45)

From (43) we know that $\widehat{\eta }=\alpha +\frac{\kappa }{\rho }y,$ therefore, the Proportional reduced order state observer for unknown dynamics of system (20)–(25) is given by

$$\begin{array}{ccc}\hfill \dot{\alpha }& =& -\kappa \alpha +\frac{y}{\rho }(\kappa \gamma -{\kappa }^2),\alpha \left(0\right)={\alpha }_0,\hfill \\ \hfill \widehat{\eta }& =& \alpha +\frac{\kappa }{\rho }y\hfill \end{array}$$ (46)

Finally, the estimates of the variables of interest, i.e., the non-measured populations are:

$$\begin{array}{cc}\hfill \widehat{s}\left(t\right)& =-{\int }_0^t\widehat{\eta }\left(x\left(\tau \right)\right)d\tau ,\hfill \\ \hfill \widehat{a}\left(t\right)& =\frac{\widehat{\eta }\left(x\right)}{\beta \widehat{s}\left(t\right)}-y\left(t\right),\widehat{s}\left(t\right)\ne 0\hfill \\ \hfill \widehat{r}\left(t\right)& =1-y\left(t\right)-\widehat{a}\left(t\right)-\widehat{s}\left(t\right)\hfill \end{array}$$

Remark 3

We define the population estimation errors as the difference between the real values and the estimates, i.e., $\tilde{s}\left(t\right)=s\left(t\right)-\widehat{s}\left(t\right),$ $\tilde{a}\left(t\right)=a\left(t\right)-\widehat{a}\left(t\right)$ and $\tilde{r}\left(t\right)=r\left(t\right)-\widehat{r}\left(t\right)$.

4.2. Model-free Proportional-Integral reduced order state observer construction

Let us consider the model-free Proportional-Integral reduced order estimation algorithm. Then, by considering (26), the observer (27)–(28) can be expressed as:

$$\dot{\widehat{\eta }}={\kappa }_p[\frac{1}{\rho }(\dot{y}+\gamma y)-\widehat{\eta }]+{\kappa }_i{\widehat{\eta }}_1,$$ (47)

$${\dot{\widehat{\eta }}}_1=\frac{1}{\rho }(\dot{y}+\gamma y)-\widehat{\eta }$$ (48)

Let us define the following artificial variables

$${\alpha }_1=\widehat{\eta }-\frac{{\kappa }_p}{\rho }y,$$ (49)

$${\alpha }_2={\widehat{\eta }}_1-\frac{{\kappa }_i}{\rho }y,$$ (50)

whose time derivatives are

$${\dot{\alpha }}_1=\dot{\widehat{\eta }}-\frac{{\kappa }_p}{\rho }\dot{y},$$ (51)

$${\dot{\alpha }}_2={\dot{\widehat{\eta }}}_1-\frac{{\kappa }_i}{\rho }\dot{y},$$ (52)

respectively. Then, from (47), (48), we can obtain the following

$${\dot{\alpha }}_1=\frac{{\kappa }_p}{\rho }\gamma y-{\kappa }_p\widehat{\eta }+{\widehat{\eta }}_1,$$ (53)

$${\dot{\alpha }}_2=\frac{{\kappa }_i}{\rho }\gamma y-{\kappa }_i\widehat{\eta }$$ (54)

Now, by considering (49), (50), and after some algebraic manipulations, we can expressed (53), (54) as

$${\dot{\alpha }}_1=-{\kappa }_p{\alpha }_1+{\alpha }_2+\frac{y}{\rho }({\kappa }_i+{\kappa }_p\gamma -{\kappa }_p^2),$$ (55)

$${\dot{\alpha }}_2=-{\kappa }_i{\alpha }_1+\frac{{\kappa }_i}{\rho }y(\gamma -{\kappa }_p),$$ (56)

such that the model-free Proportional-Integral reduced order state observer is given by:

$${\dot{\alpha }}_1=-{\kappa }_p{\alpha }_1+{\alpha }_2+\frac{y}{\rho }({\kappa }_i+{\kappa }_p\gamma -{\kappa }_p^2),{\alpha }_{1_0}={\alpha }_1\left(0\right),$$ (57)

$${\dot{\alpha }}_2=-{\kappa }_i{\alpha }_1+\frac{{\kappa }_i}{\rho }y(\gamma -{\kappa }_p),{\alpha }_{2_0}={\alpha }_2\left(0\right),$$ (58)

$$\widehat{\eta }={\alpha }_1+\frac{{\kappa }_p}{\rho }y,$$ (59)

$${\widehat{\eta }}_1={\alpha }_2+\frac{{\kappa }_i}{\rho }y$$ (60)

Finally, the unknown populations can be estimated as:

$$\widehat{s}\left(t\right)=-{\int }_0^t\widehat{\eta }\left(x\left(\tau \right)\right)d\tau ,$$ (61)

$$\widehat{a}\left(t\right)=\frac{\widehat{\eta }\left(x\right)}{\beta \widehat{s}\left(t\right)}-y,\widehat{s}\left(t\right)\ne 0$$ (62)

$$\widehat{r}\left(t\right)=1-y\left(t\right)-\widehat{a}\left(t\right)-\widehat{s}\left(t\right)$$ (63)

4.3. Populations estimation with environmental noise

So far, we have neglected the effect of environmental factors in the A-SIR model. Nonetheless, there is strong evidence that the COVID-19 pandemic is highly influenced by external conditions such as humidity or temperature. In fact, it has been reported that cold and dry conditions increase the transmission rate and therefore, boost the virus spread [28], [29].

In this section, we consider the effect of climatic fluctuations in the A-SIR model which might result in a poor measurement. To model this phenomena, let us consider the following expression [46]:

$$\omega g\left(t\right)=\omega \beta s\left(t\right){\left[a\right(t)+i(t\left)\right]}^{1/2},$$ (64)

where $\omega$ is a random number that belongs to the normal distribution, i.e., $\omega \in (0,1)$. On the other hand, $g\left(t\right)$ is an analytic function. Now, we consider that the system’s output is affected by this additive signal, such that we have

$$y\left(t\right)=i\left(t\right)+\omega g\left(t\right),$$ (65)

One can notice that $\omega g\left(t\right)$ is a non evanescent term and increases as the number of infected individuals increases. We assume the following:

• H4.The additive noise term is bounded, i.e., $\parallel \omega g\left(t\right)\parallel \le M_2$.

Theorem 3

If assumptions H1-H4 are fulfilled, then the system(27)–(28) is a model-free Proportional-Integral reduced order estimation algorithm for unknown dynamics of system (20) – (23) , with output (65) and whose estimation error is ultimate uniformly bounded.

Proof

Let us express the output (65) as $y\left(t\right)=y_1\left(t\right)+wg\left(t\right)$. Then, from assumption H1 we have that the artificial variable $\eta$ can be selected as a sum of two differential polynomials, i.e.:

$$\eta =P_1(y_1,{\dot{y}}_1,\cdots ,y_1^{\left({\mu }_1\right)})+\omega P_2(g,\dot{g},\cdots ,g^{\left({\mu }_2\right)}),{\mu }_1,{\mu }_2\in \mathbb{N},$$ (66)

where $P_1\in C^1$ and $P_2\in C^1$. Now, we express the time derivative of $\eta$ as

$$\dot{\eta }=\Omega \left(x\right)+\omega \Psi \left(t\right)$$ (67)

Then, since the estimation error is $e=\eta -\widehat{\eta },$ we have that the dynamics of the estimation error is

$$\dot{e}=\Omega \left(x\right)-{\kappa }_{p}e-{\kappa }_i{\widehat{\eta }}_1+\omega \Psi \left(t\right),$$ (68)

$${\dot{\widehat{\eta }}}_1=e$$ (69)

or in matrix form

$$\dot{E}=KE+\overline{\Omega }+\omega \overline{\Psi },$$ (70)

where

$$\begin{array}{c}\hfill K=\left(\begin{array}{cc}-{\kappa }_p& -{\kappa }_i\\ 1& 0\end{array}\right),E=\left(\begin{array}{c}e\\ {\widehat{\eta }}_1\end{array}\right),\overline{\Omega }=\left(\begin{array}{c}\Omega \\ 0\end{array}\right),\overline{\Psi }=\left(\begin{array}{c}\Psi \\ 0\end{array}\right)\end{array}$$

Let us consider the following Lyapunov candidate Function

$$V\left(E\right)=E^{T}PE+{\int }_0^t{\parallel \overline{\Omega }\left(x\left(\tau \right)\right)\parallel }^{2}d\tau$$ (71)

Where $P=P^T>0$. The derivative of the Lyapunov function is given by

$$\begin{array}{cc}\hfill \dot{V}\left(E\right)=& {\dot{E}}^{T}PE+E^{T}P\dot{E}+{\parallel \Omega \parallel }^2\hfill \\ \hfill =& [E^T{K}^T+{\overline{\Omega }}^T+\omega {\overline{\Psi }}^T]PE+E^{T}P[KE+\overline{\Omega }+\omega \overline{\Psi }]+{\overline{\Omega }}^T\overline{\Omega }\hfill \\ \hfill =& E^T{K}^{T}PE+E^{T}PKE+{\overline{\Omega }}^{T}PE+E^{T}P\overline{\Omega }\hfill \\ & +{\overline{\Omega }}^T\overline{\Omega }+\omega {\overline{\Psi }}^{T}PE+\omega E^{T}P\overline{\Psi },\hfill \end{array}$$ (72)

which we can express as

$$\dot{V}\left(E\right)={\overline{E}}^{T}Q\overline{E}+2\omega {\Phi }^{T}R\overline{E},$$ (73)

where

$$\overline{E}=\left(\begin{array}{c}E\\ \overline{\Omega }\end{array}\right),{\Phi }^T=\left(\begin{array}{c}\Psi \\ 0\\ 0\\ 0\end{array}\right),R=\left(\begin{array}{cc}P& 0_2\\ 0_2& 0_2\end{array}\right),Q=\left(\begin{array}{cc}{K}^{T}P+\mathit{PK}& P\\ P& I\end{array}\right)<0$$

Then, from the Rayleigh inequality we have

$${\lambda }_{min}\left(Q\right)\parallel \overline{E}{\parallel }^2\le {\overline{E}}^{T}Q\overline{E}\le {\lambda }_{max}\left(Q\right){\parallel \overline{E}\parallel }^2$$ (74)

From assumption H4, we have $\parallel \omega \Phi \parallel \le M_2$. Therefore, we get

$$\parallel 2\omega {\Phi }^{T}R\overline{E}\parallel \le 2M_2\parallel \overline{E}{\parallel }_R=2M_2{\left({\lambda }_{max}\left(R\right)\right)}^{1/2}\parallel \overline{E}\parallel$$ (75)

such that

$$\dot{V}\left(E\right)\le {\lambda }_{max}\left(Q\right)\parallel \overline{E}{\parallel }^2+2M_2{\left({\lambda }_{max}\left(R\right)\right)}^{1/2}\parallel \overline{E}\parallel$$ (76)

Thus, we can apply the uniform ultimate boundedness theorem [47]. This is, it follows that $\overline{E}\left(t\right)$ is uniformly bounded for any initial condition $\overline{E}\left(0\right),$ and even more, $\overline{E}\left(t\right)$ remains in the compact set $B_{\overline{\delta }}=\{\overline{E}:\parallel \overline{E}\parallel \le \overline{\delta },\overline{\delta }>0\}$. Where

$$\overline{\delta }={\left(\frac{{\lambda }_{max}\left(R\right)}{{\lambda }_{min}\left(R\right)}\right)}^{1/2}\left(\frac{2M_2{\left({\lambda }_{max}\left(R\right)\right)}^{1/2}}{{\lambda }_{max}\left(Q\right)}\right)>0.$$ (77)

Hence, the estimation error of the Proportional-Integral reduced order observer is ultimate uniformly bounded in the presence of additive noise. □

5. Numerical simulation results

In the following, we present the results obtained with the proposed methodology for two different cases. The first case consider the numerical solution of the A-SIR model. In the second case, we use real data. In both cases, we focus in the COVID-19 pandemic in Mexico City.

The numerical calculations were conducted in MatLab-SimuLink R2020b, with a fixed step size of $0.001$ and an Euler integration method.

5.1. Estimation with numerical solution

For our first case, we consider the numerical solution of the A-SIR model with no control (intervention) action, i.e., we simulate the natural progression of the COVID-19 pandemic. The A-SIR model parameters used are shown in Table 1 . According to the last census realized by the Instituto Nacional de Estadística y Geografía (INEGI), Mexico City total population is $N=9,209,944$ [48].

Table 1.

A-SIR model parameters.

Parameter Description/Reference	Symbol	Value
Disease transmission rate [21]	$\beta$	0.46
Fraction of infected symptomatic individuals [49]	$\rho$	0.15
Symptomatic removal rate (days${}^{-1}$) [21]	$\gamma$	1/4.86
Asymptomatic removal rate (days${}^{-1}$) [21]	$\nu$	1/4.86

The parameters $\beta ,$ $\gamma$ and $\nu$ in Table 1 are statistical estimates made by considering only official data from Mexico City. The parameter $\rho$ was inferred from a serological study conducted in New York City. The simulation of the A-SIR model was realized with the following initial conditions: $i\left(0\right)=1/N,$ $a\left(0\right)=1/N,$ $r\left(0\right)=0$ and $s\left(0\right)=1-i\left(0\right)-a\left(0\right)$. In Fig. 2 we can observe the numerical solution of all the normalized populations of the A-SIR model. Then, the numerical solution $i\left(t\right)$ is the available data for each observer.

Fig. 2.

A-SIR model numerical solution: Natural progression of the COVID-19 pandemic. Observe that in the endemic equilibrium point remains a considerable amount of susceptible individuals. On the other hand, infected individuals reach a maximum and then converge to zero. This behavior corresponds to a pandemic with ${\mathcal{R}}_0>1,$ just as is the case of the SARS-CoV-2 virus.

Table 2 contains the initial conditions used for the different observers, as well as the gains for each. We use the same parameters from Table 1. As a point of reference, we include the estimations made by the well known Luenberger observer [50]. This observer is extensively used due to its simplicity. However, this observer is not model-free, therefore we need to proportionate initial conditions for $\widehat{s},$ $\widehat{i},$ $\widehat{a}$ and $\widehat{r}$. The parameter $L$ is known as the Luenberger gain.

Table 2.

Initial conditions and gains.

Proportional	Proportional-Integral	Luenberger
$\alpha \left(0\right)=0.1$	${\alpha }_1\left(0\right)=1$	$\widehat{i}\left(0\right)=1/N$
$\eta \left(0\right)=1$	${\alpha }_2\left(0\right)=0$	$\widehat{a}\left(0\right)=1/N$
$\kappa =25$	$\eta \left(0\right)=1$	$\widehat{r}\left(0\right)=0$
	${\kappa }_p=25$	$\widehat{s}\left(0\right)=1-i\left(0\right)-a\left(0\right)$
	${\kappa }_i=20$	$L=250$

In Fig. 3 , we observe the estimations made by the different observers. In general, we notice that all three are capable to estimate the normalized population of asymptomatic, removed and susceptible individuals. Nonetheless, Fig. 4 reveals a considerable discrepancy between the estimate made by Luenberger observer and the numerical solution of the A-SIR model, especially in the susceptible population case. Besides, we can appreciate overshooting at the beginning of the estimates made by the Proportional and Proportional-Integral estimation algorithms. Due to this, the estimates take negative values for a time. However, these quickly converge to the numerical solutions. These overshoots are result of the initial conditions. As the initial conditions get closer to the real initial conditions, the overshoots decrease. In general, we can say that the Proportional-Integral observer has the best performance.

Fig. 3.

Population estimates. Proportional and Proportional-Integral observers exhibit overshooting at the beginning, however, these estimates quickly converge to the numerical solutions.

Fig. 4.

Population estimation errors: difference between each estimate and the corresponding numerical solution of the A-SIR model. We observe that, in general, the smallest error is shown by the Proportional-Integral observer.

In Fig. 5 is displayed the behavior of each artificial variable(s) $\eta$ and auxiliary variable(s) $\alpha$ for Proportional and Proportional-Integral observers. Remember that from these variables the populations or variables of interest are reconstructed.

Fig. 5.

New state variable(s) $\eta$ and auxiliary variable(s) $\alpha$. Proportional observer (a) and (b), Proportional-Integral observer (c) and (d).

5.1.1. Environmental noise in the output

Now, let us consider that the output of the A-SIR model contains an additive noise term $\omega g\left(t\right)$. Function $g\left(t\right)$ is as in (64). On the other hand, to generate random numbers $\omega$ we use the Band-Limited White Noise block from SimuLink. In this scenario, we omit the Luenberger observer, since this is not appropriate to deal with additive noise. The simulation parameters and initial conditions are the same from Tables 1 and 2, respectively. The only change are the values of the gains, which are: $\kappa =50,$ ${\kappa }_p=1.4$ and ${\kappa }_i=0.8$.

In Fig. 6 , we observe that the estimates of both algorithms differ considerably from the A-SIR model numerical solution when the number of infected cases is maximum. This is due to the increment of the noise term. On the other hand, although the Proportional-Integral observer presents overshooting, its estimates are better than those made by the Proportional observer. This is due to the attenuation effect given by the integral action. In Fig. 8 we can appreciate even better this effect. Notice how the auxiliary variable $\alpha$ from the Proportional estimation algorithm is clearly under the effect of the noise term. On the other hand, the auxiliary variables from the Proportional-Integral observer almost lack of this noise.

Fig. 6.

Population estimates with additive environmental noise: Both estimates oscillate near the corresponding numerical solution. We can notice that the estimates near the maximum of infected individuals are worse, since the noise magnitude increases according to these.

Fig. 8.

New state variable(s) $\eta$ and auxiliary variable(s) $\alpha$ with additive environmental noise. Proportional observer (a) and (b), Proportional-Integral observer (c) and (d). The effect of the additive noise is clearly more attenuate in the Proportional-Integral estimation algorithm.

A particular case is the susceptible population, where it seems to be a complete attenuation of the noise. This occurs since the susceptible population is obtained through an additional integration of the variable $\eta$.

We can observe the population estimation errors in Fig. 7 . As we establish in Theorem 3, these errors are ultimate uniformly bounded. Although the population estimation errors of the Proportional observer do not exhibit overshooting, it is evident that the Proportional-Integral observer performs better. Then, we conclude that the Proportional-Integral estimation algorithm is the best with and without additive environmental noise.

Fig. 7.

Population estimation errors with additive environmental noise: difference between each estimate and the corresponding numerical solution of the A-SIR model. These errors remain bounded, as we claim in Theorem 3.

5.2. Estimation with real data. Mexico City

Now, we consider official data of infected individuals reported in Mexico City from February 22nd, 2020 to March 13th, 2021 [51]. Since the testing rate in Mexico is low (0.29 test per thousand people, maximum value registred [52]), we consider that all reported cases are symptomatic. In Fig. 9, the infected cases reported in Mexico City are displayed. Mexico city has carried out two confinements during the pandemic, as a result, the behavior of the symptomatic population is quite different to that shown in Fig. 2. One can notice that the reported data is noisy, therefore, we obtain the local average of the reported cases. This is done with the movmean function from MatLab.

We use the Proportional-Integral observer to make the estimates with the real data, since this showed the best performance. For the observer, we use the initial conditions from Table 2 and the gains ${\kappa }_p=100$ and ${\kappa }_i=50$.

In Fig. 10 are shown the estimates obtained by the algorithm. Discontinuous red lines are the estimates when we fed the observer with the original data. On the other hand, the continuous blue lines are the estimates with the local average data. Both cases exhibit overshooting at the beginning, however, these seem to converge to a common trajectory. It is clear that the local average improves greatly the smoothness of the estimates. Especially, the estimation of the asymptotic population (Fig. 10.b).

Fig. 10.

Proportional-Integral estimates with real infected cases reported in Mexico City. Notice that the populations are not normalized.

In this case, we do not possess information about the evolution of susceptible, removed or asymptomatic population along the pandemic. Therefore, we cannot compare directly the obtained estimates, as we did before. However, we have additional data that might be useful to tell how precise is the estimate of removed individuals.

In Fig. 10.c, we present the accumulated deceased cases in Mexico City (discontinuous black line). Until March 13th, the number of deceased people on Mexico City was 29,047. On the other hand, the accumulated reported cases until the same date was 583,698 [51]. Meanwhile, for the same date, the reported mortality rate (ratio between confirmed deaths and confirmed cases) was $4.98\%$ [53]. Moreover, if we consider that the fraction of infected individuals that are symptomatic is 0.15, then we can multiply the number of accumulated deceased cases by the factor $f=(100/\mathcal{M})*(100/15),$ where $\mathcal{M}$ is the mortality rate.

The projected curves on Fig. 10.c, correspond to dotted lines. Notice that the estimates of removed individuals and the projected curve with a mortality rate of $4.98\%$ are similar for the last 50 days. The disparity before those days, might be a consequence of the mortality rate, which is continuously changing. As example, the maximum value registered for Mexico is $12.4\%$ (June 2020) and in March 2021, is $9\%$.

The dotted green line, indicates that the projection of removed individuals for March 13th, 2021 is 3,888,350. Meanwhile, for the same day, the estimate number of removed individuals is 3,954,610 (with local average data) and 3,934,790 (with original data), see Fig. 11 . Then, in day 384, relative error for both cases is about 0.002.

Fig. 11.

Close up, Fig. 10.a.

Finally, in Fig. 12, we observe the estimate behavior of all populations in Mexico City until March 13th, 2021. According to our estimates, around $55\%$ of the population is still susceptible to infection.

Please note that all these estimates still depend on the considered parameters. In particular, remember that the fraction of symptomatic individuals is based on a study carried out in New York City, so this value could be different in Mexico City. Therefore, all this information should be taken with caution.

6. Conclusions

We have presented two model-free reduced order estimation algorithms for the reconstruction of susceptible, asymptomatic and removed population (non-measured) of COVID-19 pandemic. Notice that, in our particular case, the proposed observers do not require certain parameters of the model. This is practical since nowadays there is still much discrepancy between the real values of these parameters. The observers are based on a simple approach which uses only simple algebraic tools like polynomials. Notice that the proposed algorithms are non redundant, since we do not reconstruct the available data of symptomatic individuals. The variables of interest are computed from an artificial variable $\eta$. In particular, we compute the asymptomatic population, which evidence suggest is considerable in the current pandemic scenario. Moreover, we have presented the stability analysis with and without additive noise in the output and proved that the estimation error is asymptotically stable or ultimate uniformly bounded, respectively. Last case is important since in the current scenario, measurements might be poor (noisy). Thus, both estimation algorithms are robust against measurement noise and at the same time, these are structurally simple and easy to implement.

To show the reliability of our methodology, we consider two cases. In the first case, we use the numerical solution of the A-SIR model to fed the proposed observers, including the well known Luenberger observer. We took into account this last observer because it is also simple and easy to implement. However, simulation results reveal that Proportional-Integral estimation algorithm is superior. This claim is based on the population estimation errors and the attenuation effect given by the integral action. In the second case, we considered official data of infected individuals in Mexico City from February 2020 to March 2021. We test the Proportional-Integral observer with the original data (noisy) and the local average data (smooth). Although the estimates made with the original data are noisy, these and the obtained with the local average seem to describe common trajectories. As we do not have official information of any other population, we did a simple analysis to argue the veracity of our estimates. By considering the mortality rate in Mexico City and the number of deceased people, we found out that the relative error in the last day (day 384) is 0.002, thus, we can say that there is at least an indicator of how good are our estimates. Notice that the first case was use as a validation case.

Despite all this, our algorithm still depends on certain parameters of the A-SIR model. Therefore, the reliability of our results depends on good estimates of these parameters. This is challenging, since estimated parameters vary over a wide range of values, depending on the considered region. As an example, Mexico City mortality rate is $4.98\%,$ meanwhile, the same rate is about $9\%$ for the whole country. On the other hand, although the A-SIR model seems to be good enough to describe the pandemic, it does not provide information that might be useful for authorities, such as hospital occupation, symptomatic individuals that do not require hospitalization or undocumented symptomatic individuals. Therefore, in a future work, we will consider a more complex model and tackle the problem of parameter estimation.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Declaration of Competing Interest

None.