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.

Keywords: A-SIR model; Asymptomatic individuals estimation; COVID-19 pandemic; Model-free estimation algorithm.

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.

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.

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.

Fig. 5

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

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. 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.

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.

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. 10

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

Fig. 11

Close up, Fig. 10.a.

Fig. 12

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