COVID-19 and other viruses: Holding back its spreading by massive testing

Highlights

• A SEIR model is introduced to analyse the efficiency of test distributions.

• It is proved that massive testing helps reducing the number of infected people.

• A heuristic method is developed to increase the number of saved infections.

• Extensive computational experience quantifies the number of saved infections.

Abstract

The experience of Singapore and South Korea makes it clear that under certain circumstances massive testing is an effective way for containing the advance of the COVID-19.

In this paper, we propose a modified SEIR model which takes into account tracing and massive testing, proving theoretically that more tracing and testing implies a reduction of the total number of infected people in the long run. We apply this model to the spreading of the first wave of the disease in Spain, obtaining numerical results.

After that, we introduce a heuristic approach in order to minimize the COVID-19 spreading by planning effective test distributions among the populations of a region over a period of time. As an application, the impact of distributing tests among the counties of New York according to this method is computed in terms of the number of saved infected individuals.

Introduction

The emergence of the coronavirus disease 2019 (COVID-19) and its mutations motivated actions on the inhabitants of several countries like isolation, social distance and others. The purpose of these restrictions is to slow down the spreading of the pandemic in order not to collapse health systems. Carrying out massive testing is a complementary way to control the pandemic. Throughout this article we will analyse the impact of testing on the number of infections and we will explain how we have developed an expert system to obtain test distributions which minimize the number of infections within a region and for a temporal horizon.

The classical SIR model and its variations are a powerful mathematical tool for predicting the evolution of epidemics around the world. The amount of literature studying the properties of solutions of such models is huge, see e.g. Brauer and Castillo-Chávez, 2012, Ji and Jiang, 2017, Jiao et al., 2020, Kuniya and Nakata, 2012, Li and Muldowney, 1995, Zhang and Teng, 2007, Zhao et al., 2017 and the references therein among many others. In the present, much effort is taken in order to estimate the evolution of the COVID-19 pandemic, see e.g. Annas et al., 2020, Arcede et al., 2020, Britton et al., 2005, Chen, Lu et al., 2020, Chen, Rui et al., 2020, Gomes et al., 2020, Ndaïrou et al., 2020, Roda et al., 2020, Sauter and Pacheco, 2020 and Xu et al. (2020).

The coefficients which are involved in the system of differential equations of a SIR model are constants in the simplest situation. However, it is more realistic in order to estimate the evolution of an epidemic to consider them as functions of time, as given in some of the above references. Moreover, when we need to take into account the impact of governmental actions like confinements, quarantines, restrictions on mobility and travelling and so on, the coefficients, especially the rate of transmission, can change abruptly. Thus, it is better to define them piecewise by choosing appropriate time intervals depending on the moments in which the governmental restrictions are more severe or, on the contrary, become more relaxed. This approach has been considered in several papers estimating the evolution of the COVID-19 pandemic in several countries, see Falco et al., 2020, Kwuimy et al., 2020, Lin et al., 2020, Mushayabasa et al., 2020, Niazi et al., 2020 and Tang, Bragazzi et al. (2020). Also, the parameters can depend on time due to seasonality (see He et al. (2020)).

Apart from tough measures leading to a lockdown, tracing, massive testing and isolation are very effective tools which help to contain the spread of the illness even without restrictive social distancing, as we can see in the examples of Singapore, South Korea and some Italian towns (see Romagnani et al. (2020)). The main drawback of massive testing is the economic burden. However, as pointed out in Eichenbaum et al. (2020) in the long run it pays off, as these policies can dramatically reduce the economic costs of the epidemic.

Thus, it is quite interesting to modify the SIR or SEIR model in order to measure their influence in the evolution of the pandemic and also to determine the best way to carry out massive testing. In Berger et al. (2020) the SEIR model is modified by taking into account testing and quarantine measures. It is shown that increasing testing the governments could relax the quarantine conditions (implementing a targeted and more efficient quarantine), so that the economic and social costs would be smaller while maintaining constant the human costs. In Wang (2020) the author makes a qualitative analysis of a SIR model in which the infected individuals are divided into two groups: those undetected and those detected by means of some tests, which have a lower rate of transmission of the disease due to quarantine measures. It is shown that testing reduces the number of infections in the long term, avoiding in this way herd immunity. In Niazi et al. (2020) a model taking into account random massive tests is given. The impact of testing is analysed, obtaining the optimal values for the number of tests to be carried out each day in two possible testing policies to control the epidemic. In these papers, detection by tracing and random testing are put together. In Tang, Wang et al. (2020) a model in which contact tracing and quarantine are considered is used in order to estimate the evolution of the epidemic in Wuhan. In Ubaru et al. (2020) the problem of optimal testing for individuals under limited testing and tracing resources is studied by using a dynamic-graph based SEIR epidemiological model.

In this paper we modify the SEIR model in such a way that the impact of tracing and random testing can be measured separately. For this aim we assume that infected people who are detected are posed in quarantine and then they cannot infect other people any more; hence, only undetected infected individuals are able to spread the epidemic (of course, this is an ideal situation, being more realistic to consider that the rate of infection of those detected is lower, as given in Li et al. (2020)). In our model we differentiate two kind of detected infected individuals: people which are detected because either they are symptomatic or are direct contacts of infected individuals (and then detected by tracing) and people which are detected by means of massive random testing. These two methods of detection are complementary, as by tracing one detects people with symptoms and their contacts, whereas by testing one detects asymptomatic individuals that otherwise would remain undetected and would continue infecting other susceptible individuals.

In Section 3, we make a qualitative analysis of the model in the case where the coefficients are constants. We show first that, as in the standard SEIR model, all the solutions converge as times increases to a fixed point. The number of susceptible individuals in the long term determines the size of the epidemic. We prove that the limit number of susceptible individuals is an increasing function of the parameters of tracing and testing. That is, the more tracing and testing, the less number of infected people in the long term.

In Section 4, we estimate first the parameters of the model during the first wave of the COVID-19 epidemic in Spain if no massive testing is used. We consider the case where the parameters are functions of time and are defined piecewise by taking into account the moments where confinement restrictions were established or relaxed in Spain between March and June of 2020. After that we study the evolution of the epidemic when a constant number of tests is carried out each day, calculating the number of infected individuals which are saved in the long term for several values for the parameters which characterize tracing and testing.

However, a constant distribution of tests per day is far to be optimal. Moreover, it could be better to distribute the tests among different populations in a non-proportional way. Thus, the main purpose of this work is to develop an expert system that provides an effective distribution of tests among the populations of a region over a period of time. So, in Section 5 a heuristic method based on the proposed modified SEIR model is introduced in order to optimize the distribution of tests. In this section it is also explained how to estimate the parameters of the model by using the Differential Evolution technique. This technique was firstly introduced in Storn (1996) as an evolutionary method for optimizing nonlinear functions. Also, it has been employed for estimating parameters of infectious diseases models such as SIR, SIS, SEIR, SEIS and others Kotyrba et al. (2015). For example, it has been applied to estimate the SIR parameters of the COVID-19 pandemic in Italy (see Iorio and Li (2020)). Several improvements, versions and applications of this technique like the ones in Wang et al. (2015) and Yi et al. (2016) or the version employed in Iorio and Li (2006), which will be the one used in this paper, can be found in the literature.

Finally, in Section 6 we study the effectiveness of the proposed heuristic approach by an extensive computational analysis of the spreading of the COVID-19 pandemic in the New York counties during the months of April, May and June of 2020. The results of our distribution are also compared with a distribution which is homogeneous in time and proportional to the size of each population.