Modelling quarantine effects on SARS-CoV-2 epidemiological dynamics in Chilean communes and their relationship with the Social Priority Index

Definition of the SUQCR model

We define the SUQCR model by including removed individuals (R) in the simplified SUQC model. However, R representing individuals who are no longer infectious, who are considered recovered, and who derive from Q or U, at a rate of 1/ D; as such, the mean time of recovery from the disease is D, in days. The factor 1/ D does not apply to individuals C because they correspond to cases counted as confirmed at any time up to t. Thus, the SUQCR model that we propose here as a generalization of the SUQC model is defined in the scheme of Fig. 1 and by the following system of ordinary differential equations:

(5)${\displaystyle \frac{dS\left(t\right)}{dt}=-\alpha U\left(t\right)S\left(t\right)/N}$ (6)${\displaystyle \frac{dU\left(t\right)}{dt}=\alpha U\left(t\right)S\left(t\right)/N-\left(\gamma +1/D\right)U\left(t\right)}$ (7)${\displaystyle \frac{dQ\left(t\right)}{dt}=\gamma U\left(t\right)-\left(\beta +1/D\right)Q\left(t\right)}$ (8)${\displaystyle \frac{dC\left(t\right)}{dt}=\beta Q\left(t\right)}$

Note that $N=S\left(t\right)+U\left(t\right)+Q\left(t\right)+C\left(t\right)+R\left(t\right)$. We assume that R(t = 0) ≈ 0.

To demonstrate the relationship between the SUQC and SUQCR models, note that Eqs. (5) and (8) are the same as Eqs. (1) and (4), respectively. In addition, Eqs. (6) and (7) converge to Eqs. (2) and (3). respectively, when D ≫ 1. This demonstrates that SUQCR is a generalization of SUQC.

The model studied did not consider the latency period of COVID-19. Through meta-analysis studies, it has been determined that the latency period of SARS-CoV-2 is about 5 days on average, within a wider range (Grudlewska-Buda et al., 2021). Considering that here we have used C(t) values reported on average at 3 days, with 25% standard deviation, in a range of 2 to 5 days, we believe that data updated daily would be more appropriate to study the effect of the latency period.

Study communes

The modeled data were the values of C(t) recorded in each of the 32 communes of the province of Santiago, in the Metropolitan Region of Chile. The data from each of these communes were grouped into two consecutive stages. The first stage (stage 1) is without quarantine imposed by the health authorities. Given the influence of national and international news about the pandemic, the constant advice of subject-matter experts and quarantines previously imposed in some communes of the country, a fraction of the population most likely followed some degree of quarantine and other social distancing measures in stage 1, in addition to hygiene protocols and the use of a masks (Howard et al., 2021), albeit voluntarily. In the second stage (stage 2), quarantine was officially imposed by the health authorities, mandating compliance with strict social distancing protocols and other self-care practices.

Total number of confirmed cases of infection (C(t)) by commune during the study period

The values of C(t) for each of the 32 communes are from the epidemiological records of the period (corresponding to stages 1 and 2 for all communes) spanned from March 30 to June 15, 2020 (Ministerio de Ciencia, Technologia, Conocimiento e Innovacion, 2020).

Calculation of epidemiological parameters of the SUQCR model

The value of α could no longer be measured in pure form through an exponential model simpler than the model that we applied here because most of the population already had enough information to adopt some self-care measures by the time the pandemic arrived in Chile. However, such a simple exponential approximation was used in China with the first data collected in Wuhan, where the first-ever cases were recorded. As a result, the value of the infection rate, α = 0.2967, calculated for Wuhan from January 20 to 27, 2020, was used in the SUQC model, as applied to a few other regions of China. In this model, the reproduction number, ℜ, was calculated considering that fluctuations in α could be compensated by fluctuations in γ because ℜ = α/γ (Zhao & Chen, 2020). However, when γ = 0 this ℜ value is indeterminate.

For the SUQCR model, defining $\mathfrak{R}\equiv \alpha /\left(\gamma +1/D\right)$, assuming that susceptible individuals almost do not decrease compared to the total population (S(t) ≈ N), and from Eq. (6), we have (9)${\displaystyle U\left(t\right)=U\left(t=0\right)e^{\left(\gamma +1/D\right)\left(\mathfrak{R}-1\right)t}}$

In this case, ℜ corresponds to an effective reproduction number, due to the quarantine effects represented by γ, such that ℜ < ℜ_basic, with ℜ_basic = αD (the basic number of reproduction) (Ridenhour, Kowalik & Shay, 2013), using D = 14 days (Byrne et al., 2020) and the aforementioned value of α (Zhao & Chen, 2020).

Each quarantine stage, 1 and 2 (i.e., voluntary and imposed), has the following time series: a time series of C(t), denoted by C, derived from the study period, and a theoretical time series, denoted by $\hat{\mathbf{C}}$, corresponding to the same days, determined by numerical integration of the Eqs. (5), (6), (7) and (8) system using the Runge–Kutta method and described as a function $f\left(\right)$ of the fitting parameters. (10)${\displaystyle \hat{\mathbf{C}}=f\left(\beta ,\gamma ,U\left(t=0\right),Q\left(t=0\right)\right)}$

The fitting parameters β, γ, U(t = 0) and Q(t = 0) were calculated by minimizing the error function err, which is equal to the mean squared deviation between the empirical and theoretical time series data: (11)${\displaystyle err\left(\beta ,\gamma ,U\left(t=0\right),Q\left(t=0\right)\right)={∥\mathbf{C}-\hat{\mathbf{C}}∥}^2}$

In fitting the parameters, to minimize err (Eq. 11), we alternatively used the MATLAB function fminsearch or fmincon (Birgin & Gentil, 2012) to optimize the fit of the model: fminsearch finds a minimum of unconstrained multivariable function. Instead, fmincon finds a minimum of constrained nonlinear multivariable function. In general, both functions were useful when used in the data adjustments to the model, thus obtaining two sets of adjustment parameters per stage and per commune. However, for each commune, we select the best adjustment results obtained.

The data and programs used are available as supplementary material.

Study of the estimated parameters by commune

After applying the SUQCR model to each commune and in each quarantine stage, the goodness-of-fit parameters were recorded, and the values of β, γ and ℜ were analyzed for stages 1 and 2. To compare the mean values per quarantine stage of epidemiologically relevant parameters (i.e., quarantine rate, γ; confirmation rate, β; and reproduction number, ℜ) of the 32 communes of the province of Santiago, Student’s t-test was applied to the mean values of two paired samples, in two-tailed tests. Specifically, in the two paired samples the values of each parameter in one commune in stage 1 is compared with the new value in stage 2 for the same parameter and commune. Moreover, the correlations between γ and the social priority index SPI were studied. The SPI is used by the Metropolitan Regional Government to support the communes with the greatest relative deficiencies, and it is an indicator developed for the Chilean Secretary of Social Development to assist in the design and assessment of public policies (Seremi de Desarrollo Social y Familia Metropolitana, 2020). The method for its estimation has changed from when it was initially launched in 1995, as the main determinants of the quality of life of the inhabitants of the Metropolitan region change along the years. However, its focus remains the three most relevant dimensions of social development (income, education, and health), and its numerical value is meant to establish a comparison between the relative standard of living of the people from each commune and targeting the resources for solving social needs. Each dimension has 2–4 indicators that are scored from 0–100, where 0 is the lowest priority and 100 is the worst relative situation or highest priority. The final score is calculated as the weighted mean of each indicator.

Assessing the robustness of results

To assess the robustness of the parameter fitting results, the following Montecarlo experiment was performed: Gaussian 100 ϵ % noise was added to data record of the confirmed daily incidences in a commune (η(t)), and new data replacing η(t) were defined as ${\eta }_{noisy}\left(t\right)=\eta \left(t\right)\times \left(1+ϵ\mathfrak{N}\left(0,1\right)\right)$, given that $\mathfrak{N}\left(0,1\right)$ is a random number from the normal distribution with mean 0 and standard deviation 1. Total confirmed daily cases in the presence of noise was calculated as $C_{noisy}\left(t\right)={\sum }_{i=t_0}^t{\eta }_{noisy}\left(i\right)$, determining the β, γ and ℜ adjustment parameters for each commune and in the corresponding quarantine stage (1 or 2). According to different noise levels (ϵ), there were 30 independent repetitions, from which means and standard deviations for above mentioned parameters were calculated. Finally, by applying the Student’s t test for paired samples from the same commune in different quarantine states, it was demonstrated that the differences between the obtained values of the same parameters are significant (p < 0.05) regardless of the used noise levels (ϵ = 0, 0.01 and 0.05).

On the other hand and following the previous experiment in case of 5% noise (ϵ = 0.05) on the original data, to assess the significance of the effects reported in A and B in Fig. 5, we have calculated the mean ($\overline{{\gamma }_i}$) and the variance (${\sigma }_i^2$) of γ per commune (i = 1, 2,…,32) and quarantine stage (1 or 2). With these statistical parameters, a lineal fitting for $\overline{{\gamma }_i}$ versus SPI_i was calculated by weighted least-squares method (Matlab curve fitting app was used, where X_data, Y_data and weights were SPI_i, $\overline{{\gamma }_i}$ and reciprocals of the variance values for each of the 32 communes, respectively). This adjustment was in cases of quarantine stages 1 and 2. In both stages, the obtained r² (r² = 0.228 and r² = 0.511, for stages 1 and 2, respectively) were higher than the corresponding values obtained with the original data, ratifying that differences between the adjustment in A and B in Fig. 5 qualitatively does not vary for small levels of Gaussian noise (5%) applied on daily incidence data.