Key Words: COVID, Models, Chile

Introduction

Since Daniel Bernoulli used a mathematical method in 1766 to evaluate the effectiveness of variation until today, a large number of models and mathematical-epidemiological concepts have been developed to study and follow the behavior of different infectious diseases in the population. Numerous authors such as J. Brownlee (1906, 1918), R. Ross (1911, 1917), McKendrick and Kermack (1927), and Reed and Frost (1928), among others, contributed to the development of this area of ​​knowledge[1]. Later there was great development that included the spatial dimension, seasonality and the role of carriers, venereal transmission and others. Among these, the contributions of Anderson and May (1978), of Hudson and Dobson from 1985 onwards, and of Roberts and Grenfell from the 90’s[1] stand out. A large number of studies highlighted concepts such as threshold population, herd immunity[2],[3],[4],[5], reproductive number, serial interval, incubation times, and spillover in the dynamics of infectious diseases. All of these concepts contribute in some way to decision-making[6],[7],[8]. In recent decades emphasis has been placed on spatial propagation[9],[10],[11],[12],[13],[14],[15],[16],[17], the role of population mobility, and the connectivity of transportation networks such as airlines during the spread of epidemics[18],[19],[20],[21],[22],[23]. These models have made great contributions to the study of different diseases such as Influenza AH1N1, AH5N1, HIV, SARS, and Ebola[18],[19],[20],[21],[22].

Numerous models have been developed for the current COVID-19 pandemic, a disease caused by the SARS-CoV-2 virus, and for SARS[24],[25],[26],[27]. Most correspond to Susceptible-Exposed-Infected-Removed (SEIR) models with different stratification types, be it spatial, age, socio-economic, etc. Given the great complexity of forms of spreading, mathematical models can offer valuable tools to synthesize information to understand epidemiological patterns and develop quantitative evidence to inform global health decisions [6].

A model has been used in Chile with good success to short-term forecast ICU occupancy, warning authorities of the possible collapse of hospital capacity[28]. SEIR-type models are also used to predict the evolution of the epidemic locally[29]. Another study uses a polynomial cubic adjustment model to estimate case number and an exponential total case model to represent the daily effort to reduce the initial daily growth rate to warn of the pandemic’s potential[30]. Another study also analyzed the minimum daily effort required for the healthcare system to not collapse during the COVID-19 outbreak. A parametric threshold condition is obtained, which involves a parameter associated with the minimum daily effort for not collapsing the system[31]. All studies have helped to highlight the importance of modeling in decision-making[32].

The COVID-19 pandemic has cost more than 12 000 deaths in Chile. Initially, it had an accelerated growth that was partially controlled with interventions, such as closing schools and universities and implementing partial quarantines. After these initial measures, a relaxation of interventions and late interventions led to a large epidemic mainly concentrated in the Metropolitan Region (Santiago), despite the warnings made by the scientific community and different epidemic modeling teams. Subsequently, massive quarantines were established that have been associated with a decrease in the number of cases, leading to a relatively stable number, between 1 000 and 2 000, of daily cases in the country[33]. Throughout this process, a series of factors have complicated the follow-up of cases, such as changes in the notification system and underreporting cases.

Considering that the simplest models can significantly contribute to the monitoring and prediction of epidemics, we have developed three simple mathematical approaches to understand and monitor the dynamics of the COVID-19 pandemic and contribute to decision-making in Chile. In this contribution, we report and discuss the usefulness, the successes, and the practical difficulties that have transpired in the development of COVID-19.

Methods

We carried out an ecological study using official daily public reports from the Chilean Ministry of Health, including new confirmed cases, cases that required admission to the ICU, and deaths attributable to COVID-19 nationally and sub-nationally (regions). With this information, we made: 1) estimates of the effective reproductive number Rt; 2) estimates of the underreporting of cases; and 3) we developed three models with different objectives to follow the epidemic in Chile.

We calculate Rt using the method developed by Cori et al[34]. We consider the last two weeks (14 days) and a serial interval τ = 5 days with variability between 2 and 8 days (based on[35],[36],[37]). The underreporting of cases was estimated according to the method proposed by Russell et al. [38],[39] adapted to the situation in Chile[33].

Maximum potential load model
The objective of this model has been to follow the evolution of the epidemic curve and establish short-term predictions (1 week) of the maximum potential loads of new cases, ICU occupancy, and deaths. This seeks to estimate the health system’s saturation level, which would allow informing decisions that aim to reduce the number of infections or increase the health system’s response capacity.

If there are C_t new cases in a week “t” and that there are I_t = (C_t + C_t-1) infected people, considering that the cases remain contagious for up to two weeks, it can be proposed that: 

C_t+1 ≈ R_t (C_t + C_t-1).

This means that all those infected in the previous two weeks are contagious and will contribute to infections the following week. However, not all infected individuals on a day “i” will become cases the following week, as this depends on the serial interval and its probability distribution. A more suitable expression is:

, t in weeks, i in days, where tf represents the last day of the week “t”, p_i represents the probability that someone infected on day “tf-i” infects someone in week t + 1. Thus a better expression is:

C_t+1 ≈ fR_t (C_t + C_t-1) (1).

where f corresponds to a correction factor as a consequence of the probability distribution of the serial interval. For a maximum load, the correction factor is estimated as the maximum p_i (i = 0-13) for a Gamma distribution with mean 5 and standard deviation 4.3 days (based on[35],[36],[37]); the correction factor is f ≈0.8. The equation (2) is a Fibbonacci like series, and requires two initial conditions (t₁ and t₂). In this study these was t₁ = 10 (March 8), t₂ =65 (March 15)(the new cases in the two first weeks).

Considering that approximately 3.5% of the cases reach an ICU (0.035Ct), that on average, an ICU is occupied for approximately two weeks, and that the latency between onset of symptoms and ICU requirement is approximately one week[38],[39], it can be proposed that occupied ICUs (U_t):

U_t+1 = 0,035 (C_t + C_t-1) (2).

To estimate the maximum number of deaths (Dt), we consider a latency of two weeks between the onset of symptoms and death of symptomatic patients of 2.6%[38],[39]. So:

D_t+1 = 0.026T_t-1 (3)

where T_t-1 corresponds to the total number of cases from two weeks ago. Thus equations (1), (2) and (3) constitute the basis of this model. This model allows to predict the maximum potential load of COVID-19 expected for the following week, based on the “history” of the two previous weeks.

Discreet SEIR model and stochastic model
The objective of this model has been to try to predict the shape of the epidemic curve and the times in which the increase in the number of cases, the peak, and the decrease would occur in the earliest possible way. We use a variation of a SEIR (Susceptible (S), Latent (E), Infectious (I), removed (deceased + recovered (Rm)) model in its discrete form:

S_i+1 = S_i – βS_iI_i (4).

E_i+1 = E_i + βS_iI_i  – νE_i (5).

I _i+1 = I _i + νE_i – (μ + γ) I_i (6).

Rm_i+1 = Rm_i + (μ + γ) I_i (7).

C_i+1 = pI_i (8).

U_i+1 = qC_i-7 (9).

D_i+1 = hT_i-14 (10).

Where β is the transmission coefficient, ν the transfer rate from E to I, µ the mortality rate, γ the recovery rate, p the proportion of infected who are notified, q the proportion of symptomatic patients requiring ICU and h the fatality rate. The sub-idex “i”represents time in days. In this study the parameters ν = 1/5 days^-1, (µ + γ) = 1/14 days^-1, p = 0.1, q = 0.035 and h = 0.026 were considered constant and the transmission coefficient β and the asymptote of the model "S*" as variable parameters. The model was adjusted as the epidemic progressed, depending on the changes in the number of reported and in the notification system. Initially, on April 15, the asymptote of the model “S*” was estimated considering a Chilean population of 19,000,000, a proportion of herd immunity 57.4%, that 5% could get sick (based on European experience), and a heterogeneity correction factor of 0.5 considering that at this time the epidemic was centered in Santiago. Thus initial S* = 19,000,000 x 0.547 x 0.5 x 0.05 = 259,825 cases. The transmission coefficient β was estimated adjusting the best model. The model was adjusted weekly (if necessary), varying S* and β. The goodness of fit of the model was studied with the coefficient of determination (R²) and its significance (F-test).

Given the great variability and stochastic fluctuations of the cases worldwide, in order to include their effects on the dynamics, we consider the simple stochastic version of the previous model, with the same parameters, based on equations (4) to (7), with the Monte Carlo kinetic method with the Gillespie algorithm[40]. In this, we consider the three events:

1. A latent occurs with rate a₁: βS(i)I(i); S = S-1; E = E+1;
2. An infected occurs with rate a₂: νE(i); E =E-1; I = I+1
3. A removed occurs with rate a₃: (μ + γ)I(i); I =I-1; Rm =Rm+1

The probability for a_j is: . The time until the next event (τ) is exponentially distributed, with a rate equal to the sum of the rates of all possible events: f(τ) = ∑ _jα _je^{-τ∑ jaj} with an expected value  [39].

Gompertz model
When a flattening in the curve of the total cases could already be established in Santiago, a generalized logistic type model, the Gompertz model, was fitted to the curve of the total cases (T(t)):  (11), where A is the asymptote and b and c are the parameters that control the curve’s position and the growth rate, respectively[41].

Results

Maximum potential load model
This model adequately reproduced the shape of the epidemic curve, maintaining values above those observed for weekly cases, the number of occupied ICUs, and the total number of deaths. When weekly cases were corrected for underreporting, they approached the maximum load predictions (Fig 1).

Figure 1. Discrete model of maximum potential load.

Discrete SEIR Models and Stochastic Model
This model was very useful in predicting the rise, peak, and decline of cases, with great predictive power until the decline phase of the initial outbreak in July and early August. (Fig 2A, Table 1). In recent months there has been a better adjustment to the corrected cases, but it has been losing adjustment, especially in ICUs and the number of deaths, which has had major changes in the reporting system (Fig 2B, Table 1). The stochastic SEIR model allowed us to observe whether the variations in the number of cases were due to a mismatch in the model or to random fluctuations (Fig. 2C).

Table 1. Adjustment of the deterministic SEIR model to the observed values.

Figure 2. Discrete deterministic SEIR model.

Gompertz Model for the Metropolitan Region (Santiago)
This model has allowed a good follow-up of the decrease in the number of cases in the Metropolitan Region and, so far, an adequate prediction of the endemic situation (Fig. 3).

Figure 3. Gompertz models for the Metropolitan Region (Santiago).

Figure 4. Evolution of the daily incidence of COVID-19 in Chile.

As the decrease in the Metropolitan Region cases was asymmetric, a Gompertz model was fitted, which has shown a very good fit in much of the process; however, in recent days, it has also been losing predictive capacity. This is probably because the decrease in the number of cases has not occurred naturally due to herd immunity but instead associated with a series of interventions that have confined a large part of the population[33]. Currently, the Metropolitan region is relatively stable at lower than 10 cases per 100 000 inhabitants.

In summary, the models used have been very useful, with different objectives at different stages of the epidemic. The short-term model is still useful, providing an upper bound on the number of cases each week; the SEIR model had a very good predictive capacity of the maximum; the stochastic model introduced variability in the prediction, and the Gompertz model has had a better predictive capacity of the decline in cases.

Notes

Authorship contributions
MC: conceptualization, methodology, formal analysis, research, writing (reviews and edits), data management, data presentation, project management. CC: conceptualization, methodology, formal analysis, research, writing (revisions and editions), supervision. AC: formal analysis, research, writing (reviews and edits), supervision.

Competing interests
The authors have completed the ICMJE declaration of conflicts of interest form, and declare that they have not received funding to prepare the report; not have financial relationships with organizations that may have an interest in the published article, in the last three years; and not having other relationships or activities that could influence the published article. The forms can be requested by contacting the responsible author or the editorial direction of the Journal.

Funding
The authors acknowledge the funding of the ANID COVID 0960 grant.

Ethics 
Our study is based on official secondary data reported by the Ministry of Health of Chile; therefore, it did not require approval from the Faculty of Medicine ethics committee.

From the editors
The original version of this manuscript was submitted in Spanish. This English version was submitted by the authors and has been copyedited by the Journal.

Figure 1. Discrete model of maximum potential load.

Table 1. Adjustment of the deterministic SEIR model to the observed values.

Figure 2. Discrete deterministic SEIR model.

Figure 3. Gompertz models for the Metropolitan Region (Santiago).

Figure 4. Evolution of the daily incidence of COVID-19 in Chile.

Esta obra de Medwave está bajo una licencia Creative Commons Atribución-NoComercial 3.0 Unported. Esta licencia permite el uso, distribución y reproducción del artículo en cualquier medio, siempre y cuando se otorgue el crédito correspondiente al autor del artículo y al medio en que se publica, en este caso, Medwave.