[Paper Review] Modelos SIR modificados para la evoluci\'on del COVID19

Nana Cabo Bizet, Alejandro Cabo Montes de|arXiv (Cornell University)|Apr 23, 2020

COVID-19 epidemiological studies5 references4 citations

TL;DR

This paper proposes modified SIR models with time-varying infection rates to predict COVID-19 dynamics in Cuba and other countries. By modeling the infection rate β*(t) as decreasing exponentially after quarantine onset—reaching near-zero by the disease’s average duration (1/γ)—the model accurately fits data from Germany and predicts a peak of 1,000–2,000 active cases in Cuba if strict quarantine is implemented by mid-May 2020, with the key insight that effective control requires β* − γ < 0 (R₀ < 1).

ABSTRACT

We study the SIR epidemiological model, with a variable contagion rate, applied to the evolution of COVID19 in Cuba. It is highlighted that an increase in the predictive character depends on understanding the dynamics for the temporal evolution of the rate of contagion $\\beta^*$. A semi-empirical model for this dynamics is formulated, where reaching $\\beta^*\\approx0$ due to isolation is achieved after the mean duration of the disease $\ au=1/\\gamma$, in which the number of infected in the confined families has decreased. It is considered that $\\beta^*(t)$ should have an abrupt decrease on the day of initiation of confinement and decrease until canceling at the end of the interval $\ au$. The analysis describes appropriately the infection curve for Germany. The model is applied to predict an infection curve for Cuba, which estimates a maximum number of infected as less than 2000 in the middle of May, depending on the rigor of the isolation. This is suggested by the ratio between the daily detected cases and the total. We consider the ratio between the observed and real infected cases (k) less than unity. The low value of k decreases the maximum obtained when $\\beta^*-\\gamma&gt;0$. The observed evolution is independent of k in the linear region. The value of $\\beta^*$ is also studied by time intervals, adjusting to the data of Cuba, Germany and South Korea. We compare the extrapolation of the evolution of Cuba with the contagion rate until 16.04.20 with that obtained by a strict quarantine at the end of April. This model with variable $\\beta^*$ correctly describes the observed infected evolution curves. We emphasize that the desired maximum of the SIR infected curve is not the maximum standard with constant $\\beta^*$, but one achieved due to quarantine when $\ ilde R_0=\\beta^*/\\gamma&lt;1$. For the countries controlling the epidemic the maxima are in the region in which SIR equations are linear.

Motivation & Objective

• To improve SIR model predictions for COVID-19 by incorporating time-dependent infection rates β*(t), reflecting real-world interventions.
• To analyze how the ratio k (observed vs. total infected) affects peak infection estimates due to nonlinearity in the SIR system.
• To determine the conditions under which observed infection curves remain linearly scalable despite k < 1, enabling reliable predictions.
• To evaluate the effectiveness of quarantine by analyzing the evolution of β*(t) and R₀ in real data from Cuba, Germany, and South Korea.
• To demonstrate that effective epidemic control requires β* − γ < 0 (i.e., R₀ < 1), not just peak reduction, and to quantify this threshold using empirical data.

Proposed method

• Formulate a semi-empirical model where β*(t) decreases sharply at quarantine onset and decays to zero within τ = 1/γ (15–20 days), the average disease duration.
• Apply the modified SIR model with time-varying β*(t) to fit daily active case data from Germany, Cuba, and South Korea.
• Use piecewise-constant β* approximations over time intervals to fit data and extrapolate future trends under different quarantine timing scenarios.
• Introduce the scaling factor k = observed infected / total infected to account for underreporting, and analyze its impact on peak prediction.
• Define effective reproduction number R₀ = β*/γ and require R₀ < 1 for epidemic control, using daily data to monitor β* and γ.
• Compare model predictions under early vs. delayed quarantine (e.g., April 16 vs. end of April) to assess policy impact on peak timing and magnitude.

Experimental results

Research questions

• RQ1 How does a time-varying infection rate β*(t), decreasing after quarantine, improve SIR model accuracy for real-world COVID-19 data?
• RQ2 What is the impact of underreporting (k < 1) on the predicted peak of active infections in nonlinear SIR models?
• RQ3 Under what conditions does the SIR system remain approximately linear, allowing k-independent predictions for observed infected and recovered cases?
• RQ4 Can the model predict the timing and magnitude of the infection peak in Cuba based on quarantine implementation dates and β* dynamics?
• RQ5 To what extent does achieving R₀ < 1 (i.e., β* − γ < 0) determine the actual peak size, rather than the standard SIR maximum?

Key findings

• The semi-empirical model with β*(t) decaying to zero within τ = 1/γ successfully reproduces the infection curve in Germany.
• For Cuba, if strict quarantine is implemented by mid-May 2020, the model predicts a peak of 1,000–2,000 active cases.
• The model predicts that a delayed quarantine (e.g., end of April) could push the peak to early June and increase the maximum to several thousand cases.
• Without effective control, with current β* − γ > 0 (R₀ ≈ 1.92), the model projects a peak of 20,000 severe cases by September 2020.
• The observed infection curve remains linearly scalable with k when R₀ < 1, meaning peak predictions for observed cases are independent of k in this regime.
• The model confirms that the true epidemic peak is not the standard SIR maximum (N), but a lower value achieved only when R₀ < 1 due to effective isolation.