A Time-dependent SIR model for COVID-19 with Undetectable Infected Persons (2003.00122v6)

Abstract: In this paper, we conduct mathematical and numerical analyses to address the following crucial questions for COVID-19: (Q1) Is it possible to contain COVID-19? (Q2) When will be the peak and the end of the epidemic? (Q3) How do the asymptomatic infections affect the spread of disease? (Q4) What is the ratio of the population that needs to be infected to achieve herd immunity? (Q5) How effective are the social distancing approaches? (Q6) What is the ratio of the population infected in the long run? For (Q1) and (Q2), we propose a time-dependent susceptible-infected-recovered (SIR) model that tracks 2 time series: (i) the transmission rate at time t and (ii) the recovering rate at time t. Such an approach is more adaptive than traditional static SIR models and more robust than direct estimation methods. Using the data provided by China, we show that the one-day prediction errors for the numbers of confirmed cases are almost in 3%, and the total number of confirmed cases is precisely predicted. Also, the turning point, defined as the day that the transmission rate is less than the recovering rate can be accurately predicted. After that day, the basic reproduction number $R_0$ is less than 1. For (Q3), we extend our SIR model by considering 2 types of infected persons: detectable and undetectable infected persons. Whether there is an outbreak in such a model is characterized by the spectral radius of a 2 by 2 matrix that is closely related to $R_0$. For (Q4), we show that herd immunity can be achieved after at least 1-1/$R_0$ fraction of individuals being infected. For (Q5) and (Q6), we analyze the independent cascade (IC) model for disease propagation in a configuration random graph. By relating the propagation probabilities in the IC model to the transmission rates and recovering rates in the SIR model, we show 2 approaches of social distancing that can lead to a reduction of $R_0$.

Analysis of a Time-dependent SIR Model for COVID-19 with Undetectable Infected Persons

The paper explores the development and evaluation of a dynamic susceptible-infected-recovered (SIR) model tailored for analyzing the COVID-19 pandemic. The primary focus is on accommodating time-varying transmission and recovery rates, alongside addressing the challenge posed by undetectable or asymptomatic carriers.

Key Contributions

1. Time-dependent SIR Model: Traditional SIR models employ constant transmission (β) and recovery (γ) rates. However, these assumptions are often overly simplistic, especially under dynamic real-world conditions influenced by policy changes and public health interventions. The authors propose a model where both rates evolve over time, offering a more flexible tool for predicting epidemic trends. The model's robustness is underscored by historical data, demonstrating a daily prediction error of under 3% in the number of cases, except during periods of data redefinition.
2. Handling Asymptomatic Infections: The paper extends the SIR framework to include both detectable and undetectable (asymptomatic) carriers. This extension is crucial given the significant role asymptomatic transmissions play in the spread of COVID-19. The model is characterized by a $2 \times 2$ matrix whose spectral radius determines the outbreak potential. Notably, if the spectral radius exceeds one, an outbreak is likely.
3. Social Distancing Measures: Through the independent cascade (IC) model adapted to the SIR framework, the authors examine the potential impact of social distancing. The model shows that these measures can effectively reduce the basic reproduction number, $R_0$, thereby impeding disease spread. Two social distancing strategies emerge: reducing interpersonal contacts across the board and specifically targeting mass gatherings, each with potential effectiveness illustrated through simulations.
4. Numerical Analysis and Real Time Adjustments: Using datasets from China, South Korea, Italy, and Iran, among others, the paper underlines the differential impacts of local policy responses and social dynamics on the pandemic's trajectory. The adaptable SIR model effectively predicts outbreak peaks and declines, aligning closely with real-world timelines in several instances.

Implications and Prospects

From a theoretical standpoint, this paper enriches the existing epidemiological toolkit, equipping researchers with a model adaptable to real-time logistics. Practically, it guides policymakers on both the timing and nature of interventions requisite for epidemic control, emphasizing the necessity of adaptable frameworks that absorb new data trends and intervention effects efficiently.

This work suggests robust avenues for future exploration. For instance, integrating stochastic elements could refine predictions when case numbers fall to low ranges. Furthermore, coupling these models with advanced machine learning techniques could automate the adjustment of parameters, enhancing real-time pandemic response.

In conclusion, the research offers significant insights into the dynamic modeling of pandemics, posing a veritable pathway for refining epidemic response strategies in the context of the ongoing global health challenge posed by COVID-19 and potential future pathogens. This work lays the groundwork for subsequent investigations aiming to leverage data-driven adaptability in epidemiological modeling, potentially shaping how health systems integrate real-time data into actionable insights.