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A Time-dependent SIR model for COVID-19 with Undetectable Infected Persons

Yi-Cheng Chen, Ping-En Lu, Cheng-Shang Chang, Tzu-Hsuan Liu

arXiv:2003.00122v6q-bio.PEcs.LGstat.ML

TL;DR

The paper asks how COVID-19 can be contained and how its spread, immunity threshold, social-distancing effects, and long-run infection depend on time variation, detectability, and network structure. It develops time-dependent SIR, two-type infection, and independent-cascade models. On China’s data, one-day prediction errors for infected and recovered counts are almost within 3%, while the analyses provide outbreak, herd-immunity, and social-distancing threshold results.

  • Problem

    The paper investigates containment, epidemic timing, asymptomatic infections, herd immunity, social distancing, and long-run infection for COVID-19.

  • Method

    The paper combines a time-dependent SIR model with a two-type infected-person model and an independent-cascade model on configuration-model networks.

  • Results

    One-day prediction errors for infected and recovered counts are almost within 3% on China’s NHC dataset, and the analyses report thresholds for outbreaks, herd immunity, and social-distancing reductions in R0.

  • Takeaways & Limitations

    The models provide a framework for tracking COVID-19 transmission over time and analyzing how detectability, immunity, social distancing, and network structure relate to epidemic spread.

Abstract

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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 $R_0$.

I. INTRODUCTION

The paper formulates six COVID-19 questions around containment, epidemic timing, asymptomatic infections, herd immunity, social distancing, and long-run infection. It combines a time-dependent SIR model with a two-type infected-person model and an independent-cascade network model to analyze these questions.

  • Motivation: COVID-19 asymptomatic infections can evade detection and increase transmission because infected persons may be unaware of their contagiousness.More than 10% of infected persons cannot be detected if body temperature is used for detection.
  • Time-dependent SIR model: The time-dependent SIR model tracks transmission and recovering rates as functions of time instead of treating them as fixed.The model uses β(t) and γ(t) to predict infected and recovered persons and assess epidemic control.
  • Containment and timing: Feb. 17, 2020 is predicted as China’s turning point, with the epidemic ending about 6 weeks later and approximately 80,000 confirmed cases if policies continue.The turning point is defined by R0(t) becoming less than 1.
  • Asymptomatic infections: The two-type infected-person model uses the spectral radius of a 2 × 2 matrix to characterize outbreaks involving detectable and undetectable infections.The spectral radius is above 1 for an outbreak and below 1 when there is no outbreak; its threshold agrees with R0’s threshold.
  • Herd immunity: Herd immunity can be achieved after at least 1 − 1/R0 of individuals are infected and recovered.The result interprets herd immunity as reducing the susceptible population.
  • Social distancing and long-run spread: The independent-cascade model links social-distancing interventions to reductions in R0 while accounting for network structure.Keeping an average fraction a of normal contacts reduces R0 by a factor of a^2; removing nodes with degree at least k0 gives a reduction involving Pk0−2.
  • Social distancing and long-run spread: In a configuration-model network, R0 > 1 implies that a certain population proportion will be infected, and the long-run ratio is obtained from a fixed-point equation.For Poisson degree distributions, the equation reduces to 1 − r = e−R0r; the paper reports this as conservative relative to power-law real networks.

B. Differential Equations for the Time-dependent SIR Model

This section derives a time-dependent SIR formulation from the traditional model and adapts it to daily COVID-19 data. Historical infected and recovered counts are used to estimate daily rates, which are then predicted with FIR filters and ridge regression.

  • SIR equations: The traditional SIR model describes susceptible, infected, and recovered populations with differential equations.Recovered includes people who recover or die and are no longer contagious.
  • Time-dependent formulation: Replacing β and γ with β(t) and γ(t) makes the transmission and recovering rates time-dependent while preserving the three SIR variables.The time-dependent rates are inserted into the differential equations governing S(t), X(t), and R(t).
  • Equation intuition: The susceptible equation reflects new infections at rate β(t)S(t)X(t)/n and removals from the susceptible state.The infection term follows the susceptible fraction S(t)/n among random contacts.
  • Discrete-time model: The differential equations are converted into discrete-time difference equations because COVID-19 data are updated daily.The discrete formulation retains the relation among S(t), X(t), and R(t).
  • Rate estimation: Assuming S(t) ≈ n during the initial spread simplifies the equations and permits daily estimation of β(t) and γ(t) from historical X(t) and R(t).The resulting rate estimates support prediction of time-varying transmission and recovering rates.
  • Rate prediction: FIR filters predict future transmission and recovering rates from lagged rate values, with coefficients estimated using ridge regression.The filters have orders J and K, and ridge regression uses regularization parameters α1 and α2.

E. Tracking the Number of Infected Persons ˆX(t) and the Number of Recovered Persons ˆR(t) of the Time-dependent SIR Model

The paper tracks and predicts infected and recovered populations using a time-dependent SIR model, while separately modeling detectable and undetectable infected persons.

  • Tracking method: Historical data are used to measure β(t) and γ(t), then ridge regression learns finite impulse response filter coefficients for prediction.The procedure estimates future rates and propagates predicted infected and recovered counts through the model equations.
  • Tracking method: At time T, predicted infected and recovered counts are computed by replacing the rates with their estimated values in the SIR equations.For later times, the method repeatedly estimates β̂(t) and γ̂(t), then predicts the next values of X̂(t) and R̂(t).
  • Tracking method: Algorithm 1 measures historical rates, trains ridge regressions, estimates the next rates, and iteratively predicts future epidemic counts.The algorithm continues until the specified prediction window is reached.
  • Model limitation: The mean-field approximation may be inaccurate when infected and recovered populations are small, requiring stochastic epidemic models such as Markov chains.This is an explicit scope limitation of the deterministic tracking model.
  • Undetectable infected persons: The two-type system is stable when the spectral radius of its transition matrix is below 1; above 1, infected counts grow exponentially and an outbreak occurs.The spectral radius is the largest absolute value of the matrix eigenvalues.

B. The Basic Reproduction Number

The two-type infected-person model uses a reproduction number and a transition-matrix spectral radius to characterize whether an outbreak occurs.

  • Definition: R0 is the expected number of additional infections caused by a newly infected person before recovery.For type I and type II persons, the expected counts are β1/γ1 and β2/γ2, respectively, weighted by w1 and w2.
  • Outbreak condition: R0 < 1 implies no outbreak, whereas R0 > 1 implies an outbreak in the two-type model.The theorem links these conditions to the transition matrix A.
  • Eigenvalue analysis: The system’s stability condition is established by analyzing the eigenvalues of the associated 2 × 2 matrix.The derivation introduces an identity matrix and transformed eigenvalue expressions before selecting the larger eigenvalue.
  • Outbreak condition: The spectral radius of A is less than 1 when R0 < 1 and larger than 1 when R0 > 1.The spectral radius is determined through the larger eigenvalue of the positive 2 × 2 transition matrix.

C. Herd Immunity

The herd-immunity analysis models immunity as a reduced susceptible fraction, which proportionally reduces transmission rates and yields a threshold for infected and recovered individuals.

  • Modeling herd immunity: Herd immunity is represented by assuming a fraction h of people remains susceptible, so 1 − h is immune.This replaces the early-stage assumption that nearly everyone is susceptible.
  • Modeling herd immunity: The governing equations differ from the original model by replacing β1 and β2 with β1h and β2h.Thus, herd immunity reduces both transmission rates by the factor h.
  • Threshold result: For the two-type model with R0 > 1, herd immunity can be achieved after at least 1 − h* of individuals become infected and recovered.This threshold is stated in Corollary 2.
  • Network extension: The network analysis considers disease propagation in a configuration random graph rather than only the population-level SIR equations.The IC model assigns type-specific propagation probabilities to infected nodes and uses degree-distribution information.
  • Network extension: In a configuration model, infected subgraphs are trees with high probability, and excess degree differs generally from the ordinary degree distribution.The two distributions coincide for Poisson degrees, yielding the Erdős-Rényi random graph.
  • Network extension: The IC analysis defines u1 and u2 as probabilities that infected trees reached through a neighbor are finite.The recursive equations use propagation probabilities, infected-person types, and excess-degree probabilities.

B. Connections to the Previous SIR Model

The paper connects network propagation probabilities to SIR transmission and recovery rates, obtaining an outbreak-stability condition for configuration-model networks.

  • Connection to SIR: The IC model’s propagation probabilities are specified so type I and type II nodes infect the same average numbers of nodes as in the SIR model.For type I, the SIR average β1/γ1 is matched using the average excess degree and φ1.
  • Stability condition: With these propagation probabilities, the configuration-model IC process has a stability condition expressed through R0.The result is derived by writing the recursive equations in matrix form and analyzing the eigenvalues of B.
  • Proof strategy: The analysis represents the recursive process with a two-dimensional vector and repeatedly expands its matrix recurrence.The limiting argument relies on the dominant eigenvalue R0 being below 1.

C. Social Distancing

The paper models two social-distancing strategies in a configuration-model network: reducing interpersonal contacts and removing high-degree nodes to suppress outbreaks.

  • Contact reduction: Allowing each person to retain a fraction a of normal contacts corresponds to randomly disconnecting each node’s edges with probability 1 − a.The resulting edge transmission probabilities become a^2φ1 and a^2φ2.
  • Contact reduction: The contact-reduction strategy reduces the two propagation probabilities to a^2φ1 and a^2φ2, respectively.This reduction supports an outbreak-prevention condition through the model’s stability result.
  • Mass-gathering cancellation: Canceling mass gatherings is modeled by removing nodes whose number of edges is at least k0, targeting superspreaders with many interpersonal contacts.The resulting stability condition guarantees a finite infected tree and no outbreak when satisfied.
  • Mass-gathering cancellation: For the mass-gathering strategy, the paper states that the infected tree is finite with probability 1 under the condition in Theorem 5.The paper notes that this condition prevents an outbreak but does not provide an explicit expression for k0.

D. The Ratio of the Population Infected in the Long Run

The paper studies the long-run infected fraction in stationary SIR and configuration-model IC networks, using fixed-point equations to characterize when a nonzero fraction is infected.

  • Stationary SIR model: The long-run infected ratio r is the population fraction infected after disease propagation, and the paper asks for it when containment fails.This question is motivated by antibody-test evidence of prior infection in New York State.
  • Stationary SIR model: In the stationary SIR model, the basic reproduction number is R0 = β/γ, and r satisfies a fixed-point equation.Here β is the stationary transmission rate and γ is the stationary recovering rate.
  • Configuration-model IC network: In the configuration-model IC network, R0 > 1 implies that a nonzero probability r exists for a randomly selected node to be infected in the long run.The probability is computed from fixed-point equations involving the degree and excess-degree distributions.
  • Configuration-model IC network: When R0 ≤ 1, the fixed-point solution is v = 0; when R0 > 1, a unique solution with v > 0 exists and can be found iteratively.The corresponding long-run infected fraction is therefore zero in the first case and nonzero in the second.
  • Configuration-model IC network: For an Erdős–Rényi network, the excess-degree distribution equals the degree distribution, so r = v and the fixed-point equation matches the stationary SIR equation.The fixed-point equation represents the probability that a node is not infected through any neighbor.

A. Dataset

The analysis uses daily COVID-19 outbreak data from China’s National Health Commission, covering confirmed cases, recoveries, and deaths from January 15 to March 2, 2020.

  • Data source: The dataset comes from the National Health Commission of the People’s Republic of China daily Outbreak Notification.The notifications report data as of 24:00 on the preceding day.
  • Data scope: The collected data include confirmed cases, recovered persons, and deaths from Jan. 15, 2020 to Mar. 2, 2020.A confirmed case is defined using a positive rRT-PCR result, with a noted definition change in Hubei.

B. Parameter Setup

The paper sets up the time-dependent SIR model from Chinese case data, evaluates its predicted epidemic trajectory and reproduction number, and reports close one-day agreement except around a case-definition change.

  • Parameter Setup: J = K = 3 FIR filters predict the time-dependent transmission and recovering rates, using post-Jan. 27 data because earlier infections were too few to show a clear trend.The stopping criterion is X(t) ≤ 0.
  • Time Evolution: The model predicts that the disease will end in 6 weeks, with roughly 80,000 total confirmed cases if China maintains its control policy.The prediction is based on city-wide lockdown and suspension of work and classes.
  • Time Evolution: β(t) decreases dramatically while γ(t) increases slightly, consistent with suppression of transmission and relatively slow recovery-rate growth.Data related to Feb. 11 are treated as having no reference value because the confirmed-case definition changed on Feb. 12.
  • Time Evolution: R0(t) = β(t)/γ(t), and the predicted turning point is Feb. 17, 2020 when R0(t) < 1.The model predicted this threshold crossing three days in advance, on Feb. 14, 2020.
  • One-day Prediction: The one-day prediction errors remain within ±3% except for recovered-person predictions on Feb. 1, Feb. 3, and Feb. 5.After Feb. 16, the model continues tracking β(t) and γ(t) despite the case-definition change.

E. Connections to the Wuhan City Lockdown

The paper connects epidemic transmission trends, undetectable infections, and network-based social distancing to outbreak control. It reports phase-transition thresholds and concrete intervention conditions across these analyses.

  • Connections to the Wuhan City Lockdown: β(t) increases before dropping dramatically from Jan. 23 to Jan. 28, 2020, peaking on Jan. 26 alongside Wuhan’s moving-out trend.The emigration trend is reported as almost the same as β(t) from Jan. 20 to Jan. 25.
  • The Effects of Type II Infected Persons: More than 90% detection and proper isolation can contain spread even when type II transmission rate β2 is as high as 0.7.Reducing β2 through measures such as wearing masks and washing hands is presented as another control approach when detection is lower.
  • The Effects of Type II Infected Persons: The phase-transition diagrams classify parameter regions above the black percolation threshold as outbreaks and regions below it as controlled disease.Figure 9 varies β2 and γ2, while the corresponding analysis describes the same threshold interpretation for undetectable infections.
  • The Effects of Social Distancing: The Facebook dataset contains 50,515 nodes and 819,306 edges, with mean degree 32.4, maximum degree 1,469, diameter 11, and clustering coefficient 0.053.Its degree distribution appears to follow a truncated Pareto distribution with exponent 1.69.
  • The Effects of Social Distancing: The minimum degree thresholds to prevent outbreaks in Italy, the U.S., and South Korea are 63, 195, and 435, respectively.The associated affected fractions of the Facebook tail distributions are 13.1%, 2.2%, and 0.4%, respectively.
  • The Effects of Social Distancing: For the U.S., removing Facebook-network nodes with degree at least 63 is reported to prevent an outbreak when R0 is 12.59.The removal might affect 13.1% of the nodes in that dataset.

I. The Ratio of the Population Infected in the Long Run

The paper relates long-run infection probability to R0 and network degree distributions in an independent cascade model. For R0 > 1, infection can persist, with network structure shaping the eventual infected fraction.

  • R0 > 1 produces a nonzero probability r that a randomly selected node remains infected in the long run.The paper plots r against R0 for several degree distributions using a Facebook dataset and random networks.
  • r increases with R0, while the two Erdős–Rényi curves overlap despite different mean degrees.For Poisson degree distributions, the long-run infected probability is independent of mean degree.
  • Power-law networks have smaller long-run infected fractions than the Erdős–Rényi model at the same R0.The paper attributes this to lower propagation probability when the mean excess degree is large.
  • Super-spreaders are more likely in power-law networks, but their neighbors tend to have fewer contacts, limiting further propagation.In the Erdős–Rényi model, super-spreaders are exponentially unlikely, yet outbreaks can eventually reach a larger population fraction when R0 > 1.
  • At R0 = 3, the Erdős–Rényi model infects more than 90% of the population in the long run.
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