# Quarantine, Contact Tracing, and Testing: Implications of an Augmented SEIR Model

• Andreas Hornstein

## Abstract

I incorporate quarantine, contact tracing, and random testing in the basic SEIR model of infectious disease diffusion. A version of the model that is calibrated to known characteristics of the spread of COVID-19 is used to estimate the transmission rate of COVID-19 in the United States in 2020. The transmission rate is then decomposed into a part that reflects observable changes in employment and social contacts, and a residual component that reflects disease properties and all other factors that affect the spread of the disease. I then construct counterfactuals for an alternative employment path that avoids the sharp employment decline in the second quarter of 2020, but also results in higher cumulative deaths due to a higher contact rate. For the simulations a modest permanent increase of quarantine effectiveness counteracts the increase in deaths, and the introduction of contact tracing and random testing further reduces deaths, although at a diminishing rate. Using a conservative assumption on the statistical value of life, the value of improved health outcomes from the alternative policies far outweighs the economic gains in terms of increased output and the potential fiscal costs of these policies.

Corresponding author: Andreas Hornstein, Federal Reserve Bank of Richmond, Richmond VA, USA, E-mail:

## Acknowledgments

I would like to thank Alex Wolman and Zhilan Feng for helpfulcomments and Elaine Wissuchek and Zach Edwards for research assistance.Any opinions expressed are mine and do not reflect those of the Federal ReserveBank of Richmond or the Federal Reserve System.

A. Appendix

Detailed derivations for the statistics defined in this appendix can be found Hornstein (2021).

### A.1 Calibration

Here we define certain statistics that are used in the calibration of the model. Non-sequentially numbered equations refer to the corresponding equations in the main text.

The probability that an exposed individual eventually becomes symptomatic is,

p E S = 0 0 τ ϕ e ϕ s β e β τ s e γ A τ s d s d τ

where the first term is the probability that the individual becomes asymptomatic infectious at s and then becomes symptomatic at τ, without recovering. This can be solved as

(A.1) p E S = β γ A + β .

The average time for an exposed individual to become symptomatic, conditional on eventually becoming symptomatic, is

T E S = 0 τ 0 τ ϕ e ϕ s β e β τ s e γ A τ s p E S d s d τ ,

where the term in curly brackets is the probability of becoming symptomatic at τ, conditional on eventually becoming symptomatic. This can be solved as

(25) T E S = 1 ϕ + 1 γ A + β .

Note that this can be rewritten as

T E S = 1 ϕ + β γ A + β 1 β = T E + p A S T A S = T E + T A ,

where the first term is the average time spent being latent, and the second term is the probability of becoming symptomatic times the average time it takes to become symptomatic from asymptomatic.

The probabilities that an infected individual becomes symptomatic or recovers before becoming symptomatic are

(A.2) p A S = β γ A + β

(26) p A R = γ A γ A + β .

The average duration of infectiousness, that is, the average time spent in states A and S is

T AS = 0 τ γ A e γ A τ e β τ + τ + 1 γ S + δ e γ A τ β e β τ d τ ,

where the first term in the integral represents being asymptomatic for duration τ, followed by a recovery, and the second term of the integral represents being asymptomatic for duration τ, followed by being symptomatic with an average duration 1 / γ S + δ . This expression simplifies to

(27) T AS = 1 β + γ A + β β + γ A 1 γ S + δ = T A + p A S T S ,

which is the average duration of being asymptomatic plus the average duration of being symptomatic times the probability of becoming symptomatic.

The infection fatality rate (IFR), that is, the probability of dying when symptomatic is

p S D = 0 δ e δ τ e γ S τ d τ = δ δ + γ S .

The probability of dying, conditional on being infected, is equal to the probability of making the transition from asymptomatic to symptomatic times the probability of dying when symptomatic

(28) p E D = p A S p S D = δ δ + γ S β β + γ A .

### A.2 Reproduction Rate with Quarantine and Contact Tracing

A symptomatic individual who is not quarantined infects on average R S = α S / N R S individuals until he recovers or dies, with

R S = 0 τ e γ S τ δ e δ τ + γ S e γ S τ e δ τ d τ = 1 γ S + δ .

An asymptomatic individual who is not quarantined infects on average R A = α S / N R A until she recovers or becomes symptomatic, with

R A = 0 σ τ e γ A τ β e β τ + γ A e γ A τ e β τ d τ = σ 1 γ A + β .

An asymptomatic individual who is quarantined with rates ɛ QA and ɛ QS infects on average R AS = α S / N R AS until he recovers or becomes symptomatic, with

R AS ε QA , ε QS = 1 ε QA R A + 1 ε QS β γ A + β R S .

Let R ̄ AS ε T , ε QA , ε QS denote the expected infection factor for a latent individual (the individual of interest) before knowing whether the individual will be traced

R ̄ AS ε T , ε QA , ε QS = ε T R AS ε QA , ε QS + 1 ε T R AS 0 , ε QS .

Consider an individual who is latent, traced with effectiveness ɛ T, and quarantined with rates ɛ QA and ɛ QS. The individual of interest may have been infected by (1) a symptomatic individual that was not quarantined, there are 1 ε QS I S of them; (2) an asymptomatic individual that was traced, but not quarantined, there are 1 ε QA I AT of them; and (3) an asymptomatic individual that was not yet traced, there are I A of them. We want to calculate the average of new infections coming from this individual until she recovers or dies.

1. Case 1 and 2: Since infectious individuals are only traced at the time they become symptomatic, the individual of interest will never be traced. Therefore, the expected number of new infections coming from the infected individual is R AS 0 , ε QS = α S / N R AS 0 , ε QS ;

2. Case 3: The expected number of new infections coming from the infected individual when the infecting individual was asymptomatic is R E = α S / N R E , with R E =

0 γ A e γ A + β τ 0 R AS 0 , ε QS d τ 0 + 0 β e γ A + β τ 0 e ϕ τ 0 R ̄ AS ε T , ε QA , ε QS d τ 0 + 0 β e γ A + β τ 0 0 τ 0 ϕ e ϕ s e γ A + β τ 0 s × σ τ 0 s + R ̄ AS ε T , ε QA , ε QS d s d τ 0 + 0 β e γ A + β τ 0 0 τ 0 ϕ e ϕ s × 0 τ 0 s γ A e γ A + β t σ t d t d s d τ 0 + 0 β e γ A + β τ 0 0 τ 0 ϕ e ϕ s 0 τ 0 s β e γ A + β t × σ t + 1 ε QS R S d t d s ,

where the five components of case 3 are

3. Case 3.1: The infecting individual recovers at τ 0, and the infected individual is not traced;

4. Case 3.2 through 3.5: The infecting individual becomes symptomatic at τ 0, and

1. Case 3.2: The infected individual never became asymptomatic;

2. Case 3.3: The infected individual became asymptomatic at s and stayed so until τ 0;

3. Case 3.4: The infected individual became asymptomatic at s and recovered at s + t;

4. Case 3.5: The infected individual became asymptomatic at s and symptomatic at s + t.

The probabilities for the components of case 3 are

p E 1 = γ A γ A + β , p E 2 = β γ A + β + ϕ , p E 3 = β ϕ 2 γ A + β + ϕ γ A + β , p E 4 = 1 2 β ϕ γ A γ A + β 2 1 γ A + β + ϕ , p E 5 = 1 2 ϕ β 2 γ A + β 2 1 γ A + β + ϕ .

The expected infections for the components of case 3 are

R E 1 = γ A γ A + β R AS 0 , ε QS , R E 2 = β γ A + β + ϕ R ̄ AS ε T , ε QA , ε QS , R E 3 = p E 3 σ 2 γ A + β + R ̄ AS ε T , ε QA , ε QS , R E 4 = β ϕ γ A σ 4 γ A + β + ϕ γ A + β 3 , R E 5 = β γ A R E 4 + p E 4 1 ε QS R S .

and

R E = i = 1 5 R E , i .

A newly infected individual then infects on average R = α S / N R individuals where

R = 1 ε QA I AT + 1 ε Q S I S R AS 0 , ε QS + I A R E 1 ε QA I AT + 1 ε QS I S + I A .

### A.3 Traceable Individuals

We consider an asymptomatic infectious individual who is quarantined once he becomes symptomatic. For this case, we calculate the average number of exposed and infectious asymptomatic individuals that this individual has created.

By the time an asymptomatic individual becomes symptomatic, on average that individual has infected α t S ( t ) / N ( t ) R AT other individuals, where

R AT = 0 σ τ β e β τ e γ A τ d τ = σ β β + γ A 2 .

The average number of individuals that the infectious individual has infected and that are not yet infectious at the time the individual becomes symptomatic is α t S ( t ) / N ( t ) R ATE , where

R ATE = 0 σ p EE τ β e β + γ A τ d τ ,

where the term in brackets is the probability that the infectious individual has been asymptomatic for duration τ, and p EE τ denotes the fraction of individuals that were infected by the infectious individual over the interval τ and that have not yet become infectious at the time the individual becomes symptomatic. The probability that an individual that was infected time s ago and has not yet become infectious is eϕs . Thus

p EE τ = 0 τ e ϕ s d s = 1 ϕ 1 e ϕ τ

and

(19) R ATE = σ β β + γ A β + γ A + ϕ .

The average number of individuals who an infectious individual has infected and that are infectious but asymptomatic at the time the individual becomes symptomatic is α t S ( t ) / N ( t ) R ATA , where

R ATA = 0 σ 0 τ p EA s d s β e β + γ A τ d τ

and p EA s denotes the fraction of individuals that were infected time s ago, have become infectious in the meantime but have not yet recovered or become symptomatic. Thus

p EA s = 0 s ϕ e ϕ v e γ A + β s v d v = ϕ ϕ γ A β 1 e ϕ γ A β s

and

(20) R ATA = σ ϕ β 2 β + γ A 2 β + γ A + ϕ .

The average number of individuals that the infectious individual has infected and that have become symptomatic or recovered at the time the individual becomes symptomatic is α t S ( t ) / N ( t ) R ATR , where

R ATR = 0 σ 0 τ p ER s d τ β e β + γ A τ d τ

and p ER s denotes the fraction of individuals that were infected time s ago and that have recovered or become symptomatic,

p ER s = 0 s ϕ e ϕ v 1 e γ A + β s v d v

and

R ATR = σ β ϕ 2 β + γ A 2 β + γ A + ϕ = R ATA .

### A.4 Robustness

In Section 5.3.2, we have shown that the effectiveness of contact tracing turns out to be limited even though the environment is consistent with a large share of asymptomatic infected individuals. The environment was calibrated to match what is known about the spread of COVID-19, but the uncertainty about how the disease spreads is large. We now consider some alternative selections of parameter values and how they affect the effectiveness of contact tracing. One would expect that contact tracing becomes relatively more effective if there are relatively more asymptomatic infectious individuals or if asymptomatic individuals are relatively more infectious. The following two experiments suggest that neither of the two alternative calibrations increase the effectiveness of contact tracing.

Table A1 reports results from alternative calibrations that increase the probability that asymptomatic infectious recover before they show symptoms. The top panel replicates the information from Table 2, columns 1 through 5, with baseline calibration p A→R = 0.4. The next two panels increase the recovery probability to 60 and 80 percent. For each of the two alternative calibrations we re-estimate the implied transmission rate. As we can see from comparing the upper left-hand side cells of each panel, ɛ Q = 0.5 and ɛ T = 0.0, the alternative employment path yields only slightly different cumulative deaths at the end of year for the different calibrations. Notice that without contact tracing, increasing quarantine effectiveness has less of an impact when the share of asymptomatic infectious is larger, moving down column 1 of each panel in Table A1. Also, for any given quarantine effectiveness, increasing contact tracing effectiveness has a smaller impact on end-of-year cumulative deaths as the recovery probability increases. Essentially, once an infectious individual becomes symptomatic, fewer of the individuals who he infected are still around to be traced if asymptomatic individuals are recovering faster.

Table A1:

Share of asymptomatic.

ɛ Q ɛ T
(1) (2) (3) (4) (5)
0.000 0.250 0.500 0.750 1.000
p A→R = 0.40
0.500 0.187 0.181 0.176 0.171 0.166
0.600 0.101 0.097 0.094 0.090 0.087
0.700 0.063 0.060 0.058 0.057 0.055
0.800 0.044 0.043 0.041 0.040 0.039
0.900 0.033 0.032 0.031 0.030 0.029
1.000 0.026 0.025 0.024 0.024 0.023
p A→R = 0.60
0.500 0.190 0.187 0.184 0.182 0.179
0.600 0.107 0.106 0.104 0.102 0.101
0.700 0.069 0.068 0.067 0.066 0.065
0.800 0.050 0.050 0.049 0.048 0.048
0.900 0.039 0.039 0.038 0.038 0.037
1.000 0.032 0.031 0.031 0.030 0.030
p A→R = 0.80
0.500 0.189 0.189 0.188 0.188 0.187
0.600 0.125 0.125 0.124 0.124 0.123
0.700 0.088 0.088 0.088 0.087 0.087
0.800 0.068 0.068 0.068 0.068 0.067
0.900 0.056 0.056 0.056 0.056 0.055
1.000 0.048 0.048 0.047 0.047 0.047
1. Note: Cumulative deaths at end of sample, 12/31/2020, as a percent of population for given recovery probability of asymptomatic individuals, p AR , its implied alternative transmission path, and variations of quarantine effectiveness, ɛ Q, and tracing effectiveness, ɛ T, and no testing, f = 0. See also Table 2.

Table A2 reports results for alternative calibrations that increase the relative infectiousness of asymptomatic individuals. The top panel replicates the information from Table 2, columns 1 through 5, with baseline calibration σ = 0.6. The next two panels increase the relative infectiousness to 80 and 100 percent. We again re-estimate the implied transmission rate for each of the two alternative calibrations. Again, the alternative employment path yields slightly different end-of-year cumulative deaths for each calibration. Notice that without contact tracing, increasing quarantine effectiveness has less of an impact when the asymptomatic are more infectious since asymptomatic individuals are not quarantined and are relatively more infectious, moving down column 1 of each panel in Table A2. On the other hand, for any given quarantine effectiveness, increasing contact tracing effectiveness now has a larger impact on end-of-year cumulative deaths as the relative infectiousness increases.

Table A2:

Infectiousness of asymptomatic.

ɛ Q ɛ T
(1) (2) (3) (4) (5)
0.000 0.250 0.500 0.750 1.000
σ = 0.60
0.500 0.187 0.181 0.176 0.171 0.166
0.600 0.101 0.097 0.094 0.090 0.087
0.700 0.063 0.060 0.058 0.057 0.055
0.800 0.044 0.043 0.041 0.040 0.039
0.900 0.033 0.032 0.031 0.030 0.029
1.000 0.026 0.025 0.024 0.024 0.023
σ = 0.80
0.500 0.188 0.180 0.173 0.166 0.159
0.600 0.109 0.103 0.098 0.094 0.090
0.700 0.070 0.067 0.064 0.061 0.059
0.800 0.050 0.048 0.046 0.044 0.042
0.900 0.039 0.037 0.035 0.034 0.033
1.000 0.031 0.029 0.028 0.027 0.026
σ = 1.00
0.500 0.188 0.178 0.170 0.162 0.154
0.600 0.115 0.108 0.102 0.096 0.091
0.700 0.076 0.072 0.068 0.064 0.061
0.800 0.056 0.053 0.050 0.047 0.045
0.900 0.043 0.041 0.039 0.037 0.035
1.000 0.035 0.033 0.031 0.030 0.028
1. Note: Cumulative deaths at end of sample, 12/31/2020, as a percent of population for given relative infectiousness of asymptomatic individuals, σ, its implied alternative transmission path, and variations of quarantine effectiveness, ɛ Q, and tracing effectiveness, ɛ T, and no testing, f = 0. See also Table 2.

### A.5 Data

Monthly employment is total monthly payroll employment and monthly GDP is the IHS estimate of nominal monthly GDP, both series downloaded from Haver 3/24/2021. The monthly work at home share in employment is from Bick et al. (2021). The monthly series are interpolated to daily series using the MATLAB function interp1 with the pchip-option for piecewise cubic splines.

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