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Precaution, Social Distancing and Tests in a Model of Epidemic Disease

  • Francesc Obiols-Homs EMAIL logo


I develop an extension of a canonical epidemiology model in which the policy in place determines the probability of transmission of an epidemic disease during economic and social interaction. I use the model to evaluate the effects of isolating symptomatic individuals, of increasing social distancing and of tests such as polymerase chain reaction – PCR – or Rapid Diagnostic Test that discriminate between currently infected agents, and its combination with a serology test like Neutralization Assay that is able to discriminate between immune and vulnerable healthy individuals, together with the role of enforcement to prevent interactions involving infected but asymptomatic agents. I find that isolating symptomatic individuals has a large effect at delaying and reducing the pick of infections. The combination of this policy with a PCR test is likely to represents only a negligible improvement in the absence of enforcement, whereas with full enforcement there is an additional delaying and reduction in the pick of infections. Social distancing alone cannot achieve similar effects without incurring in enormous output losses. I explore the combined effect of social distancing at early stages of the epidemic with a following period of tests and find that the best outcome is obtained with a light reduction of human interaction for about three months together with a subsequent test of the population over 40 days.

JEL Classification: E1; E65; H12; I1

Corresponding author: Francesc Obiols-Homs, Department of Economics and Economic History, Universitat Autonoma de Barcelona, Edifici B, Cerdanyola del Valles - 08193, Barcelona, Spain, E-mail:

Funding source: Ministry of Science, Innovation and Universities from Spain

Award Identifier / Grant number: PGC2018-094364-B-I00


This investigation was conducted under confinement, thus I thank M. Àngels Fernández, and Magalí and Marçal Obiols-Fernández for their interest, patience and excellent company. I also thank Raül Santaeulàlia-Llopis, Charles Wyplosz, the editor of this journal Arpad Abraham, an anonymous referee and seminar participants of the COVID-19 Seminar of the Barcelona GSE for helpful comments and suggestions. The usual disclaimer applies. The financial support from the Spanish Ministry of Science, Innovation and Universities and FEDER through grant PGC2018-094364-B-I00 is gratefully acknowledged.


I describe here a few additional results of a sensitivity analysis.

1. The power of test 1 under LE. The following Table 1 summarizes the effects on the fraction of infected individuals as the coverage of test 1 is increased.

Table 1:

Test 1-LE with larger coverage.

τ (%) Pick day Pick of infections (%)
4 301 3
5 312 2.8
10 394 1.8
15 551 0.91

2. The length of confinement, t 0. I check the effect of t 0 on the differences between T1-LE and T1-FE. The baseline value for t 0 is 14 days which is the recommendation of the WHO. I find that increasing t 0 to 18, 20 or 25 days reduces the differences between the tests in the magnitude of the pick of infection and it has small effects on the pick date. Still, the T1-FE performs better than the T1-LE. The results are summarized in Table 2.

Table 2:

The effect of larger t 0.

t 0 T1-LE pick day Pick of infections (%) T1-FE pick day Pick of infections (%)
18 168 9.24 192 7.27
20 160 11.71 181 9.61
25 160 16.83 179 14.59

3. The effects of N and the power of T1-LE and T1-FE: I described in the text that reducing the number of contacts has a positive effect to delay the pick of infections and to reduce it. With T1-FE covering 3% of the population the epidemic does not take place when N = 30. When N is increased to 35, 40 and 50 I obtain that the pick of infections is delayed to day 380, day 226 and day 134 respectively, and the pick of infections increases to 1.33, 3.6 and 4.5% (also respectively). The epidemic is still far from the unawareness outcome, hence T1-FE is still effective at a 3% coverage of the population. I repeat the same exercise under T1-LE. With 35, 40 and 50 contacts the pick occurs on days 189, 146 and 105, and reaches 6, 8.15 and 10.9%. The results when only precaution is in place are pick occurring at days 177, 136 and 101 and reaching 6.54, 8.6 and 11.24%. The conclusion is that the effect of a larger N goes in the expected direction, but it does not change the conclusion that test 2 is substantially more effective than test 1 to fight the epidemic. Since there are no differences in the probability of transmission of the infection across occupations changes in N do not change the shape of the optimal policy.

4. The curvature in the production function. In the calibration exercise I found that the value of a job in the “services” sector is only slightly larger than that in the “industry” sector. The effect of a reduction in n is different in the two sectors because of the curvature in f(n). When I take the two functions simultaneously closer to the linear case output losses under the optimal policy decrease at an almost constant rate from 10.58 to 2.6%, when α s = α i = 1 and thus the loss in output is proportional to n/N. Changes in the curvature do not change the optimal combination of policies.

5. I checked the effect of early end of tests under both T1-LE and T1-FE, which mainly delays the reduction in infections. I also checked that this effects persist when the fraction of asymptomatic agents is larger (ρ = 0.7), where the main innovation is that the effect of enforcement is substantially reduced. Figure 13 summarizes these findings.

Figure 13: 
The effects of early end of tests with ρ = 0.8 (solid line) and ρ = 0.7 (dashed line).
Figure 13:

The effects of early end of tests with ρ = 0.8 (solid line) and ρ = 0.7 (dashed line).

6. Finally, I checked the sensitivity of the previous results to the fraction of asymptomatic agents among those that are infected and of duration of immunity conditional on surviving the infection. I find that the differences between limited and full enforcement in the outcomes of test 1 are severely reduced when immunity is shorter and when the fraction of asymptomatic is larger. Not only this, the use of tests becomes less and less effective to prevent the spread of the disease. Figure 14 portrays an example of the dynamics over a period of three years.

Figure 14: 
Infection dynamics with shorter immunity and a larger number of asymptom-atic.
Figure 14:

Infection dynamics with shorter immunity and a larger number of asymptom-atic.

Leaving aside finite immunity, when ρ = 0 (thus all agents are asymptomatic) the infections under precaution and tests converge to the unawareness case under the benchmark coverage of 1% of infected. Thus in this case the effect of enforcement is null. When the tests cover up to 15% of the population the limited enforcement case is still very close to the precaution and unawareness, but with enforcement there is a rapid cycle of increases and decreases in the fraction of infections which again is able to eradicate the infection after about 150 periods. Figure 15 completes this picture and reveals that the combination of type 1 and 2 tests offers very marginal improvements relative to the case of test 1 with full enforcement.

Figure 15: 
Infection dynamics with infinite immunity and a 15% coverage with tests.
Figure 15:

Infection dynamics with infinite immunity and a 15% coverage with tests.


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Received: 2020-07-06
Revised: 2021-03-22
Accepted: 2021-03-22
Published Online: 2021-06-03

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