test, the present paper focuses on the improvement of the PCR test by incorporating the antibodies.

Structure of this paper is as follows. Section 2 lays out the basic model and section 3 derives the number of the infected individuals in the steady state. Concluding remarks are made in section 4.

 

2.    BASIC MODEL

Let us consider an intertemporal economy where time passes continuously and the time horizon is infinite. The economy consists of N (>0) individuals, of whom x(t) individuals in period t are infected with the coronavirus but do not to present symptoms. We assume that every individual takes the PCR test in every period, which is incomplete in a sense that p×100 % of theinfected individuals x(t), i.e., px(t), are falsely judged to be negative. We also assume that those who tested positive get treatments and are allowed to go out for work, shopping etc., but can be infected again in the same period.

As a characteristic of the present paper, we assume that, in period t, among those who are not infected, θpx(t) individuals develop antibodies, where θ is a positive constant, meaning the more the infected individuals, the more individuals develop antibodies.

By assuming each of those who are infected but judged to be negative transmits the virus to α individuals in period t, of whom a fraction of 1 is newly infected since  is infected and  has developed antibody already, we can express the motion of number of the infected individuals as.

 

x(t+1) =px(t)+px(t)α {1}.                                                                 (1)

 

3.    RESULTS AND DISCUSSIONS

Letting x* denote the steady state number of the infected individuals and substituting x(t+1) =x(t)=x* into equation (1), we have the condition x* should satisfy as

 

x* =px*[1+α {1}],                                                                                   (2)

 

and by solving x* with respect to (2), we obtain

 

.                                                                                                       (3)

 

Since x*, the steady state number of the infected people, should be less than N-θpx*, steady state number of those who can be infected, we have the condition x* should satisfy as

 

.                                                                                                                   (4)

 

If (4) is satisfied, the following inequality is also satisfied since p, the failure probability, is less than 1:

 

,                                                                                                              (5)

 

which is derived by the condition that 1 in equation (2) should be nonnegative.

Considering (3)-(5), we can draw a relationship between p and x* typically as a graph, which is the combination of (3), curve-segment AB, and  (, which is the part of (4)), curve-segment BC, as in Figure 1 on the p-x* plane.

 

Figure 1 Relationship between failure probability and steady state number of the infected

 

By paying attention to the region around point C, we have the following proposition.

 

Proposition:

The steady state number of the infected people, x*, increases if the failure probability of PCR test decreases from 1

From this proposition, we can say that we should improve the accuracy of the PCR test much enough to reduce the number of the infected individuals since the improvement of the PCR test increases the number of the infected individuals at first.

 

4.    CONCLUSIONS

The present study theoretically investigated the effects of improvement of the PCR test, to obtain the result that accuracy of the PCR test should be improved much enough to reduce the number of the infected individuals since the improvement of the PCR test increases the number of the infected individuals at first.

It is necessary to examine the robustness of our results in a more general framework that incorporates the economic activities. It is also of interest to investigate the compound effects of the PCR test, the vaccination and the treatment of COVID-19. We will conduct such analyses in the next paper.

 

 

REFERENCES

Baker S. R., Bloom, N., Davis, S.J., Kost, K.J., Sammon, M.C. and Viratyosin, T (2020) The Unprecedented Stock Market Impact of Covid-19, NBER Working Paper Series 26945. Retrieved from  https://doi.org/10.3386/w26945

Baker, S.R., Bloom, N., Davis, S.J. and Terry, S.J. (2020) COVID-induced Economic Uncertainty, NBER Working Paper Series 26983. Retrieved from https://doi.org/10.3386/w26983

Baker, S.R., Farrokhnia R.A., Meyer, S. Pagel, M. and Yannelis, C. (2020) How does Household Spending Respond to an Epidemic? Consumption during the 2020 Covid-19 Pandemic, NBER Working Paper Series 26949. Retrieved from https://doi.org/10.3386/w26949

Bandara, K. G. A. G. and Weerasooriya, W. M. R. B. (2021) A Study between the Taxpayers' Perception toward the Tax Policy Changes and the Tax Compliance in Sri Lanka: A Tax Relief in Response to the Impact of Covid-19 Epidemic, and its Implications, Business, Management and Economics Research, vol. 7(2), 52-64. Retrieved from https://doi.org/10.32861/bmer.72.52.64

Fujita, Y. (2020a) When Should We Start the Lockdown and How Long Should it be?, Modern Economy,　11, 1007-1011. Retrieved from  https://doi.org/10.4236/me.2020.115075

Fujita, Y. (2020b) How Much should Government Compensate Firms for Suspension of their Businesses in ORDER to Fight Off the New Coronavirus?, Theoretical Economics Letters, 10, 600-606. Retrieved from https://doi.org/10.4236/tel.2020.103038

Fujita, Y. (2020c) How could We Prevent Spread of the Coronavirus without Deteriorating Economy?, Modern Economy, 11, 1280-1287. Retrieved from https://doi.org/10.4236/me.2020.117091

Fujita, Y. (2021a) Does Vaccinations Reduce the COVID-19 Infection?, Archives of Business Research, forthcoming. Retrieved from https://doi.org/10.14738/abr.98.10608

Fujita, Y. (2021b) Better than Nothing?: Consequence of Incomplete Treatment of COVID-19, Business Management and Economics Research, forthcoming. Retrieved from https://doi.org/10.5430/bmr.v10n3p11

Peto, J., et al. (2020) Stopping the Lockdown and Ending the Epidemic by Universal Weekly Testing as the Exit Strategy, Retrieved from https://ephg-covid-19.org/

Ramadhanti, T., Singh, J. S. K. and Kularajasingham, J. (2021) Transactional and Transformational Leadership Styles as Predictors of Employee Performance during the Covid-19 Crisis and the Mediating Role of Organisational Culture, Business, Management and Economics Research, vol. 7(2), 39-51. Retrieved from https://doi.org/10.32861/bmer.72.39.51

Romer, P. (2020) Roadmap to Responsibly Reopen America, Retrieved from https://roadmap.paulromer.net/

Watanabe, T. (2020) The Responses of Consumption and Prices in Japan to the COVID-19 Crisis and the Tohoku Earthquake, Working Paper Series, CARF-F-476. Retrieved from https://doi.org/10.7916/d8-qs4v-q792