key: cord-0906644-7eomhj6j
authors: Odagaki, T.
title: Estimation of the onset rate and the number of asymptomatic patients of COVID-19 from the proportion of untraceable patients
date: 2021-07-30
journal: nan
DOI: 10.1101/2021.07.28.21261241
sha: de3f2764f2051e23378fabb3652b28c9def9b685
doc_id: 906644
cord_uid: 7eomhj6j

A simple method is devised to estimate the onset rate of COVID-19 from the proportion of untraceable patients tested positive, which allows us to obtain the number of asymptomatic patients, the number of infectious patients and the effective reproduction number. The recent data in Tokyo indicate that there are about six times as many infectious patients in the city as the daily confirmed new cases. It is shown that a quarantine measure on non-symptomatic patients is critically important in controlling the pandemic.

The pandemic COVID-19 is still prevalent all over the world despite of continuous efforts by governments to control it. Difficulty of the control lies in the fact that patients of COVID-19 take a route different from common epidemics.

In common epidemics, infected individuals show symptoms and become infectious after an incubation period. Then they are treated and recover from the disease. In COVID-19, infected individuals become infectious before symptom-onset and asymptomatic patients who do not show any symptoms before their recovery are infectious. In order to formulate proper strategies in controlling COVID-19, it is important to know the proportion of asymptomatic and pre-symptomatic patients (1) , whom I call non-symptomatic patients collectively. However, it is unpractical to identify all non-symptomatic patients in the entire population by PCR tests. Therefore, it is an important problem to devise a method for estimating the number of infectious patients from data reported daily such as the confirmed new cases and the proportion of untraceable patients tested positive.

In this paper, I propose a simple method by which the onset rate of COVID-19 patients can be estimated from the proportion of untraceable patients tested positive and show that the proportion of the infectious patients can be obtained from the onset rate. I first analyze the infection process on the basis of the SIQR model (2) and find a relation between and . Then, I argue that the number of infectious patients and the number of new patients on a given day can be related to the proportion . I also discuss the effective reproduction number which depends on a quarantine rate of non-symptomatic patients and show that the quarantine of non-symptomatic patients is critically important in controlling COVID-19. I analyze the situation in Tokyo and discuss why COVID-19 does not converge in Tokyo.

I classify infected individuals on the basis of the SIQR model (2) and the mean field approach as follows. I first . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted July 30, 2021. ; https://doi.org/10.1101/2021.07.28.21261241 doi: medRxiv preprint assume that patients of COVID-19 follow the same disease progression day by day at the same pace on average since their infection. On a particular day which I call day zero, there are many infected individuals who can be classified by the number of days since their infection. I denote by 0 the number of infected symptomatic individuals who are identified as patients by PCR tests and quarantined on day zero. Here, is the average onset rate of COVID-19, and therefore there are (1 − ) 0 asymptomatic patients who got infected on the same day as the quarantined patients got infected. I denote by −1 , −2 , ⋯, − 0 the number of infected individuals who got infected one, two, ⋯, 0 days later than the day when the 0 individuals got infected. Similarly, I denote by 1 , 2 , ⋯ , 1 −1 the number of individuals who got infected one, two, ⋯ , 1 − 1 days earlier than the day when the 0 individuals got infected (3) . Here, 0 is the sum of the infectious period before symptom-onset and the period between the onset and getting quarantined, and 1 denotes the infectious period of asymptomatic patients after symptomatic patients in the same group are quarantined. I assume that the proportion 1 − of asymptomatic patients is common in every group of patients. Figure 1 shows schematically the breakdown of patients into these three groups, where patients in the shaded area are infectious. Note that in the present analysis, the latent and incubation periods do not play any roles.

I assume that all symptomatic patients will be tested intentionally some days after the symptom-onset and quarantined. I also assume that PCR tests are conducted on the general population and denote the quarantine rate of patients as . If the quarantine measure on infected individuals is taken effectively, these infected patients decrease by a factor (1 − ) every day. Therefore, the number of infectious patients ℑ on day zero is given by . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted July 30, 2021. ; https://doi.org/10.1101/2021.07.28.21261241 doi: medRxiv preprint and the new patients Δ infected on day zero who will be identified some days later are given by

where is the period that PCR test is effective before day zero, is the infection coefficient of patient group , µ is the fraction of immunized individuals who are no longer susceptible, and represents the reduction rate of social contacts among people due to lockdown measures.

In order to make the following description transparent, I define an average of and a weighted average of as follows:

and write Eqs. (1) and (2) as

and

Out of the newly infected individuals ∆ , ∆ ≡ ∆ will show symptoms some days later and (1 − )∆ will not show any symptoms. Patients ∆ showing symptoms will be identified by PCR tests and be listed as daily confirmed new cases. I assume that they are classified into two groups ∆ and ∆ , where ∆ are traceable patients who are infected from symptomatic and pre-symptomatic patients and ∆ are untraceable patients who are infected from asymptomatic patients. Namely, ∆ and ∆ are given by

Therefore, the proportion of untraceable cases in the daily confirmed new cases is given by

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted July 30, 2021. ; https://doi.org/10.1101/2021.07.28.21261241 doi: medRxiv preprint which does not depend on µ, and explicitly. This relation can be inverted to get

which indicates that the onset rate can be obtained from the proportion of untraceable patients once other parameters are known.

In order to incorporate the trend of infection status (4) into the present analysis, I first define trend parameters 0 and 1 by 0 = 〈 〉 0 / 0 and 1 = 〈 〉 1 / 0 . It is apparent that the infection status after day zero will be increasing, stationary and decreasing when 0 > 1, 0 = 1 and 0 < 1, respectively. Similarly, 1 > 1, 1 = 1 and 1 < 1

indicate that the infection status before day zero has been increasing, stationary and decreasing, respectively. It is straight forward to express in terms of 0 and 1 as

Therefore, Δ is written as

Since the confirmed new cases on day zero (∆ ) 0 is given by (∆ ) 0 = 0 , ℑ can be expressed in terms of 0 , 

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted July 30, 2021. ; https://doi.org/10.1101/2021.07.28.21261241 doi: medRxiv preprint which is shown in Fig.2 In Tokyo, the proportion of untraceable patients is = 50 ~60%. Using the lower value f = 50 %, I find = 77%. This value becomes = 69% if = 60% is used. These values are consistent with observations of the onset rate = 76% (7) or 75% (8).

On the same conditions, the number of infectious patients Eq. (15) can be written as

If there were no untraceable patients, all infected individuals would be symptomatic and, therefore, the number of infectious patients will be given by the number of new cases times days during which they are infectious, i.e. ℑ/ (∆ ) 0 = 0 when = 0 as Eq. (18) indicates. Equation (18) shows ℑ/(∆ ) 0 = 2 0 and 2.5 0 when = 50% and 60% respectively. Figure 3 shows dependence of ℑ (∆ ) 0 ⁄ when 0 = 3 days and 1 = 7 days. It is important to note that in Tokyo = 50~60% and 0 = 3 days, and thus there are 6~7.5 times more infectious patients than the daily confirmed new cases. It should also be mentioned that the number of asymptomatic infectious patients ℑ excluding pre-symptomatic patients is given by

and thus ℑ as (∆ ) 0 ⁄ = 3 when 0 = 3 days, 1 = 7 days and = 50%. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted July 30, 2021. ; https://doi.org/10.1101/2021.07.28.21261241 doi: medRxiv preprint Effective reproduction number and assessment of policies I first define an effective infectious period eff and an effective infection coefficient eff as follows:

Equation (8) can now be expressed as

and the effective reproduction number eff on day zero is given by

It should be remarked that when eff is equal to the recovery time or the inverse of the recovery rate γ and eff is a constant, the expression for eff is identical to that defined in the standard SIR model. It is important to note that, in contrast to the standard SIR model, the effective reproduction number eff defined by Eq. (23) depends on the quarantine rate of non-symptomatic patients through eff .

The basic strategy against COVID-19 is to bring the effective reproduction number smaller than unity so that the number of patients decreases. Equation (23) In Tokyo, the PCR test has been used only to confirm the infection of novel corona virus for people who show some symptoms, which means = 0, and thus it has not been contributing to the battle against COVID-19.

Furthermore, lockdown measures have been very sloppy. Instead of enforcing social distancing among the entire population it has been applied only specific targets, like the night life district last year and restaurants serving alcohol beverage this year. The policy targeting on certain shops and opening hours has only limited effects on social distancing since people gather together in a park or on a street. Furthermore, the policy has been enforced and lifted every one or two months, which has caused the wavy infection curve (9, 10) . It could be possible to increase by, for example, promoting Telework, limiting working days, reducing crowd in commuter trains and banning gatherings.

In the present analysis, traceable and untraceable patients tested positive are related to infection from pre-symptomatic and symptomatic patients and from asymptomatic patients, respectively. Although this assumption may not be rigorous, it is a good assumption if people cooperate in the investigation of infection route by the health . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted July 30, 2021. ; https://doi.org/10.1101/2021.07.28.21261241 doi: medRxiv preprint department of local government. It should also be mentioned that, since the important point is if patients are quarantined or not, the present results do not depend on the accuracy of PCR test at all.

In the analysis of asymptomatic patients in Tokyo, I assumed that the infection status is stationary. The infection status can also be increasing or decreasing (4) . However, if the condition 〈 〉 0 = 〈 〉 1 is satisfied, the relation Eq.

(15) between the number of infectious patients and the number of the newly confirmed new cases reduces to

which depends only on 0 0 when = 0. This ratio becomes large in the increasing status and small in the decreasing status compared to Eq. (18). In this case, the effective reproduction number depends on the infection status through eff . In general, eff is an increasing function of 0 and 1 . Since the increasing state corresponds to 0 > 1 > 1 and the decreasing state to 0 < 1 < 1 , eff can increase or decrease depending on the relation between 0 and 1 .

The infection coefficient of SARS-CoV-2 depends on variants. When a new variant with stronger infection coefficient emerges, 〈 〉 0 becomes larger than 〈 〉 1 . The number of infectious patients will increases for a stronger variant when other parameters are the same.

In 2021, vaccination has been progressed in many countries and the infection status seems to be improving at least in the reduction of serious cases. The effect of vaccination appears through µ in Eq. 

Quantifying asymptomatic infection and transmission of COVID-19 in New York City using observed cases, serology, and testing capacity

Analysis of the outbreak of COVID-19 in Japan by SIQR model

COVID-19 overview of the period of communicability -what we know so far. Toronto, ON: Queen's Printer for Ontario

Classification of the infection status of COVID-19 in 190 countries

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Updates on COVID-19 in Tokyo

SARS-CoV-2 Transmission From People Without COVID-19 Symptoms

Mengist et al, Magnitude of asymptomatic COVID-19 cases throughout the course of infection: A systematic review and meta-analysis

Self-organized wavy infection curve of COVID-19

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