Estimation of the Onset Ratio and the Number of Asymptomatic Patients of COVID-19 from the Proportion of Untraceable Patients
Takashi Odagaki^1,2
1Kyushu University, Fukuoka, 819-0395, Japan
²Research Institute for Science Education, Inc., Kyoto, 603-8346, Japan

Correspondence to: Takashi Odagaki, Kyushu University, Fukuoka, 819-0395, Japan, E-mail: t.odagaki@kb4.so-net.ne.jp. 

Abstract

A simple method is devised to estimate the onset ratio of COVID-19 patients from the proportion of untraceable patients who 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 to ten times as many infectious patients in the city as the daily confirmed new cases. It is shown that, besides social distancing and the use of effective masks, a quarantine measure on non-symptomatic patients is critically important in controlling the pandemic.

Introduction

The pandemic COVID-19 is still prevalent in most parts of the world despite continuous efforts by governments to control it. The 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 [1-3], who will be called non-symptomatic patients collectively, in addition to the number of symptomatic patients. 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 non-symptomatic infectious patients from data reported daily such as the confirmed new cases and the proportion of untraceable patients tested positive. Here, symptomatic patients are divided into two groups; one is traceable patients who know from whom they have gotten infected, and the other is untraceable patients who have had no contact with pre-symptomatic and symptomatic patients.
In this paper, I propose a simple method by which the onset ratio x of COVID-19 patients can be estimated from the proportion ƒ of untraceable patients tested positive and show that the proportion of the infectious patients can be obtained from the proportion of untraceable patients. I first analyze the infection process on the basis of the SIQR (Susceptible-Infectious-Quarantined-Removed) model [4] and find a relation between x and ƒ. 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 ƒ. 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.

Classification of infectious patients

I classify infected individuals based on the SIQR model [4] and the mean-field approach as follows. I first assume that patients of COVID-19 follow the same disease progression day by day at the same pace on average since their infection. Namely, I assume that the timeline of infection follows infected, infectious, symptom/quarantined or asymptomatic and that the time between these events is deterministic. 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. The number of patients, some of whom are identified as patients by PCR test and quarantined on day zero, is denoted by n₀. I denote the average onset ratio of COVID-19 by, and then the number of infected symptomatic individuals in the group of patients is given by x, the number of asymptomatic individuals in the group of patients is given by (1 – x)n₀. I denote by n_–1,n_–2,...,n_–k the number of infected individuals who got infected one, two,…, k days later than the day when the n₀ individuals got infected. Similarly, I denote by n₁,n₂,...,n_l-1 the number of individuals who got infected one, two, ..., l-1 days earlier than the day when the n₀ individuals got infected [5]. Here, k is the sum of the infectious period before symptom onset and the period between the onset date and the quarantined date, and l denotes the infectious period of asymptomatic patients after symptomatic patients in the same group are quarantined. I assume that the proportion of 1–x of asymptomatic patients is common in every group of patients. Figure 1 shows the breakdown of patients into these three groups, where patients in the shaded area are infectious. Note that the latent and incubation periods do not play any roles in the present analysis.

I assume that all symptomatic patients will be tested intentionally some days after the symptom-onset and quarantined. I also assume that PCR tests, which is effective since T days before day zero, are conducted on the general population and denote the quarantine rate of patients by q. If the quarantine measure on infected individuals is taken effectively, these infected patients decrease by a factor  (1-q) every day. Therefore, the number of infectious patients ℑ on day zero is given by

and the new patients  ΔI infected on day zero who will be identified some days later are given by

where β_i is the infection coefficient of patient group n_i, µ is the fraction of immunized individuals who are no longer susceptible, and a represents the reduction ratio of social contacts among people due to lockdown measures. Here, I assume T ≥ k  for simplicity.

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

Relation between the proportion of untraceable patients and the onset ratio

In this section, I argue that the traceability of patients can be related to infection from asymptomatic patients and derive a relation between the onset ratio and the proportion of untraceable patients. Out of the newly infected individuals ∆I, ∆Q ≡ x∆I will show symptoms some days later and (1-x)∆I will not show any symptoms. Patients ∆Q 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, ∆Q_t and ∆Q_unt, where ∆Q_t are traceable patients who are infected from symptomatic and pre-symptomatic patients, and ∆Q_unt are untraceable patients who are infected from asymptomatic patients. Namely, ∆Q_t and ∆Q_unt are given by

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

which does not depend on µ and a explicitly. This relation can be inverted to get

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

In order to incorporate the trend of infection status [6] into the present analysis, I first define trend parameters τ_k and τ_l by τ_k=⟨n⟩_k⁄(1-q)^Tn₀ and τ_l=⟨n⟩_l⁄(1-q)^Tn₀. It is apparent that the infection status after day zero will be increasing, stationary and decreasing when τ_k>1, τ_k=1 and τ_k<1, respectively. Similarly, τ_l>1, τ_l=1 and τ_l<1 indicate that the infection status before day zero has been decreasing, stationary and increasing, respectively. It is straightforward to express x in terms of τ₀ and τ₁ as

Since the confirmed new cases on day zero (∆Q)₀ is given by (∆Q)₀=x(1-q)^Tn₀, ℑ can be expressed in terms of τ_k, τ_l and (∆Q)₀ as

Number of infectious patients and asymptomatic patients in Tokyo

As an example of analysis, I apply the present analysis to the infection status in Tokyo in the middle of June 2021. In Tokyo, PCR tests have not been conducted on non-symptomatic patients, and I can set q = 0. I assume that the infection status was stationary in the middle of June 2021 and set τ_k = τ_l = 1 and ⟨β⟩_k = ⟨β⟩_l. Then, Eq. (13) reduces to

According to the Ministry of Health, Labour and Welfare of Japan (MHLW), transmission occurs 2 days before symptom-onset and in the 7~10 days post-symptom-onset [7]. These estimations are consistent with those reported in Ontario, Canada: Ontario Agency for Health Protection and Promotion reported that the transmission occurs in the 3~5 days before onset and 8~10 days post onset [5]. Assuming, on average, a patient will be tested and quarantined the next day of symptom-onset, I set as a model case k = 3 days and l = 7 days. Then, Eq. (16) is reducible to

which is shown in figure 2.  In Tokyo, the proportion of untraceable patients is ƒ = 50~60% [8]. Using the lower value ƒ = 50%, I find x = 77%. This value becomes x = 69% if ƒ = 60% is used. These values are consistent with observations of the onset ratio x = 76% [9] or 75% [10].

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/(∆Q)₀ = k when ƒ = 0 as Eq. (18) indicates. Equation (18) shows ℑ/(∆Q)₀ = 2k and 2.5k when ƒ = 50~60% respectively. Figure 3 shows ƒ dependence of  ℑ/(∆Q)₀ when k = 3 days and l = 7 days. It is important to note that in Tokyo, ƒ = 50~60%  and k = 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_as excluding pre-symptomatic patients is given by

and thus ℑ/(∆Q)₀ = 3 when k = 3 days, l = 7 days and  ƒ = 50%.

In January 2022, the number of patients in Tokyo increased due to the omicron variant. I can assume that the difference between ⟨β⟩_ｋ and ⟨β⟩_l is small and q is negligible in Eq. (15). From the infection curve increasing exponentially with a rate 0.2, τ_ｋ is estimated to be τ_ｋ = 1.24. In this period, the proportion of untraceable patients was 65.5% [8], and thus, setting k = 3 days, I get ℑ/(∆Q)₀ ≈ 11.

Around the end of February 2022, the sixth wave in Tokyo [8] seems to be in the stationary state with ƒ = 61%, and thus I can set τ_ｋ = 1. Therefore, ℑ/(∆Q)₀  is estimated to be ℑ/(∆Q)₀ ≈ 7.7.

Effective reproduction number and assessment of policies

I first define an effective infectious period d_eff and an effective infection coefficient β_eff as follows:

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

It should be remarked that when d_eff is equal to the recovery time or the inverse of the recovery rate γ and β_eff is a constant, the expression for R_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 R_eff defined by Eq. (23) depends on the quarantine rate of non-symptomatic patients through d_eff, which has a significant effect on reducing the effective reproduction number.

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) indicates that there are three tactics: (1) increase immunized people by vaccination, if it is effective in preventing transmission of SARS-CoV-2, or by natural immunity to make (1-μ) smaller, (2) enforce social distancing by various lockdown measures to make (1-a) smaller, and (3) quarantine asymptomatic patients by PCR tests to make d_eff smaller. In countries like Taiwan, Australia and New Zeeland who have succeeded in controlling COVID-19 before the vaccination was started, PCR tests have been conducted more than 100 times per positive patient in a well-designed manner.

In Tokyo, the PCR test has been used only to confirm the infection of SARS-CoV-2 for people who show some symptoms, which means q ≈ 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 to specific targets, like the nightlife district in 2020 and restaurants serving alcoholic beverages in 2021 and 2022. The policy targeting certain shops and opening hours has only limited effects on social distancing since people gather together in parks or on the streets. Furthermore, the policy has been enforced and lifted every one or two months, which has caused the wavy infection curve [11,12]. It could be possible to increase a by, for example, promoting Telework, limiting working days, reducing crowd in commuter trains and banning gatherings.

Discussion

In the present analysis, traceable and untraceable patients who tested positive are related to infection from pre-symptomatic and symptomatic patients and asymptomatic patients, respectively. Although this assumption may not be rigorous, it is a good assumption if people cooperate in investigating the infection routes by the health department of the local government. 

When the infection status is increasing or decreasing [6], the relation Eq. (15) between the number of infectious patients and the number of the newly confirmed new cases reduces to

if the condition ⟨β⟩_k = ⟨β⟩_l is satisfied. 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 d_eff. In general, d_eff is an increasing function of τ_ｋ and τ_l. Since the increasing state corresponds to τ_ｋ>1> τ_l and the decreasing state to τ_ｋ<1< τ_l, d_eff can increase or decrease depending on the relation between τ_ｋ and τ_l. 

The infection coefficient of SARS-CoV-2 depends on variants. When a new variant with a stronger infection coefficient emerges, ⟨β⟩_k becomes larger than ⟨β⟩_l. The number of infectious patients will increase for a stronger variant when other parameters are the same.

In 2021, vaccination progressed in many countries, and the infection status seems to be improving, at least in reducing serious cases. The effect of vaccination appears through μ in Eq. (2), which depends on variants of the virus. Therefore, policies for controlling COVID-19 should not rely only on the vaccination, and a proper combination of policies on vaccination, social distancing and quarantine of non-symptomatic patients must be designed in each country around the world [13].

Conclusion

It has been shown that the onset ratio and the number of infectious patients can be estimated from the proportion of untraceable patients who tested positive. The effective reproduction number depends not only on the vaccination rate and effects of social distancing but also on d_eff which is controlled by quarantine measures on pre-symptomatic and asymptomatic patients. The effect of the quarantine measure appears as a reduction of the effective reproduction number by (1-q)^T, which could be significantly large compared to the effects of the lockdown measures and the vaccination.

Acknowledgments

This work was supported in part by JSPS KAKENHI Grant Number 18K03573.