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COVID19 currently poses a global public health threat. Although Tokyo, Japan, is no exception to this, it was initially affected by only a smalllevel epidemic. Nevertheless, medical collapse nearly happened since no predictive methods were available to assess infection counts. A standard susceptibleinfectiousremoved (SIR) epidemiological model has been widely used, but its applicability is limited often to the early phase of an epidemic in the case of a large collective population. A full numerical simulation of the entire period from beginning until end would be helpful for understanding COVID19 trends in (separate) counts of inpatient and infectious cases and can also aid the preparation of hospital beds and development of quarantine strategies.
This study aimed to develop an epidemiological model that considers the isolation period to simulate a comprehensive trend of the initial epidemic in Tokyo that yields separate counts of inpatient and infectious cases. It was also intended to induce important corollaries of governing equations (ie, effective reproductive number) and equations for the final count.
Timeseries data related to SARSCoV2 from February 28 to May 23, 2020, from Tokyo and antibody testing conducted by the Japanese government were adopted for this study. A novel epidemiological model based on a discrete delay differential equation (apparent timelag model [ATLM]) was introduced. The model can predict trends in inpatient and infectious cases in the field. Various data such as daily new confirmed cases, cumulative infections, inpatients, and PCR (polymerase chain reaction) test positivity ratios were used to verify the model. This approach also derived an alternative formulation equivalent to the standard SIR model.
In a typical parameter setting, the present ATLM provided 20% less infectious cases in the field compared to the standard SIR model prediction owing to isolation. The basic reproductive number was inferred as 2.30 under the condition that the time lag
A novel mathematical model was proposed and examined using SARSCoV2 data for Tokyo. The simulation agreed with data from the beginning of the pandemic. Shortening the period from infection to hospitalization is effective against outbreaks without rigorous public health interventions and control.
COVID19 currently represents a global public health threat. Tokyo, Japan, is no exception, but its epidemic was small despite lacking rigorous public health intervention. The thorough behavior changes of individuals, with social distancing and avoidance of the 3 Cs [
From a clinical perspective, SARSCoV2 has an incubation period of 7 days, according to the World Health Organization (WHO) [
SIR/SEIR models including a compartment for quarantined (
The SIR model essentially suits analysis for a shortterm epidemic in local districts [
The limitation of these SIR derivative models lie in the fact that in the case that the collective population is large (eg, Tokyo, Japan, or Wuhan, China), previous works solved only a part of the equation rather than the whole governing equation, with the assumption that susceptible individuals are replaced by the collective population (N) [
This paper attempts to propose a new epidemic model that provides not a combination of piecewise solutions but a direct simulation based on a discrete delay differential equation that includes the isolation period (hospitalization). This model is unique because of its inclusion of delay time
The relation between the fundamental reproduction number
For this study, we used a publicly available COVID19 data set provided by the public health authority of the Tokyo Metropolitan Government in Japan [
The average number of treatment days in hospital was estimated from data on the cumulative sum of discharge and deaths [
Concept of the present epidemic model. Two paths from newly infected until removed (recovered/died) are shown, with symptomatic and asymptomatic paths. The portion of the former is ε and the latter 1–ε. The former remains infectious in the time period [
where
Given the number of initial infectors
Equation 1 is featured in an explicit inclusion of time lag parameters,
The model for ε=0 treats clinical symptoms alone and does not take asymptomatic infections (subclinical patients) into consideration. Hence, all infections are to be eventually detected and hospitalized, and equation 1 becomes equation 3, which we refer to as the epidemic model PART1:
In the case ε=1, an alternative formulation equivalent to a standard SIR model is obtained. In the case 0<ε<1, we call equation 1 the epidemic model PART2 (see
In the present model, the whole epidemic trend was simulated by a single delay differential equation in terms of cumulative infections
Once
In the subsequent section, we will focus our attention on model PART1.
Expressions for infection variables in terms of cumulative infections x.
Item  Variable  Expression 
Cumulative onset 

—^{a} 
Hospitalized 


Recovered/died 


Inpatients 


Infectious cases in the field 


Positivity ratio for PCR^{b} testing 

^{a}Not applicable.
^{b}PCR: polymerase chain reaction.
The value of time interval
Equation 3 can be simplified to respective time spans for 0<
This equation has an analytic solution, a socalled logistic function:
At the beginning, equation 5 shows an exponential epidemic growth
For example, with α being 0.164, the equivalent value of τ would be 4.22 days; x/x(0) would become 10 in 2 weeks.
When
Normalization of equation 7 provides the following:
where
Here, we have derived a final size equation from equation 7:
Solving for final size
It is interesting to note that once α
The effective reproductive number
Combining equation 10 with equation 7 yields:
This is consistent with the literature [
Since no preventive measure was taken at an initial stage of the first wave of the COVID19 pandemic from February 14 until May 23 in Tokyo, the value of
Now that dimensionless parameter α
Recalling β
Behavior of solution x with a change in the value of the parameter (α
Cumulative confirmed cases between February 28 and April 28 in the first wave of COVID19 in Tokyo were adopted for the fitting of parameters, transmission rate α, and virtual collective population
Comparison of simulation results with data in the entire period of the first wave of COVID19 in Tokyo. Axis represents the number of cumulative hospitalized cases (Y), the number of inpatients (P), and the number of recovered/deaths (Z). Solid lines represent simulations by model PART1. Dotted points show data observed in Tokyo, Japan.
Simulation of daily new confirmed cases compared with observed data. Model PART1 with the same parameter values as in
Simulation of positivity ratio compared with observed PCR (polymerase chain reaction) tests. Model PART1 with the same parameter values as in
Data in the early stage might include large statistical error because of the fewer inspections conducted. Except for this period, however, data trends are well simulated in spite of the model simplicity. That is reasonable because the model counted infections with the onset of clinical symptoms, which suits the attribute of tested data. It is important to note that the agreement of numerical results with observed data implies correctness of counting infectious cases at large (
Trends of the reproductive number,
The vaccine ratio needed to avoid an outbreak in Tokyo was estimated using equation 11. To calculate the condition
Rearrangement provides the vaccine ratio as:
This estimate is reasonable since it is close to the value of 0.63 that Wu et al [
Comparison of simulations by the present apparent timelag model (ATLM) with that of the standard susceptibleinfectiousremoved (SIR) equivalent. The results obtainable by SIR are based on equation 1 with ε=1 shown as a broken line and those by ATLM model PART1 as a solid line. Variables for infectious, removed and susceptible in SIR model correspond to those P+Q, Z and Mx, respectively in ATLM model PART1.
According to the report of antibody prevalence tests conducted by the Ministry of Health, Labor and Welfare from June 17, 2020, just after the end of the first wave in Tokyo, the antibody ratio was found to be 0.10%. Since the metropolitan population is 14 million people, the number of individuals having antibodies is estimated to be 14,000. As the number of removed cases in the first wave was 5236 as of May 3 in Tokyo, this leaves 9000 in the field. Taking ε as 9000/14,200=0.634 as the first estimate, simulation of the first wave was conducted using PART2 to reproduce 5200 for (1–ε)
Model calculation with subclinical patients considered. Model PART2 (equation 1) was used. Model parameter M was determined to be 14,200 with the aid of the antibody prevalence test results. Parameter ε was optimized so as to give cumulative symptomatic infections, (1ε)x is the same as that computed by PART1. The remaining parameter values of x(0),
Toward the end of the first wave in Tokyo, strong public health interventions were conducted to prevent contact with others by 80% (stayathome orders issued for 80% of residents). The actual reduction in the number of contacts was estimated to be 50%60%. The official stayathome announcement was declared on day 72 (April 27) after the onset of the epidemic. The simulation was tailored to assess its effects on daily new confirmed cases with and without stayathome actions. Results are presented in
A posteriori assessment of the impact of restrained contact to reduce transmission rate, α. Stepwise change in the value of transmission rate α from α0 to 0.2α0 (action A) or to 0.43α0 (action B) was applied at day 86 when the effect of the action should appear on daily new confirmed cases. Solid lines represent daily new confirmed cases (left axis) while the red broken line (infectious cases at large) to the right. It was found that “real” action A was too late in suppressing spread. If “imaginary” action C had been implemented 1 month earlier, it would have had a greater impact.
In
In
The Japanese government issued a criterion for the rescission of the stayathome order for the purpose of early economic recovery, stipulating that daily infections decrease to no more than 0.5 person/day per 100,000 population. Applied to Tokyo, the criterion would yield 10 infections per day. To confirm its validity, a posterior assessment was conducted.
Trends of infectious variables as SARSCoV2 transmission declines.
The criterion of 10 infections/day can be represented as
Effect of transmission rate α on cumulative infections. With a reduced transmission rate α, the final size x(∞) was observed to be smaller and took longer to be attained.
Effect of time lag
A similar effect is expected by reduction of time lag
Effect of α
Another point of checking is to prepare the necessary number of hospital beds.
Relationship between attack rate p(∞) and α
Effect of
The present epidemiological model ATLM is very simple in terms of mathematics and comprises a small number of fitting parameters, that is, the governing equation is described by only one dependent variable
Among them it is noteworthy that the simulation matched the trend of the positivity ratio of PCR testing. This implies that the number of infectious cases at large was counted properly by ATLM. This fact may essentially be a base to apply simulation results to the assessment or proposal of strategies for public control of the epidemic (ie, public health interventions at the right magnitude and timing). We demonstrate two examples below.
One is the assessment of the stayathome order to reduce persontoperson contact by 80% declared by the metropolitan authority on April 27 (72 days after the onset of the first wave of the epidemic). Based on the prediction of infectious cases at large by the present model, the declaration should have been made 1 month earlier. If so, moderate reduction of contact by 43% would have been effective enough to reduce both inpatient and impact on social and economic activities.
The second example is the timing of the rescission of the state of emergency issued on April 7. It was actually done on May 25 based on PCR test results. However, according to the behavior of the infectious cases at large, it should have been postponed by 1 week when the calculated infectious cases at large would reach below one at which point the epidemic would cease.
As a corollary, we induced a single dimensionless parameter (α
As for the timedependent effective reproductive number
Three parametric survey calculations for the preparation of a coming wave clarified the combination of
The present model (ATLM) has limitations. First, it is assumed that all new cases spread the infection from the time of infection
After June 2020, PCR testing was enhanced in Japan in order to suppress infectious subclinical patients as a measure of intensive cluster intervention. As a result, the number of positive PCR tests increased compared with the first wave of the epidemic and a similar number of subclinical patients as inpatients was found. Nevertheless, the maximum number of hospital beds required were less than that of the first wave in Tokyo. This may be owing to improved medical care as well as the larger portion of young patients. Postanalysis of the epidemic from June to October 2020 using general ATLM PART2 (equation 1) with information mentioned above will be our next task.
Secondly, the characteristics of the onset of clinical symptoms are stochastic. Therefore, the modeling of a preonset period and a statistical process are needed for accurate prediction. Thirdly, we assumed in the model PART2 that subclinical patients continue to be infectious from infection to removal, which must overestimate the number of infectious cases. With more information on period
Fourthly, input parameter
In summary, this study is the first complete simulation of the first wave of the epidemic in terms of the trends associated with various SARSCoV2 infection parameters in Tokyo, Japan. Existing data and outbreak patterns in other countries may be better understood via the present model.
A novel epidemiological model (ATLM) was developed using a single delayed differential equation with explicit inclusion of the time lag associated with the isolation of infectious cases. It provides a full simulation of the various infection variables in the entire span from onset to endpoint with a small number of calculation parameters. The model was verified by various epidemic trend data (including the PCR positivity ratio) published by the Tokyo Metropolitan Government. The validity of counting infectious cases at large was checked indirectly by the coincidence of data for the PCR positivity ratio. Based on this, two practical issues about public health control of SARSCoV2 surfaced. One of them is the mitigation of infections by reducing social contact, declared on April 27, 2020. Based on the trend of infectious cases at large predicted by ATLM PART1, this order should have been issued 1 month earlier, which would have led to less infection as well as a reduced slowdown of social activities. The other issue is the timing of the declaration of the rescission of the state of emergency, which was issued on April 7 and rescinded on May 25. However, according to the predicted behavior of the infectious cases at large, this should have been done 1 week later when infectious cases are at <1 and the epidemic would fade out. Finally, as a control measure for a coming second wave, the combination of
Derivation of the general delay differential equation: consideration of silent spreaders (model PART2).
Derivation of the final size equation.
apparent time lag model
polymerase chain reaction
susceptible, exposed, infectious, and removed
susceptible, infectious, and removed
susceptible, infectious, quarantine, and removed
susceptible, exposed, infectious, quarantine, and removed
World Health Organization
We are indebted to Mr Kazushige Tohei for his valuable comments on interpreting the numerical results. MU is a former professor of the Tokyo Institute of Technology, MK a former researcher of Hitachi Research Laboratory, Hitachi Ltd, and SK is a former engineer of Hitachi Works, Hitachi Ltd.
MU developed and verified the epidemiological model. MK developed the computational method. SK was responsible for the mathematical aspect of ATLM. All authors have read and approve the version of the manuscript to be published.
None declared.