Introduction

Measures to prevent the spread of the new coronavirus infection (COVID-19) include urban blockade, self-restraint of behavior and sales, RT-PCR inspection and cluster measures, contact tracking app, infected person behavior monitoring app, etc. It is being tackled in various countries. Until now, Japan has focused on PCR testing, tracking by cluster countermeasures teams, and self-restraint in actions and sales by declaring an emergency, but as of May 6, 2020, in China, Italy, Iran, the United States, etc. It has not led to the explosive spread of infection seen. However, continuing to refrain from behavior and business as it is has harmful effects on various aspects such as economy, education, and culture, and we are at the stage of searching for an exit strategy. Vaccines are being actively developed, but considering the confirmation of safety and efficacy and the establishment of a production system, it is expected that a period of at least one year will be required at the earliest. More efficient quarantine measures for infected people are needed to prevent the collapse of medical care and gradually lift restrictions on behavior and sales. As an important item, I would like to verify the ** effect of introducing the contact tracking app ** by simulation.

Situation around the world

First, let's take a look at the current medical status of COVID-19 around the world. In the figure below, in countries with 10,000 or more infected people, the horizontal axis is the recovery rate (number of recoverers / number of people requiring treatment), and the vertical axis is the mortality rate (number of deaths / number of people requiring treatment). It is a graph. See this article for the calculation method.

Japan is in a position with a recovery rate of 1.5% and a mortality rate of around 0.25%. In terms of recovery rate, it is equivalent to France, Belgium and Singapore, and in terms of mortality rate, it is equivalent to Germany, Austria, Pakistan and Belarus. Among them, corona tracking apps such as China, Israel, Singapore, and South Korea are said to have been successful, but with the exception of China, which is considered to be the source of infection and the initial response was delayed, Israel, Singapore, and South Korea have certainly died. The rate is low. More than 10 types of corona tracking apps have already been developed around the world. (Reference: Top 10 popular smartphone apps to track Covid-19) It seems that Japan is also developing an application using the API jointly developed by Apple and Google. (Reference: About the development of contact tracing apps). Apple and Google require the approval of national authorities to register the Corona Tracking app in the store, so I don't think there will be multiple apps in a mess. (Reference: How do the corona countermeasure apps in each country implement privacy measures?)

Calculation assumptions

By the way, the corona tracking app that is being introduced in this way has the expected effect.

** Effect of quickly detecting close contacts and promptly promoting examination, isolation, and treatment (especially asymptomatic infected persons) **
Reduction of cases where "infection route is unknown" during interview survey
Early detection of clusters
As a passport to prevent infected people from entering high-risk areas
In addition, accumulation of epidemiologically useful knowledge, etc.

Various effects are expected, but this time I would like to focus on 1 and evaluate the effects quantitatively.

At that time, the Cluster-based SEIR model shown in the previous article is used as the calculation model. The main reasons for using this model are:

Consistent with the finding that COVID-19 infection is caused mainly by clusters
The time change of the effective reproduction number $ R_t $ is estimated from the observed value and incorporated.
Do not depend on the initial value of S, which is a problem when applying the SEIR model to reality.

In particular, in the normal SEIR model, when the basic reproduction number is $ R_0 = 2.5 , it is said that the peak out occurs in about 60% of the population ( 1-1 / 2.5 = 0.6 $). (Acquisition of herd immunity) Even if you look at the countries where the number of infected people per million is high, it is 5,000 to 6,000 (/ 1 million) in Qatar and Spain, so the infection rate is at most 0.6% ** Is estimated to be. ** Only 0.01% in Japan. ** Therefore, it seems reasonable at this point to think that the observed peak-out phenomenon of COVID-19 cannot be explained by a simple SEIR model.

Computational model

The calculation formula is shown. First, introduce the variables.

S: Those who are not immune to infectious diseases (Susceptible)
E: Infectious disease is in the incubation period (Exposed)
I: Infectious
R1: Those who recovered from infectious diseases and acquired immunity by routes other than the app (Recovered 1)
** R2: Person quarantined by app detection ** (Recovered2)
R: Those who have recovered from an infectious disease and acquired immunity (Recovered). ($ R = R_1 + R_2 $)

Introduce the following as parameters.

** $ \ alpha $: App quarantine rate. Percentage and definition of infected individuals during and during the incubation period per day isolated by the app **
Rp (t): Effective number of reproductions at time t
lp: Incubation period (5 days adopted)
ip: Infection period (8 days adopted)
Th: Half-life of effective reproduction number (7.5 days adopted)

The following is a cluster-based SEIR model that incorporates the isolation effect of the app.

\begin{eqnarray}
\frac{dRp}{dt} &=& - \frac{ln2}{T_h}Rp \\
\frac{dS}{dt} &=& -\frac{1}{ip} Rp \cdot I \\
\frac{dE}{dt} &=& -\frac{dS}{dt} - \frac{1}{lp} E - \alpha E\\
\frac{dI}{dt} &=& \frac{1}{lp} E - \frac{1}{ip} I - \alpha I\\
\frac{dR_1}{dt} &=& \frac{1}{ip} I \\
\frac{dR_2}{dt} &=& \alpha E + \alpha I \\
\frac{dR}{dt} &=& \frac{dR_1}{dt} +\frac{dR_2}{dt} 
\end{eqnarray}

By the way, the following equation holds.

\frac{d}{dt}(S+E+I+R_1+R_2)=\frac{d}{dt}(S+E+I+R)=0

The point is that the app effect isolates you from E and I at a constant rate.

Calculation result

This time, I would like to show the results first. The initial value of the effective reproduction number is $ R_p (0) = 10 $. Also, let the initial state be $ (E, I, R) = (0,1,0) $.

When the application isolation rate α = 0

There is no quarantine by the app. Eventually 113 people will be infected. m8_1_Rp0_10_Al_0.0.jpg

When the application isolation rate α = 0.1

When the quarantine rate by the app is 10 [% / day]. In other words, assume that 10% of infected people are quarantined per day. Eventually 26 people will be infected (R1 + R2). The number of infected people has decreased by 77%. m8_1_Rp0_10_Al_0.1.jpg

When the app quarantine rate α = 0.2

When the quarantine rate by the app is 20 [% / day]. Eventually 11 people will be infected (R1 + R2). ** The number of infected people has decreased by 90%. ** ** m8_1_Rp0_10_Al_0.2.jpg

Relationship between app quarantine rate α and final number of infected people

Let's graph the relationship between the quarantine rate by the app $ \ alpha $ and the final number of infected people $ R (T_ {max}), R_1 (T_ {max}), R_2 (T_ {max}) $. At around ** $ \ alpha = 0.07 $ or more, app quarantines outnumber non-app quarantines **.

Consideration

From the above, the following trends can be derived from the simulation regarding the effect of early isolation by the corona tracking app based on the cluster-based SEIR model.

** If the quarantine rate by the application is 20 [% / day] or more, the effect of reducing the final number of infected people by 90% or more ** can be expected.
** If the app quarantine rate is 7 [% / day] or higher, the app quarantine is higher than the non-app quarantine. ** **

Furthermore ...

The main reason why the number of infected people in South Korea is kept low may be that the app effect is greater than the number of PCR tests **.
Similarly, a major factor in keeping Japanese infected low is likely to be the effectiveness of cluster tracking by cluster response teams and health centers. Other factors such as national character can be fully considered, but it is necessary to fully consider the disadvantageous condition that ** Japan is one of the most densely populated countries in the world **.
Considering that the human resources (doctors, nurses, laboratory engineers, health center staff, etc.) and physical resources (protective clothing, masks, test kits, etc.) for PCR tests are limited, antibody tests and antigens Inspection, etc. ** It is desired to develop a non-contact, simple, highly sensitive and highly specific inspection **. The Corona Tracking App is ** beneficial in that you can isolate asymptomatic people, even if you don't test them **.
** The Corona Tracking App can be expected to have a tremendous effect as shown in this article, so I think it should be actively introduced while giving due consideration to privacy. If it also has a passport function to events and facilities, it can be expected to prevent clusters, and I think it can be a key item for resuming economic activities.

Reference link

I referred to the following page.

Python code

Finally, I'll paste the Python code used in this article.

import numpy as np
import matplotlib.pyplot as plt

# ODE solver
def my_odeint(deq, ini_state, tseq, keys):
    sim = None
    v = np.array(ini_state).astype(np.float64)
    dt = (tseq[1] - tseq[0])*1.0
    for t in tseq:
        dv = deq(v,t, keys)
        v = v + np.array(dv) * dt
        if sim is None:
            sim = v
        else:
            sim = np.vstack((sim, v))
    return sim

#define differencial equation of seir model
def seir_eq8(v, t, keys):
    Th = keys['Th']
    lp = keys['lp']
    ip = keys['ip']
    al = keys['al']
    #
    Rp  = v[0];
    s   = v[1];
    e   = v[2];
    i   = v[3];
    r1  = v[4];
    r2  = v[5];
    r   = v[6];
    #
    dRp  = - np.log(2)/Th * Rp
    ds   = - Rp/ip * i
    de   = - ds     - (1/lp) * e - al * e
    di   = (1/lp)*e - (1/ip) * i - al * i
    dr1  = (1/ip)*i
    dr2  = al * e + al * i
    dr   = dr1 + dr2
    return [dRp, ds, de, di, dr1, dr2, dr]

# simulation 1
def calcsim(Rp0, keys):
    # solve seir model
    #          Rp  S  E  I  R1 R2  R
    ini_state=[Rp0, 0, 0, 1, 0, 0, 0]
    t_max=180
    dt=0.01
    t=np.arange(0,t_max,dt)
    #
    sim = my_odeint(seir_eq8, ini_state, t, keys)
    #
    plt.rcParams["font.size"] = 12
    fig, ax = plt.subplots(figsize=(10,5))
    ax.plot(t,sim[:,[2,3,4,5]]) # extract E I R1 R2
    ax.set_xticks(np.linspace(0,t_max,19))
    yw = 10; yn = int(120/yw)+1
    ax.set_yticks(np.linspace(0,yw*(yn-1), yn))
    ax.grid(which='both')
    ax.legend([ 'Exposed', 'Infected','Recoverd1(normal)', 'Recoverd2(App-based)'])
    ax.set_xlabel('date')
    ax.set_ylim(0,)
    plt.show()
    print("R(tmax):{}".format(sim[-1,6]))

# do simulation 1
keys = {'lp':5, 'ip':8, 'Th':7.5, 'al':0.0 }
calcsim(10, keys)
keys = {'lp':5, 'ip':8, 'Th':7.5, 'al':0.1 }
calcsim(10, keys)
keys = {'lp':5, 'ip':8, 'Th':7.5, 'al':0.2 }
calcsim(10, keys)


# simulation 2
def calcAltoRfin(Rp0):
    #          Rp  S  E  I  R1 R2  R
    ini_state=[Rp0, 0, 0, 1, 0, 0, 0]
    t_max=180
    dt=0.01
    t=np.arange(0,t_max,dt)
    #
    lp = 5
    ip = 8
    keys = {'lp':lp, 'ip':ip, 'Th':7.5, 'al':(0.) }
    #
    mklist = lambda e, l : np.append(np.array(e),l)
    #
    rslt = []
    for i in np.linspace(0,0.4,20):
        keys['al'] = i
        sim = my_odeint(seir_eq8, ini_state, t, keys)
        r = mklist(keys['al'], sim[-1][4:])  # (i, R(tmax), R1(tmax), R2(tmax))
        rslt.append(r)
    #
    rslt = np.array(rslt)
    ymax = max(rslt[:,2])
    plt.rcParams["font.size"] = 12
    fig, ax = plt.subplots(figsize=(10,5))
    #
    ax.plot( rslt[:,0], rslt[:,3], 'b')
    ax.plot( rslt[:,0], rslt[:,1], 'g')
    ax.plot( rslt[:,0], rslt[:,2], 'r')
    ax.legend([ 'R(Total infected)', 'R1(normal purge)', 'R2(App-based purge)'], loc='upper right')
    ax.grid(which='both')
    ax.set_xlabel('application-based purge rate [/day]')
    ax.set_ylabel('total infected cases at tmax')
    yw = 10; yn = int(120/yw)+1
    ax.set_yticks(np.linspace(0,yw*(yn-1), yn))
    #
    for tat in [0,0.1,0.2,0.3,0.4]:
        idx = [i for i in range(len(rslt[:,0])) if rslt[i,0] >= tat][0]
        print("R_fin with Al:{} is {}".format(rslt[idx,0], rslt[idx,3]))
    #
    ax.set_xlim(0,)
    ax.set_ylim(0,)
    plt.show()

# do simulation 2
calcAltoRfin(10)

Let's simulate the effect of introducing a contact tracking app as a countermeasure against the new coronavirus