Introduction

As of March 5, 2020, Japan's situation regarding the new coronavirus (COVID-19) has been taken very seriously, including immigration restrictions from other countries. Although the government has requested that all elementary, junior high and high schools be closed all over the country, and some have questioned the effect, it is also understood that political judgment is necessary because it is difficult to show scientific grounds. I can do it. This article uses the SEIR model, which is one of the mathematical models of infectious diseases, to verify whether it is possible to control the peak of infection by taking a certain period of time off from a school or business establishment (with a scale of 1000 people). is. However, this article is a simulation under limited assumptions and does not provide scientific confirmation or academic evidence. We do not take any responsibility as it may be helpful or incorrect. For most of the programs, I referred to this article.

Prerequisites

Details of the SEIR model are omitted in this article, but in Wikipedia According to

S: Susceptible who is not immune to infectious diseases
E: Those with an infectious disease during the incubation period (Exposed)
I: Infectious
R: Those who have recovered from infectious diseases and acquired immunity (Recovered)

It is composed of, and its acronym is called SEIR model. It seems. SEIR model parameters include

β: Infection rate
lp: Incubation period
ip: Infection period

there is. According to Ministry of Health, Labor and Welfare, the incubation period is "according to WHO's knowledge, the incubation period is currently Is 1-12.5 days (mostly 5-6 days), "so let's assume 5.5 days. The duration of infection is unknown, but it is said that fever will continue for about 4 days after the onset and then recover or become severe, so it is assumed to be 8 days. (If the condition becomes severe, it is considered that the patient will be hospitalized, but the mildly ill person shall recover while living a normal life without being aware of it.) The most important is the infection rate, but based on the data provided by the Ministry of Health, Labor and Welfare, we have done the following.

Infection rate

According to Q & A of the Ministry of Health, Labor and Welfare, the frequency with which one infected person causes secondary infection is shown in the figure below. It seems. From this data, using uniform random numbers, Define a function that calculates the rate R0 (basic reproduction number) that produces a secondary infection.

def COVID19R0(er):
    if np.random.rand() < er:
        # good environment
        if np.random.rand() < 0.8:
            R0 = 0
        else:
            R0 = np.random.randint(1,4)*1.0
    else:
        # bad environment
        R0 = np.random.randint(0,12)*1.0
    return R0

Here, er is defined as the probability that the source of infection was in a poorly ventilated environment.

er: Probability that the source of infection was in a poorly ventilated environment

Additional parameters

We also want to evaluate the impact of the leave, so we will introduce the following parameters.

tb: Holiday start date
te: Holiday end date

Based on R0 and these parameters, I modified the SEIR model as follows:

#define differencial equation of seir model
def seir_eq5(v,t, keys):
    er = keys['er']
    lp = keys['lp']
    ip = keys['ip']
    tb = keys['tb']
    te = keys['te']
    #
    if t >= tb and t <= te: #Holiday period
        R0 = 0 #Completely closed (when there are no people in the school / business office)
    else:
        R0 = COVID19R0(er)
    #
    ds = - R0/ip * v[2] if v[0] >= 0 else 0 # note: s >= 0
    de = -ds - (1/lp) * v[1]
    di = (1/lp)*v[1] - (1/ip)*v[2]
    dr = (1/ip)*v[2]
    return [ds, de, di, dr]

Other programs

Import the library to run the simulation.

import numpy as np
import matplotlib.pyplot as plt

Define a function to solve the ODE.

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

Code that calculates and graphs the results.

#solve seir model
ini_state=[999,0,1,0]
t_max=180
dt=0.01
t=np.arange(0,t_max,dt)
#
keys = {'er':0.5, 'lp':5.5, 'ip':8, 'tb':0, 'te':0}
sim = my_odeint(seir_eq5, ini_state, t, keys)
#
plt.rcParams["font.size"] = 12
fig, ax = plt.subplots(figsize=(10,5))
ax.plot(t,sim)
ax.set_xticks(np.linspace(0,t_max,19))
ax.set_yticks(np.linspace(0,1000,11))
ax.grid(which='both')
ax.legend(['Susceptible','Exposed','Infected','Recovered'])
plt.show()

Here, er (probability that the source of infection was in a poorly ventilated environment) was assumed to be 0.5.

simulation result

We will start with one infected person on the 0th day out of 1000 offices.

The first is when you do not take any leave.

The peak is about the 60th day, and there are days when there are up to about 320 infected people.

This is the case when we are closed for 20 days from a very early stage (30th day).

The peak is about the 95th day, and there are days when there are up to about 310 infected people. It is interesting that the peak delay days (35 = 95-60) are longer than the holidays (20 days).

This is the case when the patient is closed for 20 days from the stage when the infection begins to rise rapidly (day 50).

Peaks are dispersed on days 52 and 90, with up to about 200 infected days occurring on day 90.

This is the case when the patient is closed for 20 days from the stage when the infection begins to subside (60th day).

The result is the same as if you do not take any leave. This is thought to be because on the 60th day, there were no people who were not immune to S: infection (Susceptible).

Consideration

From the above, the following trends can be derived from the simulation regarding the peak control of infection by taking a certain period (20 days) of absence from schools and business establishments (with a scale of 1000 people).

If you take a leave of absence when there are few infected people, you can expect the effect of delaying the peak, but the number of peak infected people does not change much.
If you take a leave of absence when the infection starts to rise rapidly, you can expect the effect of spreading the peak and the effect of reducing the number of peak infected people.
Taking a leave of absence when the infection has begun to subside has no effect.

Therefore, if you want to take a leave of absence, it may be most effective to do it from "the stage when the infection starts to rise rapidly" to "before the infection starts to subside".

Furthermore ...

If even one out of 1,000 people is infected, the infection will reach its peak in about 60 days, so I think that strict border measures cannot be expected to be very effective except for delays.
Since the Chinese New Year in 2020 starts on January 24th, one of the guidelines for the peak infection in Japan is from the end of March to the beginning of April.
So, about 10 days before the peak, about March 20th is the crucial point for countermeasures ...
On the SEIR model, the infection is completely resolved when R: the number of people who have recovered from the infection and acquired immunity has increased. Therefore, urgent vaccine development and therapeutic drug development are strongly desired.
Infection occurs only when S: a person who is not immune to the infectious disease and I: an onset person meet, so I think it is meaningful to reduce the chances of such encounters (opportunities for clusters to occur). I will.
Tokyo Metropolitan Government and others publish Infection Trends in a graph. It is important to determine which infection phase the local government you live in is in, so I would like each local government to publish similar data.

Reference link

I referred to the following page.

[PYTHON] Verify the effect of leave as a countermeasure against the new coronavirus with the SEIR model