Introduction

We simulated the spread of COVID-19 infection using the SIR model in Python. We created each of the 47 prefectures for the model and visualized it with a heat map.

github page https://github.com/chihina/analysis_covid_19_python

Visualized video (Some watermarks are included for editing with free software. I'm sorry !!) https://www.youtube.com/watch?v=7XvPKcasL1U

Development environment

Interpreter: python 3.71 Special library: folium == 0.11.0 Others such as matplotlib and numpy.

About the SIR model

The ordinary differential equations of the SIR model are as follows. (Simply speaking, β represents the infection rate and γ represents the recovery rate.)

\frac{dS}{dt}(t) = -\beta S(t)I(t)\\  
\frac{dI}{dt}(t) = \beta S(t)I(t) - \gamma I(t)\\
\frac{dR}{dt}(t) = \gamma I(t)\\

S (susceptible) ... Susceptible (potentially infected)

I (Infected) ... Infected person

R (Recovered) ... Immune carrier (or quarantine)

First equation

\frac{dS}{dt}(t) = -\beta S(t)I(t)\\

The left side represents the rate of change of susceptibility carriers per unit time. This is expressed by multiplying the susceptibility carrier and the immune carrier by the infection rate.

In other words, S is reduced by the number of infected people per unit time.

Second equation

\frac{dI}{dt}(t) = \beta S(t)I(t) - \gamma I(t)\\

The left side shows the rate of change of infected persons per unit time. The first term on the right side is the one that appeared earlier. The second term on the right side shows the person who recovers per unit time.

In other words, it is the number of people who get infected per unit time minus those who recover.

Third equation

\frac{dR}{dt}(t) = \gamma I(t)\\

The left side shows the rate of change of the immune carrier per unit time. The right side is the same as the previous one, showing the person who recovers per unit time.

In other words, R increases by the amount of people who recover per unit time.

Improvement of SIR model

Considering the SIR model described so far for 47 prefectures, it is meaningless because each one becomes independent. Therefore, the influence term due to the infection status of other prefectures is added using the population ratio data. Did.

\frac{dS}{dt}(t) = -\beta S(t)I(t) - I_{pref} × PR_{pref} × 0.1\\  
\frac{dI}{dt}(t) = \beta S(t)I(t) - \gamma I(t) + I_{pref} × PR_{pref} × 0.1 \\
\frac{dR}{dt}(t) = \gamma I(t)\\

About the added item

I_{pref} × PR_{pref} × 0.1

The first letter expresses the situation of infected people other than the prefecture you are thinking about at that time.

The second letter indicates the population ratio of Tokyo in the prefecture you are thinking of at that time.

Therefore, according to this section, if the number of infected people increases in other prefectures, the number of infected people will increase in their own prefectures. By using the information on the population ratio, the population is large in Tokyo, Osaka, Aichi, etc. I am trying to make the influence of the prefecture stronger.

Implementation method

I will solve the improved SIR model with python. Originally it was easy to implement using scipy, but I implemented it steadily for understanding. For details, see the code and ask a question Please!!

Implementation method: 4th-order Runge-Kutta method

There are many explanations about the Runge-Kutta method, so please search for "4th-order Runge-Kutta method".

Visualization method

Draw a time series heatmap with python folium https://sammi-baba.hatenablog.com/entry/2018/12/25/074017

After creating the infected person information for each prefecture with the pandas dataframe, I referred to. I read and used what I exported to csv. Please see the actual video uploaded to youtube !!

Visualized video (Some watermarks are included for editing with free software. I'm sorry !!) https://www.youtube.com/watch?v=7XvPKcasL1U

Summary

I couldn't get a perfect grasp of the current infection situation, but I have a relatively simple ODE. I was very surprised to be able to reproduce the spread of infection to some extent.

If you have any questions about more detailed content, please do not hesitate to contact us !!

References

I referred to the following articles and statistical data. Thank you.

[1] Draw a time series heatmap with python folium https://sammi-baba.hatenablog.com/entry/2018/12/25/074017

[2] Population estimation As of October 1, 2017 Population estimation census site (e-stat) https://www.e-stat.go.jp/dbview?sid=0003215844

[3] Data of Japan github page (accessed July 28, 2020) https://www.e-stat.go.jp/dbview?sid=0003215844

COVID-19 simulation with python (SIR model) ~~ with prefectural heat map ~~