IEEE Workshop on Nonlinear Circuit Networks
December 4-5, 2020

Simulation of Infectious Number of COVID-19in JAPAN by Using SIR Modelwith Echo State Networks
Aika Ohno, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering of Tokushima University
2-1 Minami-Josanjima, Tokushima-shi, Tokushima-ken 770-8506, Japan
Email: {a-ohno, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract—In this study, we propose a model to simulate infectious number of COVID-19 in Japan by using a mathematicalmodel that is called SIR model. We use time variables for SIR
model, and Echo State Networks (ESN) is applied to simulate
parameters of SIR model. Then, we use the simulated parameters,
and infectious number in Japan is estimated.

who pass away. In SIR model, susceptible people move tothe statement of infectious with the infectious rate α, and theinfectious people move to the statement of removed with theremoved rate γ. Figure 1 shows the statement structure of SIRmodel.

I. I NTRODUCTION
COVID-19 is confirmed for the first time in the city ofWuhan on December 2019. As everyone knows, COVID19 spread all over the world, and still now on confirmed
cases are increasing day by day. According to Jonhs Hopskins
University’s report, more than 50,000,000 infected cases and1,200,000 deaths had been confirmed globally until November
11th 2020 [1]. Then, many cities and countries are locked
down to prevent spreading COVID-19, and some of cities areenforced lock down before pandemic or outbreak situation.
As a result, they could prevent pandemic. Therefore, it isimportant to predict infectious numbers, and to know how theinfectious number would go on.
Mathematical models for infectious diseases such as SIR
model and SEIR model have been used for the simulation ofinfectious diseases. SIR model and SEIR model are used for
Severe Acute Respiratory Syndrome and seasonal influenza as
well [2],[3].
In addition, Echo State Networks (ESN) is one kind of neural networks, and ESN is used for time-series data computing.
In this study, we simulate infectious number of COVID-19
in Japan by using SIR model. Then, ESN is used to train andtest the parameters of SIR model, and we aim at simulating
the infectious graph that has more realistic waves.

Fig. 1.

This SIR model can be expressed by the following differential equations. S(t), I(t) and R(t) mean the population ofsusceptible, the population of infectious and the populationof removed at time t, respectively. Equation (1) shows thedifferential equations of SIR model.
dS= −αS(t)I(t)
dt


dI
= αS(t)I(t) − γI(t)dt
 dR = γI(t)dtII. SIR MODELSIR model is a popular mathematical model for infectious
diseases. SIR is the initial letters of susceptible, infectious
and removed. Susceptible indicates the statement of the people
who do not have antibodies of that infection. In this study, asto COVID-19, we regard all of the people at the first point
as susceptible. Infectious means the statement of the people
who have infectious capacity. Removed means the statementof the people who recover from that infection and also people

The statement structure of SIR model.

(1)

In SIR model, all of the population are divided in thesethree statements which are susceptible, infectious and removed. Then, whole population N is the addition of the threestatements as shown in Eq. (2), and we regard N as constant.

-3-The update equations are shown in Eq. (3).
N = S(t) + I(t) + R(t){(2)Then, Eq. (1) is solved by the Runge-Kutta method. Figure 2
shows a typical graph of SIR model.

x̃(n) = tanh(Win [1; u(n)] + W x(n − 1)x(n) = (1 − a)x(n − 1) + ax̃(n)(3)

Root Mean Squared Percentage Error (MSE) is used forvalidation. N , Ysimulated and Y t means the number of thedata, the simulation value and the target value, respectively.
1 ∑ t(Y − Ysimulated )2
N i=1
NM SE =(4)Wout is the output weights, reg is the regularization coefficient, and I is the identity matrix in Eq. (5). X indicates[1;u(n);x(n)].

Fig. 2.

Wout = Y t X T (XX T + regI)−1

A graph of SIR model.

As shown in Fig. 2, the infectious graph of SIR model
has only one peak. SIR model has the advantage that isquick simulation and the simple structure. However, SIR
model has the disadvantage that SIR model can not express
multiple waves with constant parameters. Therefore, we aim
at expressing multiple peaks in infectious simulation graph byapplying ESN which is one kind of neural networks and whichis introduced in the following chapter.
III. E CHO S TATE N ETWORKS
ESN is one kind of reservoir computing, and it can handle
time-series data [4]. ESN has the feature of quick learning.
Therefore, the advantage of SIR model would not be lose byusing this network. ESN has reservoir layer between input andoutput. Figure 3 shows the structure of ESN.

(5)In ESN, Wout is learned and updated to minimize M SE.
In this study, ESN is used for simulating of the parametersα(t) and γ(t).
IV. DATA
We use open data from Ministry of Health, Labour andWelfare of Japanese government [5]. From the open data
source, positive cases, recovered cases and death toll in Japanare obtained.
Then, S(t), I(t) and R(t) are calculated by the followingequations. P (t), Re (t) and D(t) means the reported positivecases, reported recovered cases and reported death toll at timet.
S(t) = N − I(t) − R(t)tt∑∑ I(t) =P (t) −R(t)i=1i=1tt
∑∑R(t)=R(t)−D(t)e
i=1(6)

i=1

In this study, N is 126,000,000 that is the whole populationof Japan.

Fig. 3.

V. P ROPOSED M ETHODFirst, the parameters α(t) and γ(t) are calculated from
SIR model differential equations by using difference equations
method.

The structure of Echo State Networks.

In the reservoir layer there is a lot of reservoir nodes, andthe reservoir nodes are connected randomly. The input weights
and reservoir weights are not updated, only the output weights
are updated.
x(n) is a vector of reservoir neuron activation states at time
step n, and n is the discrete time. Tanh function is used as
the activation function as shown in Eq. (3). u(n) is the input
data, a is the leaking rate and Win is the input weight.

S(t) − S(t + 1)α(t) =
S(t)I(t)R(t + 1) − R(t) γ(t) =I(t)(7)

Second, the parameters α(t) and γ(t) are learned and tested
by ESN. It is iterated 30 times with random number for initial
weight, and average output is used.

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Third, I(t) is calculated by using simulated parameters α(t)and γ(t).
 S(t + 1) = S(t) − α(t)S(t)I(t)R(t + 1) = γ(t)I(t) + R(t)I(t + 1) = N − S(t + 1) − R(t + 1)

(8)

As to real data, we use first 100 data for training and thenext 100 data for testing.
VI. S IMULATION R ESULTS
Figure 4 shows the real data of α(t) which is calculated byEq. (7).

Fig. 6.

The calculated real data of γ(t).

The following parameters are used for ESN. The size ofreservoir is 50, the leaking rate is 0.8, the weight is 1.1 andthe regularization coefficient is 1.0e-2. As a result, M SE ofγ(t) is 0.0030813.

Fig. 4.

The calculated real data of α(t).

Figure 5 shows the test result of α(t).
The following parameters are used for ESN. The size ofreservoir is 50, the leaking rate is 0.15, the weight is 1.5 andthe regularization coefficient is 0.5. As a result, M SE of α(t)is 2.17192e-19.

Fig. 7.

The test result of γ(t).

The test graph seems to be following with the real datagraph. The serrate waveform which is shown in the real graphis expressed in the test graph. In addition, though the valuesof test graph have differences with the real data, it seems tobe big differences compared with the result of α(t).
Figure 8 shows the simulation result of infectious number.

Fig. 5.

The test result of α(t).

The test graph seems not to be following with the real datagraph enough. Between 100th day to 140th day, the test graphseems to be following with the real data, but after 140th day
the test graph shows different waveform and values with thereal data.
Figure 6 shows the real data of γ(t) which is calculated byEq. (7).

Fig. 8.

-5-The simulation result of infectious number for 300th days.

The waveform is not like a curb, it is serrate waveform.
These features are not shown in the typical infectious graph ofSIR model. Moreover, the valley of the simulation test graph isthe same with the valley of real data. However the simulation
test result is not following with the real data, especially as tothe infectious number.
VII. C ONCLUSIONSIn this study, infectious number was simulated with time
variables α(t) and γ(t) by using SIR model method and ESN
to simulate the parameters α(t) and γ(t). Then, we could
express serrate waveform in the test part of infectious number
simulation.
For future works, we need to improve the test accuracy ofthe parameters α(t) and γ(t). Especially, we need to improveα(t) simulation. In this study, though we used 100 data fortraining and other 100 data for testing, these data size might
be too small data size for ESN. However, to increase the datasize is not good method. We will keep using small data size
to predict for unknown infectious diseases besides COVID-19.
To improve α(t) simulation, we will try to use other neuralnetworks such as Convolutional Neural Networks and other
Recurrent Neural Networks as well.
ACKNOWLEDGEMENT
This work was supported by Center for Design-Oriented AI
Education and Research Grant-in-Aid for Students.
R EFERENCES[1] JOHNS HOPSKINS UNVERSITY OF MEDICINE, “COVID-19 Map”,
https://coronavirus.jhu.edu/map.html, 2020-11-11.
[2] Ng, Tuen Wai, Gabriel Turinici and Antoine Danchin, “A double
epidemic model for the SARS propagation”, BMC Infectious Diseases
3.1 (2003): 19.
[3] Osthus, Dave, et al, “Forecasting seasonal influenza with a state-space
SIR model”, The annals of applied statistics 11.1 (2017): 202.
[4] Lukoeviius, Mantas, “A practical guide to applying echo state networks”,Neural networks: Tricks of the trade. Springer, Berlin, Heidelberg, 2012.
[5] Ministry of Health, Labour and Welfare, “Open data”,
https://www.mhlw.go.jp/stf/covid-19/open-data.html, 2020-10-03.

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