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Abstract—This paper presents open-source computer

simulation programs developed for simulating, tracking, and

estimating the COVID-19 outbreak. The programs consist of

two separate parts: one set of programs built in Simulink with a

block diagram display, and another one coded in MATLAB as

scripts. The mathematical model used in this package is the

SIR, SEIR, and SEIRD models represented by a set of

differential-algebraic equations. It can be easily modified to

develop new models for the problem. A generalized method is

adopted to simulate worldwide outbreaks in an efficient, fast,

and simple way. To get a good tracking of the virus spread, a

sum of sigmoid functions was proposed to capture any dynamic

changes in the data. The parameters used for the input (infection

and recovery rate functions) are computed using the parameter

estimation tool in MATLAB. Several statistic methods were

applied for the rate function including linear, mean, root-mean-

square, and standard deviation. In addition, an adaptive neuro-

fuzzy inference system is employed and proposed to train the

model and predict its output. The programs can be used as a

teaching tool and for research studies.

Index Terms— Coronavirus, COVID-19, Virus Spread,

Epidemiology, Program, MATLAB, Simulink.

1.1 Introduction

oronavirus disease 2019 (well-known as COVID-19 or

2019-nCoV) is a disease caused by a novel virus called

SARS-CoV-2 [1]. The first reported case of this disease

was on Dec. 31, 2019, in Wuhan, China. The outbreak has been

declared as a public health emergency of international concern

by the World Health Organization (WHO) on Jan. 30, 2020 [2],

and as a pandemic on March 11 [3]. The virus spread rapidly

around the world and several large-size clusters of the spread

have been observed worldwide [4]. According to Johns

Hopkins University, the total number of confirmed cases

surpassed one million cases on April 2, 2020 [5].

An essential part of minimizing the spread of the virus is to

monitor, track, and estimate the outbreak. This is extremely

useful for decision making against the public health crises [6].

One way to predict the dynamic spread of the epidemic is by

using computer simulation following the mathematical model

of an epidemic. In the literature, several analytical approaches

have been proposed to model the pandemic including

Susceptible-Infected-Removed (SIR) model [6-7], Susceptible-

Exposed-Infected-Removed (SEIR) model [1], Susceptible-

Infected-Recovered-Dead (SIRD) model [8,9], fractional-

derivative SEIR [10], and SEIRD [11]. While some recent

studies are addressing this pandemic by developing simulation

codes [6-11], there is an increasing need to develop open-source

computer programs to perform a time-domain simulation of the

dynamic spread of the virus. References [12-13] present

MATLAB code scripts to achieve this objective. Reference [14]

presents a Python-based program called CHIME (COVID-19

Hospital Impact Model for Epidemics) for hospital uses.

It is advantageous to have an educational program that

displays the mathematical model of the virus spread in a

visualized block diagram in addition to coded scripts. One of

the widely-used platforms for studying the dynamic behavior of

a system is MATLAB\Simulink. It has been used by many

academic researchers in different fields for simulating dynamic

systems using time-domain simulations. A dynamic system,

such as the COVID-19 spread, can be mathematically modeled

by a set of differential equations (DEs) or a set of differential-

algebraic equations (DAEs) depending on the employed model.

The main challenge with such a simulation is to estimate the

parameters in the model–for instance, the infection and

recovery rates. For the SIR model, these two parameters are the

inputs to the model, whereas the outputs of the model are the

state variables of the system. Since we have initial data to

compare (reported confirmed cases of infection), we can

estimate the parameters of the infection and recovery rates so

that the two outputs (reported and simulated cases) are equal

and their difference is near zero. However, parameter

estimation could fail in many situations [12–13] due to

restricted limits applied to the parameter, unknown initial

values of the parameters, and sudden changes in the reported

data.

One observation from the proposed models in the literature

is that the exponential function is mathematically the major part

of the estimation. This is because the problem is nonlinear and

the exponential function can represent any increasing or

decreasing rates depending on the sign and magnitude of the

exponent. A challenge emerges when there are multiple

dynamic changes in the reported-cases data or another wave of

the pandemic that the simulation struggles to accurately track

these data and estimate the future outbreak. This requires

variable inputs with multiple step functions to vary its

parameters at each time point the data changes its rate

dramatically. By applying this strategy, the program solves the

problem even when the infection data changes its rate at various

levels.

With several open-source programs, this paper introduces a

generalized method to track and estimate the virus spread

worldwide in an easy, efficient, and fast way. The proposed

method is implemented using the SIR and SEIR models and

coded under the MATLAB\Simulink platform. The time points

SimCOVID: Open-Source Simulation Programs for the

COVID-19 Outbreak

C

Ismael Abdulrahman, ismael.abdulrahman@epu.edu.iq

Department of Information System Engineering, Erbil Technical Engineering College, Erbil Polytechnic University, Iraq

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is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted June 22, 2020. .https://doi.org/10.1101/2020.04.13.20063354doi: medRxiv preprint

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where the data changes dynamically are extracted from the data

automatically based on some statistics measures. The number

of exponential branches for the rate functions are left for the

user to choose since each data has its own needs of parameters.

The data is taken from [15] and is updated daily for all countries

worldwide. The user enters the country name and runs the

model with optional editable settings. With the application of

the adaptive neuro-fuzzy inference system (ANFIS), the

outbreak of China and Italy are implemented in Simulink using

both standard mathematical model and ANFIS system.

1.2 Introduction to SIR, SEIR, and SIRD models

The SIR model is a basic representation used widely to

describe a disease spread, and it is the fundamental model for

the other models such as SEIR and SIRD. SIR model consists

of three-compartment levels: Susceptible (S), Infectious (I), and

Removed (R). Any individual belongs to one of these groups.

A brief description of these compartments is given below.

Susceptible individuals are those people who have no immunity

to the disease but they are not infectious. Since there is no

vaccine yet developed for this disease, we can say that the entire

community is exposed to get infected by this disease and hence,

the “Susceptible” compartment can be represented by the entire

population. An individual in the “Susceptible” level can move

into the next level of the model (Infectious) through contact

with an infectious person. By this single transmission, the

number of susceptible\infectious people reduces\increases by

one, respectively. The next compartment is for the infectious

people who have the disease and can spread it to susceptible

people. Infectious people can move to the “Removed”

compartment by recovering from the disease. The removed

compartment includes those who are no longer infectious and

the ones who have dies from the disease (closed cases). The

summation of these three compartments in the SIR model

remains constant and equals the initial number of population. A

basic SIR model is shown in Fig. 1, where denotes the

infection rate or the transmission rate, and denotes the

recovery or removed rate. Generally speaking, these parameters

are not constant; they are functions of the size of

infectious and recovery compartments. These are the

parameters that we want to optimize and estimate so that the

reported and simulated cases are approximately equals. To

solve this set of differential equations, we need initial values for

the three-state variables , , and namely , , and .

The initial value is the community population impacted by

the outbreak [6], whereas, is the number of confirmed cases

that can be any value but not zero. We can set to zero, if

the start times of the spread and simulation are equals. The

transmission rate reduces monotonically with time [6].

Mathematically, a standard SIR model can be represented using

the following differential equations:

where denotes the total population size, and

the natural birth and death rates are ignored.

For the SEIR model, an additional compartment is added

between the Susceptible and Infectious compartment called

“Exposed”. This compartment is dedicated to those people who

are infectious but they do not infect others for a period of time

namely incubation or latent period. Note that the summation of

the four state-variables at any time stays constant and equals the

initial population, and in the equations is the reciprocal of the

incubation period (which can vary between 2 to 14 days). For

the SIRD model, an additional compartment is added at the end

of the SIR model to distinguish between recovered and death

cases in the “Removed” compartment. It is worth mentioning

that the

ratio in the SIR model gives us an important metric

called “basic reproductive number”, or . This ratio is a

measure of how contagious a disease is, in other words, how

many people are infected by a single infected person. If

, there is a spread of disease which is a strong sign of a

pandemic. If , there is a decline in the spread [16].

2. THE PROPOSED METHODOLOGY

Since the programs developed in this paper are for simulating

any outbreak around the world with potential multiple dynamic

changes in the reported cases, we need to look for a generalized

method that is applicable for all possible scenarios. The

reported data used for comparison is either the daily

confirmed\measured infection cases () or its cumulative

infection cases (). These plots are nonlinear curves and can

be represented by an exponential formula such as the sigmoid

function expressed in (4):

where “” refers to power gain (exponent) which could be

positive or negative, and “” refers to a time constant. The

above sigmoid function is always positive regardless of the

signs of its parameters which satisfies the representation of

physical components such as rate variables in this problem. At

each time “” when there is a noticeable change in the reported

data, the sigmoid function changes its magnitude to find new

values for the parameters. In other words, the proposed rate

function consists of multiple branches of sigmoid functions;

each has different gain and time constant. The final rate

function is the aggregation of these branch functions as in

expressed in (5).

Fig. 1a SIR model

Fig. 1b SEIR model

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where “” is the gain of branch–.

For simplicity, it is assumed the parameter “” is a constant

value to be estimated. The time parameters “” is represented

by a vector which its elements can be manually pre-defined or

estimated. The length of this vector is the number of sigmoid

branches–that is, the number of iterations in (5). The aggregated

sigmoid function can be further generalized by subtracting the

previous sigmoid function from the one in the consideration.

The following rate function is for the latter concept with a

generalized infection\recovery rate function used for tracking

the problem. The higher the number of branches (), the

smoother and better matching between the reported data and the

simulated plot.

(6)

(1)

(3)

(2)

For the estimation problem, we need to multiply the above

function with an exponential component having a negative

exponent in order to downturn the curve if the pandemic has not

yet passed the peak. The parameters of the new component are

also estimated by the solver. If the exponent is zero, the function

would be as the above equation. However, any nonzero

exponent leads to different case studies such as standard,

optimistic, and pessimist estimations. The following equation

can be used for estimating the parameters with any optional

scenario.

where and are the parameters that need to be estimated.

It should be noted that since the recovery function does not

change its rate rapidly–unlike the infection function– due to

lack of vaccine at the current time, we can represent this

function with only one sigmoid function or even simpler as a

constant parameter to be estimated by the program. It is worth

mentioning that if in the aggregated rate function, the

sigmoid functions become an aggregated step function. The

lower the power gains of the sigmoid function, the steeper the

curve is, and vice versa. Figure 2 illustrates the proposed

concept assuming different values of , where in the figure

refers to a step function.

For the initial values of the parameters, we need a

methodology that satisfies all potential outbreaks. One efficient

way is to use the confirmed infection ratio

and

normalizing the ratio with a range between zero and one. This

quantity is employed for the gain parameters of sigmoid

branches–that is, in (5). For the time parameters– in (5)– we

can use one of these methods: (1) a set of predefined time

locations (2) a statistical approach to automatically find the best

values for these parameters to optimally fit the curve. Due to its

generality and simplicity, the second method is considered as

the standard method for the programs. In MATLAB, we have a

function called “MaxNumChanges” intended to determine

some useful statistical information with the following

measures: (1) “” (2) root-mean-square “” (3)

“” (4) standard deviation “”. The first technique is

considered as the basic tool for the initial time calculation but it

can be easily switched to the other techniques by changing its

name. Figure 3 illustrates this concept assuming the maximum

number of changes 2, 3, 5, 10. Clearly, increasing this quantity

provides better output matching and parameter estimation. For

many cases, it is found that 3–5 change points (which gives the

number of sigmoid branches as well) are sufficient for this

problem. However, the choice of the number of change points

is left for the user to tune the estimation for each outbreak.

3. SIMULATION EXAMPLE

To see how the problem is modeled in Simulink using the

differential equations (1)-(3), a simulation example is shown in

Fig. 3 with the proposed concept of rate function assuming three

sigmoid branches. A similar program is coded in MATLAB. In

the diagram, different colors are used to distinguish the

equations (blue for “Susceptible”, red for “Infectious”, and

green for “Removed” compartment). Note that the control

Fig. 2 Illustration of the proposed rate function with an aggregated step function (left) and sigmoid function (right).

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measure parameters () are functions of time to be found and

estimated by the program. All the numerical data for the test

cases were taken from [16]. The test outbreaks used in this

section are the cases of China, Italy, and Iran–which has several

fast-dynamic changes in the data– assuming multiple sigmoid

branches in the models. Figure 5 displays part of the results for

the outbreak of China using SimCOVID. The plot on the left is

for the beta-gamma function which varies over time and

provides some useful information about the reproduction

number and the overall pandemic. The plot on the right is a two-

y-axis plot showing the infection and its cumulative cases. If

the number of sigmoid input changes, these plots change as well

and exhibit more detail that is hidden in the simple model. The

same program is used to plot the graphs in Fig. 6a which is

related to the Italy outbreak and the estimation is carried out on

April 7th, 2020. The estimation ends by the end of July 2020

which shows good matching between the measured cases

(released later after this estimation) and the simulation results

given in Fig. 6a–b. The gray area in the plots is for confidence

interval on the same date of estimation with Figure 7

displays part of the results obtained for Iran outbreak and the

estimation occurs on June 21th, 2020 using the MATLAB codes

(added later to the package). This outbreak is chosen due to its

multiple dynamic changes in the reported cases to show the

Fig. 3 Changing maximum number of change points with values 2, 3, 5, 10, respectively.

Fig. 4 SIR model in Simulink for simulating the virus spread with three sigmoid branches– blue is for “Susceptible” with its input (), red is for “Infectious”, and

green is for “Recovered” with its input ()

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capability of the programs to simulate such a problem. As can

be observed, the programs can be used for simulating the

epidemic assuming various scenarios and confidence intervals

at different times. Applying new measures is easy to see the

impact of these control efforts on future estimation.

4. Detailed Model of SIR (SEIR, SIRD, and SEIRD)

The SIR model described above does not give information on

the exposed people who are infected but not detected yet (not

confirmed yet). It does not provide any knowledge of the closed

cases of infectious people who have passed away. An exposed

variable can be added to the SIR model to form a SEIR model,

whereas a closed (death) variable can be added to the SIR model

to form a SIRD model. These two variables can also be added

together to the SIR model and form a new model called SEIRD.

This model is more general and is adopted in this section [11].

The differential equations of this model are as follows:

where refers to the exposed state variable, refers to the

closed or death case, refers to the exposed parameter, and

refers to the dead variable. Summing (8)–(12) should give zero

whereas the total sum of the state variables should be constant

and equals the population:

Several programs are added to SimCOVID including SEIR,

SEIRD, and SEIRD designed and coded in Simulink and

MATLAB. An example in Simulink is shown in Fig. 8 using

the SEIRD model. Note that the actual collected data was used

as input to this model. This data is stored in Excel sheets and

there is a block in Simulink that allows us to import data from

external sources and build a customized signal. This block is

called “signal builder”. The ratio between the daily dead and

daily infectious is used to represent the mortality rate. This

signal represents an actual data-based signal which is multiplied

by the simulated infectious signal to be integrated and form the

cumulative dead variable using an integrator block in Simulink.

In the figure, the orange color is used for “Exposed” and dark

red is used for “Dead” compartments. The parameter

optimization process and the numerical outputs are not plotted

here owing to its similarity to the previous model and for space

limitation reasons. However, all these models and programs are

provided with this paper. The objective here in this section is to

show how to edit the SIR and build a new model. Note that in

the initial design of this model, a set of unit steps is used instead

of sigmoid functions which can be replaced to give smoother

plots.

5. SIMULATION WITH CONTROL MEASURES

Total or partial lockdown, social distancing, and stay-at-

home control steps can influence the spread of the virus and

flatten the curve earlier if these steps were applied in advance.

In the Simulink model, we can represent all these control

measures by a step function (or sigmoid function) and simulate

the system under these conditions. These controls affect the

infection rate giving a decline in the beta function. On the other

hand, developing a vaccine or involving any other ways to

recover or slow down the disease (such as providing the

hospitals with all the necessary ventilators) affect the recovery-

rate parameter by increasing this value. As a result, the

reproduction ratio reduces by any change in these two

parameters. It is also easy in Simulink to see the impact of a

response delay on the curve. This can be achieved by adding a

delay block to the infection function. However, more details

about the model are needed for this problem.

6. SimCOVID Capabilities and Updates

SimCOVID went through several updates and the new

version of the package is Version 5. The package started with

Simulink as the platform and the first preprint was published

based on this update. It was realized that some researchers like

to work on MATLAB coding instead of Simulink diagrams for

some reasons. For instance, unlike Simulink files, MATLAB

codes can be run on different releases of the software. Another

reason is that the codes can be used to build a graphical user

interface. Further, the for-loop programming–which is used for

the rate functions– is easier and faster in MATLAB. To keep

the package useful for both researcher groups, a new set of

MATLAB-based programs is added to SimCOVID. These

programs have been updated several times. In the beginning,

SimCOVID used three-step functions for the rate functions with

a manual time setting. Step function leads to a sharp change in

the simulation plots which can be avoided by replacing it with

a sigmoid function providing smooth outputs. The number of

sigmoid branches is also updated to a number specified by the

user. Another challenge was to find the initial values of the

parameters to be estimated. An update was added to

automatically determine optimal initial values of the parameters

that led to solve the problem even for data with multiple

dynamic changes and reduce the overall simulation time. It is

only required to enter the country name and run the model to

simulate the outbreak for that specific country. Several

statistical tools were added to choose the optimal time location

for the sigmoid branches.

7. THE EDUCATIONAL value of SimCOVID

SimCOVID is an open-source package used for simulating,

tracking, and estimating an outbreak that comes with editable

files and codes. The MATLAB programs were coded in a

simple way; there is only one main script for everything

(reading data, parameter estimation, solving DEs, and plotting).

The data itself comes from the source as an Excel sheet. A

generalized method is adopted to reduce the user’s actions to

solve the problem. Changing the model from one to another is

also straightforward; the new equations, initial values, and their

limits are appended to the existing ones.

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Fig. 5 Estimated parameter values over the time using three sigmoid branches (left) and the results of infection and its cumulative plots–China outbreak

Fig. 6a Infection plot estimated on April 7th, 2020 (left) and its cumulative plot (right)– 180 days period and three sigmoid branches

Fig. 6b Infection plot simulated on June 22th, 2020 (left) and its cumulative plot (right)– 180 days period and ten sigmoid branches

Fig. 7 Simulation of an outbreak with multiple-dynamic changes in the reported data (including second wave of outbreak)– example of Iran, with 30 sigmoid

branches

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8. Adaptive Neuro-Fuzzy SIR MODEL

The models used in the previous sections was based on the

mathematical model of the problem. We can also build a

machine-learning program to simulate the same system using

the input and output data of the model. Simulink provides the

user with an adaptive neuro-fuzzy inference system (ANFIS)

toolbox to generate if-then fuzzy rules automatically based on

training the given data. References [17] present a detailed

description of this technique in which the same methodology is

employed in this paper. Using a basic SIR model built in

Simulink with a variable infection rate and constant recovery

rate, the model is trained using input-out data. The input could

be the infection rate, recovery rate, or a combination of the two.

The output could be the infectious output or its cumulative

function. In this study, the beta function and its derivative are

used as input variables to the ANFIS model whereas the

infectious and cumulative infectious variables are chosen for

the output in two different training processes. ANFIS allows us

to use only one output for each block and for this reason, two

separate processes are used to generate two ANFIS blocks for

the two outputs. More outputs require more ANFIS blocks.

Figure 9 shows a simple Simulink program used in this

training. The recovery function is treated as constant as

proposed in [6] whereas the infection rate is treated as variable

[6]. With the parameters optimized, the ANFIS toolbox is used

to generate seven fuzzy rules for each output. The membership

function used in this training is the gaussian function. Figure

10a-c shows the training iterations, fuzzy rules, and the ANFIS

outputs for the cumulative and infectious variables. Figure 11

shows the Simulink model for the ANFIS blocks used in this

simulation. These if-then rules are used to simulate the case of

China outbreak and the results are shown in Fig. 11a. whereas

Fig. 11b shows the infections and recovery parameter values.

Notably, the results show some good matching but it needs

improvement. A more detailed and complicated beta function

formula can be used to improve the accuracy of the results.

9. Conclusion

This paper presented an open-source toolbox to model,

simulate, and predict the coronavirus COVID-19 outbreak

using Simulink and MATLAB. The programs are easy to edit

for new models, simple in their structure, generalized, fast, and

can be used for simulating worldwide outbreaks with only

inputting the country name. Several models were presented

including SIR, SEIR, SIRD, and SEIRD models. The rate

functions were treated as variables with multiple sigmoid

branches to give smooth and good tracking and estimation. The

initial values for the parameters to be estimated are

automatically calculated based on some efficient criteria

extracted from the given data. Several statistical measures were

used for determining the optimal time parameters for the

sigmoid functions. In addition, an adaptive neuro-fuzzy

inference system was used to generate the output of the model

based on some training tasks applied to the system. This paper

promises some lasting values in the field of the coronavirus

spread. The program can be used as an educational tool or for

research studies.

Fig. 8 SEIRD detailed model

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https://www.dropbox.com/s/5xm9699r7dnvw7m/SimCOVID.

Fig. 9 A simple Simulink program used for building an ANFIS block

Fig. 10a Training errors for the infectious variable (left) and the cumulative infectious variable (right)

Fig. 10b Fuzzy rules generated by ANFIS for the infectious variable (left) and the cumulative infectious variable (right)

Fig. 10c Testing data against ANFIS output for the infectious variable (left) and the cumulative infectious variable (right)

Fig. 4 Simulated and reported cases (daily on left y-axis and cumulative on right y-axis) – case of China

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zip?dl=0

Fig. 11 The developed ANFIS model in Simulink

Fig. 11a ANFIS outputs and the reported cases (daily on the left y-axis and cumulative on the right y-axis) – case of China

Fig. 11b Optimized parameters used in building the ANFIS model

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Appendix I

The programs are available at:

https://www.mathworks.com/matlabcentral/fileexchange/75

025-simcovid-an-open-source-simulation-program-for-the-

covid-19

Appendix II

Demonstrating Videos

[1]. https://www.dropbox.com/s/sbnn1784a00hiw7/ANFIS.webm?d

l=0

[2]. https://www.dropbox.com/s/mxbz8j9ogoom65g/China_Wuhan

_SEIRD.webm?dl=0

[3]. https://www.dropbox.com/s/myi95d6dwnttoc8/ChinaOneStep.

webm?dl=0

[4]. https://www.dropbox.com/s/mj6bi1jbiik9c03/Italy_SEIRD.web

m?dl=0

[5]. https://www.dropbox.com/s/ncihp1dbqx1p35m/Italy_Optimiza

tion.webm?dl=0

[6]. https://www.dropbox.com/s/illjl95v4r603og/Italy.webm?dl=0

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