Predictive Research on COVID-19 using the Compartmental Model

Haotian Guo

Department of Statistics and Operations Research, University of North Carolina at Chapel Hill,

27514, Chapel Hill, NC, U.S.A.

Keywords: Compartmental Model, COVID-19, SIR Model, Epidemiology.

Abstract: Compartmental models appeared in our field of vision in the early 20th century, with the contributions mainly

of Ronald Ross, Hilda Hudson, David Kendall and other scientists. These models turned out to be one of the

most important techniques to model real world issues and to provide most of the answers that people seek. In

epidemiology, in particular, these are the essential modeling techniques to the modeling of infectious diseases.

Compartmental models have helped epidemiologists solve problems like the initial spread of the disease, the

number of people infected, and the level of risk the disease would bring to people. In the early 2019, a highly

infectious kind of pneumonia, later named COVID-19 by the World Health Organization, was firstly

discovered in China, and later the doctors and epidemiologists would find out that this disease was actually

due to a new kind of virus that had never been studied or seen before. In study this “new” virus,

epidemiologists revealed to us how important and effective compartmental models can be even when dealing

with a virus that scientists did not have full knowledge on its properties. For this project, several models were

also developed with the toll of compartmental model based on the data extracted from the website of the

World Health Organization of the spread of COVID-19 in countries like Vietnam, Spain, and the United States

of America. The goal was trying to do was to simulate the spread of the data and predict the trend of the spread

in such countries. Based on the analysis, it’s concluded that COVID may be still prevalent in countries like

these for a very long period of time, and because of that, there is a possibility that COVID may never end

worldwide.

1 INTRODUCTION

Compartmental models have greatly affected the

world of epidemiology and scientists of other

aspects, and even people like us since they were first

invented and used in the 1910s. Compartmental

models, as indicated by their name, break the issue

of one’s interests into different compartments and

then evaluate and analyze people or other elements

in different compartments and how do they change

between compartments.

Epidemiologists would use ODEs (ordinary

differential equations) and stochastic models and

framework as a platform when trying to analyze

diseases with compartmental models. The

advantages of ODEs would be that the process of

solving equations would be faster, and it would give

a deterministic solution to the equations scientists

found. Then we can use these differential equations

to plot graphs that would help visualize the models.

On the other hand, the platform of stochastic

framework is more suited for analysis of real-world

problems. Stochastic, which means random, would

assume that everything happens randomly following

current incident. For example, if there are three

states, A, B, and C, that a person can be in an event

and he’s currently in state A, then it’s completely

random whether he would be in state B, state C, or

remain in state A in the next time slot, and the choices

would not be affected by the states he had been in in

the previous state, which is also known as the

Markov property (Alshomrani et al., 2021). ODEs

would rather assume that things would happen

according to the will of scientists.

With the outbreak of COVID-19 and the pressing

demand for analysis of such disease because of the

damage it brought to the world, it is realized that we

can also analyze COVID-19 using SIR model,

although the pathogenic mechanism remained

unclear to scientists. That’s one of the many

advantages of such a model, which is that we can

analyze epidemic with just the presence of the data

of the spread of the disease. Now that COVID-19 has

Guo, H.

Predictive Research on COVID-19 using the Compartmental Model.

DOI: 10.5220/0011154000003437

In Proceedings of the 1st International Conference on Public Management and Big Data Analysis (PMBDA 2021), pages 187-190

ISBN: 978-989-758-589-0

187

been affecting us for more than a year, the question

that most scientists and general public would want

answers for is how long such epidemic would persist

and more radically. Would it ever end? So several

SIR models were created in order to find an answer

to this question using ordinary differential equations

to simulate the progress of COVID-19 in several

countries with data from the website of the World

Health Organization, and a prediction of the trend of

development of the disease was finally successfully

made.

Thus, the knowledge of compartmental models

and SIR model in particular provided a useful tool to

analyze the disease and predict number of people

would be infected and how long the disease would

last. However, it is unsettling that the results tended

to support the idea that COVID-19 may not end

worldwide when predicting using the SIR model

(Merchant, 2020). Such prediction is extremely

important in that epidemiologists and health

scientists could focus on how to deal with this

disease in a long-term fashion, and not as an

emergent outbreak anymore. Also, general public

would need to realize that COVID would eventually

just be like flu and common cold, and they need to

take action like vaccinate and wear masks as well in

order to cope with this disease.

2 RESULTS AND ANALYSIS

Fig. 1 below showed what a typical SIR model would

look like. The blue, red, and green lines represent

susceptible (S), infected (I), and recovered (R)

individuals as time evolves. In the short term, we

would assume that S+I+R would equal a fixed

number, N, the total population size. In other words,

it means that the whole population in a given area

would be divided into the three groups as identified.

However, when it comes to longer terms like several

months or years, it would not be so accurate to

assume that these are the only three groups that

existed. For example, in the case of COVID-19,

millions of people died after infection. Although it is

rare for people to be infected again and thus

susceptible after recovery, several people did show

signs of infection after recovery, and in this case it

might be better to use SIS model. But for purposes of

simplicity, most epidemiologists assume for a SIR

model at the initial stages of research.

Figure 1: SIR Model.

As we can see in the diagram, the red line, or the

infected individuals, has a right skewed distribution,

which means that the disease can be highly infectious

because people infect quickly at first, but it takes

time to recover, so the infected amount would

decrease with a smaller rate than the increasing rate.

And as assumed by the model, once infected

individuals are recovered, they would not be

susceptible to the disease again, so when no one is

susceptible to the disease, the epidemic or pandemic

is considered to be finished (Bailey, 1975). A typical

example would be the spread of COVID-19 in China,

as shown below (Fig. 2). It is easy to see that by the

time 100, which roughly corresponds to 220 days

after the first case, the red curve shows no sign of

increasing anymore and roughly no more infected

individuals present. It is then safe to say that COVID-

19, in a short period of time, would not surge in

China again (Yang et al., 2020).

Figure 2: COVID-19 Spread in China.

However, China might be one of the few cases

that would show a sign of ending COVID-19. For

example, Fig. 3, 4, and 5 below show the change of

infected population over time in the United States of

America, Spain, and Vietnam respectively. In the

case of the United States, we can see that there are

multiple peaks in the graph, which would correspond

to multiple SIR models. However, if we split the full

graph into several models, we would discover that

each SIR model does not indicate a full recovery. In

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other words, the disease did not quench before the

next breakout happened. In such countries, they need

to first achieve the final SIR and then finally end the

disease, and that would cost even years to come. For

Spain, it resembles the case of the US, so one

particular peak was cut out to be analyzed. In this

case, it shows a left skewed distribution, and also, if

we focus on the right tail, we can see that it has not

achieved a flat tail, meaning that it also lacks the

stability required by the SIR model to consider the

pandemic to be finished. In the case of Vietnam, at

first it seemed like it showed a complete peak and

showed signs of dying down, but it suddenly had

more cases surging out. So, we need to make sure

that we have a really flat right tail, which means the

newly infected individuals become mostly zero for a

long time before saying that this epidemic ended in

this area. These three countries were chosen in a way

to represent the three of the five continents on earth,

America, Europe, and Asia. And more countries

displayed similar traits of these countries, which

would finally lead to the conclusion that it might be

years before COVID-19 end worldwide, and it is

even possible that it may never end.

Figure 3: COVID-19 Spread in the United States of

America.

Figure 4: COVID-19 Spread in Spain.

Figure 5: COVID-19 Spread in Vietnam.

3 CONCLUSION

Based on the analysis of these models, it has also

been found that the R0 is less than one for all cases

analyzed, which means that the virus should not

spread as quickly as before and such value does not

signify an outbreak of the epidemic, but it certainly

was not the case shown by these models. So

alternatively it can be concluded that a short period

of a small R0 achieved is not full proof that the

disease would end permanently. We also need to see

a clear sign of dying out for a long period of time

before having such a conclusion. Thus, as the

statistics have shown, for countries like Vietnam,

Spain, and the United States of America, we don’t

have proof that COVID would subside in a short

time. Also, it is clear that even if there was only one

country that still had COVID going on, it would be a

disaster to the whole world because of the high

infectivity of COVID. Actually that was how the

breakout first happened. After its first emergence in

Wuhan, China, although China had done great effort

to quarantine people and implement lockdown, the

virus failed to be locked down and quickly spread to

other countries. There is no way to isolate virus in

modern world. So, in our case where most countries

or even continents showed signs that COVID would

still affect people there, it would be months and even

years to reach the state where only one or few

countries are still affected by COVID, and even if we

reached that state it would still cost great effort to

eliminate it. This is where we made such a

conclusion that epidemiologists, scientists, and

common people need to be aware of the possibility

that we may need to coexist with this virus in the end.

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ACKNOWLEDGMENT

Through the writing of this paper I received a great

deal of help and assistance. I would first like to thank

Dr. Otto X. Cordero for being my supervisor and

mentor. Your expertise had given me great insight on

how to approach problems of my interest and I could

never finish my project without your methodology. I

would also like to acknowledge my tutor Shiqi Wen

and Mandy for their excellent guidance throughout

my studies. With the tools they provided I was able

to conduct my research in a more timely and effective

manner.

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