Prediction of Epidemic and Pandemic Outbreaks Using
Mathematical Models
Review Report
Submitted by
Ms. Bijila B
M.Tech First Semester
Biomedical Engineering
Department of ICE
Submitted To
Ms. Priyanka C P
Assistant Professor
Department of ME
NSS College of Engineering
Palakkad - 678008, Kerala, India
January 2, 2025
Abstract
Mathematical models are powerful tools for the prediction of infectious
diseases. The main objective of using mathematical models is to understand the complex dynamics of disease that useful for the predictions,
planning and control of epidemic and pandemic outbreaks.
Here,
propose a literature review of ten peer-reviewed articles that introduce
ten different models. Articles were collected from Google Scholar that
were published between 2020 to 2024. The type of model used, year of
study, data collection and method of solving of each article are analyzed
and identified the advantages and limitations of each model compared
with others. The literature review suggests best models for prediction
and prevention of infectious disease.
i
Contents
Abstract
i
1
Introduction
1
2
Literature Review
2
2.1
Basic Compartmental Model . . . . . . . . . . . . . . .
2
2.2
Extended SIR model . . . . . . . . . . . . . . . . . . .
4
2.2.1
SIR Model with Vital Dynamics . . . . . . . . .
4
2.2.2
SIR Model with Machine Learning . . . . . . .
4
Extended SEIR model . . . . . . . . . . . . . . . . . . .
4
2.3.1
SEIR Model with Vaccination and Quarantine . .
4
2.3.2
SEIR Model with Neural Network and Deep
2.3
2.4
3
4
Learning . . . . . . . . . . . . . . . . . . . . .
5
Comparison of Models . . . . . . . . . . . . . . . . . .
5
Discussion
7
3.1
9
Advantages and Limitations . . . . . . . . . . . . . . .
Conclusion
10
References
11
ii
1.
Introduction
When a disease spread rapidly for a large population within a
specified geographical area or region is considered as a epidemic
condition. Yellow fever, smallpox, malaria, and mpox are examples
of epidemic cases. When the spread crosses different countries or
continents, then the condition will become a pandemic. COVID in 2019
and HINI influenza in 2009 are pandemic diseases and many people lost
their lives. It is necessary to control the effect of diseases on human life.
Mathematical models are the best solutions to reduce the effect of
infectious diseases to certain extent. It helps to understand the disease
dynamics such as infection rate, recovery rate, and death rate. This
useful for the prediction of severity of infectious disease. The prediction
about uncontrolled outbreak of disease provides a warning for the public
and for the government to take necessary steps and policies such as
quarantine and lockdown to prevent outbreaks.
There are many mathematical models that are used for these objectives, including deterministic models, stochastic models, and statistical
models. The most simple and widely used are compartmental models. A
total of ten articles have been reviewed in this paper. The goal of review
is to familiarize the reader with different compartment mathematical
models and to compare the strengths, drawbacks, and challenges of each
model with respect to the other.
1
2.
Literature Review
There are different types of classification for mathematical modeling.
The main classifications studied here are based on :
1. Basic Compartmental
2. Extended Compartmental
2.1
Basic Compartmental Model
The basic compartmental/deterministic model includes two types: SusceptibleInfected-Recovered (SIR) model and Susceptible-Exposed-Infected-Recovered
(SEIR) model.
SIR model was developed by Kermack and McKendrick in 1927 [1],
which is considered as the simple mathematical model of epidemics. It
includes three separate groups:
Figure 2.1: SIR Model
The group of individuals that are not currently infected but may be
infected are included in the susceptible(S) category. Those are currently
infectious are in the infected(I) category and those are no longer
infectious are entered in the recovered or removed(R) category [2]. The
2
SIR model is represented by some ordinary differential equations:
dS
SI
= −β
dt
N
dI
SI
= β − γI
dt
N
dR
= γI
dt
(2.1)
(2.2)
(2.3)
where β is the transmission rate and γ is the recovery rate. The N is the
total population which can be represented as:
S (t) + I(t) + R(t) = N
(2.4)
SEIR model is a modified version of the SIR model that proposes
a new category, Exposed(E), which indicates the individuals who are
infected by pathogen but not yet infected [3].
Figure 2.2: SEIR Model
The differential equations for SEIR model can be written as:
dS
SI
= −β
dt
N
dE
SI
= β − ϵE
dt
N
dI
= ϵE − γI
dt
dR
= γI
dt
3
(2.5)
(2.6)
(2.7)
(2.8)
where β, ϵ and γ indicates the infection, progression and recovery rates.
2.2
Extended SIR model
2.2.1
SIR Model with Vital Dynamics
The SIR model assumes that sex, age, death, birth and some social
behaviours have no effect in the case of infectious disease. SIRD model
extends the SIR categories with the division of Deceased or Death (D)
[4]. It creates a four set of ordinary differential equations and the total
population can be represented here as N = S + I + R + D.
Infected quarantined individuals do not contribute to the spread of
disease. In order to afford quarantine cases in SIR model, the infected
people is divided into two groups: Quarantined and Unquarantined [5].
2.2.2
SIR Model with Machine Learning
For a dynamic and non-homogeneous population, it is better to use
machine learning(ML) methods together with SIR model to handle complex spatial interactions and to provide accurate predictions. SIMLR
model uses SIR model with time-varying parameters that utilizes machine learning methods [6].
2.3
Extended SEIR model
2.3.1
SEIR Model with Vaccination and Quarantine
There are several categories are included in the SEIR model for better
predictions of epidemic and pandemic outbreak. SEIRV model includes
4
the category of Vaccinated(V) people [7].
Vaccination is the only
measure to stop the death rate and acquire immunity against infectious
diseases.
SEIQRDP model divides the population into Suscepted,
Exposed, Infected, Quarantined, Recovered, Death and Protected (or
insusceptible) people [8]. metapopulation or mSEIR model is used in
the cases of interactions of the population between different regions [9].
2.3.2
SEIR Model with Neural Network and Deep Learning
Neural-SEIR model approximates the core parameters through neural
networks [10]. It uses long short-term memory (LSTM) neural network
to predict complex epidemics like coronavirus. It also preserves the
traditional SEIR structure.
The metapopulation SEIR model is combined with hybrid deep
learning(DL) [11]. This model helps to minimize the gap between real
and estimated status of infectious diseases.
2.4
Comparison of Models
The articles are searched using Google Scholar, those are published
between 2020 and 2024. The review includes ten articles that describe
different epidemic models. The key parameter of epidemic models is the
reproduction number R0 and can be defined it in case of SIR model as:
R0 =
β
γ
(2.9)
When R0 > 1, virus is currently spreading and when R0 < 1, virus begins
to stop spreading.
5
6
mSEIR
NeuralSEIR
SEIR+DL
8.
9.
10.
SIMLR
5.
SEIQRDP
SIR+Q
4.
7.
SIRD
3.
SEIRV
SEIR
2.
6.
2020
SIR
1.
2020
2023
2020
2024
2022
2022
2020
2020
2020
Year of Study
Sl.No Model
Collected data of different months (2020) in two cities, key parameters R0 and RMS E.
Collected data from Dec 2019 to Feb 2020, RMS E, MAE and S MAPE, NVIDIA
GeForce RTX 3090 hardware and encoded with Python Tensorflow
Collected data from 10 Jan 2020 to 16 Feb 2020, key parameter is MAPE, window
sliding algorithm (optimization)
Global prediction of Mpox cases in 31 Dec 2022, key parameter is RE(relative error),
least square method and simulation.
Analyze collected data for one year, key parameters are R0 and V(vaccination rate)
Predict new infections one to four weeks in advance, key parameters are CT (change in
trend) and MAPE(Mean absolute percentage error)
Collected data from 26 April 2020 to 12 May 2020, Key parameters are R0 and f t(factor
of testing), Python simulation
Results based on data of 116 days, Key parameters are R0 and R2 (regression coefficient),
MATLAB simulation
Collected data from 15 Jan 2020 to 29 Feb 2020, R0 is the key parameter, Optimization
Toolbox of MATLAB software.
Collected data from 2 March 2020 to 15 May 2020, total population(N) is constant, no
vital dynamics, R0 is the key parameter,MATLAB simulation.
Description
Table 2.1: Brief characteristics of each model
COVID cases in south Korea
COVID cases in China
COVID cases in China
Mpox data of six countries
collected from WHO
COVID data of vaccinated
countries from WHO
COVID data form Canada and
United States
COVID cases in the state of
Kerala and India
COVID cases in Kuwait
COVID cases in Japan
COVID cases in Saudi Arabia
Data Collection
3.
Discussion
The commonly used mathematical models for pandemic or epidemic
prediction are the SIR [2] and SEIR [3] models. These predict based on
certain assumptions. Reproductive number R0 is the key parameter for
both models that define as the average number of susceptible individuals
infected by a single infected person. It indicates the transmissivity and
impact of the infectious disease. The remaining studied models are the
extension of SIR and SEIR models.
The SIR model assumes that vital dynamics such as birth, death, age,
sex and social behaviour has no effect on the spreading of disease. Also,
the population is taken as constant, so the recovered people are included
only in immune category, not in the death category. That is, the death
rate is taken as constant. But, the death is a major concept in case of
infectious disease. The SIRD model [3] helps to avoid this limitation of
SIR model and is the best model to detects the severity of the disease.
Testing and quarantine are necessary factors for controlling the
spread of infectious disease by mutual contact which can proposed by
SIR+Q model [4]. Fraction of testing( f t) is the fraction of infected
people who are tested and quarantine. If f t increases, more people get
quarantined and controls the spreading of disease. SIR model uses the
static parameters only. To performing time-varying parameter, combine
SIR model with machine learning called as SIMLR model [5].
7
It
predicts the number of new infections for one to four week in advance.
The SEIR model have also some limitations which can overcome
by its extensions. Vaccination helps to increase the immunity of each
individuals internally and to defend the infectious disease. SEIRV model
[6] includes different compartments such as vaccination rate, death rate
and birth rate. Social distancing and vaccination rate have major role in
reducing R0 and to overcome infectious disease. SEIQRDP model [7]
include more compartments to enhance the accuracy of results than
SEIR model.
SEIR model focus on fixed transmission rate. mSEIR model [8]
used for metapopulation that considers the mobility and interactions of
population in different regions. It helps for heterogeneous transmission,
that is, to predict the spread of disease between different areas. mSEIR
model need more accuracy, so, it is combinated with hybrid deep
learning models [10]. To prove the efficiency and effectiveness, it
compared with other forecasting models such as LSTM and DNN
networks.
By combining SEIR with neural network structure [9],
achieve high accuracy and efficiency on the prediction of disease using
time-series data. Long short-term memory(LSTM) network provides
accurate predictions not only in case of disease, but only several
applications such as weather predictions, energy prediction and drug
prediction.
8
3.1
Advantages and Limitations
Sl.No Model
Advantages
Limitations
1.
SIR
Relatively simple and easy to
implement
Assumes no vital dynamics such as
birth, death, age and sex, population
size as constant
2.
SEIR
Useful in case of incubation
period
Limited scope for heterogeneous population, assumptions similar to SIR
model
3.
SIRD
Detect severity within limited days
Not very accurate and error percentage
remain high
4.
SIR+Q
Understand the usage of testing and quarantine
Limited to homogeneity, cannot apply
for complexity real-world epidemics
5.
SIMLR
Apply for heterogeneous
world, describe complex
dynamics
Require large and high-quality
datasets, complex model
6.
SEIRV
Analyze effectiveness of vaccination, consider reinfection
possibility
Homogeneity, Effect of vaccine considered constant for all age groups and in
all regions.
7.
SEIQRDP
Includes
more
compartments,
analyze
disease on global scale
Homogeneity, oversimplification in dynamics
8.
mSEIR
Consider mobility and interaction of urban population
Requires up-to-date dataset, assume
homogeneity within subpopulation
9.
NeuralSEIR
Elimiate the need of huge
and high-quality data, reduce
complexity
Long-term prediction may undergo less
accuracy
10.
SEIR+DL
Applicable
for
heteropopulation, Provide real-time
predictions
Not good in generalizability, hybrid
model is risky and complex
Table 3.1: Advantages and limitation of each model
9
4.
Conclusion
This review paper presents different mathematical models for the prediction and control of epidemic and pandemic situations. It includes
ten different compartmental models from ten published articles. Each
model has its own advantages and some limitations. All models are
best for the application of epidemic prediction based on availability of
data, geographical area, homogeneous or heterogeneous population, and
severity of the disease. Even though, for the new artificial intelligence
world, it is better to use the hybrid models using machine learning, deep
learning and neural network. The advancements in such hybrid models
can be considered as the future scope of epidemic predictions.
10
References
[1] Kermack William Ogilvy and McKendrick A. G. 1927 A
contribution to the mathematical theory of epidemics Proc. R. Soc.
Lond. A115700–721 http://doi.org/10.1098/rspa.1927.
0118
[2] Alboaneen, Dabiah, et al. Predicting the epidemiological outbreak of the coronavirus disease 2019 (COVID-19) in Saudi
Arabia. International journal of environmental research and
public health 17.12 (2020): 4568. https://doi.org/10.3390/
ijerph17124568
[3] Kuniya, Toshikazu. Prediction of the epidemic peak of coronavirus
disease in Japan, 2020 Journal of clinical medicine 9.3 (2020):
789. https://doi.org/10.3390/jcm9030789
[4] Sedaghat, Ahmad, et al. Predicting Trends of Coronavirus Disease
(COVID-19) Using SIRD and Gaussian-SIRD Models 2020 IEEE
3rd International Conference and Workshop in Óbuda on Electrical
and Power Engineering (CANDO-EPE). IEEE, 2020. doi:10.
1109/CANDO-EPE51100.2020.9337783.
[5] Anand, Nikhil, et al. Predicting the spread of COVID-19 using SIR
model augmented to incorporate quarantine and testing Transactions of the Indian National Academy of Engineering 5.2 (2020):
141-148. https://doi.org/10.1007/s41403-020-00151-5
11
[6] Vega, Roberto, Leonardo Flores, and Russell Greiner. SIMLR:
Machine Learning inside the SIR model for COVID-19 Forecasting
Forecasting 4.1 (2022):
72-94.https://doi.org/10.3390/
forecast4010005
[7] Poonia, Ramesh Chandra, et al. An enhanced SEIR model for
prediction of COVID-19 with vaccination effect Life 12.5 (2022):
647. https://doi.org/10.3390/life12050647
[8] Zhang, Li, et al. Global prediction for mpox epidemic Environmental Research 243 (2024): 117748. https://doi.org/10.1016/
j.envres.2023.117748
[9] Chen, Duxin, et al. Prediction of COVID-19 spread by sliding
mSEIR observer Science China Information Sciences 63 (2020):
1-13. https://doi.org/10.1007/s11432-020-3034-y
[10] Wang, Haoyu, et al. Neural-SEIR: A flexible data-driven
framework for precise prediction of epidemic disease Mathematical
Biosciences and Engineering 20.9 (2023): 16807-16823. http:
//www.aimspress.com/journal/mbe
[11] Rahmadani, Firda, and Hyunsoo Lee. Hybrid deep learningbased epidemic prediction framework of COVID-19: South Korea
case Applied Sciences 10.23 (2020):
app10238539
12
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