COVID-19 Overview in Saudi Arabia Using the SIRV Model
Sadiqah Al Marzooq 1
1 Al
Yamamah University, Department of Mathematics and Natural
Sciences, College of Engineering and Architecture, Riyadh, Saudi Arabia
|
ABSTRACT |
||
In this paper,
we propose a modified SIR model with the consideration of vaccinated
individuals called SIRV. We provide a proof that the model’s solution is
non-negative and derive the model reproduction number and steady state.
Finally, we apply the model to analyze COVID -19 pandemic in Saudi Arabia
over the last three years. |
|||
Received 13 February 2023 Accepted 12 March 2023 Published 31 March 2023 Corresponding Author Sadiqah Al Marzooq, S_Almarzooq@yu.edu.sa DOI 10.29121/granthaalayah.v11.i3.2023.5079 Funding: This research
received no specific grant from any funding agency in the public, commercial,
or not-for-profit sectors. Copyright: © 2023 The
Author(s). This work is licensed under a Creative Commons
Attribution 4.0 International License. With the
license CC-BY, authors retain the copyright, allowing anyone to download,
reuse, re-print, modify, distribute, and/or copy their contribution. The work
must be properly attributed to its author. |
|||
Keywords: SIRV, COVID-19,
Model, Vaccine, Outbreak, Pandemic |
1. INTRODUCTION
The COVID-19 pandemic is considered the most signifcant pandemic in this century, Alboaneen et al. (2020). To date, according to WHO dashboard, there are all most 652, millions confirmed cases including about 6.5 million deaths with almost thirteen billion vaccinated individuals. The virus was recognized on December 8, 2019, in Wuhan, China and then it spread worldwide. It’s been considered as a pandemic on March 11, 2020, by the World Health Organization and called COVID-19. Saudi Arabia was one of the suffered by the pandemic Barry et al. (2020). The first COVID- 19 case in the kingdom was reported on March 2nd, 2020, and then the number has been increased rapidly to reach 4919 cases by the mid of June 2020. On December 17, 2020, the ministry of health in Saudi Arabia has offered the first dose of the vaccine without fees to all citizens and residents in the kingdom, and the second dose in February 2021 which led to decrease the cases by the end of 2021.
One of the most significant methods that's been used to analyze the COVID-19 outbreak is mathematical modeling. Researchers have developed the susceptible - infected SI model by considering other compartments such as exposed, recovered, quarantined, and vaccinated individuals to predict the outbreak long term behavior applied them to their studies based on real data in different countries.
In Alboaneen et al. (2020), they apply
the logistic growth model and the susceptible-infected-recovered
SIR on real- time data of COVID-19 in Saudi Arabia during the first
three months of the pandemic. The result of this paper predicts that the
outbreak's end point is the end of June 2020.
In Riyapan et al. (2021), a new
mathematical model called ""
formulated based on the seven compartments; the authors considered exposed, quarantined and death individuals beside (symptomatically and asymptomatically)
infected, and recovered ones to
study and analyze the long term behaviour of the
COVID-19 in Thailand by
calculating the equilibrium of the nonlinear system and the related
reproduction number, the result provides the threshold at which the pandemic
steady state is stable;
that is if the reproduction number is greater than 1 which indicates that the
outbreak won't die -out.
In Kozioł et al. (2020). a generalization of SIR model is presented based on the Grunwald- Letnikov derivative and discretization. The new model predicts the effects of fractional orders of the model derivatives on the dynamics of COVID-19. the simulations of the model have applied for two countries, namely Italy and Spain. The result of Italy indicates the effectiveness of this proposed model while it is limited in Spain.
In this paper, we introduce a model to study the dynamics of the COVID-19 outbreak with consideration of four compartments, set the model assumptions and prove that the proposed model has nonnegative solutions in Section 2. The model's steady state and the reproduction number along with the steady state stability analysis are presented in Section 3. Results and simulations are given for two different scenarios in Saudi Arabia in Section 4 and we discuss conclusions in Section 5.
2. THE CONTINUOUS SIRV MODEL
In
this section, we introduce a classic model denoted by analyze the dynamic of the COVID-19 virus
based on four sub-populations which are susceptible, infected, recovered, and
vaccinated individuals denoted by
and
, respectively.
The model is governed by the system of non- linear ODEs
and subject to non-
negative initial values
with
consideration of the following assumption:
·
All compartments are functions in time with
·
The four sub- populations are mixing around and they are equally
at risk of getting infected by the virus.
·
At time t, new births and
residents are denoted by .
·
The virus is transmitted from susceptible individuals to infected
ones with a constant rate α.
·
The recovery rate is also constant denoted by β.
·
Due to loss of
immunity, recovered individuals might have the virus again which means they
return to the susceptible statue with a constant rate μ.
·
The rate of natural death is represented by σ.
·
The parameter 0 ≤ γ ≤ 1 represents the vaccination
rate.
Figure 1 represents the
model of the four compartments.
Figure 1
Figure 1 SIRV Model Flowchart |
2.1. Theorem
Under non-negative initial
conditions, the system has non- negative and bounded solutions.
Proof:
Let's prove that system has non-negative solutions for all
.
Starting with the first equation, let since
π and μ are positive, we get
By integrating both sides
and applying the IC, we get , we have
Similarly, we can prove that other solutions
are non-negative for all t > 0.
Now, let's prove that the solutions are bounded.
By the assumption
we get
and hence
Therefore, the solution of
the linear equation is
Apply the initial condition
we get
and then
As we deduce
that
which proves that the solutions are bounded.
3. Stability Analysis of the Model's Steady State
3.1. the Model's Steady State
The
goal in section is to obtain the steady state of the proposed of the system
by solving the homogeneous system governed by setting
as a result, we get then non- trivial steady state given by
3.2. the Model's Steady State
The
dynamic of the infectious disease depends on , the
reproduction number, which is denoted by the number
of secondary cases can be caused by a single case. To calculate
, consider the
inequality
at
which leads to
and
hence,
Inequality
indicates that if
then
. Here, the
ratio
represents the
reproduction number which controls the pandemic if
, otherwise the
number of infected individuals will grow.
3.3. Stability of the Steady State
To obtain the
condition at which the steady state is stable, we calculate the linearized
Jacobian matrix of system about the fixed point
which given by the matrix
The
eigenvalues of are
Since
we assumed that the parameters and
are non-negative, it is clear that
and
Now, we Notice that
So, we can
conclude that the steady state is stable if
4. Simulation and Results
In this section
we provide approximated solutions of system by solving the associated
discrete system numerically
We
apply the model to data sets in Saudi Arabia on three periods of time; year 1, year 2 and year 3 which represent the years 2020, 2021 and 2022,
respectively, with
a total number of populations and a constant new birth
. The parameters
used in this study are as follow:
and
based on Ghostine et al. (2021). Figure 1 shows the result of the year 1 when
and
which provides a high
value of the reproduction
due to the large number of infected individuals with non-vaccinated
ones. In Figure 2, we study the
case of the year 2 when
and
, the result shows a significant decrease in
as the number of vaccinated
people exceeds the number of infected individuals. Figure 3 shows the
result of the year 3 when
and
and this
provides
which provides that the number of
infected people vanished and the virus die-out.
Figure 2
Figure 2 The Dynamic of COVID-19 in Year 1 |
Figure
3
Figure 3 The Dynamic of COVID-19 in Year 2 |
Figure 4
Figure 4 The Dynamic of COVID-19 in Year 3 |
5. Conclusion
In this paper, we present a classical model
called the in which we study four compartments to analyze
the dynamic of the COVID-19 disease. The model is an extended model of the
with consideration of vaccinated individuals
to study the impact of the vaccine on the virus's dynamic. We derived the
model reproduction number along with the non-zero steady state theoretically.
Simulations have been applied by solving the system numerically using three
sets of real-time data in Saudi Arabia which provide a general overview of the
pandemic in KSA.
CONFLICT OF INTERESTS
None.
ACKNOWLEDGMENTS
A sincere thanks to Al Yamamah university to support this research.
REFERENCES
Adiga, A., Dubhashi, D., Lewis, B., Marathe, M., Venkatramanan, S., & Vullikanti, A. (2020). Mathematical Models for Covid-19 Pandemic : A Comparative Analysis. Journal of the Indian Institute of Science, 100(4), 793-807. https://doi.org/10.1007/s41745-020-00200-6.
Alboaneen, D., Pranggono, B., Alshammari, D., Alqahtani, N., & Alyaffer, R. (2020). Predicting the Epidemiological Outbreak of the Coronavirus Disease 2019 (Covid-19) In Saudi Arabia. International Journal of Environmental Research and Public Health, 17(12). https://doi.org/10.3390%2Fijerph17124568.
Ameen, I. G., Ali, H. M., Alharthi, M. R., Abdel-Aty, A. H., & Elshehabey, H. M. (2021). Investigation of the Dynamics of Covid-19 With a Fractional Mathematical Model : A Comparative Study with Actual Data. Results in Physics, 23. https://doi.org/10.1016/j.rinp.2021.103976.
Anand, N., Sabarinath, A., Geetha, S., & Somanath, S. (2020). 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), 141–148. https://doi.org/10.1007/s41403-020-00151-5.
Barry, M., Ghonem, L., Alsharidi. A., Alanazi, A., Alotaibi, N.H., Al-Shahrani, F.S., Al Majid, F., BaHammam, A.S. (2020). Coronavirus Disease-2019 Pandemic in the Kingdom of Saudi Arabia : Mitigation Measures and Hospital Preparedness. Journal of Nature and Science of Medicine, 3(3), 155. https://www.jnsmonline.org/text.asp?2020/3/3/155/282983.
Ghostine, R., Gharamti, M., Hassrouny, S., & Hoteit, I. (2021). An Extended Seir Model with Vaccination for Forecasting the Covid-19 Pandemic in Saudi Arabia Using an Ensemble Kalman Filter. Mathematics, 9(6), 636. https://doi.org/10.3390/math9060636.
Jiang, Y. X., Xiong, X., Zhang, S., Wang, J. X., Li, J. C., & Du, L. (2021). Modeling and Prediction of the Transmission Dynamics of Covid-19 Based on the SINDy-LM Method. Nonlinear Dynamics, 105(3), 2775-2794. https://doi.org/10.1007%2Fs11071-021-06707-6.
Kozioł, K., Stanisławski, R., & Bialic, G. (2020). Fractional-Order Sir Epidemic Model for Transmission Prediction of Covid-19 Disease. Applied Sciences, 10(23), 8316. https://doi.org/10.3390/app10238316.
Lounis, M., & Bagal, D. K. (2020). Estimation of SIR Model’s Parameters of COVID-19 in Algeria. Bulletin of the National Research Centre, 44(1), 1-6. https://doi.org/10.1186/s42269-020-00434-5.
Mwalili, S., Kimathi, M., Ojiambo, V., Gathungu, D., & Mbogo, R. (2020). SEIR Model for Covid-19 Dynamics Incorporating the Environment and Social Distancing. BMC Research Notes, 13(1), 1-5. https://doi.org/10.1186%2Fs13104-020-05192-1.
Oliveira, J. F., Jorge, D. C., Veiga, R. V., Rodrigues, M. S., Torquato, M. F., da Silva, N. B., Fiaccone, R. L., Cardim, L. L., Pereira, F. A. C., Castro, C. P. D., Paiva, A. S. S., Amad, A. A. S., Lima, E. A. B. F., Souza, D. S., Pinho, S. T. R., Ramos, P. I. P., Andrade, R. F. S. (2021). Mathematical Modeling of COVID-19 in 14.8 Million Individuals in Bahia, Brazil. Nature communications, 12(1), 333. https://doi.org/10.1038/s41467-020-19798-3.
Riyapan, P., Shuaib, S. E., & Intarasit, A. (2021). A Mathematical Model of Covid-19 Pandemic : A Case Study of Bangkok, Thailand. Computational and Mathematical Methods in Medicine. https://doi.org/10.1155/2021/6664483.
Youssef, H. M., Alghamdi, N. A., Ezzat, M. A., El-Bary, A. A., & Shawky, A. M. (2020). A New Dynamical Modeling Seir with Global Analysis Applied to the Real Data of Spreading COVID-19 in Saudi Arabia. Math. Biosci. Eng, 17(6), 7018-7044. https://doi.org/10.3934/mbe.2020362.
This work is licensed under a: Creative Commons Attribution 4.0 International License
© Granthaalayah 2014-2023. All Rights Reserved.