A Study of a Class Continuous SIR Epidemic Model with History
DOI: https://doi.org/10.33003/jobasr-2023-v1i1-19
Olopade I. A.
Mohammed I. T.
Philemon M. E.
Akinwumi T. O.
Sangoniyi S. O.
Adeniran G. A.
Ajao S. O.
Adewale S. O.
Abstract
The SIR model, an epidemiological model, divides a population into three
compartments: Susceptible (S), Infected (I), and Recovered (R). It is widely used
to understand the spread of infectious diseases and predict epidemic outcomes
based on factors such as transmission rates and population dynamics. A
deterministic epidemic mathematical model to describe the transmission
dynamics of an infectious disease was constructed and analyzed by incorporating
memory term which provides information on the current and past disease states.
The model revealed two key equilibria: a disease-free equilibrium and an endemic
equilibrium. The calculated basic reproductive numberđť‘…0
, was employed to establish that when đť‘…0 < 1, the disease-free equilibrium is locally asymptotically
stable, while the endemic equilibrium is locally asymptotically stable when đť‘…0 >
1. Additionally, we explored the global stability of these equilibria using
Lyapunov functions and Dulac's method, respectively. To validate our analytical
findings, we conducted numerical simulations of the model which show the
importance of history in the dynamic spread and elimination of disease.
References
Adeniran G. A., Olopade I. A., Ajao S. O., Akinrinmade
V. A., Aderele O. R., & Adewale S. O. (2022). Sensitivity
and Mathematical Analysis of Malaria and Cholera CoInfection. Asian Journal of Pure and Applied
Mathematics, 4(1), 425–452.
Adesanya A. O., Olopade I. A., Akinwumi T. O. &
Adesanya A. A. (2016). Mathematical Analysis of Early
Treatment of Gonorrhea Infection. American
International Journal of Research in Science,
Technology, Engineering & Mathematics, 15(2).
Adesola O. I., Oloruntoyin S. S., Emmanuel P. M.,
Temilade M. I., Adeyemi A. G., Oladele A. S., Mamman
A. U., & Kareem A. A. (2024). Mathematical Modelling
and Analyzing the Dynamics of Condom Efficacy and
Compliance in the Spread of HIV/AIDS. Asian Research
Journal of Current Science. 6(1): 54–65.
Adesola O. I., Temilade M. I., Emmanuel P. M., Oladele
A. S., Adeyemi A. G., Sunday S, & Olumuyiwa A. S.
(2024). Mathematical Analysis of Optimal Control of
Human Immunodeficiency Virus (HIV) Co-infection
with Tuberculosis (TB). Asian Research Journal of
Current Science. 6(1), 23–53.
Adewale S. O., Olopade I. A., Ajao S. O., & Adeniran G.
A. (2015a). Mathematical Analysis of Diarrhea in the
Presence of Vaccine. International Journal of Scientific
and Engineering Research. 6: 396-404.
Adewale S. O., Olopade I. A. & Adeniran G. A. (2015b).
Mathematical Analysis of Effects of Isolation on Ebola
Transmission Dynamics. Researchjouranli’s Journal of
Mathematics. 2: 1-21.
Ajao S., Olopade I., Akinwumi T., & Adesanya O.
(2023). Understanding the Transmission Dynamics and
Control of HIV Infection: A Mathematical Model
Approach. Journal of the Nigerian Society of Physical
Sciences. 5(2): 1389.
DOI:https://doi.org/10.46481/jnsps.2023.1389.
Akinwumi T. O., Olopade I. A., Adesanya A. O. & Alabi
M. O. (2021). A Mathematical Model for the Transmission of HIV/AIDS with Early
Treatment. Journal of Advances in Mathematics and
Computer Science; 36(5): 35-51.
Almuqrin M. A., Goswami P. & Sharma S. (2021).
Fractional model of Ebola virus in population of bats in
frame of Atangana- Baleanu fractional
derivative,” Results in Physics. 26: 104295.
Alzaid S. S., Alkahtani B. S. T. & Sharma S. (2021).
Numerical solution of fractional model of HIV-1
infection in framework of different fractional
derivatives,” Journal of Function Spaces. 10: 20-21.
Beretta E. & Takeuchi Y. (2010). The class of discrete
SIR epidemic models with distributed delay. Nonlinear
Analysis. 28:1909–1921.
Brauer F. & Castillo-Chávez C. (2001). Mathematical
Models in Population Biology and Epidemiology,
Springer-Verlag, New York.
Capasso V. (2008). Mathematical Structures of Epidemic
Systems, Springer Verlag, Berlin, 2nd edition.
Cooke K. L. (1979). Stability analysis for a vector disease
model, Rocky Mountain J. Math. 9: 31–42.
Diekmann O., Heesterbeek H. & Britton T. (2013).
Mathematical Tools for Understanding Infectious
Disease Dynamics. Princeton University Press.
Princeton.
Foy B. H., Wahl B., & Mehta K. (2021). Comparing
COVID-19 vaccine allocation strategies in India: a
mathematical modelling study. International Journal of
Infectious Diseases. 103: 431–438.
Ibrahim M. O., Akinyemi S. T. & Dago M. M. (2015).
Mathematical Modelling of a Staged Progression
HIV/AIDS Model with Control Measures. Journal of
Nigerian Association of Mathematical Physics. 29: 163-
166.
Kermack W. O. & McKendrick A. G. (1927). A
contribution to the mathematical theory of
epidemics,” Proceedings of the Royal Society of London.
115:700–721.
LaSalle J. P. (1987). The stability of dynamical systems.
Society for Industrial and Applied Mathematics. 25.
Olopade I. A., Adesanya A. O. & Mohammed I. T.
(2017). Mathematical Analysis of the Global Dynamics
of an SVEIR Epidemic Model with Herd Immunity.
International Journal of Science and Engineering
Investigations. (IJSEI). 6(69) :141-148.
Olopade I. A., Adesanya A. O. & Akinwumi T. O.
(2021a). Mathematical Transmission of SEIR Epidemic
Model with Natural Immunity. Asian Journal of Pure and
Applied Mathematics. 3(1) :19-29.
Olopade I. A., Adewale S. O., Mohammed I. T.,
Adeniran G. A., Ajao S. O. & Ogunsola A. W. (2021b).
Effect of Effective Contact Tracing in Curtaining the
Spread of Covid-19. Asian Journal of Research in
Biosciences. 3(2): 118-134.
Olopade I. A., Ajao S. O., Adeniran G. A., Adamu A. K.,
Adewale S. O., & Aderele O. R. (2022). Mathematical
Transmission of Tuberculosis (TB) with Detection of
Infected Undetectected. Asian Journal of Research in
Medicine and Medical Sciences. 4(1): 100-119.
Olopade I. A., Akinola E. I., Philemon M. E., Mohammed
I. T., Ajao S. O., Sangoniyi S. O., Adeniran G. A.
(2024c). Modeling the Mathematical Transmission of a
Pneumonia Epidemic Model with Awareness. J. Appl.
Sci. Environ. Manage. 28(2): 403-413.
Philemon M. E., Olopade I. A., & Ogbaji E. O. (2023).
Mathematical Analysis of the Effect of Quarantine on the
Dynamical Transmission of Monkey-Pox. Asian Journal
of Pure and Applied Mathematics. 5(1): 473–492.
Ramos A. M., Ferrández M. R., & Vela-Pérez M. (2021).
A simple but complex enough θ-SIR type model to be
used with COVID-19 real data. Application to the case of
Italy. Physica D. 421, article 132839.
Rao F., Mandal P. S., & Kang Y. (2019). Complicated
endemics of an SIRS model with a generalized incidence
under preventive vaccination and treatment
controls. Applied Mathematical Modelling. 67: 38–61.
Safiel R., Massawe E. S., & Makinde O. D. (2012).
Modelling the Effect of Screening and Treatment on
Transmission of HIV/AIDS Infection in a Population.
American Journal of Mathematics and Statistics. 2(4):
75-88. DOI: 10.5923/j.ajms.20120204.03.
Sajid M., Abbas Z., Ali N., & Javed T. A. (2013). Note
on Solutions of the SIR Models of Epidemics Using Ham.
ISRN Applied Mathematics; Article ID 457072, 4 pages.
Srivastava H. M., Shanker D. R., & Jain M. (2019). A
study of the fractional-order mathematical model of
diabetes and its resulting complications. Mathematical
Methods in Applied Sciences 42
PDF