Asymptotic behavior and semi-analytic solution of a novel compartmental biological model

Muhammad Sinan; Jinsong Leng; Misbah Anjum; Mudassar Fiaz

doi:10.53391/mmnsa.2022.008

EN

Asymptotic behavior and semi-analytic solution of a novel compartmental biological model

Abstract

This study proposes a novel mathematical model of COVID-19 and its qualitative properties. Asymptotic behavior of the proposed model with local and global stability analysis is investigated by considering the Lyapunov function. The mentioned model is globally stable around the disease-endemic equilibrium point conditionally. For a better understanding of the disease propagation with vaccination in the population, we split the population into five compartments: susceptible, exposed, infected, vaccinated, and recovered based on the fundamental Kermack-McKendrick model. He's homotopy perturbation technique is used for the semi-analytical solution of the suggested model. For the sake of justification, we present the numerical simulation with graphical results.

Keywords

Local asymptotic stability, global asymptotic stability, Routh-Hurwitz criterion, COVID-19, infectious disease modeling

References

https://www.worldometers.info/coronavirus/, Coronavirus cases. Access Date: 20.06.2022
Sinan, M., Ali, A., Shah, K., Assiri, T.A., & Nofal, T.A. Stability analysis and optimal control of COVID-19 pandemic SEIQR fractional mathematical model with harmonic mean type incidence rate and treatment. Results in Physics, 22, 103873, (2021).
Ali, A., Khan, M.Y., Sinan, M., Allehiany, F.M., Mahmoud, E.E., Abdel-Aty, A.H., & Ali, G. Theoretical and numerical analysis of novel COVID-19 via fractional order mathematical model. Results in physics, 20, 103676, (2021).
Asamoah, J.K.K., Owusu, M.A., Jin, Z., Oduro, F.T., Abidemi, A., & Gyasi, E.O. Global stability and cost-effectiveness analysis of COVID-19 considering the impact of the environment: using data from Ghana. Chaos, Solitons & Fractals, 140, 110103, (2020).
Pe´ni, T., Csutak, B., Szederke´nyi, G., & Ro¨st, G. Nonlinear model predictive control with logic constraints for COVID-19 management. Nonlinear Dynamics, 102(4), 1965-1986, (2020).
Matouk, A.E. Complex dynamics in susceptible-infected models for COVID-19 with multi-drug resistance. Chaos, Solitons & Fractals, 140, 110257, (2020).
Sun, D., Duan, L., Xiong, J., & Wang, D. Modeling and forecasting the spread tendency of the COVID-19 in China. Advances in Difference Equations, 2020(1), 1-16, (2020).
Ahmed, I., Baba, I.A., Yusuf, A., Kumam, P., & Kumam, W. Analysis of Caputo fractional-order model for COVID-19 with lockdown. Advances in difference equations, 2020(1), 1-14, (2020).
Alqahtani, R.T. Mathematical model of SIR epidemic system (COVID-19) with fractional derivative: stability and numerical analysis. Advances in Difference Equations, 2021(1), 1-16, (2021).
Naik, P.A., Zu, J., & Owolabi, K.M. Global dynamics of a fractional order model for the transmission of HIV epidemic with optimal control. Chaos, Solitons & Fractals, 138, 109826, (2020).

Zhu, L., Zhou, X., Li, Y., & Zhu, Y. Stability and bifurcation analysis on a delayed epidemic model with information-dependent vaccination. Physica scripta, 94(12), 125202, (2019).
Allegretti, S., Bulai, I.M., Marino, R., Menandro, M.A., & Parisi, K. Vaccination effect conjoint to fraction of avoided contacts for a Sars-Cov-2 mathematical model. Mathematical Modelling and Numerical Simulation with Applications, 1(2), 56-66, (2021).
Shah, K., Abdeljawad, T., Mahariq, I., & Jarad, F. Qualitative analysis of a mathematical model in the time of COVID-19. BioMed Research International, 2020, (2020).
Pais, R.J., & Taveira, N. Predicting the evolution and control of the COVID-19 pandemic in Portugal. F1000Research, 9, (2020).
Martcheva, M. An introduction to mathematical epidemiology (Vol. 61, pp. 9-31). New York: Springer, (2015).
Samji, H., Wu, J., Ladak, A., Vossen, C., Stewart, E., Dove, N., ... & Snell, G. Mental health impacts of the COVID-19 pandemic on children and youth–a systematic review. Child and Adolescent Mental Health, 27(2), 173-189, (2022).
Li, X.P., Gul, N., Khan, M.A., Bilal, R., Ali, A., Alshahrani, M.Y., ... & Islam, S. A new Hepatitis B model in light of asymptomatic carriers and vaccination study through Atangana–Baleanu derivative. Results in Physics, 29, 104603, (2021).
Din, A., & Li, Y. Stationary distribution extinction and optimal control for the stochastic hepatitis B epidemic model with partial immunity. Physica Scripta, 96(7), 074005, (2021).
Din, A., & Li, Y. Lévy noise impact on a stochastic hepatitis B epidemic model under real statistical data and its fractal–fractional Atangana–Baleanu order model. Physica Scripta, 96(12), 124008, (2021).
Din, A., Li, Y., Khan, F.M., Khan, Z.U., & Liu, P. On Analysis of fractional order mathematical model of Hepatitis B using Atangana–Baleanu Caputo (ABC) derivative. Fractals, 30(01), 2240017, (2022).
Din, A., Li, Y., Yusuf, A., & Ali, A.I. Caputo type fractional operator applied to Hepatitis B system. Fractals, 30(01), 2240023, (2022).
Din, A., & Abidin, M.Z. Analysis of fractional-order vaccinated Hepatitis-B epidemic model with Mittag-Leffler kernels. Mathematical Modelling and Numerical Simulation with Applications, 2(2), 59-72, (2022).
Ali, A., Alshammari, F.S., Islam, S., Khan, M.A., & Ullah, S. Modeling and analysis of the dynamics of novel coronavirus (COVID-19) with Caputo fractional derivative. Results in Physics, 20, 103669, (2021).
Daşbaşı, B. Stability analysis of an incommensurate fractional-order SIR model. Mathematical Modelling and Numerical Simulation with Applications, 1(1), 44-55, (2021).
Ali, A., Islam, S., Khan, M.R., Rasheed, S., Allehiany, F.M., Baili, J., ... & Ahmad, H. Dynamics of a fractional order Zika virus model with mutant. Alexandria Engineering Journal, 61(6), 4821-4836, (2022).
Ali, A., Iqbal, Q., Asamoah, J.K.K., & Islam, S. Mathematical modeling for the transmission potential of Zika virus with optimal control strategies. The European Physical Journal Plus, 137(1), 1-30, (2022).
Zhang, X.H., Ali, A., Khan, M.A., Alshahrani, M.Y., Muhammad, T., & Islam, S. Mathematical analysis of the TB model with treatment via Caputo-type fractional derivative. Discrete Dynamics in Nature and Society, 2021, (2021).
Faniran, T., Ali, A., Adewole, M.O., Adebo, B., & Akanni, O.O. Asymptotic behavior of tuberculosis between smokers and non-smokers. Partial Differential Equations in Applied Mathematics, 5, 100244, (2022).
Din, A., & Li, Y. The extinction and persistence of a stochastic model of drinking alcohol. Results in Physics, 28, 104649, (2021).
Naik, P.A., Eskandari, Z., Yavuz, M., & Zu, J. Complex dynamics of a discrete-time Bazykin–Berezovskaya prey-predator model with a strong Allee effect. Journal of Computational and Applied Mathematics, 413, 114401, (2022).
Abdy, M., Side, S., Annas, S., Nur, W., & Sanusi, W. An SIR epidemic model for COVID-19 spread with fuzzy parameter: the case of Indonesia. Advances in difference equations, 2021(1), 1-17, (2021).
Yavuz, M., Coşar, F.Ö., Günay, F., & Özdemir, F.N. A new mathematical modeling of the COVID-19 pandemic including the vaccination campaign. Open Journal of Modelling and Simulation, 9(3), 299-321, (2021).
Özköse, F., Yavuz, M., Şenel, M.T., & Habbireeh, R. Fractional order modelling of omicron SARS-CoV-2 variant containing heart attack effect using real data from the United Kingdom. Chaos, Solitons & Fractals, 157, 111954, (2022).
Shen, Z.H., Chu, Y.M., Khan, M.A., Muhammad, S., Al-Hartomy, O.A., & Higazy, M. Mathematical modeling and optimal control of the COVID-19 dynamics. Results in Physics, 31, 105028, (2021).
Ahmad, S., Ullah, A., Al-Mdallal, Q.M., Khan, H., Shah, K., & Khan, A. Fractional order mathematical modeling of COVID-19 transmission. Chaos, Solitons & Fractals, 139, 110256, (2020).
Abdo, M.S., Shah, K., Wahash, H.A., & Panchal, S.K. On a comprehensive model of the novel coronavirus (COVID-19) under MittagLeffler derivative. Chaos, Solitons & Fractals, 135, 109867, (2020).
Robinson, E., Sutin, A.R., Daly, M., & Jones, A. A systematic review and meta-analysis of longitudinal cohort studies comparing mental health before versus during the COVID-19 pandemic in 2020. Journal of affective disorders, 296, 567-576, (2022).
Oud, M.A.A., Ali, A., Alrabaiah, H., Ullah, S., Khan, M.A., & Islam, S. A fractional order mathematical model for COVID-19 dynamics with quarantine, isolation, and environmental viral load. Advances in Difference Equations, 2021(1), 1-19, (2021).
Mathieu, E., et al. A global database of COVID-19 vaccinations. Nature Human Behaviour, 5(7), 947-953, (2021).
https://www.who.int/ Access Date: 20.06.2022
He, J.H. Homotopy perturbation technique. Computer methods in applied mechanics and engineering, 178(3-4), 257-262, (1999).
He, J.H. Comparison of homotopy perturbation method and homotopy analysis method. Applied Mathematics and Computation, 156(2), 527-539, (2004).
Sinan, M., Shah, K., Khan, Z.A., Al-Mdallal, Q., & Rihan, F. On Semianalytical Study of Fractional-Order Kawahara Partial Differential Equation with the Homotopy Perturbation Method. Journal of Mathematics, 2021, (2021).
Mohyud-Din, S.T., & Noor, M.A. Homotopy perturbation method for solving partial differential equations. Zeitschrift für Naturforschung A, 64(3-4), 157-170, (2009).
Sinan, M. Analytic approximate solution of rabies transmission dynamics using homotopy perturbation method. Matrix Science Mathematics (MSMK), 4(1), 01-05, (2020).
Zedan, H.A., & El Adrous, E. The application of the homotopy perturbation method and the homotopy analysis method to the generalized Zakharov equations. In Abstract and Applied Analysis, 2012, (2012).
Demir, A., Erman, S., O¨zgu¨r, B., & Korkmaz, E. Analysis of the new homotopy perturbation method for linear and nonlinear problems. Boundary Value Problems, 2013(1), 1-11, (2013).
Zhang, Z., ur Rahman, G., Gómez-Aguilar, J.F., & Torres-Jiménez, J. Dynamical aspects of a delayed epidemic model with subdivision of susceptible population and control strategies. Chaos, Solitons & Fractals, 160, 112194, (2022).
Wang, Z., Nie, X., & Liao, M. Stability Analysis of a Fractional-Order SEIR-KS Computer Virus-Spreading Model with Two Delays. Journal of Mathematics, 2021, (2021).

Details

Primary Language

English

Subjects

Bioinformatics and Computational Biology, Applied Mathematics

Journal Section

Research Article

Authors

Muhammad Sinan ^* This is me
0000-0003-2177-3806
China

Jinsong Leng This is me
0000-0001-6916-1062
China

Misbah Anjum This is me
0000-0002-1415-8760
China

Mudassar Fiaz This is me
0000-0002-0176-2758
Pakistan

Publication Date

June 30, 2022

Submission Date

March 24, 2022

Acceptance Date

June 21, 2022

Published in Issue

Year 2022 Volume: 2 Number: 2

DOI

https://doi.org/10.53391/mmnsa.2022.008

IZ

https://izlik.org/JA96HL38NU

APA

Sinan, M., Leng, J., Anjum, M., & Fiaz, M. (2022). Asymptotic behavior and semi-analytic solution of a novel compartmental biological model. Mathematical Modelling and Numerical Simulation With Applications, 2(2), 88-107. https://doi.org/10.53391/mmnsa.2022.008

AMA

1.Sinan M, Leng J, Anjum M, Fiaz M. Asymptotic behavior and semi-analytic solution of a novel compartmental biological model. MMNSA. 2022;2(2):88-107. doi:10.53391/mmnsa.2022.008

Chicago

Sinan, Muhammad, Jinsong Leng, Misbah Anjum, and Mudassar Fiaz. 2022. “Asymptotic Behavior and Semi-Analytic Solution of a Novel Compartmental Biological Model”. Mathematical Modelling and Numerical Simulation With Applications 2 (2): 88-107. https://doi.org/10.53391/mmnsa.2022.008.

EndNote

Sinan M, Leng J, Anjum M, Fiaz M (June 1, 2022) Asymptotic behavior and semi-analytic solution of a novel compartmental biological model. Mathematical Modelling and Numerical Simulation with Applications 2 2 88–107.

IEEE

[1]M. Sinan, J. Leng, M. Anjum, and M. Fiaz, “Asymptotic behavior and semi-analytic solution of a novel compartmental biological model”, MMNSA, vol. 2, no. 2, pp. 88–107, June 2022, doi: 10.53391/mmnsa.2022.008.

ISNAD

Sinan, Muhammad - Leng, Jinsong - Anjum, Misbah - Fiaz, Mudassar. “Asymptotic Behavior and Semi-Analytic Solution of a Novel Compartmental Biological Model”. Mathematical Modelling and Numerical Simulation with Applications 2/2 (June 1, 2022): 88-107. https://doi.org/10.53391/mmnsa.2022.008.

JAMA

1.Sinan M, Leng J, Anjum M, Fiaz M. Asymptotic behavior and semi-analytic solution of a novel compartmental biological model. MMNSA. 2022;2:88–107.

MLA

Sinan, Muhammad, et al. “Asymptotic Behavior and Semi-Analytic Solution of a Novel Compartmental Biological Model”. Mathematical Modelling and Numerical Simulation With Applications, vol. 2, no. 2, June 2022, pp. 88-107, doi:10.53391/mmnsa.2022.008.

Vancouver

1.Muhammad Sinan, Jinsong Leng, Misbah Anjum, Mudassar Fiaz. Asymptotic behavior and semi-analytic solution of a novel compartmental biological model. MMNSA. 2022 Jun. 1;2(2):88-107. doi:10.53391/mmnsa.2022.008