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Abstract
In this work, we investigate the Global Stability of a Mathematical model that describes the impact of vaccination on the dynamics of COVID-19 disease transmission in a human population. The model, represented by a system of ordinary differential equations explains how infection from an index case, which could potentially lead to endemic state, can be averted through effective vaccination. The global stability analysis shows that, the diseases free state is globally asymptotically stable, when the basic reproduction number, in the absence of disease associated death. This is supported by numerical simulation which suggests the combination of vaccination and non-pharmaceutical measures in the disease control. We also show numerically that the disease invades when and that there is a transcritical bifurcation at .
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References
B. Y. Ahmad, and H. Zeaid, (2020). COVID-19: Modeling, Prediction, and Control. Applied Sciences. 10(11), Appl. Sci. 10(11),3666; https://doi.org/10.3390/app10113666
M. Y. Hyun, and P. L. Luis, (2021). Mathematical modeling of the transmission of SARS-CoV-Evaluating the impact of isolation in São Paulo State (Brazil) and lockdown in Spain associated with protective measures on the epidemic of Covid-19.
O. Amanso, N. Agueboh, P. U. Achimugwu, C. Okeke, B. E. Chukwuemeka, K. C. Nnamaga, E. Oshilim, (2020) Mathematical model of the early incidence and spread of COVID-19 in Nigeria combined with control measure. International Journal of Scientific and Engineering Research. 11(4):1110.
Y. Chayu, and W.Jin, (2020). A mathematical model for the novel coronavirus epidemic in Wuhan, China. Math Biosci Eng. 17(3):2708–2724. DOI: 10.3934/mbe.2020148
M. A. Khan, and A. Atangana, (2020). Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative.Alexandria Eng. J; 59(4): Available:https://doi.org/10.1016/j.aej.2020.02.033
A. B. Okrinya, and E. Esekhaigbe, (2021). Mathematical modelling of the dynamics of COVID-19 disease transmission. Asian Research Journal of Mathematics.;17(1):123-137.
F. Ndarou, I. Area, J. Nieto, (2020). Torres D. Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan. Chaos, Solitons, and Fractals. ;135(1046). Available:https://doi.org/10.1016/j.chaos.2020.109846.
Center for Disease Control and Prevention; 2019. Available:cdc.gov/coronavirus/2019-ncov/symptoms
A. B. Okrinya and C. N. Timinibife. (2021). The Impact of Vaccination on Covid-19 Disease Transmission Patterns in a Human Population: A Theoretical Analysis. Asian Research Journal of Mathematics.;17(1):123-137.
G. Nezihal, and K. Bilgen. Mathematical modelling of Covid-19 with the effect of vaccine AIP Conference Proceedings. 2021;2325:020065. Available: https://doi.org/10.1063/5.0040301
D. Wang, B. Hu, and C. Hu, (2020). Clinical characteristics of 138 hospitalized patients with 2019 Novel Corona virus infected Pneumonia in Wuhan, China, JAMA.;323(11). Available: https://jamanetwork.com
A. B. Okrinya, and J. I. Consul, (2019). Logistic mathematical model of Ebola virus disease with convalescence. International Journal of Applied Scientific and Research.;2(6):1-14.
F. Fatemeh, V. Karimi, C. O. Puig, (2021). Economic model predictive control of nonlinear systems using a linear parameter varying approach. International Journal of Robust and Nonlinear Control;.
C. Aoun, N. Boutabba, H. Eleuch, (2021). Alpha Model: A Mathematical Modeling Approach Applied to an Air Quality Monitoring Network Applied Mathematics & Information Sciences. 27 (1). DOI: 10.12785/amis/010104
A. Dokoumetzidis, A. Iliadis, P. Macheras, (2001). Nonlinear Dynamics and Chaos Theory: Concepts and Applications Relevant to Pharmacodynamics. 18:415-426.
D. S. Glass, X. Jin, I. H. Riedel-Kruse, (2021). Nonlinear delay differential equations and their application to modeling biological network motifs. 12(1788).
M. Rahman, and F. Gómez-Aguilarc, (2020). Investigating a nonlinear dynamical model of COVID-19 disease under fuzzy caputo, random and ABC fractional order derivative. 140:110232. doi: 10.1016/j.chaos.2020.110232
B. M. Ndiaye, and L. S. Tendeng, (2020). Analysis of the COVID-19 pandemic by SIR model and machine learning techniques for forecasting.
T. Chen, J. Rui, Q. Wang, Z. Zhao, J. Cui, L. Yin, (2020). A mathematical model for simulating the phase-based transmissibility of a novel coronavirus. Infectious Diseases of Poverty. 9(24). Available: https//doi.0rg/10.1186/s40249-020-00640-3.
W. Wael, A. E. Mohammed, E. S. Aly, C. D. Matouk, S. A. Albosaily, E. M. Elabbasy, (2021). An analytical study of the dynamic behavior of Lotka-Volterra based models of COVID-19. 2021; 26(1).
A. Elazzouzi, A. Lamrani Alaoui, M. Tilioua, A. Tridane, (2019). Global stability analysis for a generalized delayed SIR model with vaccination and treatment. Adv Differ Equ. 2019;2019(1):532. Doi:10.1186/s13662-019-2447-z. Epub 2019 Dec 21. PMID: 32226453; pmcid: PMC7100696.
Y. Enatsu, (2012). Lyapunov functional techniques on the global stability of equilibria of SIS epidemic models with delays. Kyoto Univ. Res. Inf. Repos. 1792, 118–130.
Z. Lu, X. Chi, L. Chen, (2002). The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission. Math. Comput. Model. 36, 1039–1057.
Makinde, O. D.(2007). Adomian decomposition approach to a SIR epidemic model with constant vaccination strategy. Appl. Math. Comput. 184(2), 842–848.
B. Shulgin, L. Stone, Z. Agur, (1998). Pulse vaccination strategy in the SIR epidemic model. Bull. Math. Biol. 60(6), 1123–1148.
Y. Ma, J. B. Liu, H. Li, (2018). Global dynamics of an SIQR model with vaccination and elimination hybrid strategies. Mathematics 6(12), 1–12.
N. Chitnis, (2005). Using mathematical models in controlling the spread of malaria. PhD thesis, University of Arizona, Tucson, Arizona, USA;.
N. Chitnis, J. M. Cushing, J. M. Hyman, (2006). Bifurcation analysis of a mathematical model for malaria transmission. SIAM J. Appl. Math. 67:24-45.
A. B. Okrinya, and J. L. Consul, (2019). Logistic mathematical model of Ebola virus disease with convalescence. International Journal of Applied Scientific and Research. 2(6):1-14.
J. S. Allen, (2007). An introduction to mathematical biology. Pearson Education Inc., Prentice Hall. Upper Saddle River, NJ07458.
M. O. Onuorah, and N. I. Akinwande, (2016). Sensitivity analysis of Lassa fever model. European Centre for Research Training and Development UK (www.eajournals.org).14(1)
and its Role on Global Stability. Biometrics, Statistics and Theoretical and Applied Mechanics Departments, Cornell University, Ithaca, NY 14853, USA.
S. Busenberg, P. Van den Driessche. Analysis of disease transmission model in a population with varying size, J. Maths.Biol. 28 (1990) 257-270.
M. Y. Li, H. I. Smith, L. Wang, Global stability of an SEIR epidemic model with vertical transmission, SIAMm. Appl. Math. 62 (1) (2001) 69-80.
S. M. Moghadas, A. B. Gumel, Global stability of a two-stage epidemic model with generalized non-linear incidence. Elsevier, Mathematics and Computers in Simulation 60 (2002) 107-118.
A. B. Okrinya, Mathematical modelling of Malaria Transmission and Pathogenesis, PhD Thesis, Loughborough University, 2015.