MATHEMATICAL TRANSMISSION OF SEITR EPIDEMIC MODEL IN THE PRESENCE OF AWARENESS
DOI:
https://doi.org/10.55766/sujst-2024-06-e05186Keywords:
Equilibrium Points, Effective Basic Reproduction Number, Stability, Sensitivity, Numerical SimulationsAbstract
A model for the dynamic transmission of SEITR pandemics is proposed, incorporating awareness and a mass action incidence function. The stability of both endemic and disease-free equilibrium points was examined, revealing that the disease-free equilibrium is locally and globally asymptotically stable when the Basic Reproduction Number is less than 1, while the endemic equilibrium is stable when . Additionally, it's shown that the Basic Reproduction Number (without awareness) is greater than (with awareness), this means that when , then . A Lyapunov function is constructed to demonstrate the global stability of both disease-free equilibrium when and the endemic equilibrium when . Numerical simulations illustrate the impact of awareness on disease spread, highlighting the importance of awareness in informing and educating individuals about infection risks during outbreaks, thereby reducing the health burden of the epidemic by lowering its peak.
References
Adesanya, A.O., Olopade, I.A., Akinwumi, T.O., and Adesanya, A.A. (2016). Mathematical analysis of early treatment of gonorrhea infection. American International Journal of Research in Science, Technology, Engineering and Mathematics, 15(2), June-August. Available online at http://www.iasir.net
Adesola, I.O., Mohammed, I.T., Philemon, M.E., Akinwumi, T.O., Sangoniyi, S.O., Adeniran, G.A., Ajao, S.O., and Adewale, S.O. (2024). A study of a class continuous SIR epidemic model with history. Journal of Basics and Applied Sciences Research, 2(1):54-60.
Ajao, S.O., Olopade, I.A., Adewale, S.O., and Adesanya, O.A. (2023). Global stability analysis of malaria model with prophylactic treatment. Advances in Systems Science and Applications, 3(23):91-107.
Akinola, E.I., Akinpelu, F.O., Areo, A.O., and Oladejo, J.K. (2017). Elzaki decomposition method for solving epidemic model. International journal of Chemistry, Mathematics and Physics (IJCMP), 1(1):1-5
Akinola, E.I., Awoyemi, B.E., Olopade, I.A., Falowo, O.D., and Akinwumi, T.O. (2021). Mathematical analysis of
a diarrhea model in the presence of vaccination and treatment waves with sensitivity analysis. Journal of Applied Sciences and Environmental Management, 25(7):1,077-1,084. https://doi.org/10.4314/jasem.v25i7.2
Akinwumi, T.O., Olopade, I.A., Adesanya, A.O., and 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. https://doi.org/10.9734/jamcs/2021/v36i530362
Bernoulli, D. and Blower, S. (2004). An attempt at a new analysis of the mortality caused by smallpox and of the advantages of inoculation to prevent it. Reviews in Medical Virology, 14:275-288. https://doi.org/10.1002/rmv.443
Cui, J., Tao, X., and Zhu, H. (2008). An SIS infection model incorporating media coverage. Rocky Mountain Journal of Mathematics, 38(5):1,323-1,334. https://doi.org/10.1216/RMJ-2008-38-5-1323
Daley, D.J. and Gani, J. (2005). Epidemic modeling: An introduction. Cambridge University Press.
Das, D.K., Khajanchi, S., and Kar, T.K. (2019). The impact of media awareness and optimal strategy on the prevalence of tuberculosis. Applied Mathematics and Computation, 366(2):124732. https://doi.org/10.1016/j.amc.2019.124732
Driessche, P.V. and Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180:29-48. https://doi.org/10.1016/S0025-5564(02)00108-6
Etbaigha, F., Willms, A.R., and Poljak, Z. (2018). An SEIR model of influenza A virus infection and reinfection within a farrow-to-finish swine farm. PLoS One, 13(9). https://doi.org/10.1371/journal.pone.0202493
Feng, Z., Towers, S., and Yang, Y. (2011). Modeling the effects of vaccination and treatment on pandemic influenza. The AAPS Journal, 13(3):427-436. https://doi.org/10.1208/s12248-011-9284-7
Huang, J. (2020). A study of the effect of contact rate and asymptomatic infected in an epidemiology model of 2019-Ncov (Master’s thesis, Rutgers, The State University of New Jersey).
Goel, K., Kumar, A., and Nilam. (2020). Nonlinear dynamics of a time-delayed epidemic model with two explicit aware classes, saturated incidences, and treatment. Nonlinear Dynamics, 101:1,693-1,715. https://doi.org/10.1007/s11071-020-05762-9
Goel, K., and Nilam. (2019). A mathematical and numerical study of a SIR epidemic model with time delay, nonlinear incidence, and treatment rates. Theory in Biosciences, 138(2):203-213. https://doi.org/10.1007/s12064-019-00275-5
Huo, H., Chen, R., and Wang, X. (2016). Modelling and stability of HIV/AIDS epidemic model with treatment. Applied Mathematical Modelling, 40:6,550-6,559. https://doi.org/10.1016/j.apm.2016.01.054
Just, W., Saldana, J., and Xin, Y. (2018). Oscillations in epidemic models with spread of awareness. Mathematical Biology, 76:1,027-1,057. https://doi.org/10.1007/s00285-017-1166-x
Korobeinikov, A. (2009). Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages. Bulletin of Mathematical Biology, 71:75-83. https://doi.org/10.1007/s11538-008-9352-z
Kumar, A., and Nilam. (2019). Stability of a delayed SIR epidemic model by introducing two explicit treatment classes along with nonlinear incidence rate and Holling type treatment. Computational and Applied Mathematics, 38:130. https://doi.org/10.1007/s40314-019-0866-9
Kumar, A., Goel, K., and Nilam. (2020). Deterministic time-delayed SIR epidemic model: Mathematical modeling and analysis. Theory in Biosciences, 139:67-76. https://doi.org/10.1007/s12064-019-00300-7
Kumar, A., Nilam, and Kishor, R. (2019). A short study of an SIR model with inclusion of an alert class, two explicit nonlinear incidence rates, and saturated treatment rate. SeMA Journal, 76:505-519. https://doi.org/10.1007/s40324-019-00189-8
Lacitignola, D. and Diele, F. (2021). Using awareness to Z-control a SEIR model with overexposure: Insights on COVID-19 pandemic. Chaos, Solitons and Fractals, 150:111063. https://doi.org/10.1016/j.chaos.2021.111063
LaSalle, J.P. (1976). The stability of dynamical systems: Regional conference series in applied mathematics. SIAM.
Li, J., Xiao, Y., Zhang, F., and Yang, Y. (2012). An algebraic approach to proving the global stability of a class of epidemic models. Nonlinear Analysis: Real World Applications, 13:2,006-2,016. https://doi.org/10.1016/j.nonrwa.2011.12.022
Mummert, A. and Otunuga, O. (2019). Parameter identification for a stochastic SEIRS epidemic model: Case study influenza. Journal of Mathematical Biology, 79(2):1-25. https://doi.org/10.1007/s00285-019-01374-z
Musa, S.S., Zhao, S., Wang, M.H., Habib, A.G. Mustapha, U.T., He, D. (2020). Estimation of exponential growth rate and basic reproduction number of the coronavirus disease 2019 (COVID-19) in Africa. Infectious Diseases of Poverty, 9(2020):96. https://doi.org/10.1186/s40249-020-00718-y
Odebiyi, O.A., Oladejo, J.K., Yahaya, A.A., and Elijah, E.O. (2024). Analysis of HIV/AIDS model with nonlinear incidence function. International Journal of Research and Innovation in Applied Science, 317-341. https://doi.org/10.51584/IJRIAS.2024.904023
Olaniyi, S., Falowo, O.D., Okosun, K.O., Mukamuri, M., Obabiyi, O.S., and Adepoju, O.A. (2023). Effects of saturated treatment on malaria spread with optimal control intervention. Alexandria Engineering Journal, 65:443-459. https://doi.org/10.1016/j.aej.2022.09.024
Olopade, I.A., Adesanya, A.O., Mohammed, I.T., Afolabi, M.A., and Oladapo, A.O. (2017). Mathematical analysis of the global dynamics of an SVEIR epidemic model with herd immunity. International Journal of Science and Engineering Investigations, (69):141-148.
Olopade, A.I., Adesanya, A.O., and Akinwumi, T.O. (2021). Mathematical transmission of SEIR epidemic model with natural immunity. Asian Journal of Pure and Applied Mathematics, 3(1):19-29.
Omale, D., Atokolo, W., and Akpa, M. (2019). Mathematical modeling of the transmission dynamics of tuberculosis and its control: A case study of Ika General Hospital, Ankpa, Kogi State. IOSR Journal of Mathematics, 15(6):37-57.
Otunuga, O.M. (2017). Global stability of nonlinear stochastic SEI epidemic model. International Journal of Stochastic Analysis, 1-7. https://doi.org/10.1155/2017/6313620
Qiu, Z. (2008). Dynamical behavior of a vector-host epidemic model with demographic structure. Journal of Computers and Mathematics with Applications, 56(12):3,118-3,129. https://doi.org/10.1016/j.camwa.2008.09.002
Roosa, K. and Chowell, G. (2019). Assessing parameter identifiability in compartmental dynamic models using a computational approach: Application to infectious disease transmission models. Theoretical Biology and Medical Modelling, 1(2019). https://doi.org/10.1186/s12976-018-0097-6
Yang, Q. and Mao, X. (2013). Extinction and recurrence of multi-group SEIR epidemic models with stochastic perturbations. Nonlinear Analysis: Real World Applications, 14:1,434-1,456. https://doi.org/10.1016/j.nonrwa.2012.10.007
Zarroug, I.M., Hashim, K., ElMubark, W.A., Shumo, Z.A., Salih, K.A., ElNojomi, N.A., Awad, H.A., Aziz, N., Katabarwa, M., Hassan, H.K., and Unnasch, T.R. (2016). The first confirmed elimination of an onchocerciasis focus in Africa: Abu Hamed, Sudan. The American Journal of Tropical Medicine and Hygiene, 95(5):1,037-1,040. https://doi.org/10.4269/ajtmh.16-0274
