Abstract
Vaccination is widely recognized as a powerful tool in controlling the spread of diseases. However, it is important to acknowledge that some diseases are not completely preventable by vaccines, meaning that the protective effect of certain vaccines is not absolute. Additionally, the effectiveness of vaccines may diminish over time, which implies that even with completed vaccinations, there remains a potential risk of infection. Therefore, conducting comprehensive studies on how these imperfect vaccines impact the dynamics of disease transmission is crucial. In this study, we formulate and analyze a two-strain epidemic model with imperfect vaccination on complex networks. We derive the basic reproduction number and introduce the invasion reproduction numbers corresponding to each strain. Using the Lyapunov function method and LaSalle’s invariance principle, we demonstrate the global asymptotical stability of the disease-free equilibrium and dominant equilibrium. Furthermore, we apply persistence theory to verify the existence of coexistence equilibrium and the persistence of the disease. To validate our theoretical findings, we conduct numerical simulations. Through these simulations, we confirm that vaccination plays a significant role in controlling the spread of diseases, despite the imperfect efficacy of certain vaccines in specific cases. These findings highlight the importance of continued efforts to improve vaccine effectiveness and explore alternative strategies to enhance disease control measures.
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The data that support the findings of this study are available from the corresponding author upon reasonable request.
References
Jabbari, A., Lotfi, M., Kheiri, H., et al.: Mathematical analysis of the dynamics of a fractional-order tuberculosis epidemic in a patchy environment under the influence of re-infection. Math. Meth. Appl. Sci. 46, 17798–17817 (2023)
Dwivedi, A., et al. Modeling optimal vaccination strategy for dengue epidemic model: a case study of India. Physica Scripta 97 (2022)
Cheng, X., Wang, Y., Huang, G.: Dynamics of cholera transmission model with imperfect vaccination and demographics on complex networks. J. Franklin Inst. 360(2), 1077–1105 (2023)
Hassani, H., Avazzadeh, Z., Machado, J.A.T., et al.: Optimal solution of a fractional HIV/AIDS epidemic mathematical model. J. Comput. Biol. 29(3), 276–291 (2022)
Rehman, A.U., Singh, R., Agarwal, P.: Modeling, analysis and prediction of new variants of covid-19 and dengue co-infection on complex network. Chaos Solitons Fractals 150(1), 111008 (2021)
Agarwal, P., Nieto, J., Ruzhansky, M., et al.: Analysis of infectious disease problems (Covid-19) and their global impact. Infosys Sci. Found. Ser. (2021)
Chowdhury, S.M.E.K., Chowdhury, J.T., Shams, F.A., Praveen, A., et al.: Mathematical modelling of COVID-19 disease dynamics: interaction between immune system and SARS-CoV-2 within host. AIMS Math. 7(2), 2618–2633 (2022)
Morales-Delgado, V.F., Gomez-Aguilar, J.F., et al.: Analytic solution for oxygen diffusion from capillary to tissues involving external force effects: a fractional calculus approach. Stat. Mech. Appl. Physica A (2019)
Habenom, H., Aychluh, M., Suthar, D.L., Al-Mdallal, Q., Purohit, S.D.: Modeling and analysis on the transmission of covid-19 pandemic in Ethiopia. Alex. Eng. J. 61(7), 5323–5342 (2022)
Baba, I. A., Ahmed, I., Al-Mdallal, Q. M., Jarad, F., Yunusa, S.: Numerical and theoretical analysis of an awareness COVID-19 epidemic model via generalized Atangana-Baleanu fractional derivative. J. Appl. Math. Comput. Mech. 21(1) (2022)
Tiwari, P.K., Rai, R.K., Khajanchi, S., Gupta, R.K., Misra, A.K.: Dynamics of coronavirus pandemic: effects of community awareness and global information campaigns. Eur. Phys. J. Plus 136(10), 994 (2021)
Bera, S., Khajanchi, S., Roy, T.K.: Dynamics of an HTLV-I infection model with delayed CTLs immune response. Appl. Math. Comput. 430, 127206 (2023)
Ullah, R., Mdallal, Q.A., Khan, T., et al.: The dynamics of novel corona virus disease via stochastic epidemiological model with vaccination. Sci. Rep. 13(1), 3805 (2023)
Baba, I.A., Kaymakamzade, B., Hincal, E.: Two-strain epidemic model with two vaccinations. Chaos Solitons Fractals 106, 342–348 (2018)
Rahman, S.M.A., Zou, X.: Flu epidemics: a two-strain flu model with a single vaccination. J. Biol. Dyn. 5(5), 376–390 (2011)
de Leon, U.A.P., Avila-Vales, E., Huang, K.L.: Modeling COVID-19 dynamic using a two-strain model with vaccination. Chaos Solitons Fractals 157, 111927 (2022)
Cai, L., Xiang, J., Li, X., Lashari, A.A.: A two-strain epidemic model with mutant strain and vaccination. J. Appl. Math. Comput. 40, 125–142 (2012)
May, A.J.N., Vales, E.J.A.: Global dynamics of a two-strain flu model with a single vaccination and general incidence rate. (2020)
Kumar, M., Abbas, S., Tridane, A.: Optimal control and stability analysis of an age-structured SEIRV model with imperfect vaccination. Math. Biosci. Eng. 20(8), 14438–14463 (2023)
Tchoumi, S.Y., Rwezaura, H., Tchuenche, J.M.: Dynamic of a two-strain COVID-19 model with vaccination. Results Phys. 39, 105777 (2021)
Li, T., Guo, Y.: Modeling and optimal control of mutated COVID-19 (Delta strain) with imperfect vaccination. Chaos Solitons Fractals 156, 111825 (2022)
Arefin, M.R., Kabir, K.M.A., Tanimoto, J.: A mean-field vaccination game scheme to analyze the effect of a single vaccination strategy on a two-strain epidemic spreading[J]. J. Stat. Mech: Theory Exp. 2020(3), 033501 (2020)
Bugalia, S., Tripathi, J.P., Wang, H.: Mutations make pandemics worse or better: modeling SARS-CoV-2 variants and imperfect vaccination. (2022)
Kuga, K., Tanimoto, J.: Impact of imperfect vaccination and defense against contagion on vaccination behavior in complex networks. J. Stat. Mech: Theory Exp. 2018(11), 113402 (2018)
Chen, S., Small, M., Fu, X.: Global stability of epidemic models with imperfect vaccination and quarantine on scale-free networks. IEEE Trans. Netw. Sci. Eng. 7(3), 1583–1596 (2018)
Li, C.L., Li, C.H.: Dynamics of an epidemic model with imperfect vaccinations on complex networks. J. Phys. A: Math. Theor. 53(46), 464001 (2020)
Lv, W., Ke, Q., Li, K.: Dynamic stability of an SIVS epidemic model with imperfect vaccination on scale-free networks and its control strategy. J. Franklin Inst. 357(11), 7092–7121 (2020)
Huang, S., Chen, F., Chen, L.: Global dynamics of a network-based SIQRS epidemic model with demographics and vaccination. Commun. Nonlinear Sci. Numer. Simul. 43, 296–310 (2017)
Li, C.L., Cheng, C.Y., Li, C.H.: Global dynamics of two-strain epidemic model with single-strain vaccination in complex networks. Nonlinear Anal. Real World Appl. 69, 103738 (2023)
Yao, Y., Zhang, J.: A two-strain epidemic model on complex networks with demographics. J. Biol. Syst. 24(04), 577–609 (2016)
Wang, X., Yang, J., Luo, X.: Competitive exclusion and coexistence phenomena of a two-strain SIS model on complex networks from global perspectives. J. Appl. Math. Comput. 68(6), 4415–4433 (2022)
Cheng, X., Wang, Y., Huang, G.: Dynamics of a competing two-strain SIS epidemic model with general infection force on complex networks. Nonlinear Anal. Real World Appl. 59, 103247 (2021)
Dreessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Bio 180(1–2), 29–48 (2002)
Zhao, X.Q., Jing, Z.J.: Global asymptotic behavior in some cooperative systems of functional differential equations. Can. Appl. Math. Q. 4, 421–444 (1996)
Thieme, H.R.: Persistence under relaxed point-dissipativity (with application to an endemic model). SIAM J. Math. Anal. 24, 407–435 (1993)
Zhao, X.Q.: Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflows with applications. Canad. Appl. Math. Quart. 3, 473–495 (1995)
Al-Mdallal, Q.M.: Mathematical modeling and simulation of SEIR model for COVID-19 outbreak: a case study of Trivandrum. Front. Appl. Math. Stat. 9, 1124897 (2023)
Wu, Y., Zhang, Z., Song, L., Xia, C.: Global stability analysis of two strains epidemic model with imperfect vaccination and immunity waning in a complex network. Chaos Solitons Fractals 179, 114414 (2024)
Wu, Q.C., Fu, X.C., Yang, M.: Epidemic thresholds in a heterogenous population with competing strains. Chin. Phys. B 20(4), 046401 (2011)
Acknowledgements
This work is partially supported by the National Natural Science Foundation of China (No.12101574), and the Shanxi Province Science foundation (20210302124621).
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Li, S., Yuan, Y. Dynamics of two-strain epidemic model with imperfect vaccination on complex networks. J. Appl. Math. Comput. (2024). https://doi.org/10.1007/s12190-024-02025-3
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DOI: https://doi.org/10.1007/s12190-024-02025-3