In the paper, we study a kind of time-delayed novel coronavirus pneumonia dynamical model with vaccination. This model considers that people are vaccinated, and the human immune system has a series of processes, which need a certain time. We first obtain the disease-free equilibrium and the basic reproduction number R 0 , and the system has a unique endemic equilibrium when R 0 > 1. Then we discuss the stability of the disease-free equilibrium and the endemic equilibrium with different delays τ . For τ = 0, using the Lyapunov function approach, we obtained the stability of disease-free equilibrium and the endemic equilibrium, respectively. For any delay τ ≠ 0, using the Routh-Hurwitz Criteria, we obtained that the disease-free equilibrium is locally asymptotically stable. We also find the critical value τ 0 at the endemic equilibrium, and obtain the condition that the system has a Hopf bifurcation at the endemic equilibrium. Finally, with the suitable choices of the parameters, some numerical simulations are presented in order to verify the effectiveness of the obtained theoretical results.
Wuhan Municipal Health Commission (2020) Wuhan Municipal Commission of Health and Health on Pneumonia of New Coronavirus Infection. http://www.wuhan.gov.cn/gsgg/202012/t20201229_1572409.shtml
Health Commission of Hubei Province (2020) Epidemic Situation of New Coronavirus Infection in Hubei Province. https://www.hubei.gov.cn/
National Health Commission of the P.R. China (2020) Update on Pneumonia Outbreak of New Coronavirus Infection. http://www.nhc.gov.cn/xcs/yqtb/list_gzbd.shtml
World Health Organization (WHO) (2020) Novel Coronavirus (2019-nCoV). Situation Report 1, 21 January. https://www.who.int/docs/default-source/coronaviruse/situation-reports/20200121-sitrep-1-2019-ncov.pdf?sfvrsn=20a99c10_4
Roser, M., Ritchie, H. and Ortiz-Ospina, E. (2020) Coronavirus Disease (COVID-19)-Research and Statistics. https://ourworldindata.org/coronavirus
Lopez, A.D., Mathers, C.D., Ezzati, M., Jamison, D.T., Murray, C.J., Lamptey, P.R., Johnson, J.L., Khan, M., Nayeri, F. and Amini, E. (2006) Infectious Diseases. Changes in Individual Behavior Could Limit the Spread of Infectious Diseases. Population Bulletin, 61.
Mameli, C. and Zuccotti, G.V. (2013) The Impact of Viral Infections in Children with Community-Acquired Pneumonia. Current Infectious Disease Reports, 15, 197-202. https://doi.org/10.1007/s11908-013-0339-z
Diekmann, O., Heesterbeek, H. and Britton, T. (2013) Mathematical Tools for Understanding Infectious Disease Dynamics. Princeton Series in Theoretical and Computational Biology. Princeton University Press, Princeton.
Hethcote, H.W. (2000) The Mathematics of Infectious Diseases. SIAM Review, 42, 599-653. https://doi.org/10.1137/S0036144500371907
Keeling, M.J. and Rohani, P. (2007) Modeling Infectious Diseases in Humans and Animals. Princeton University Press, Princeton. https://doi.org/10.1515/9781400841035
Donnelly, C.A., Ghani, A., Leung, G., et al. (2003) Epidemiological Determinants of Spread of Causal Agent of Severe Acute Respiratory Syndrome in Hong Kong. The Lancet, 361, 1761-1766. https://doi.org/10.1016/S0140-6736(03)13410-1
https://www.yicai.com/news/101285104.html
Lipsitch, M., Cohen, T., Cooper, B., Robins, J., et al. (2003) Transmission Dynamics and Control of Severe Acute Respiratory Syndrome. Science (New York, N.Y.), 300, 1966-1970. https://doi.org/10.1126/science.1086616
Sun, C., Wei, Y., Arino, J. and Khan, K. (2011) Effect of Media-Induced Social Distancing on Disease Transmission in a Two Patch Setting. Mathematical Biosciences, 230, 87-95. https://doi.org/10.1016/j.mbs.2011.01.005
Pereda, A., Vielva, L.A., Vegas, A. and Prieto, A. (2001) Analyzing the Stability of the FDTD Technique by Combining the von Neumann Method with the Routh-Hurwitz Criterion. IEEE Transactions on Microwave Theory Techniques, 49, 377-381. https://doi.org/10.1109/22.903100
Kim, A.V. (1999) The Lyapunov Function Method. Springer, Berlin. https://doi.org/10.1007/978-94-017-1630-7_8
Hassard, D.D., Kazarinoff, N.D. and Wan, Y.-H. (2006) Theory and Applications of Hopf Bifurcation. SIAM Review, 24, 498-499. https://doi.org/10.1137/1024123
Jiang, Z.C., Ma, W.B. and Wei, J.J. (2016) Global Hopf Bifurcation and Permanence of a Delayed SEIRS Epidemic Model. Mathematics Computers in Simulation, 122, 35-54. https://doi.org/10.1016/j.matcom.2015.11.002
Kermack, W.O. and McKendrick, A.G. (1991) Contributions to the Mathematical Theory of Epidemics. II. The Problem of Endemicity. Bulletin of Mathematical Biology, 53, 57-87. https://doi.org/10.1016/S0092-8240(05)80041-2
Grossman, Z. (1980) Oscillatory Phenomena in a Model of Infectious Diseases. Theoretical Population Biology, 18, 204-243. https://doi.org/10.1016/0040-5809(80)90050-7
Rihan, F.A. and Anwar, M.N. (2012) Qualitative Analysis of Delayed SIR Epidemic Model with a Saturated Incidence Rate. International Journal of Differential Equations, 2012, Article ID: 408637. https://doi.org/10.1155/2012/408637
Elazzouzi, A., Alaoui, A.L., Tilioua, M. and Tridane, A. (2019) Global Stability Analysis for a Generalized Delayed SIR Model with Vaccination and Treatment. Advances in Difference Equations, 2019, Article No. 532. https://doi.org/10.1186/s13662-019-2447-z
Zhang, X.H., Jiang, D.Q., Hayat, T. and Ahmad, B. (2017) Dynamical Behavior of a Stochastic SVIR Epidemic Model with Vaccination. Physica A: Statistical Mechanics and Its Applications, 483, 94-108. https://doi.org/10.1016/j.physa.2017.04.173
Li, X.Z., Wang, J. and Ghosh, M. (2010) Stability and Bifurcation of an SIVS Epidemic Model with Treatment and Age of Vaccination. Applied Mathematical Modelling, 34, 437-450. https://doi.org/10.1016/j.apm.2009.06.002
Kumar, A., Srivastava, P.K. and Gupta, R.P. (2019) Nonlinear Dynamics of Infectious Diseases via Information-Induced Vaccination and Saturated Treatment. Mathematics and Computers in Simulation, 157, 77-99. https://doi.org/10.1016/j.matcom.2018.09.024
Buri, N., Mudrinic, M. and Vasovi, N. (2001) Time Delay in a Basic Model of the Immune Response. Chaos Solitons Fractals, 12, 483-489. https://doi.org/10.1016/S0960-0779(99)00205-2
Wang, K., Wang, W., Pang, H. and Liu, X. (2007) Complex Dynamic Behavior in a Viral Model with Delayed Immune Response. Physica D: Nonlinear Phenomena, 226, 197-208. https://doi.org/10.1016/j.physd.2006.12.001
Liu, X., Takeuchi, Y. and Iwami, S. (2008) SVIR Epidemic Models with Vaccination Strategies. Journal of Theoretical Biology, 253, 1-11. https://doi.org/10.1016/j.jtbi.2007.10.014
Sda, B. and Sba, C. (2021) Global Dynamics of SVIR Epidemic Model with Distributed Delay and Imperfect Vaccine. Results in Physics, 25, Article ID: 104245.
Diekmann, O., Heesterbeek, J. and Metz, J. (1990) On the Definition and the Computation of the Basic Reproduction Ratio R0 in Models for Infectious Diseases in Heterogeneous Populations. Journal of Mathematical Biology, 28, 365-382. https://doi.org/10.1007/BF00178324
Driessche, P. 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
Kuniya, T. (2013) Global Stability of a Multi-Group SVIR Epidemic Model. Nonlinear Analysis: Real World Applications, 14, 1135-1143. https://doi.org/10.1016/j.nonrwa.2012.09.004
Wiggins, S. (2003) Introduction to Applied Nonlinear Dynamical Systems and Chaos. 2nd Edition, Springer-Verlag, New York.