Dynamic modeling and analysis for the periodic spread of influenza: A case study in Shanghai

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

In this paper, anS IRS epidemic model with periodic transmission and vaccination is proposed to study the periodic transmission phenomenon of influenza. The threshold dynamics of the model are completely determined by the basic reproduction number $\mathcal{R}_{0}$, i.e., the disease-free periodic steady state is globally asymptotically stable if $\mathcal{R}_{0}$ < 1, while the disease is uniformly persistent if $\mathcal{R}_{0}$ > 1. With the help of control theory, the resonance frequency ω_r of our model is obtained, which is numerically validated in that there exists a resonance phenomenon of the disease spread (i.e., there is a maximum amplitude of the oscillatory spread of diseases) when the frequency ω = ω_r. By fitting the real data of influenza from Shanghai, we estimate the parameter values of the model. Results show that there is no resonance phenomenon in the spread of influenza in Shanghai. Based on the expression of resonance frequency ω_r, the effects of the mean transmission rate β and the transition rate δ on the resonance frequency ω_r are displayed numerically.

Keywords

influenza / S IRS model / oscillation / uniform persistence / resonance frequency

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] K. G. Nicholson, R. G. Webster, A. J. Hay, Textbook of Influenza, Trans. R. Soc. Trop. Med. Hyg., 93 (1999), 31. https://doi.org/10.1016/S0035-9203(99)90169-4

[2] J. N. Barr, R. Fearns, Chapter 2 - Genetic instability of RNA viruses, Genome Stab., (2016), 21-35. https://doi.org/10.1016/B978-0-12-803309-8.00002-1

[3] S. Cauchemez, A. Valleron, P. Boëlle, A. Flahault, N. M. Ferguson, Estimating the impact of school closure on influenza transmission from Sentinel data, Nature, 452 (2008), 750-754. https://doi.org/10.1038/nature06732

[4] L. Shi, L. Wang, Z. Jin, The seasonal influenza transmission dynamic and the cor- relation analysis with meteorology in Beijing, Int. J. Biomath., 17 (2024), 2350071. https://doi.org/10.1142/S1793524523500717

[5] J. Shaman, E. Goldstein, M. Lipsitch, Absolute humidity and pandemic versus epidemic influenza, Am. J. Epidemiol., 173 (2011), 127-135. https://doi.org/10.1093/aje/kwq347

[6] G. Neumann, Y. Kawaoka, Seasonality of influenza and other respiratory viruses, EMBO Mol. Med., 14 (2022), e15352. https://doi.org/10.15252/emmm.202115352

[7] Y. L. Shu, W. Liu, S. Vlas, Y. Gao, J. H. Richardus, W. C. Cao, Dual seasonal patterns for influenza, China, Emerging Infect. Dis., 16 (2010), 725-726. https://doi.org/10.3201/eid1604.091578

[8] Y. Z. Zhang, C. C. Ye, J. X. Yu, W. P. Zhu, Y. P. Wang, Z. J. Li, et al., The complex associations of climate variability with seasonal influenza A and B virus transmission in subtropical Shanghai, China, Sci. Total Environ., 701 (2020), 134607. https://doi.org/10.1016/j.scitotenv.2019.134607

[9] J. L. Ma, J. Dushoff, D. Earn, Age-specific mortality risk from pandemic influenza, J. Theor. Biol., 288 (2011), 29-34. https://doi.org/10.1016/j.jtbi.2011.08.003

[10] J. Arino, C. Bauch, F. Brauer, S. M. Driedger, A. L. Greer, S. M. Moghadas, et al., Pandemic influenza: Modelling and public health perspectives, Math. Biosci. Eng. , 8 (2011), 1-20. https://doi.org/10.3934/mbe.2011.8.1

[11] S. M. Asaduzzaman, J. L. Ma, P. Driessche, Estimation of cross-immunity between drifted strains of influenza A/H3N2, Bull. Math. Biol., 80 (2018), 657-669. https://doi.org/10.1007/s11538-018-0395-5

[12] D. H. He, J. Dushoff, T. Day, J. L. Ma, D. Earn, Mechanistic modelling of the three waves of the 1918 influenza pandemic, Theor. Ecol., 4 (2011), 283-288. https://doi.org/10.1007/s12080-011-0123-3

[13] S. M. Asaduzzaman, J. L. Ma, P. Driessche, The coexistence or replacement of two subtypes of influenza, Math. Biosci., 270 (2015), 1-9. https://doi.org/10.1016/j.mbs.2015.09.006

[14] Y. Furuse, H. Oshitani, Mechanisms of replacement of circulating viruses by seasonal and pandemic influenza A viruses, Int. J. Infect. Dis., 51 (2016), 6-14. https://doi.org/10.1016/j.ijid.2016.08.012

[15] Z. McCarthy, S. Athar, M. Alavinejad, C. Chow, I. Moyles, K. Nah, et al., Quantifying the annual incidence and underestimation of seasonal influenza: a modelling approach, Theor. Biol. Med. Modell., 17 (2020), 11. https://doi.org/10.1186/s12976-020-00129-4

[16] X. C. Duan, J. F. Yin, X. Z. Li, Global Hopf bifurcation of an SIRS epidemic model with age-dependent recovery, Chaos, Solitons Fractals, 104 (2017), 613-624. https://doi.org/10.1016/j.chaos.2017.09.029

[17] T. L. Zhang, J. L. Liu, Z. D. Teng, Stability of Hopf bifurcation of a delayed SIRS epidemic model with stage structure, Nonlinear Anal. Real World Appl., 11 (2010), 293-306. https://doi.org/10.1016/j.nonrwa.2008.10.059

[18] H. Cao, X. Y. Gao, J. Q. Li, D. X. Yan, Z. M. Yue, The bifurcation analysis of an SIRS epidemic model with immunity age and constant treatment, Appl. Anal., 100 (2021), 2844-2866. https://doi.org/10.1080/00036811.2019.1698728

[19] S. Gakkhar, K. Negi, Pulse vaccination in SIRS epidemic model with non-monotonic incidence rate, Chaos, Solitons Fractals, 35 (2008), 626-638. https://doi.org/10.1016/j.chaos.2006.05.054

[20] S. L. Jing, H. F. Huo, H. Xiang, Modelling the effects of ozone concentration and pulse vaccination on seasonal influenza outbreaks in Gansu province, China, Discrete Contin. Dyn. Syst. - Ser. B, 27 (2022), 1877-1911. https://doi.org/10.3934/dcdsb.2021113

[21] Z. D. Teng, Y. P. Liu, L. Zhang, Persistence and extinction of disease in non-autonomous SIRS epidemic models with disease-induced mortality, Nonlinear Anal. Theory Methods Appl., 69 (2008), 2599-2614. https://doi.org/10.1016/j.na.2007.08.036

[22] B. Chaihao, S. Khomrutai, Extinction and permanence of a general non-autonomous discrete-time SIRS epidemic model, AIMS Math., 8 (2023), 9624-9646. https://doi.org/10.3934/math.2023486

[23] A. Kumar, N. Kumari, S. Manda, P. K. Tiwari, Autonomous and non-autonomous dynamics of an SIRS model with convex incidence rate, J. Franklin Inst., 362 (2025), 108236. https://doi.org/10.1016/j.jfranklin.2025.108236

[24] W. C. Zhao, J. L. Liu, M. N. Chi, F. F. Bian, Dynamics analysis of stochastic epidemic models with standard incidence, Adv. Differ. Equations, 2019 (2019), 22. https://doi.org/10.1186/s13662-019-1972-0

[25] I. A. Moneim, Modeling and simulation of the spread of H1N1 flu with periodic vaccination, Int. J. Biomath., 9 (2016), 1650003. https://doi.org/10.1142/S1793524516500030

[26] J. Dushoff, J. B. Plotkin, S. Levin, D. Earn, Dynamical resonance can account for seasonality of influenza epidemics, Proc. Natl. Acad. Sci. U.S.A., 101 (2004), 16915-16916. https://doi.org/10.1073/pnas.0407293101

[27] G. F. Franklin, J. D. Powell, A. Emami-Naeini, Feedback Control of Dynamic Systems , Pearson, 2019.

[28] Shanghai Municipal Health Commission, Report on the Epidemic Situation of Legally notifiable Infectious Diseases in Shanghai. Available from: https://wsjkw.sh.gov.cn/yqxx/index.html.

[29] N. Bacaër, S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436. https://doi.org/10.1007/s00285-006-0015-0

[30] W. Wang, X. Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Differ. Equations, 20 (2008), 699-717. https://doi.org/10.1007/s10884-008-9111-8

[31] X. Q. Zhao, Dynamical Systems in Population Biology , Springer International Publishing, 2017. https://doi.org/10.1007/978-3-319-56433-3

[32] H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407-435. https://doi.org/10.1137/0524026

[33] H. Haario, M. Laine, A. Mira, E. Saksman, Dram: efficient adaptive MCMC, Stat. Comput., 16 (2006), 339-354. https://doi.org/10.1007/s11222-006-9438-0