Ergodic stationary distribution and extinction of stochastic pertussis model with immune and Markov switching

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This article was retracted on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

Temperature, humidity, and other environmental factors can influence the spread of diseases. To investigate the impact of environmental perturbations and state changes on pertussis, this study established a random pertussis model with immunity and Markov switching. This stochastic model presented a global positive solution. Subsequently, using Itô’s lemma and Lyapunov function, we concluded that the disease will become extinct. Then, a critical value R^e₀ was introduced. It was established that the stochastic model with Markov switching has an ergodic stationary distribution when R^e₀ > 1, which implies that this infectious disease will persist and remain prevalent. Some examples are presented to further substantiate our theoretical conclusions.

Keywords

stochastic pertussis model / Markov switching / ergodic stationary distribution

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This article has been retracted.Please see the Retraction Notice for more detail: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] R. M. Jena, S. Chakraverty, S. K. Jena, Analysis of the dynamics of phytoplankton nutrient and whooping cough models with nonsingular kernel arising in the biological system, Chaos Solitons Fractals, 141 (2020), 110373. https://doi.org/10.1016/j.chaos.2020.110373

[2] G. Rozhnova, A. Nunes, Modeling the long-term dynamics of pre-vaccination pertussis, J. R. Soc. Interface, 9 (2012), 2959-2970. https://doi.org/10.1098/rsif.2012.0432

[3] D. M. Skowronski, G. D. Serres, D. MacDonald, W. Wu, C. Shaw, J. Macnabb, et al., The changing age and seasonal profile of pertussis in Canada, J. Infect. Dis., 185 (2002), 1448-1453. https://doi.org/10.1086/340280

[4] H. E. D. Melker, J. F. P. Schellekens, S. E. Neppelenbroek, F. R. Mooi, H. C. Rümkе, M. A. E. Conyn-van Spaendonck, Reemergence of pertussis in the highly vaccinated population of the Netherlands: observations on surveillance data, Emerging Infect. Dis., 6 (2000), 348-357. https://doi.org/10.3201/eid0604.000404

[5] A. Mughal, Y. F. Kazi, H. A. Bukhari, M. Ali, Pertussis resurgence among vaccinated children in Khairpur, Sindh, Pakistan, Public Health, 126 (2012), 518-522. https://doi.org/10.1016/j.puhe.2012.02.001

[6] T. Tan, T. Dalby, K. Forsyth, S. A. Scott, U. Heininger, D. Hozbor, et al., Pertussis across the globe: recent epidemiologic trends from 2000 to 2013, Pediatr Infect Dis. J., 34 (2015), e222-e232. https://doi.org/10.1097/INF.0000000000000795

[7] J. Zhang, J. Deng, Y. Yang, Pertussis vaccination in Chinese children with increasing reported pertussis cases, Lancet Infect. Dis., 22 (2022), 21-22. https://doi.org/10.1016/S1473-3099(21)00752-0

[8] C. R. Maclntyre, J. C. D. Sousa, U. Heininger, P. Kardos, A. Konstantopoulos, D. Middleton, et al., Public health management of pertussis in adults: Practical challenges and future strategies, Hum. Vaccines Immunother., 20 (2024), 2377904. https://doi.org/10.1080/21645515.2024.2377904

[9] K. H. T. Yeung, P. Duclos, E. A. S. Nelson, D. R. C. W. Hutubessy, An update of the global burden of pertussis in children younger than 5 years: a modelling study, Lancet Infect. Dis., 17 (2017), 974-980. https://doi.org/10.1016/S1473-3099(17)30390-0

[10] J. S. Lavine, A. A. King, O. N. Bjørnstad, Natural immune boosting in pertussis dynamics and the potential for long-trem vaccine failure, PNAS, 108 (2011), 7259-7264. https://doi.org/10.1073/pnas.1014394108

[11] Y. Chen, J. Li, S. Zou, Global dynamics of an epidemic model with relapse and nonlinear incidence, Math. Method. Appl. Sci., 42 (2019), 1283-1291. https://doi.org/10.1002/mma.5439

[12] X. Tian, W. Wang, Dynamical analysis of age-structured pertussis model with cover infection, Math. Method. Appl. Sci., 43 (2020), 1631-1645. https://doi.org/10.1002/mma.5989

[13] Q. Han, An age-structured model for pertussis transmission with multiple infections studying the effects of childhood DTaP and adolescent Tdap vaccines, J. Biol. Syst., 30 (2022), 761-797. https://doi.org/10.1142/S0218339022500280

[14] M. Safan, M. Kretzschmar, K. P. Hadeler, Vaccination based control of infections in SIRS models with reinfection: special reference to pertussis, J. Math. Biol., 67 (2013), 1083-1110. https://doi.org/10.1007/s00285-012-0582-1

[15] R. Águas, G. Gonçalves, M. G. M. Gomes, Pertussis: increasing disease as a consequence of reducing transmission, Lancet Infect Dis., 6 (2006), 112-117. https://doi.org/10.1016/s1473-3099(06)70384-x

[16] A. Din, Y. Li, A. Omame, A stochastic stability analysis of an HBV -COVID-19 co-infection model in resource limitation settings, Waves Random Complex Media, (2022), 1-33. https://doi.org/10.1080/17455030.2022.2147598

[17] Y. Zhang, H. Bambrick, K. Mengersen, S. Tong, L. Fenf, G. Liu, et al., Association of weather variability with resurging pertussis infections among different age groups: A non-linear approach, Sci. Total Environ., 719 (2020), 137510. https://doi.org/10.1016/j.scitotenv.2020.137510

[18] X. Mao, Stochastic differential equations and applications, in Stochastic Differential Equations , Springer, 77 (1997), 75-148. https://doi.org/10.1007/978-3-642-11079-5_2

[19] G. P. Sahu, J. Dhar, Dynamics of an SEQIHRS epidemic model with media coverage, quarantine and isolation in a community with pre-existing immunity, J. Math. Anal. Appl., 421 (2015), 1651-1672. https://doi.org/10.1016/j.jmaa.2014.08.019

[20] N. Gunasekaran, R. Vadivel, G. Zhai, S. Vinoth, Finite-time stability analysis and control of stochastic SIR epidemic model: A study of COVID-19, Biomed. Signal Process. Control, 86 (2023), 105123. https://doi.org/10.1016/j.bspc.2023.105123

[21] Q. Liu, D. Jiang, Stationary distribution and extinction of a stochastic SIR model with nonlinear perturbation, Appl. Math. Lett., 73 (2017), 8-15. https://doi.org/10.1016/j.aml.2017.04.021

[22] R. Z. Has'minskii, Stochastic Stability of Differential Equations , Springer Science and Business Media, 2011.

[23] B. Han, D. Jiang, H. Tasawar, A. Ahmed, A. Bashir, Stationary distribution and extinction of a stochastic staged progression AIDS model with staged treatment and second-order perturbation, Chaos Solitons Fractals, 140 (2020), 110238. https://doi.org/10.1016/j.chaos.2020.110238

[24] L. Zu, D. Jiang, D. O'Regan, T. Hayat, B. Ahmad, Ergodic property of a lotka-volterra predator-prey model with white noise higher order perturbation under regime switching, Appl. Math. Comput., 330 (2018), 93-102. https://doi.org/10.1016/j.amc.2018.02.035

[25] B. Zhou, B. Han, D. Jiang, T. Hayat, A. Alsaedi, Ergodic stationary distribution and extinction of a hybrid stochastic SEQIHR epidemic model with media coverage, quarantine strategies and pre-existing immunity under diecrete Markov switching, Appl. Math. Comput., 410 (2021), 126388. https://doi.org/10.1016/j.amc.2021.126388

[26] G. R. Ghorbani, S. M. Zahraei, M. Moosazadeh, M. Afshari, F. Doosti, Comparing seasonal pattern of laboratory confirmed cases of pertussis with clinically suspected cases, Osong Public Health Res. Perspect., 7 (2016), 131-137. https://doi.org/10.1016/j.phrp.2016.02.004

[27] H. T. H. Nguyen, P. Rohani, Noise, nonlinearity and seasonality: the epidemics of whooping cough revisited, J. R. Soc. Interface, 5 (2008), 403-413. https://doi.org/10.1098/rsif.2007.1168

[28] X. Chen, X. Li, Y. Ma, C. Yuan, The threshold of stochastic tumor-immune model with regime switching, J. Math. Anal. Appl., 522 (2023), 126956. https://doi.org/10.1016/j.jmaa.2022.126956

[29] G. Lan, S. Yuan, B. Song, The impact of hospital resources and environmental perturbations to the dynamics of SIRS model, J. Franklin Inst., 358 (2021), 2405-2433. https://doi.org/10.1016/j.jfranklin.2021.01.015

[30] T. Caraballo, M. E. Fatini, M. E. Khaliff, R. Geriach, R. Pettersson, Analysis of a stochastic distributed delay epidemic model with relapse and Gamma distribution kernel, Chaos Solitons Fractals, 133 (2020), 109643. http://dx.doi.org/10.1016/j.chaos.2020.109643

[31] T. Khan, A. Khan, Z. Gui, The extinction and persistence of the stochastic hepatitis B epidemic model, Chaos Solitons Fractals, 108 (2018), 123-128. https://doi.org/10.1016/j.chaos.2018.01.036

[32] B. Zhou, D. Jiang, B. Han, T. Hayat, Ergodic stationary distribution and practical application of a hybrid stochastic cholera transmission model with waning vaccine-induced immunity under nonlinear regime switching, Math. Methods Appl. Sci., 45 (2022), 423-455. https://doi.org/10.1002/mma.7785

[33] A. C. Lowen, S. Mubareka, J. Steel, P. Palese, Influenza virus transmission is dependent on relative humidity and temperature, PLoS Pathog., 3 (2007), e151. https://doi.org/10.1371/journal.ppat.0030151

[34] T. D. Tuong, N. N. Nguyen, Characterization of long-term behavior of stochastic NP ecological model under regime switching, Commun. Nonlinear Sci. Numer. Simul., 93 (2021), 105497. https://doi.org/10.1016/j.cnsns.2020.105497

[35] D. Greenhalgh, Y. Liang, X. Mao, Modelling the effect of telegraph noise in the SIRS epidemic model using Markovian switching, Physica A, 462 (2016), 684-704. https://doi.org/10.1016/j.physa.2016.06.125

[36] X. Li, D. Jiang, X. Mao, Population dynamical behavior of Lotka-Volterra system under regime switching, J. Comput. Appl. Math., 232 (2009), 427-448. https://doi.org/10.1016/j.cam.2009.06.021

[37] Q. Liu, D. Jiang, T. Hayat, A. Alsaedi, Dynamical behavior of stochastic multigroup S-DIA epidemic models for the transmission of HIV, J. Franklin Inst., 355 (2018), 5830-5865. https://doi.org/10.1016/j.jfranklin.2018.05.047

[38] Y. Zhao, S. Yuan, T. Zhang, The stationary distribution and ergodicity of a stochastic phytoplankton allelopathy model under regime switching, Commun. Nonlinear Sci. Numer. Simul., 37 (2016), 131-142. https://doi.org/10.1016/j.cnsns.2016.01.013

[39] A. Omame, M. Abbas, A. Din, Global asymptotic stability, extinction and ergodic stationary distribution in a stochastic model for dual variants of SARS-CoV-2, Math. Comput. Simul., 204 (2023), 302-336. https://doi.org/10.1016/j.matcom.2022.08.012

[40] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546. https://doi.org/10.1137/S0036144500378302

[41] K. Mamis, M. Farazmand, Modeling correlated uncertainties in stochastic compartmental models, Math. Biosci., 374 (2024), 109226. https://doi.org/10.1016/j.mbs.2024.109226