Determining the global threshold of an epidemic model with general interference function and high-order perturbation

Yassine Sabbar; Asad Khan; Anwarud Din; Driss Kiouach; S. P. Rajasekar; Yassine Sabbar; Asad Khan; Anwarud Din; Driss Kiouach; S. P. Rajasekar

doi:10.3934/math.20221088

AIMS Mathematics

2022, Volume 7, Issue 11: 19865-19890. doi: 10.3934/math.20221088

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Research article

Determining the global threshold of an epidemic model with general interference function and high-order perturbation

1.
LPAIS Laboratory, Faculty of Sciences Dhar El Mahraz, Sidi Mohamed Ben Abdellah University, Fez, Morocco
2.
School of Computer Science and Cyber Engineering, Guangzhou University, Guangzhou 510006, China
3.
Department of Mathematics, Sun Yat-Sen University, Guangzhou, China
4.
Department of Mathematics, Government Arts College for Women, Nilakottai-624202, Tamilnadu, India

Received: 02 July 2022 Revised: 16 August 2022 Accepted: 26 August 2022 Published: 08 September 2022
MSC : 34A26, 34A12, 92D30, 37C10, 60H30

This research provides an improved theoretical framework of the Kermack-McKendrick system. By considering the general interference function and the polynomial perturbation, we give the sharp threshold between two situations: the disappearance of the illness and the ergodicity of the higher-order perturbed system. Obviously, the ergodic characteristic indicates the continuation of the infection in the population over time. Our study upgrades and enhances the work of Zhou et al. (2021) and suggests a new path of research that will serve as a basis for future investigations. As an illustrative application, we discuss some special cases of the polynomial perturbation to examine the precision of our outcomes. We deduce that higher order fluctuations positively affect the illness extinction time and lead to its rapid disappearance.
- epidemic system,
- stochastic process,
- extinction,
- ergodicity,
- stationarity
Citation: Yassine Sabbar, Asad Khan, Anwarud Din, Driss Kiouach, S. P. Rajasekar. Determining the global threshold of an epidemic model with general interference function and high-order perturbation[J]. AIMS Mathematics, 2022, 7(11): 19865-19890. doi: 10.3934/math.20221088

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Abstract

This research provides an improved theoretical framework of the Kermack-McKendrick system. By considering the general interference function and the polynomial perturbation, we give the sharp threshold between two situations: the disappearance of the illness and the ergodicity of the higher-order perturbed system. Obviously, the ergodic characteristic indicates the continuation of the infection in the population over time. Our study upgrades and enhances the work of Zhou et al. (2021) and suggests a new path of research that will serve as a basis for future investigations. As an illustrative application, we discuss some special cases of the polynomial perturbation to examine the precision of our outcomes. We deduce that higher order fluctuations positively affect the illness extinction time and lead to its rapid disappearance.

References

[1]	Z. C. Wang, K. Tang, Combating COVID-19: health equity matters, Nat. Med., 26 (2020), 458. https://doi.org/10.1038/s41591-020-0823-6 doi: 10.1038/s41591-020-0823-6
[2]	S. Djilali, L. Benahmadi, A. Tridane, K. Niri, Modeling the impact of unreported cases of the COVID-19 in the North African countries, Biology, 9 (2020), 1–18. https://doi.org/10.3390/biology9110373 doi: 10.3390/biology9110373
[3]	S. Bentout, A. Tridane, S. Djilali, T. M. Touaoula, Age-structured modeling of COVID-19 epidemic in the USA, UAE and {A}lgeria, Alex. Eng. J., 60 (2021), 401–411.
[4]	M. Abdy, S. Side, S. Annas, W. Nur, W. Sanusi, An SIR epidemic model for COVID-19 spread with fuzzy parameter: the case of Indonesia, Adv. Differ. Equ., 2021 (2021), 1–17. https://doi.org/10.1186/s13662-021-03263-6 doi: 10.1186/s13662-021-03263-6
[5]	P. Brodin, Immune determinants of COVID-19 disease presentation and severity, Nat. Med., 27 (2021), 28–33. https://doi.org/10.1038/s41591-020-01202-8 doi: 10.1038/s41591-020-01202-8
[6]	W. J. Li, J. C. Ji, L. H. Huang, Z. Y. Guo, Global dynamics of a controlled discontinuous diffusive SIR epidemic system, Appl. Math. Lett., 121 (2021), 107420. https://doi.org/10.1016/j.aml.2021.107420 doi: 10.1016/j.aml.2021.107420
[7]	W. O. Kermack, A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. A, 115 (1927), 700–721. https://doi.org/10.1098/rspa.1927.0118 doi: 10.1098/rspa.1927.0118
[8]	N. A. Kudryashov, M. A. Chmykhov, M. Vigdorowitsch, Analytical features of the SIR model and their applications to COVID-19, Appl. Math. Model., 90 (2021), 466–473. https://doi.org/10.1016/j.apm.2020.08.057 doi: 10.1016/j.apm.2020.08.057
[9]	B. Q. Zhou, B. T. Han, D. Q. Jiang, Ergodic property, extinction and density function of a stochastic SIR epidemic model with nonlinear incidence and general stochastic perturbations, Chaos Solitons Fract., 152 (2021), 111338. https://doi.org/10.1016/j.chaos.2021.111338 doi: 10.1016/j.chaos.2021.111338
[10]	J. J. Wang, J. Z. Zhang, Z. Jin, Analysis of an SIR model with bilinear incidence rate, Nonlinear Anal., 11 (2010), 2390–2402. https://doi.org/10.1016/j.nonrwa.2009.07.012 doi: 10.1016/j.nonrwa.2009.07.012
[11]	Y. G. Lin, D. Q. Jiang, M. L. Jin, Stationary distribution of a stochastic SIR model with saturated incidence and its asymptotic stability, Acta Math. Sci., 35 (2015), 619–629. https://doi.org/10.1016/S0252-9602(15)30008-4 doi: 10.1016/S0252-9602(15)30008-4
[12]	C. J. Sun, W. Yang, J. Arino, K. Khan, Effect of media-induced social distancing on disease transmission in a two patch setting, Math. Biosci., 230 (2011), 87–95. https://doi.org/10.1016/j.mbs.2011.01.005 doi: 10.1016/j.mbs.2011.01.005
[13]	D. Kiouach, Y. Sabbar, Stability and threshold of a stochastic SIRS epidemic model with vertical transmission and transfer from infectious to susceptible individuals, Discrete Dyn. Nat. Soc., 2018 (2018), 1–13. https://doi.org/10.1155/2018/7570296 doi: 10.1155/2018/7570296
[14]	N. H. Du, N. N. Nhu, Permanence and extinction of certain stochastic SIR models perturbed by a complex type of noises, Appl. Math. Lett., 64 (2017), 223–230. https://doi.org/10.1016/j.aml.2016.09.012 doi: 10.1016/j.aml.2016.09.012
[15]	A. Kumar, Nilam, Dynamic behavior of an SIR epidemic model along with time delay; Crowley-Martin type incidence rate and Holling type II treatment rate, Int. J. Nonlinear Sci. Numer. Simul., 20 (2019), 757–771. https://doi.org/10.1515/ijnsns-2018-0208 doi: 10.1515/ijnsns-2018-0208
[16]	M. J. Faddy, Nonlinear stochastic compartmental models, Math. Med. Biol., 2 (1985), 287–297. https://doi.org/10.1093/imammb/2.4.287 doi: 10.1093/imammb/2.4.287
[17]	Y. Sabbar, A. Din, D. Kiouach, Predicting potential scenarios for wastewater treatment under unstable physical and chemical laboratory conditions: a mathematical study, Results Phys., 39 (2022), 105717. https://doi.org/10.1016/j.rinp.2022.105717 doi: 10.1016/j.rinp.2022.105717
[18]	Y. Sabbar, A. Zeb, D. Kiouach, N. Gul, T. Sitthiwirattham, D. Baleanu, et al., Dynamical bifurcation of a sewage treatment model with general higher-order perturbation, Results Phys., 39 (2022), 105799. https://doi.org/10.1016/j.rinp.2022.105799 doi: 10.1016/j.rinp.2022.105799
[19]	Y. Sabbar, A. Khan, A. Din, Probabilistic analysis of a marine ecological system with intense variability, Mathematics, 10 (2022), 1–19. https://doi.org/10.3390/math10132262 doi: 10.3390/math10132262
[20]	Y. Sabbar, D. Kiouach, New method to obtain the acute sill of an ecological model with complex polynomial perturbation, Math. Methods Appl. Sci., 2022. https://doi.org/10.1002/mma.8654
[21]	S. Ditlevsen, A. Samson, Introduction to stochastic models in biology, In: Stochastic biomathematical models, Berlin, Heidelber: Springer, 2013, 3–35. https://doi.org/10.1007/978-3-642-32157-3_1
[22]	S. P. Rajasekar, M. Pitchaimani, Q. X. Zhu, Dynamic threshold probe of stochastic SIR model with saturated incidence rate and saturated treatment function, Phys. A, 535 (2019), 122300. https://doi.org/10.1016/j.physa.2019.122300 doi: 10.1016/j.physa.2019.122300
[23]	D. Kiouach, Y. Sabbar, The threshold of a stochastic SIQR epidemic model with Lévy jumps, In: Trends in biomathematics: mathematical modeling for health, harvesting, and population dynamics, Cham: Springer, 2019, 87–105. https://doi.org/10.1007/978-3-030-23433-1_7
[24]	S. Winkelmann, C. Schütte, Stochastic dynamics in computational biology, Cham: Springer, 2020. https://doi.org/10.1007/978-3-030-62387-6
[25]	S. P. Rajasekar, M. Pitchaimani, Q. X. Zhu, K. B. Shi, Exploring the stochastic host-pathogen tuberculosis model with adaptive immune response, Math. Probl. Eng., 2021 (2021), 1–23. https://doi.org/10.1155/2021/8879538 doi: 10.1155/2021/8879538
[26]	S. P. Rajasekar, M. Pitchaimani, Q. X. Zhu, Probing a stochastic epidemic hepatitis C virus model with a chronically infected treated population, Acta Math. Sci., 42 (2022), 2087–2112. https://doi.org/10.1007/s10473-022-0521-1 doi: 10.1007/s10473-022-0521-1
[27]	N. S. Goel, N. Richter-Dyn, Stochastic models in biology, Academic Press, 1974.
[28]	D. Kiouach, Y. Sabbar, S. E. A. El-idrissi, New results on the asymptotic behavior of an SIS epidemiological model with quarantine strategy, stochastic transmission, and Lévy disturbance, Math. Methods Appl. Sci., 44 (2021), 13468–13492. https://doi.org/10.1002/mma.7638 doi: 10.1002/mma.7638
[29]	D. Kiouach, Y. Sabbar, Developing new techniques for obtaining the threshold of a stochastic SIR epidemic model with 3-dimensional Levy process, J. Appl. Nonlinear Dyn., 11 (2022), 401–414. https://doi.org/10.5890/JAND.2022.06.010 doi: 10.5890/JAND.2022.06.010
[30]	Y. Sabbar, D. Kiouach, S. P. Rajasekar, S. E. A. El-idrissi, The influence of quadratic Lévy noise on the dynamic of an SIC contagious illness model: new framework, critical comparison and an application to COVID-19 (SARS-CoV-2) case, Chaos Solitons Fract., 159 (2022), 112110. https://doi.org/10.1016/j.chaos.2022.112110 doi: 10.1016/j.chaos.2022.112110
[31]	D. Kiouach, Y. Sabbar, Threshold analysis of the stochastic Hepatitis B epidemic model with successful vaccination and Lévy jumps, In: 2019 4th World Conference on Complex Systems (WCCS), 2019, 1–6. https://doi.org/10.1109/ICoCS.2019.8930709
[32]	S. P. Rajasekar, M. Pitchaimani, Q. X. Zhu, Higher order stochastically perturbed SIRS epidemic model with relapse and media impact, Math. Methods Appl. Sci., 45 (2022), 843–863. https://doi.org/10.1002/mma.7817 doi: 10.1002/mma.7817
[33]	D. Kiouach, Y. Sabbar, The long-time behavior of a stochastic SIR epidemic model with distributed delay and multidimensional Lévy jumps, Int. J. Biomath., 15 (2021), 2250004. https://doi.org/10.1142/S1793524522500048 doi: 10.1142/S1793524522500048
[34]	D. Kiouach, Y. Sabbar, Dynamic characterization of a stochastic SIR infectious disease model with dual perturbation, Int. J. Biomath., 14 (2021), 2150016. https://doi.org/10.1142/S1793524521500169 doi: 10.1142/S1793524521500169
[35]	D. Kiouach, Y. Sabbar, Ergodic stationary distribution of a stochastic Hepatitis B epidemic model with interval-valued parameters and compensated Poisson process, Comput. Math. Methods Med., 2020 (2020), 1–12. https://doi.org/10.1155/2020/9676501 doi: 10.1155/2020/9676501
[36]	Q. Liu, D. Jiang, Stationary distribution and extinction of a stochastic SIR modelwith nonlinear perturbation, Appl. Math. Lett., 73 (2017), 8–15. https://doi.org/10.1016/j.aml.2017.04.021 doi: 10.1016/j.aml.2017.04.021
[37]	Q. Liu, D. Q. Jiang, Dynamical behavior of a higher order stochastically perturbed HIV/AIDS model with differential infectivity and amelioration, Chaos Solitons Fract., 141 (2020), 110333. https://doi.org/10.1016/j.chaos.2020.110333 doi: 10.1016/j.chaos.2020.110333
[38]	B. T. Han, D. Q. Jiang, T. Hayat, A. Alsaedi, B. Ahmed, Stationary distribution and extinction of a stochastic staged progression AIDS model with staged treatment and second-order perturbation, Chaos Solitons Fract., 140 (2020), 110238. https://doi.org/10.1016/j.chaos.2020.110238 doi: 10.1016/j.chaos.2020.110238
[39]	Y. Sabbar, D. Kiouach, S. P. Rajasekar, Acute threshold dynamics of an epidemic system with quarantine strategy driven by correlated white noises and Lévy jumps associated with infinite measure, Int. J. Dyn. Control, 2022. https://doi.org/10.1007/s40435-022-00981-x
[40]	N. Ikeda, S. Watanabe, A comparison theorem for solutions of stochastic differential equations and its applications, Osaka J. Math., 14 (1977), 619–633.
[41]	X. R. Mao, Stochastic differential equations and applications, Elsevier, 2007.
[42]	J. Y. Tong, Z. Z. Zhang, J. H. Bao, The stationary distribution of the facultative population model with a degenerate noise, Stat. Probabil. Lett., 83 (2013), 655–664. https://doi.org/10.1016/j.spl.2012.11.003 doi: 10.1016/j.spl.2012.11.003
[43]	D. L. Zhao, S. L. Yuan, Sharp conditions for the existence of a stationary distribution in one classical stochastic chemostat, Appl. Math. Comput., 339 (2018), 199–205. https://doi.org/10.1016/j.amc.2018.07.020 doi: 10.1016/j.amc.2018.07.020
[44]	N. T. Dieu, V. H. Sam, N. H. Du, Threshold of a stochastic SIQS epidemic model with isolation, Discrete Cont. Dyn. Syst. B, 27 (2022), 5009–5028. https://doi.org/10.3934/dcdsb.2021262 doi: 10.3934/dcdsb.2021262

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