Research ArticleNo Access

A regime-switching stochastic SIR epidemic model with a saturated incidence and limited medical resources

Wei Wei

Department of Applied Mathematics, Northwestern Polytechnical University Xi’an, 710072, Shaanxi, P. R. China

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Wei Xu

Department of Applied Mathematics, Northwestern Polytechnical University Xi’an, 710072, Shaanxi, P. R. China

E-mail Address: weixunpu@nwpu.edu.cn

Corresponding author.

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, and

Jiankang Liu

Department of Applied Mathematics, Northwestern Polytechnical University Xi’an, 710072, Shaanxi, P. R. China

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https://doi.org/10.1142/S1793524522501248Cited by:4 (Source: Crossref)

Abstract

The stochastic switching SIR epidemic model with saturated incidence and limited medical treatment is investigated in this paper. By using Lyapunov methods and Itô formula, we first prove that the system has a unique global positive solution with any positive initial value. Then combining inequality technique and the ergodic property of Markov switching, the sufficient conditions for extinction and persistence in the mean of the disease are established. The results demonstrate that increasing medical resources and improving supply efficiency can accelerate the transition from the persistent state to the extinct state. Meanwhile, the high incidence rate will slow down the extinction of the disease. Specially, the switching noise can induce the system to toggle between the extinct and persistent states. Finally, some numerical simulations are carried out to confirm the analytical results.

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References

1. W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. London Ser. A, Contain. Papers Math. Phys. Character 115(772) (1927) 700–721. Crossref, Google Scholar
2. W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics-II. The problem of endemicity, Bull. Math. Biol. 53(1–2) (1991) 57–87. Web of Science, Google Scholar
3. G. Zaman, H. K. Yong and I. H. Jung, Stability analysis and optimal vaccination of an SIR epidemic model, Biosystems 93(3) (2008) 240–249. Crossref, Web of Science, Google Scholar
4. C. Ji, D. Jiang and N. Shi, The behavior of an SIR epidemic model with stochastic perturbation, Stoch. Anal. Appl. 30(5) (2012) 755–773. Crossref, Web of Science, Google Scholar
5. Q. Liu and D. Jiang, The threshold of a stochastic delayed SIR epidemic model with vaccination, Phys. A: Stat. Mech. Appl. 461 (2016) 140–147. Crossref, Web of Science, Google Scholar
6. L. Qi and X. Mao, Stochastic population dynamics under regime switching II, J. Math. Anal. Appl. 334(1) (2007) 69–84. Crossref, Web of Science, Google Scholar
7. Y. Lin and D. Jiang, Long-time behaviour of a perturbed SIR model by white noise, Discrete Contin. Dyn. Syst. B 18(7) (2013) 1873. Crossref, Web of Science, Google Scholar
8. Y. Zhou, W. Zhang and S. Yuan, Survival and stationary distribution of a SIR epidemic model with stochastic perturbations, Appl. Math. Comput. 244 (2014) 118–131. Crossref, Web of Science, Google Scholar
9. A. Lahrouz and A. Settati, Necessary and sufficient condition for extinction and persistence of SIRS system with random perturbation, Appl. Math. Comput. 233 (2014) 10–19. Crossref, Web of Science, Google Scholar
10. H. Alsakaji, F. Rihan and A. Hashish, Dynamics of a stochastic epidemic model with vaccination and multiple time-delays for COVID-19 in the UAE, Complexity 33 (2022) 4247800. Google Scholar
11. C. Ji and D. Jiang, Threshold behaviour of a stochastic SIR model, Appl. Math. Model. 38(21–22) (2014) 5067–5079. Crossref, Web of Science, Google Scholar
12. J.-J. Wang, J.-Z. Zhang and Z. Jin, Analysis of an SIR model with bilinear incidence rate, Nonlinear Anal.: Real World Appl. 11(4) (2010) 2390–2402. Crossref, Web of Science, Google Scholar
13. F. A. Rihan, H. J. Alsakaji and C. Rajivganthi, Stochastic SIRC epidemic model with time-delay for COVID-19, Adv. Difference Equations 2020(1) (2020) 1–20. Crossref, Web of Science, Google Scholar
14. Y. Zhao and D. Jiang, The threshold of a stochastic SIRS epidemic model with saturated incidence, Appl. Math. Lett. 34 (2014) 90–93. Crossref, Web of Science, Google Scholar
15. X. Song, Y. Jiang and H. Wei, Analysis of a saturation incidence SVEIRS epidemic model with pulse and two time delays, Appl. Math. Comput. 214(2) (2009) 381–390. Crossref, Web of Science, Google Scholar
16. Q. Yang, D. Jiang, N. Shi and C. Ji, The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence, J. Math. Anal. Appl. 388(1) (2012) 248–271. Crossref, Web of Science, Google Scholar
17. Q. Liu and Q. Chen, Dynamics of a stochastic SIR epidemic model with saturated incidence, Appl. Math. Comput. 282 (2016) 155–166. Web of Science, Google Scholar
18. G. Lan, Z. Chen, C. Wei and S. Zhang, Stationary distribution of a stochastic SIQR epidemic model with saturated incidence and degenerate diffusion, Phys. A: Stat. Mech. Appl. 511 (2018) 61–77. Crossref, Web of Science, Google Scholar
19. Z. Feng and H. R. Thieme, Recurrent outbreaks of childhood diseases revisited: The impact of isolation, Math. Biosci. 128(1–2) (1995) 93–130. Crossref, Web of Science, Google Scholar
20. J. M. Hyman and J. Li, Modeling the effectiveness of isolation strategies in preventing STD epidemics, SIAM J. Appl. Math. 58(3) (1998) 912–925. Crossref, Web of Science, Google Scholar
21. L.-I. Wu and Z. Feng, Homoclinic bifurcation in an SIQR model for childhood diseases, J. Differential Equations 168(1) (2000) 150–167. Crossref, Web of Science, Google Scholar
22. W. Wang and S. Ruan, Bifurcations in an epidemic model with constant removal rate of the infectives, J. Math. Anal. Appl. 291(2) (2004) 775–793. Crossref, Web of Science, Google Scholar
23. W. Wang, Backward bifurcation of an epidemic model with treatment, Math. Biosci. 201(1–2) (2006) 58–71. Crossref, Web of Science, Google Scholar
24. X. Zhang and X. Liu, Backward bifurcation of an epidemic model with saturated treatment function, J. Math. Anal. Appl. 348(1) (2008) 433–443. Crossref, Web of Science, Google Scholar
25. L. Zhou and F. Meng, Dynamics of an SIR epidemic model with limited medical resources revisited, Nonlinear Anal. Real World Appl. 13(1) (2012) 312–324. Crossref, Web of Science, Google Scholar
26. A. Gray, D. Greenhalgh, L. Hu, X. Mao and J. Pan, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math. 71(3) (2011) 876–902. Crossref, Web of Science, Google Scholar
27. D. Zhao, Study on the threshold of a stochastic SIR epidemic model and its extensions, Commun. Nonlinear Sci. Numer. Simul. 38 (2016) 172–177. Crossref, Web of Science, Google Scholar
28. A. Lahrouz and L. Omari, Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence, Stat. Probab. Lett. 83(4) (2013) 960–968. Crossref, Web of Science, Google Scholar
29. R. M. May, Stability and complexity in model ecosystems, in Stability and Complexity in Model Ecosystems (Princeton University Press, 2019). Crossref, Google Scholar
30. C. Ji and D. Jiang, The extinction and persistence of a stochastic SIR model, Adv. Difference Equations 2017(1) (2017) 30. Crossref, Web of Science, Google Scholar
31. A. Raza, M. S. Arif and M. Rafiq, A reliable numerical analysis for stochastic gonorrhea epidemic model with treatment effect, Int. J. Biomath. 12(6) (2019) 1950072. Link, Web of Science, Google Scholar
32. L. Xin, Y. Guo and Q. Zhu, Stochastic epidemic dynamics based on the association between susceptible and recovered individuals, Int. J. Biomath. 14(1) (2021) 2050085. Link, Web of Science, Google Scholar
33. D. Jiang, J. Yu, C. Ji and N. Shi, Asymptotic behavior of global positive solution to a stochastic SIR model, Math. Comput. Model. 54(1–2) (2011) 221–232. Crossref, Web of Science, Google Scholar
34. R. Xu and Z. Ma, Stability of a delayed SIRS epidemic model with a nonlinear incidence rate, Chaos Solitons Fractals 41(5) (2009) 2319–2325. Crossref, Web of Science, Google Scholar
35. Q. Han, L. Chen and D. Jiang, A note on the stationary distribution of stochastic SEIR epidemic model with saturated incidence rate, Sci. Rep. 7(1) (2017) 3996. Crossref, Web of Science, Google Scholar
36. F. Wei and F. Chen, Stochastic permanence of an SIQS epidemic model with saturated incidence and independent random perturbations, Phys. A: Stat. Mech. Appl. 453 (2016) 99–107. Crossref, Web of Science, Google Scholar
37. Q. Yang, D. Jiang, N. Shi and C. Ji, The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence, J. Math. Anal. Appl. 388(1) (2017) 248–271. Crossref, Web of Science, Google Scholar
38. A. Lahrouz and L. Omari, Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence, Stat. Probab. Lett. 83(4) (2013) 960–968. Crossref, Web of Science, Google Scholar
39. C. Liu, L. Wang, D. He and M. Li, Stochastic dynamical analysis in a hybrid bioeconomic system with telephone noise and distributed delay, J. Franklin Inst. 357(8) (2020) 4922–4948. Crossref, Web of Science, Google Scholar
40. Q. Luo and X. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl. 334(1) (2007) 69–84. Crossref, Web of Science, Google Scholar
41. Q. Liu and D. Jiang, Dynamics of a multigroup SIS epidemic model with standard incidence rates and Markovian switching, Phys. A: Stat. Mech. Appl. 527 (2019) 121270. Crossref, Web of Science, Google Scholar
42. N. Dang, N. Du and G. Yin, Existence of stationary distributions for Kolmogorov systems of competitive type under telegraph noise, J. Differential Equations 257(6) (2014) 2078–2101. Crossref, Web of Science, Google Scholar
43. A. Settati and A. Lahrouz, Stationary distribution of stochastic population systems under regime switching, Appl. Math. Comput. 244 (2014) 235–243. Crossref, Web of Science, Google Scholar
44. A. Gray, D. Greenhalgh, X. Mao and J. Pan, The SIS epidemic model with Markovian switching, J. Math. Anal. Appl. 394(2) (2012) 496–516. Crossref, Web of Science, Google Scholar
45. Z. Chang, X. Meng and X. Lu, Analysis of a novel stochastic SIRS epidemic model with two different saturated incidence rates, Phys. A: Stat. Mech. Appl. 472 (2017) 103–116. Crossref, Web of Science, Google Scholar
46. Q. Liu and Q. Chen, Dynamics of a stochastic SIR epidemic model with saturated incidence, Appl. Math. Comput. 282 (2016) 155–166. Web of Science, Google Scholar
47. D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev. 43(3) (2001) 525–546. Crossref, Web of Science, Google Scholar
48. Z. Xu and X. Liu, Backward bifurcation of an epidemic model with saturated treatment function, J. Math. Anal. Appl. 348(1) (2008) 433–443. Crossref, Web of Science, Google Scholar
49. F. Wang, X. Wang, S. Zhang and C. Ding, On pulse vaccine strategy in a periodic stochastic SIR epidemic model, Chaos Solitons Fractals 66 (2014) 127–135. Crossref, Web of Science, Google Scholar
50. H. Guo, L. Chen and X. Song, Dynamical properties of a kind of SIR model with constant vaccination rate and impulsive state feedback control, Int. J. Biomath. 10(7) (2017) 1750093. Link, Web of Science, Google Scholar