Research ArticleNo Access

Global dynamics of deterministic and stochastic epidemic systems with nonmonotone incidence rate

Tao Feng

http://orcid.org/0000-0001-8550-0626

School of Science, Nanjing University of Science and Technology, Nanjing 210094, P. R. China

E-mail Address: mathfengtao@163.com

Search for more papers by this author

and

Zhipeng Qiu

http://orcid.org/0000-0001-9041-4140

School of Science, Nanjing University of Science and Technology, Nanjing 210094, P. R. China

E-mail Address: smoller_1@163.com

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S1793524518501012Cited by:9 (Source: Crossref)

Abstract

This paper is devoted to studying the dynamics of a susceptible-infective-latent-infective (SILI) epidemic model that is subject to the combined effects of environmental noise and intervention strategy. We extend the classical SILI epidemic model from a deterministic framework to a stochastic one. For the deterministic case, the global stability analysis of the solution is carried out in terms of the basic reproduction number. For the stochastic case, sufficient conditions for the extinction of diseases are obtained. Then, the existence of stationary distribution and asymptotic behavior of the solution are further studied to illustrate the cycling phenomena of recurrent diseases. Numerical simulations are conducted to verify these analytical results. It is shown that both stochastic noise and intervention strategy contribute to the control of diseases.

Keywords:

Remember to check out the Most Cited Articles in IJB!

Check out new Biomathematics books in our Mathematics 2018 catalogue!
Featuring author Frederic Y M Wan and more!

References

1. L. Bolzoni, E. Bonacini, C. Soresina and M. Groppi, Time-optimal control strategies in SIR epidemic models, Math. Biosci. 292 (2017) 86–96. Web of Science, Google Scholar
2. S. Funk, M. Salathé and V. A. Jansen, Modeling the influence of human behaviour on the spread of infectious diseases: A review, J. Royal Soc. Int. Google Scholar
3. D. M. Morens, G. K. Folkers and A. S. Fauci, The challenge of emerging and re-emerging infectious diseases, Nature 430(6996) (2004) 242–249. Web of Science, Google Scholar
4. A. G. Murray, R. J. Smith and R. M. Stagg, Shipping and the spread of infectious salmon anemia in scottish aquaculture, Emerg. Infect. Dis. 8(1) (2002) 1–5. Web of Science, Google Scholar
5. H. W. Hethcote, The mathematics of infectious diseases, SIAM Review 42(4) (2000) 599–653. Web of Science, Google Scholar
6. T. Kypraios, P. Neal and D. Prangle, A tutorial introduction to bayesian inference for stochastic epidemic models using approximate bayesian computation, Math. Biosci. 287 (2017) 42–53. Web of Science, Google Scholar
7. J. Mena-Lorcat and H. W. Hethcote, Dynamic models of infectious diseases as regulators of population sizes, J. Math. Biol. 30(7) (1992) 693–716. Web of Science, Google Scholar
8. Z. Qiu and Z. Feng, Transmission dynamics of an influenza model with age of infection and antiviral treatment, J. Dyn. Diff. Eqs. 22(4) (2010) 823–851. Web of Science, Google Scholar
9. Z. Qiu and Z. Feng, Transmission dynamics of an influenza model with vaccination and antiviral treatment, Bull. Math. Biol. 72(1) (2010) 1–33. Web of Science, Google Scholar
10. Q. Wu, Y. Lou and W. Zhu, Epidemic outbreak for an sis model in multiplex networks with immunization, Math. Biosci. 277 (2016) 38–46. Web of Science, Google Scholar
11. P. van den Driessche and X. Zou, Modeling relapse in infectious diseases, Math. Biosci. 207(1) (2007) 89–103. Web of Science, Google Scholar
12. S. Blower, T. Porco and G. Darby, Predicting and preventing the emergence of antiviral drug resistance in HSV-2, Nature Med. 4(6) (1998) 673–678. Web of Science, Google Scholar
13. World health organization, herpes simplex virus (2017), http://www.who.int/ mediacentre/factsheets/fs400/en/#hsv2. Google Scholar
14. D. Tudor, A deterministic model for herpes infections in human and animal populations, SIAM Rev. 32(1) (1990) 136–139. Web of Science, Google Scholar
15. C. J. Silva and D. F. Torres, Optimal control for a tuberculosis model with reinfection and post-exposure interventions, Math. Biosci. 244(2) (2013) 154–164. Web of Science, Google Scholar
16. P. van den Driessche and X. Zou, Modeling diseases with latency and relapse, Math. Biosci. Eng. 4(2) (2007) 205–219. Web of Science, Google Scholar
17. Z. Feng, Z. Qiu, Z. Sang, C. Lorenzo and J. Glasser, Modeling the synergy between HSV-2 and HIV and potential impact of hsv-2 therapy, Math. Biosci. 245(2) (2013) 171–187. Web of Science, Google Scholar
18. Z. Feng, Z. Qiu and Z. Sang, Global stability of a 9-dimensional HSV-2 epidemic model, Can. Appl. Math. Quart. 19 (2011) 319–342. Google Scholar
19. T. Britton and A. Traoré, A stochastic vector-borne epidemic model: Quasi-stationarity and extinction, Math. Biosci. 289 (2017) 89–95. Web of Science, Google Scholar
20. N. Dalal, D. Greenhalgh and X. Mao, A stochastic model for internal HIV dynamics, J. Math. Anal. Appl. 341(2) (2008) 1084–1101. Web of Science, Google Scholar
21. D. Jiang, C. Ji, N. Shi and J. Yu, The long time behavior of di SIR epidemic model with stochastic perturbation, J. Math. Anal. Appl. 372(1) (2010) 162–180. Web of Science, Google Scholar
22. C. Ji and D. Jiang, The asymptotic behavior of a stochastic multigroup SIS model, Int. J. Biomath. 11(03) (2018) 1850037. Link, Web of Science, Google Scholar
23. N. Kirupaharan and L. J. S. Allen, Coexistence of multiple pathogen strains in stochastic epidemic models with density-dependent mortality, Bull. Math. Biol. 66(4) (2004) 841–864. Web of Science, Google Scholar
24. Q. Yang and X. Mao, Extinction and recurrence of multi-group seir epidemic models with stochastic perturbations, Nonlin. Anal.: Real World Appl. 14(3) (2013) 1434–1456. Web of Science, Google Scholar
25. B. Berrhazi, M. E. Fatini, A. Lahrouz, A. Settati and R. Taki, A stochastic SIRS epidemic model with a general awareness-induced incidence, Physica A: Stat. Mech. Appl. 512 (2018) 968–980. Web of Science, Google Scholar
26. B. Berrhazi, M. E. Fatini, A. Laaribi and R. Pettersson, A stochastic viral infection model driven by lévy noise, Chaos, Solitons Fractals 114 (2018) 446–452. Web of Science, Google Scholar
27. W. Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of sirs epidemiological models, J. Math. Biol. 23(2) (1986) 187–204. Web of Science, Google Scholar
28. S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Diff. Eqs. 188(1) (2003) 135–163. Web of Science, Google Scholar
29. X. Meng, S. Zhao, T. Feng and T. Zhang, Dynamics of a novel nonlinear stochastic sis epidemic model with double epidemic hypothesis, J. Math. Anal. Appl. 433(1) (2016) 227–242. Web of Science, Google Scholar
30. B. Soufiane and T. M. Touaoula, Global analysis of an infection age model with a class of nonlinear incidence rates, J. Math. Anal. Appl. 434(2) (2016) 1211–1239. Web of Science, Google Scholar
31. D. Li, S. Liu and J. Cui, Threshold dynamics and ergodicity of an SIRS epidemic model with markovian switching, J. Diff. Eqs. 263(12) (2017) 8873–8915. Web of Science, Google Scholar
32. R. Liu, J. Wu and H. Zhu, Media/psychological impact on multiple outbreaks of emerging infectious diseases, Comput. Math. Methods Med. 8(3) (2007) 153–164. Google Scholar
33. A. Misra, A. Sharma and J. Shukla, Modeling and analysis of effects of awareness programs by media on the spread of infectious diseases, Math. Comput. Model. 53(5) (2011) 1221–1228. Web of Science, Google Scholar
34. B. Berrhazi, M. El Fatini, A. Laaribi, R. Pettersson and R. Taki, A stochastic SIRS epidemic model incorporating media coverage and driven by lévy noise, Chaos, Solitons Fractals 105 (2017) 60–68. Web of Science, Google Scholar
35. Y. Cai, Y. Kang, M. Banerjee and W. Wang, A stochastic SIRS epidemic model with infectious force under intervention strategies, J. Diff. Eqs. 259(12) (2015) 7463–7502. Web of Science, Google Scholar
36. 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. Web of Science, Google Scholar
37. M. Liu and K. Wang, Survival analysis of a stochastic single-species population model with jumps in a polluted environment, Int. J. Biomath. 9(01) (2016) 1650011. Link, Web of Science, Google Scholar
38. A. Lahrouz, H. E. Mahjour, A. Settati and A. Bernoussi, Dynamics and optimal control of a nonlinear epidemic model with relapse and cure, Physica A: Stat. Mech. Appl. 496 (2018) 299–317. Web of Science, Google Scholar
39. P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180(1) (2002) 29–48. Web of Science, Google Scholar
40. J. Li, Y. Yang, Y. Xiao and S. Liu, A class of lyapunov functions and the global stability of some epidemic models with nonlinear incidence, J. Appl. Anal. Comput. 6 (2016) 38–46. Web of Science, Google Scholar
41. R. Khasminskii, Stochastic Stability of Differential Equations, Vol. 66 (Springer Science and Business Media, 2011). Google Scholar
42. D. Xu, Y. Huang and Z. Yang, Existence theorems for periodic markov process and stochastic functional differential equations, Discr. Cont. Dyn. Syst.-A 24(3) (2009) 1005–1023. Web of Science, Google Scholar
43. X. Mao, Stochastic Differential Equations and Applications (Elsevier, 2007). Google Scholar
44. M. E. Fatini, A. Lahrouz, R. Pettersson, A. Settati and R. Taki, Stochastic stability and instability of an epidemic model with relapse, Appl. Math. Comput. 316 (2018) 326–341. Web of Science, Google Scholar
45. D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev. 43(3) (2001) 525–546. Web of Science, Google Scholar
46. C. Vargas-De-León, On the global stability of infectious diseases models with relapse, Abstr. Appl. Mag. 9 (2014) 50–61. Google Scholar