 ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Nonlinear Dynamical Analysis of a Stochastic SIRS Epidemic System with Vertical Dissemination and Switch from Infectious to Susceptible Individuals

Journal of Applied Nonlinear Dynamics 11(3) (2022) 605--633 | DOI:10.5890/JAND.2022.09.007

Driss Kiouach, Yassine Sabbar

LPAIS Laboratory, Faculty of Sciences Dhar El Mahraz, Sidi Mohamed Ben Abdellah University, Fez, Morocco

Abstract

This study presents a Susceptible-Infected-Recovered-Susceptible (SIRS) system for the dynamics of epidemics in a non-closed human population. We consider the vertical transmission of the epidemic and incorporate varying periods of the individuals' immunity. The stochastic version of our new epidemic model is obtained by simultaneously introducing the stochastic transmission and the proportional perturbation. For the permanence case, the ergodicity of the perturbed system is proved by employing a new approach different from the Lyapunov method. Under the same condition of the existence of the unique ergodic stationary distribution, the persistence in the mean of the epidemic is shown. When the basic reproduction number $\mathcal{R}_0>1$, the asymptotic behavior of the solution around the endemic equilibrium of the deterministic model is provided. For the extinction case, sufficient conditions for the disappearance of the epidemic are obtained. When $\mathcal{R}_0\leq1$, the analysis of the asymptotic behavior around the disease-free equilibrium of the deterministic model is also investigated. In order to make the readers understand our study better, we present some numerical simulations to illustrate our theoretical results.

References

1.   Anderson, R. and May, R. (1991), Infectious Disease of Humans: Dynamics and Control, Oxford University Press.
2.   Anderson, R. and May, R. (1982), Population Biological of Infectious Disease, Springer.
3.   Kermack, W., MCKendrick, A., and Walker, G.T. (1927), A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society A, 115, 700-721.
4.   Guo, H., Li, M., and Shuai, Z.S. (2006), Global stability of the endemic equilibrium of multigroup SIR epidemic models, Canadian Applied Mathematics Quarterly, 14, 259-284.
5.   Meng, X.Z. and Chen, L.S. (2008), The dynamics of a new SIR epidemic model concerning pulse vaccination strategy, Applied Mathematics and Computation, 197, 528-597.
6.   Roy, M. and Holt, R.D. (2008), Effects of predation on host-pathogen dynamics in SIR models, Theoretical Population Biology, 73, 319-331.
7.   Yang, J. and Wang, X. (2018), Threshold dynamics of an SIR model with nonlinear incidence rate and age dependent susceptibility. Complexity, 2018, 1-15.
8.   Rashidinia, J., Sajjadian, M., Duarte, J., Januario, C., and Martins, N. (2018), On the dynamical complexity of a seasonally forced discrete SIR epidemic model with a constant vaccination strategy, Complexity, 2018, 1-11.
9.   Hethcote, H. (1976), Qualitative analyses of communicable disease models. Mathematical Biosciences, 28, 335-356.
10.   Lorca, J.M. and Hethcote, H. (1992), Dynamic models of infectious diseases as regulators of population size, Journal of Mathematical Biology, 30, 639-716.
11.   Muroya, Y., Li, H., and Kuniya, T. (2014), Complete global analysis of an {SIRS} epidemic model with graded cure and incomplete recovery rates, The Journal of Mathematical Analysis and Applications, 410, 719-732.
12.   Zhao, J., Wang, L., and Han, Z. (2018), Stability analysis of two new {SIRS} models with two viruses, Journal International Journal of Computer Mathematics, 95, 2026-2035.
13.   Capasso, V. and Serio, G. (2012), A generalization of the kermack mckendrick deterministic epidemic model, Mathematical Biosciences, 42, 43-61.
14.   Ma, Z., Zhou, Y., and Wu, J. (2009), Modeling and Dynamics of Infectious Diseases, Higher Education Press.
15.   Greenhalgh, D. and Moneim, I. (2003), SIRS epidemic model and simulations using different types of seasonal contact rate, Systems Analysis Modelling Simulation, 42, 573-600.
16.   Chen, J. (2004) An SIRS epidemic model, Applied Mathematics-A Journal of Chinese Universities, 19, 101-108.
17.   Li, T., Zhang, F., liu, H., and Chen, Y. (2017), Threshold dynamics of an {SIRS} model with nonlinear incidence rate and transfer from infectious to susceptible, Applied Mathematics Letters, 70, 52-57.
18.   Song, Y., Miao, A., Zhang, T., Wang, X., and Liu, J. (2018), Extinction and persistence of a stochastic SIRS epidemic model with saturated incidence rate and transfer from infectious to susceptible, Advances in Differential Equations, 2018, 1-11.
19.   Sprecher, S., Soumenkoff, G., Puissant, F., and Degueldre, M. (1986), Vertical transmission of HIV in 15-week fetus, The Lancet, 328, 288-289.
20.   Meng, X. and Chen, L. (2008), The dynamics of a new SIR epidemic model concerning pulse vaccination strategy, Applied Mathematics Letters, 197, 582-597.
21.   Lu, Z., Chi, X., and Chen, L. (2002), The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission, Mathematical and Computer Modelling, 36, 1039-1057.
22.   Bellenir, K. and Dresser, P. (1996), Contagious and Non-Contagious Infectious Diseases Source, Health Science Series.
23.   Busenberg, S. and Cooke, K. (1993), Vertically Transmitted Diseases, Springer-Verlag.
24.   Martcheva, M. (2015), An introduction to mathematical epidemiology, Springer.
25.   Driessche, P.V. and Watmough, J. (2002), Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical biosciences, 180, 29-48.
26.   Kiouach, D. and Sabbar, Y. (2018), Stability and threshold of a stochastic {SIRS} epidemic model with vertical transmission and transfer from infectious to susceptible individuals, Discrete Dynamics in Nature and Society, 2018, 1-13.
27.   Gray, A., Greenhalgh, D., Hu, L., Mao, X., and Pan, J. (2011), A stochastic differential equation SIS epidemic model, SIAM Journal on Applied Mathematics, 71, 876-902.
28.   Allen, L. (2008), An introduction to stochastic epidemic models, Mathematical Epidemiology, 144, 81-130.
29.   Kiouach, D. and Sabbar, Y. (2019), Modeling the impact of media intervention on controlling the diseases with stochastic perturbations, AIP Conference Proceedings, 2074, 1-15.
30.   Qi, H., Liu, L., and Meng, X. (2017), Dynamics of a nonautonomous stochastic {SIS} epidemic model with double epidemic hypothesis, Complexity, 2017, 1-14.
31.   Zhao, Y.,Li, Y., and Chen, Q. (2019), Analysis of a stochastic susceptible-infective epidemic model in a polluted atmospheric environment, Complexity, 2019, 1-14.
32.   Kiouach, D. and Sabbar, Y. (2020), Ergodic stationary distribution of a stochastic hepatitis B epidemic model with interval valued parameters and compensated poisson process, Computational and Mathematical Methods in Medicine, 2020, 1-12.
33.   Zhao, Y. and Jiang, D. (2014), The threshold of a stochastic {SIRS} epidemic model with saturated incidence, Applied Mathematics Letters, 34, 90-93.
34.   Cai, Y., Kang, Y., and Wang, W. (2017), A stochastic SIRS epidemic model with nonlinear incidence rate, Applied Mathematics and Computation, 305, 221-240.
35.   Zhao, Y., Jiang, D., and Mao, X. (2015), The threshold of a stochastic SIRS epidemic model in a population with varying size, Discrete and Continuous Dynamical Systems B, 20, 1289-1307.
36.   Liu, W. (2013), A SIRS epidemic model incorporating media coverage with random perturbation, Abstract and Applied Analysis.
37.   Ji, C. and Jiang, D. (2016), The threshold of a non-autonomous SIRS epidemic model with stochastic perturbations, Mathematical Methods in the Applied Sciences,
38.   Cai, Y., Kang, Y., and Wang, W. (2017), A stochastic sirs epidemic model with nonlinear incidence rate, Applied Mathematics and Computation, 305, 221-240.
39.   Zhang, X., Wang, X., and Huo, H. (2019), Extinction and stationary distribution of a stochastic sirs epidemic model with standard incidence rate and partial immunity, Physica A, 531, 1-14.
40.   Chen, C. and Kang, Y. (2016), The asymptotic behavior of a stochastic vaccination model with backward bifurcation, Applied Mathematical Modelling, 000, 1-18.
41.   Khasminskii, R. (1980), Stochastic stability of differential equations, A Monographs and Textbooks on Mechanics of Solids and Fluids, 7.
42.   Jiang, D., Yu, J., Ji, C., and Shi, N. (2011), Asymptotic behavior of global positive solution to a stochastic SIR model, Mathematical and Computer Modelling, 54, 221-232.
43.   Ji, C., Jiang, D., and Shi, N. (2012), The behavior of an SIR epidemic model with stochastic perturbation, Stochastic Analysis and Applications, 30, 221-232.
44.   Wu, B. and Jia, J. (2020), Asymptotic behavior of a stochastic delayed model for chronic hepatitis B infection, Complexity, 2020, 1-19.
45.   Mao, X. (1997), Stochastic Differential Equations and Applications, Horwoodl.
46.   Zhao, Y. and Jiang, D. (2014), The threshold of a stochastic sis epidemic model with vaccination, Applied Mathematics and Computation, 143, 718-727.
47.   Wang, Y. and Jiang, D. (2017), Stationary distribution and extinction of a stochastic viral infection model, Discrete Dynamics in Nature and Society, 2017, 1-13.
48.   Chen, Y., Wen, B., and Teng, Z. (2018), The global dynamics for a stochastic SIS epidemic model with isolation, Physica A, 492, 1604-1624.
49.   Stettner, L. (1986), On the existence and uniqueness of invariant measure for continuous-time markov processes, Technical Report, LCDS, Brown University, province, RI, pp. 18-86.
50.   Tong, J., Zhang, Z., and Bao, J. (2013), The stationary distribution of the facultative population model with a degenerate noise, Statistics and Probability Letters, 83, 655-664.
51.   Zhang, Q. and Zhou, K. (2019), Extinction and persistence of a stochastic sirs model with nonlinear incidence rate and transfer from infectious to susceptible, Journal of Physics: Conference Series, 1324, 1-10.
52.   Song, Y., Miao, A., Zhang, T., Wang, X., and Liu, J. (2018), Extinction and persistence of a stochastic sirs epidemic model with saturated incidence rate and transfer from infectious to susceptible, Advances in Difference Equations, 2018, 1-11.
53.   Wang, Y. and Liu, G. (2020), Stability of a nonlinear stochastic epidemic model with transfer from infectious to susceptible, Complexity, 2020, 1-12.
54.   Wang, Y. and Liu, G. (2019), Dynamics analysis of a stochastic sirs epidemic model with nonlinear incidence rate and transfer from infectious to susceptible, Mathematical biosciences and engineering, 16, 6074-6070.
55.   Li, T., Zhang, F., Liu, H., and Chen, Y. (2017), Threshold dynamics of an sirs model with nonlinear incidence rate and transfer from infectious to susceptible, Applied Mathematics Letters, 70, 52-57.
56.   Bentout, S., Kumar, S., and Djilali, S. (2021), Hopf bifurcation analysis in an age-structured heroin model, The European Physical Journal Plus, 136.
57.   Bentout, S., Ghanbari, B., Djilali, S., and Guin, L.N. (2021), Impact of predation in the spread of an infectious disease with time fractional derivative and social behavior, International Journal of Modeling, Simulation, and Scientific Computing, https://doi.org/10.1142/S1793962321500239.