Asymptotic behavior of a stochastic delayed avian influenza model with saturated incidence rate

Yanyan Du; Ting Kang; Qimin Zhang; Yanyan Du; Ting Kang; Qimin Zhang

doi:10.3934/mbe.2020289

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 5: 5341-5368. doi: 10.3934/mbe.2020289

Previous Article Next Article

Research article Special Issues

Asymptotic behavior of a stochastic delayed avian influenza model with saturated incidence rate

1.
School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China
2.
Xinhua College, Ningxia University, Yinchuan 750021, China

Received: 18 May 2020 Accepted: 03 August 2020 Published: 11 August 2020

In this paper, we establish a stochastic delayed avian influenza model with saturated incidence rate. Firstly, we prove the existence and uniqueness of the global positive solution with any positive initial value. Then, we study the asymptotic behaviors of the disease-free equilibrium and the endemic equilibrium by constructing some suitable Lyapunov functions and applying the Young's inequality and H?lder's inequality. If $\mathscr{R}_0 < 1$, then the solution of stochastic system is going around disease-free equilibrium while the solution of stochastic system is going around endemic equilibrium as $\mathscr{R}_0 >1$. Finally, some numerical examples are carried out to illustrate the accuracy of the theoretical results.
- stochastic avian influenza,
- time delays,
- saturated incidence rate,
- Lyapunov function,
- asymptotic behavior
Citation: Yanyan Du, Ting Kang, Qimin Zhang. Asymptotic behavior of a stochastic delayed avian influenza model with saturated incidence rate[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 5341-5368. doi: 10.3934/mbe.2020289

Related Papers:

Abstract

In this paper, we establish a stochastic delayed avian influenza model with saturated incidence rate. Firstly, we prove the existence and uniqueness of the global positive solution with any positive initial value. Then, we study the asymptotic behaviors of the disease-free equilibrium and the endemic equilibrium by constructing some suitable Lyapunov functions and applying the Young's inequality and H?lder's inequality. If $\mathscr{R}_0 < 1$, then the solution of stochastic system is going around disease-free equilibrium while the solution of stochastic system is going around endemic equilibrium as $\mathscr{R}_0 >1$. Finally, some numerical examples are carried out to illustrate the accuracy of the theoretical results.

References

[1]	Centers for Disease Control and Prevention (CDC), Avian influenza, 2017. Available from: https://www.cdc.gov/flu/avianflu/influenza-a-virus-subtypes.htm.
[2]	S. Liu, S. Ruan, X. Zhang, Nonlinear dynamics of avian influenza epidemic models, Math. Biosci., 283 (2017), 118-135.
[3]	S. Iwami, Y. Takeuchi, X. Liu, Avian-human influenza epidemic model, Math. Biosci., 207 (2007), 1-25.
[4]	A. B. Gumel, Global dynamics of a two-strain avian influenza model, Int. J. Comput. Math., 86 (2009), 85-108.
[5]	Q. Tang, J. Ge, Z. Lin, An SEI-SI avian-human influenza model with diffusion and nonlocal delay, Appl. Math. Comput., 247 (2014), 753-761.
[6]	Y. Chen, Y. Wen, Global dynamic analysis of a H7N9 avian-human influenza model in an outbreak region, J. Theor. Biol., 367 (2015), 180-188.
[7]	S. Liu, S. Ruan, X. Zhang, On avian influenza epidemic models with time delay, Theor. Biosci., 134 (2015), 75-82.
[8]	N. S. Chong, J. M. Tchuenche, J. S. Robert, A mathematical model of avian influenza with halfsaturated incidence, Theor. Biosci., 133 (2013), 23-38.
[9]	Z. Liu, C. T. Fang, A modeling study of human infections with avian influenza A H7N9 virus in mainland China, Int. J. Infect. Dis., 41 (2015), 73-78.
[10]	C. Modnak, J. Wang, An avian influenza model with latency and vaccination, Dynam. Syst., 34 (2018), 195-217.
[11]	A. K. Muhammad, F. Muhammad, I. Saeed, B. Ebenezer, Modeling the transmission dynamics of avian influenza with saturation and psychological effect, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 455-474.
[12]	T. Kang, Q. Zhang, L. Rong, A delayed avian influenza model with avian slaughter: Stability analysis and optimal control, Physica A, 529 (2019), 121544.
[13]	Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993.
[14]	E. Beretta, Y. Takeuchi, Global stability of an sir epidemic model with time delays, J. Math. Biol., 33 (1995), 250-260.
[15]	M. Liu, C. Bai, K. Wang, Asymptotic stability of a two-group stochastic SEIR model with infinite delays, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 3444-3453.
[16]	Q. Liu, D. Jiang, N. Shi, T. Hayat, A. Alsaedi, Asymptotic behavior of a stochastic delayed seir epidemic model with nonlinear incidence, Physica A, 462 (2016), 870-882.
[17]	G. P. Samanta, Permanence and extinction for a nonautonomous avian-human influenza epidemic model with distributed time delay, Math. Comput. Model., 52 (2010), 1794-1811.
[18]	J. Semenza, B. Menne, Climate change and infectious diseases in europe, Lancet. Infect. Dis., 9 (2009), 365-375.
[19]	A. Lowen, J. Steel, Roles of humidity and temperature in shaping influenza seasonality, J. Virol., 88 (2014), 7692-7705.
[20]	X. Zhang, Global dynamics of a stochastic avian-human influenza epidemic model with logistic growth for avian population, Nonlinear Dynam., 90 (2017), 2331-2343.
[21]	F. Zhang, X. Zhang, The threshold of a stochastic avian-human influenza epidemic model with psychological effect, Physica A, 492 (2018), 485-495.
[22]	X. Zhang, Z. Shi, Y. Wang, Dynamics of a stochastic avian-human influenza epidemic model with mutation, Physica A, 534 (2019), 12940.
[23]	D. Jiang, J. Yu, C. Ji, N. Shi, Asymptotic behavior of global positive solution to a stochastic sir model, Math. Comput. Model., 54 (2011), 221-232.
[24]	Q. Liu, D. Jiang, T. Hayat, A. Alsaedi, Threshold behavior in a stochastic delayed sis epidemic model with vaccination and double diseases, J. Franklin I., 356 (2019), 7466-7485.
[25]	J. H. Desmond, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)