Global analysis of SIRI knowledge dissemination model with recalling rate

Fang Liu; Zhen Jin; Cai-Yun Wang

doi:10.3934/dcdss.2020116

Article Contents

2020, Volume 13, Issue 11: 3099-3114. Doi: 10.3934/dcdss.2020116

This issue Previous Article Bifurcations and exact traveling wave solutions for the nonlinear Schrödinger equation with fourth-order dispersion and dual power law nonlinearity Next Article Bifurcation of limit cycles in a family of piecewise smooth systems via averaging theory

Global analysis of SIRI knowledge dissemination model with recalling rate

1.
Department of Mathematics, Xinzhou Teachers University, Xinzhou, Shan'xi 034000, China
2.
Complex Systems Research Center, Shanxi University, Taiyuan Shan'xi 030006, China
3.
Shanxi Key Laboratory of Mathematical Techniques and, Big Data Analysis on Disease Control and Prevention, Taiyuan, Shan'xi 030006, China

^* Corresponding author: Zhen Jin
^* Corresponding author: Zhen Jin

Received: December 2018

Revised: March 2019

Early access: October 2019

Published: July 2020

The first author is supported by the College Scientific Research Project of XinZhou Normal University 2018KY14. The third author is supported by the key Construction Disciplines Project of Xinzhou Teachers University XK201501

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Abstract

In order to study the dissemination mechanism of knowledge, a SIRI dynamics model with the learning rate, the forgetting rate and the recalling rate is constructed in this paper. Stability of equilibria and global dynamics of the SIRI model are analyzed. Two thresholds that determine whether knowledge is disseminated are given. We describe the stability of the equilibria for the SIRI model in which there are an equilibrium and a line of equilibria. In particular, we find the dividing curve function which is used to partition invariant set in order to discuss the local stability, and obtain the equation of the wave peak value or wave trough value in the process of knowledge dissemination. Numerical simulations are provided to support the theoretical results. The complicated dynamics properties exhibit that the model is very sensitive to variation of parameters, which play an important role on controlling and administering the knowledge dissemination.

Keywords:

Mathematics Subject Classification: Primary: 34D23, 34D05; Secondary: 91E40, 91B52.

Citation:

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Figure 1. Schematic diagram for the knowledge dissemination model

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Figure 2. Transfer diagram for knowledge dissemination SIRI model

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Figure 3. When $ R_0 = 1 $ and keeping the fixed parameter $ \alpha = 0.72 $, the phase portraits with different others parameters are given. (a) Parameters are $ \lambda = \beta = 0.0036 $, $ K = 200 $. (b) Parameters are $ \lambda = 0.0072 $, $ \beta = 0.0172 $, $ K = 100 $. (c) Parameters are $ \lambda = 0.0072 $, $ \beta = 0.0033 $, $ K = 100 $

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Figure 4. When $ R_0<1 $ and keeping the fixed parameter $ \alpha = 0.72 $, the phase portraits with different others parameters are given. (a) Parameters are $ \lambda = \beta = 0.0033 $, $ K = 200 $. (b1) Parameters are $ \lambda = 0.0033 $, $ \beta = 0.005 $, $ K = 200 $. (b2) Parameters are $ \lambda = 0.0033 $, $ \beta = 0.00357 $, $ K = 200 $. (c) Parameters are $ \lambda = 0.005 $, $ \beta = 0.0033 $, $ K = 130 $

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Figure 5. When $ R_0>1 $ and keeping the fixed parameter $ \alpha = 0.72 $, the phase portraits with different others parameters are given. (a) Parameters are $ \lambda = \beta = 0.0032 $, $ K = 300 $. (b) Parameters are $ \lambda = 0.0033 $, $ \beta = 0.005 $, $ K = 250 $. (c1) Parameters are $ \lambda = 0.004 $, $ \beta = 0.0033 $, $ K = 230 $. (c2) Parameters are $ \lambda = 0.004 $, $ \beta = 0.0033 $, $ K = 200 $. (c3) Parameters are $ \lambda = 0.005 $, $ \beta = 0.0034 $, $ K = 150 $

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Figure 6. The Numerical simulation for system(2) with $ K = 61 $, $ \alpha = 0.4 $, $ \beta = 0.01 $. (a) Parameters are $ S(0) = 60 $, $ I(0) = 1 $, $ R(0) = 0 $, $ \lambda = 0.004 $. (b) Parameters are $ S(0) = 6 $, $ I(0) = 55 $, $ R(0) = 0 $, $ \lambda = 0.04 $

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Figure 7. The Numerical simulation for system (2) with $ \beta = 0.01 $, $ \lambda = 0.004 $, $ \alpha_1 = 0.2 $, $ \alpha_2 = 0.4 $, $ \alpha_3 = 0.8 $

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Figure 8. The Numerical simulation for system (2) with $ \alpha = 0.4 $, $ \lambda = 0.004 $, $ \beta_1 = 0.005 $, $ \beta_2 = 0.01 $, $ \beta_3 = 0.02 $

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Figure 9. The Numerical simulation for system (2) with $ \alpha = 0.4 $, $ \beta = 0.01 $, $ \lambda_1 = 0.002 $, $ \lambda_2 = 0.004 $, $ \lambda_3 = 0.008 $

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