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November  2020, 13(11): 3099-3114. doi: 10.3934/dcdss.2020116

## 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

Received  December 2018 Revised  March 2019 Published  November 2020 Early access  October 2019

Fund Project: 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

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.

Citation: Fang Liu, Zhen Jin, Cai-Yun Wang. Global analysis of SIRI knowledge dissemination model with recalling rate. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3099-3114. doi: 10.3934/dcdss.2020116
##### References:
 [1] M. T. Dora, H. Hussin and S. Sidek, Impacts of training on knowledge dissemination and application among academics in malaysian institutions of higher education, Asian Social Science, 8 (2012). doi: 10.5539/ass.v8n1p146. [2] B. van den Hooff and J. A. de Ridder, Knowledge sharing in context: The influence of organizational commitment, communication climate and cmc use on knowledge sharing, Journal of Knowledge Management, 8 (2004), 117-130.  doi: 10.1108/13673270410567675. [3] F. Llorens, J. J. Bayona, J. Gómez and F. Sanguino., The University of Alicante's institutional strategy to promote the open dissemination of knowledge, Online Information Review, 34 (2010), 565-582.  doi: 10.1108/14684521011072981. [4] R. Monclar, A. Tecla, J. Oliveira and J. M. de Souza, MEK: Using spatial-temporal information to improve social networks and knowledge dissemination, Information Sciences An International Journal, 179 (2009), 2524-2537.  doi: 10.1016/j.ins.2009.01.032. [5] M. M. Parent, D. MacDonald and G. Goulet, The theory and practice of knowledge management and transfer: The case of the Olympic Games, Sport Management Review, 17 (2014), 205-218.  doi: 10.1016/j.smr.2013.06.002. [6] M. Song, H. Berends, H. van der Bij and M. Weggeman, The effect of it and co-location on knowledge dissemination, Journal of Product Innovation Management, 24 (2007), 52-68.  doi: 10.1111/j.1540-5885.2006.00232.x.

show all references

##### References:
 [1] M. T. Dora, H. Hussin and S. Sidek, Impacts of training on knowledge dissemination and application among academics in malaysian institutions of higher education, Asian Social Science, 8 (2012). doi: 10.5539/ass.v8n1p146. [2] B. van den Hooff and J. A. de Ridder, Knowledge sharing in context: The influence of organizational commitment, communication climate and cmc use on knowledge sharing, Journal of Knowledge Management, 8 (2004), 117-130.  doi: 10.1108/13673270410567675. [3] F. Llorens, J. J. Bayona, J. Gómez and F. Sanguino., The University of Alicante's institutional strategy to promote the open dissemination of knowledge, Online Information Review, 34 (2010), 565-582.  doi: 10.1108/14684521011072981. [4] R. Monclar, A. Tecla, J. Oliveira and J. M. de Souza, MEK: Using spatial-temporal information to improve social networks and knowledge dissemination, Information Sciences An International Journal, 179 (2009), 2524-2537.  doi: 10.1016/j.ins.2009.01.032. [5] M. M. Parent, D. MacDonald and G. Goulet, The theory and practice of knowledge management and transfer: The case of the Olympic Games, Sport Management Review, 17 (2014), 205-218.  doi: 10.1016/j.smr.2013.06.002. [6] M. Song, H. Berends, H. van der Bij and M. Weggeman, The effect of it and co-location on knowledge dissemination, Journal of Product Innovation Management, 24 (2007), 52-68.  doi: 10.1111/j.1540-5885.2006.00232.x.
Schematic diagram for the knowledge dissemination model
Transfer diagram for knowledge dissemination SIRI model
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 $\lambda = 0.0072$, $\beta = 0.0033$, $K = 100$
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 $\lambda = 0.005$, $\beta = 0.0033$, $K = 130$
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 $\lambda = 0.005$, $\beta = 0.0034$, $K = 150$
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 $S(0) = 6$, $I(0) = 55$, $R(0) = 0$, $\lambda = 0.04$
The Numerical simulation for system (2) with $\beta = 0.01$, $\lambda = 0.004$, $\alpha_1 = 0.2$, $\alpha_2 = 0.4$, $\alpha_3 = 0.8$
The Numerical simulation for system (2) with $\alpha = 0.4$, $\lambda = 0.004$, $\beta_1 = 0.005$, $\beta_2 = 0.01$, $\beta_3 = 0.02$
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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