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Abstract

In the process of rumor propagation, people who know the truth or judge the truth can spread true information about rumors. Therefore, on the rumor propagation, it is significant to introduce the spreaders who spread true information in the rumor propagation. But the previous studies did not take into consideration the influence of true information spreading on the rumor propagation. In this paper, the susceptible-infective-true-removed (SITR) rumor propagation model with the true information spreader and the forgetting factor of rumors is established. The threshold K ₀is obtained by using the method of the next generation matrix. If K ₀< 1, the balance between no rumor and no true information spreader is locally asymptotically stable. The existence and stability of two boundary balance (that is, there are rumor spreaders but no true information spreaders, and there are no rumor spreaders but true information spreaders) are proved. The bistable region of two-boundary balance is given. Further, under different conditions we obtain the existence and locally asymptotical stability of positive balance (rumor spreaders and true information spreaders coexist). Finally, the theoretical results are verified by numerical simulations. We find that the initial value of the true information spreaders affects the peak value of the rumor spreaders and the duration of the rumor. The bigger the initial value of the true information spreaders, the smaller the peak value of the rumor spreaders is and the shorter the duration of the rumor is. The initial value of the rumor spreaders affects the peak value of the rumor spreaders, and the time when the rumor spreaders reach the peak value. The larger the initial value of the rumor spreaders, the larger the peak value of the rumor spreaders is and the earlier the peak value appears. But the initial value of the rumor spreaders does not affect the duration of the rumor. Therefore, in the process of rumor propagation, according to mathematical analysis of the rumor propagation model, we find that the rumor spread is a very complicated process. The results of mathematical analysis can provide theoretical basis to control the rumor propagation and reduce the negative effects of rumors.

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Authors and contacts

Corresponding author:Jin Zhen,jinzhn@263.net

Funds:Project supported by the National Natural Science Foundation of China (Grant Nos. 61873154, 11601294) and the Natural Science Foundation of Shanxi Province, China (Grant Nos. 201801D121008, 201801D121206).

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系统(2)的平衡点
$ \epsilon< \left( {1 - {\alpha _1}} \right){\rm{\beta }}$	${K_2} > {K_1}$	${K_2} > \dfrac{{{K_1}\left( {\mu {R_1} + \theta } \right)}}{{\mu + \theta }}$	${K_2} < 1$	图3(a)中黄色区域${E_0},\;E_1^*$
			${K_2} > 1 > {K_1}$	图3(a)中绿色区域${E_0},\;{E_2},\;E_1^*$
			${K_1} > 1$	图3(a)中紫色区域${E_0},\;{E_1},\;{E_2},\;E_1^*$
		${K_2} < \dfrac{{{K_1}\left( {\mu {K_1} + \theta } \right)}}{{\mu + \theta }}$	$\epsilon < \left( {1 - 2{\alpha _1} } \right)\beta + \dfrac{ {\delta \theta } }{\mu },{\varDelta _1} \geqslant 0$	图3(a)中蓝色区域${E_0},\;{E_1},\;{E_2},\;E_6^,\;E_7^$
	${K_2} < {K_1}$	${K_1} < 1$		图3(a)中橙色区域$E_0$
		${K_1} > 1 > {K_2}$		图3(a)中红色区域$E_0,\;E_1$
		${K_1} > {K_2} > 1$		图3(a)中空白区域$E_0,\;E_1,\;E_2$
$\epsilon > \left( {1 - {\alpha _1}} \right){\rm{\beta }}$	${K_2} > {K_{1,}}$	${K_2} > \dfrac{{{K_1}\left( {\mu {K_1} + \theta } \right)}}{{\mu + \theta }}$	${K_2} < 1$	图3(b)中黄色区域$E_0,\;E_2^*$
			${K_2} > 1 > {K_1}$	图3(b)中绿色区域$E_0,\;E_2,\;E_2^*$
			${K_1} > 1$	图3(b)中紫色区域$E_0,\;E_1,\;E_2,\;E_2^*$
	$K_2	$K_1<1$		图3(b)中橙色区域$E_0$
	$K_2	$K_1>1>K_2$		图3(b)中红色区域$E_0,\;E_1$
	$1 < {K_2} < \dfrac{{{K_1}\left( {\mu {K_1} + \theta } \right)}}{{\mu + \theta }}$			图3(b)中空白区域$E_0,\;E_1,\;E_2$

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Vol. 68, Issue 15 - 2019-08-05

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