American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdsb.2020274

A theoretical approach to understanding rumor propagation dynamics in a spatially heterogeneous environment

Linhe Zhu 1,, , Wenshan Liu 1,2, and  Zhengdi Zhang 1,

1. 

School of Mathematical Sciences, Jiangsu University, Zhenjiang, 212013, China
2. 

School of Mathematical Sciences, Nanjing Normal University Nanjing, 210023, China

* Corresponding author: Linhe Zhu

Received  March 2020 Revised  July 2020 Published  September 2020

Fund Project: The first author is supported by National Natural Science Foundation of China (Grant No.12002135), China Postdoctoral Science Foundation (Grant No.2019M661732), Natural Science Foundation of Jiangsu Province (Grant No.BK20190836) and Natural Science Research of Jiangsu Higher Education Institutions of China (Grant No.19KJB110001). The third author is supported by National Natural Science Foundation of China (Grant No.11872189)

Most of the previous work on rumor propagation either focus on ordinary differential equations with temporal dimension or partial differential equations (PDE) with only consideration of spatially independent parameters. Little attention has been given to rumor propagation models in a spatiotemporally heterogeneous environment. This paper is dedicated to investigating a SCIR reaction-diffusion rumor propagation model with a general nonlinear incidence rate in both heterogeneous and homogeneous environments. In spatially heterogeneous case, the well-posedness of global solutions is established first. The basic reproduction number $ R_0 $ is introduced, which can be used to reveal the threshold-type dynamics of rumor propagation: if $ R_0 < 1 $, the rumor-free steady state is globally asymptotically stable, while $ R_0 > 1 $, the rumor is uniformly persistent. In spatially homogeneous case, after introducing the time delay, the stability properties have been extensively studied. Finally, numerical simulations are presented to illustrate the validity of the theoretical analysis and the influence of spatial heterogeneity on rumor propagation is further demonstrated.

Keywords: Spatial heterogeneity, Reaction-diffusion model, Basic reproduction number, Stability, Uniform persistence.
Mathematics Subject Classification: Primary:35K57;Secondary:92D25.
Citation: Linhe Zhu, Wenshan Liu, Zhengdi Zhang. A theoretical approach to understanding rumor propagation dynamics in a spatially heterogeneous environment. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020274
References:
[1]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in: H.J. Schmeisser, H. Triebel (Eds.), Function Spaces, Differential Operators and Nonlinear Analysis (Friedrichroda, 1992), in: Teubner-Texte zur Mathematik, vol 133, Teubner, Stuttgart, 1993, 9-126. doi: 10.1007/978-3-663-11336-2_1.  Google Scholar

[2]

L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic patch model, Siam Journal on Applied Mathematics, 67 (2007), 1283-1309.  doi: 10.1137/060672522.  Google Scholar

[3]

Y. L. Cai, Y. Kang, M. Banerjee and W. M. Wang, Complex Dynamics of a host-parasite model with both horizontal and vertical transmissions in a spatial heterogeneous environment, Nonlinear Analysis: Real World Applications, 40 (2018), 444-465.  doi: 10.1016/j.nonrwa.2017.10.001.  Google Scholar

[4]

Y. L. Cai, X. Z. Lian, Z. H. Peng and W. M. Wang, Spatiotemporal transmission dynamics for influenza disease in a heterogenous environment, Nonlinear Analysis: Real World Applications, 46 (2019), 178-194.  doi: 10.1016/j.nonrwa.2018.09.006.  Google Scholar

[5]

T. Chen, L. Chen, X. Xu, Y. F. Cai, H. B. Jiang and X. Q. Sun, Reliable sideslip angle estimation of four-wheel independent drive electric vehicle by information iteration and fusion, Mathematical Problems in Engineering, 2018 (2018), 9075372, 14pp. doi: 10.1155/2018/9075372.  Google Scholar

[6]

D. J. Daley and D. G. Kendall, Epidemic and rumors, Nature, 204 (1964), 1118. Google Scholar

[7]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, Journal of Mathematical Biology, 28 (1990), 365-382.  doi: 10.1007/BF00178324.  Google Scholar

[8]

Z. M. Guo, F. B. Wang and X. F. Zou, Threshold dynamics of an infective disease model with a fixed latent period and non-local infections, Journal of Mathematical Biology, 65 (2012), 1387-1410.  doi: 10.1007/s00285-011-0500-y.  Google Scholar

[9]

J. Groeger, Divergence theorems and the supersphere, Journal of Geometry And Physics, 77 (2014), 13-29.  doi: 10.1016/j.geomphys.2013.11.004.  Google Scholar

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J. K. Hale, Asymptotic Behavior of Dissipative Systems, , American Mathematical Society, Providence, RI, 1988.  Google Scholar

[11]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[12]

H. W. Hethcote, The mathematical of infectious diseases, SIAM Review, 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar

[13]

X. L. Lai and X. F. Zou, Repulsion effect on superinfecting virions by infected cells, Bulletin of Mathematical Biology, 76 (2014), 2806-2833.  doi: 10.1007/s11538-014-0033-9.  Google Scholar

[14]

J. R. Li, H. J. Jiang, Z. Y. Yu and C. Hu, Dynamical analysis of rumor spreading model in homogeneous complex networks, Applied Mathematics and Computation, 359 (2019), 374-385.  doi: 10.1016/j.amc.2019.04.076.  Google Scholar

[15]

X. Liang, L. Zhang and X. Q. Zhao, Basic reproduction ratios for periodic abstract functional differential equations (with application to a spatial model for lyme disease), Journal of Dynamic and Differential Equations, 31 (2019), 1247-1278.  doi: 10.1007/s10884-017-9601-7.  Google Scholar

[16]

Y. J. Lou and X. Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, Journal of Mathematical Biology, 62 (2011), 543-568.  doi: 10.1007/s00285-010-0346-8.  Google Scholar

[17]

Y. T. Luo, L. Zhang, T. T. Zheng and Z. D. Teng, Analysis of a diffusive virus infection model with humoral immunity, Cell-to-cell Transmission and Nonlinear Incidence. Physica A, 535 (2019), 122415, 20pp. doi: 10.1016/j.physa.2019.122415.  Google Scholar

[18]

D. Maki and M. Thomson, Mathematical Models and Applications, Prentice-Hall, Englewood Cliffs, 1973.  Google Scholar

[19]

P. Miao, Z. D. Zhang, C. W. Lim and X. D. Wang, Hopf bifurcation and hybrid control of a delayed ecoepidemiological model with nonlinear incidence rate and Holling type Ⅱ functional response, Mathematical Problems in Engineering, 2018 (2018), 6052503, 12pp. doi: 10.1155/2018/6052503.  Google Scholar

[20]

R. Peng and X. Q. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471.  doi: 10.1088/0951-7715/25/5/1451.  Google Scholar

[21]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Pren-tice Hall, Englewood Cliffs, 1967.  Google Scholar

[22]

X. Ren, Y. Tian, L. Liu and X. Liu, A reaction-diffusion within-host HIV model with cell-to-cell transmission, Journal of Mathematical Biology, 76 (2018), 1831-1872.  doi: 10.1007/s00285-017-1202-x.  Google Scholar

[23]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, in: Math. Surveys Monger. vol. 41, American Mathematical Society, Providence, RI, 1995.  Google Scholar

[24]

H. L. Smith and X. Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Analysis: Theory Methods & Applications, 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2.  Google Scholar

[25]

S. T. Tang, Z. D. Teng and H. Miao, Global dynamics of a reaction-diffusion virus infection model with humoral immunity and nonlinear incidence, Computers and Mathematics with Applications, 78 (2019), 786-806.  doi: 10.1016/j.camwa.2019.03.004.  Google Scholar

[26]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM Journal on Applied Mathematics, 70 (2009), 188-211.  doi: 10.1137/080732870.  Google Scholar

[27]

H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, Journal of Mathematical Biology, 30 (1992), 755-763.  doi: 10.1007/BF00173267.  Google Scholar

[28]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[29]

N. K. Vaidya, F. B. Wang and X. F. Zou, Avian influenza dynamics in wild birds with bird mobility and spatial heterogeneous environment, Discrete and Continuous Dynamical Systems, 17 (2012), 2829-2848.  doi: 10.3934/dcdsb.2012.17.2829.  Google Scholar

[30]

J. Wang, L. Zhao and R. Huang, SIRaRu rumor spreading model in complex networks, Physica A, 398 (2014), 43-55.  doi: 10.1016/j.physa.2013.12.004.  Google Scholar

[31]

W. Wang and X. Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM Journal on Applied Dynamical Systems, 11 (2012), 1652-1673.  doi: 10.1137/120872942.  Google Scholar

[32]

W. Wang and X. Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM Journal on Applied Mathematics, 71 (2011), 147-168.  doi: 10.1137/090775890.  Google Scholar

[33]

J. L. Wang, F. L. Xie and T. Kuniya, Analysis of a reaction-diffusion cholera epidemic model in a spatially heterogeneous environment, Communications in Nonlinear Science and Numerical Simulation, 80 (2020), 104951, 20pp. doi: 10.1016/j.cnsns.2019.104951.  Google Scholar

[34]

W. Wang, W. B. Ma and X. L. Lai, Repulsion effect on superinfecting virions by infected cells for virus infection dynamic model with absorption effect and chemotaxis, Nonlinear Analysis: Real World Applications, 33 (2017), 253-283.  doi: 10.1016/j.nonrwa.2016.04.013.  Google Scholar

[35]

R. Wu and X. Q. Zhao, A reaction-diffusion model of vector-borne disease with periodic delays, Journal of Nonlinear Science, 29 (2019), 29-64.  doi: 10.1007/s00332-018-9475-9.  Google Scholar

[36]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar

[37]

D. M. Xiao and S. G. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Mathematical Biosciences, 208 (2007), 419-429.  doi: 10.1016/j.mbs.2006.09.025.  Google Scholar

[38]

Y. Yu, Z. D. Zhang and Q. S. Bi, Multistability and fast-slow analysis for van der Pol-Duffing oscillator with varying exponential delay feedback factor, Applied Mathematical Modelling, 57 (2018), 448-458.  doi: 10.1016/j.apm.2018.01.010.  Google Scholar

[39]

R. Zhang, Y. Wang, Z. D. Zhang and Q. S. Bi, Nonlinear behaviors as well as the bifurcation mechanism in switched dynamical systems, Nonlinear Dynamics, 79 (2015), 465-471.   Google Scholar

[40]

C. Zhang, J. G. Gao, H. Q. Sun and J. L. Wang, Dynamics of a reaction-diffusion SVIR model in a spatial heterogeneous environment, Physica A, 533 (2019), 122049, 15pp. doi: 10.1016/j.physa.2019.122049.  Google Scholar

[41]

X. Q. Zhao, Basic reproduction ratios for periodic compartmental models with time delay, Journal of Dynamic and Differential Equations, 29 (2017), 67-82.  doi: 10.1007/s10884-015-9425-2.  Google Scholar

[42]

X. Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

[43]

L. H. Zhu, G. Guan and Y. M. Li, Nonlinear dynamical analysis and control strategies of a network-based SIS epidemic model with time delay, Applied Mathematical Modelling, 70 (2019), 512-531.  doi: 10.1016/j.apm.2019.01.037.  Google Scholar

[44]

L. H. Zhu, W. S. Liu and Z. D. Zhang, Delay differential equations modeling of rumor propagation in both homogeneous and heterogeneous networks with a forced silence function, Applied Mathematics and Computation, 370 (2020), 124925, 22pp. doi: 10.1016/j.amc.2019.124925.  Google Scholar

[45]

L. H. Zhu and X. Y. Huang, SIRaRu rumor spreading model in complex networks, Communications in Theoretical Physics, 72 (2020), 015002. Google Scholar

[46]

L. H. Zhu, M. X. Liu and Y. M. Li, The dynamics analysis of a rumor propagation model in online social networks, Physica A, 520 (2019), 118-137.  doi: 10.1016/j.physa.2019.01.013.  Google Scholar

[47]

L. H. Zhu, H. Y. Zhao and H. Y. Wang, Partial differential equation modeling of rumor propagation in complex networks with higher order of organization, Chaos, 29 (2019), 053106, 23pp. doi: 10.1063/1.5090268.  Google Scholar

[48]

L. H. Zhu, X. Zhou, Y. M. Li and Y. X. Zhu, Stability and bifurcation analysis on a delayed epidemic model with information dependent vaccination, Physica Scripta, 94 (2019), 125202. doi: 10.1088/1402-4896/ab2f04.  Google Scholar

[49]

L. H. Zhu, H. Y. Zhao and H. Y. Wang, Stability and spatial patterns of an epidemi-like rumor propagation model with diffusions, Physica Scripta, 94 (2019), 085007. Google Scholar

[50]

M. Zhu and Y. Xu, A time-periodic dengue fever model in a heterogeneous environment, Mathematics and Computers in Simulation, 155 (2019), 115-129.  doi: 10.1016/j.matcom.2017.12.008.  Google Scholar

show all references

References:
[1]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in: H.J. Schmeisser, H. Triebel (Eds.), Function Spaces, Differential Operators and Nonlinear Analysis (Friedrichroda, 1992), in: Teubner-Texte zur Mathematik, vol 133, Teubner, Stuttgart, 1993, 9-126. doi: 10.1007/978-3-663-11336-2_1.  Google Scholar

[2]

L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic patch model, Siam Journal on Applied Mathematics, 67 (2007), 1283-1309.  doi: 10.1137/060672522.  Google Scholar

[3]

Y. L. Cai, Y. Kang, M. Banerjee and W. M. Wang, Complex Dynamics of a host-parasite model with both horizontal and vertical transmissions in a spatial heterogeneous environment, Nonlinear Analysis: Real World Applications, 40 (2018), 444-465.  doi: 10.1016/j.nonrwa.2017.10.001.  Google Scholar

[4]

Y. L. Cai, X. Z. Lian, Z. H. Peng and W. M. Wang, Spatiotemporal transmission dynamics for influenza disease in a heterogenous environment, Nonlinear Analysis: Real World Applications, 46 (2019), 178-194.  doi: 10.1016/j.nonrwa.2018.09.006.  Google Scholar

[5]

T. Chen, L. Chen, X. Xu, Y. F. Cai, H. B. Jiang and X. Q. Sun, Reliable sideslip angle estimation of four-wheel independent drive electric vehicle by information iteration and fusion, Mathematical Problems in Engineering, 2018 (2018), 9075372, 14pp. doi: 10.1155/2018/9075372.  Google Scholar

[6]

D. J. Daley and D. G. Kendall, Epidemic and rumors, Nature, 204 (1964), 1118. Google Scholar

[7]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, Journal of Mathematical Biology, 28 (1990), 365-382.  doi: 10.1007/BF00178324.  Google Scholar

[8]

Z. M. Guo, F. B. Wang and X. F. Zou, Threshold dynamics of an infective disease model with a fixed latent period and non-local infections, Journal of Mathematical Biology, 65 (2012), 1387-1410.  doi: 10.1007/s00285-011-0500-y.  Google Scholar

[9]

J. Groeger, Divergence theorems and the supersphere, Journal of Geometry And Physics, 77 (2014), 13-29.  doi: 10.1016/j.geomphys.2013.11.004.  Google Scholar

[10]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, , American Mathematical Society, Providence, RI, 1988.  Google Scholar

[11]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[12]

H. W. Hethcote, The mathematical of infectious diseases, SIAM Review, 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar

[13]

X. L. Lai and X. F. Zou, Repulsion effect on superinfecting virions by infected cells, Bulletin of Mathematical Biology, 76 (2014), 2806-2833.  doi: 10.1007/s11538-014-0033-9.  Google Scholar

[14]

J. R. Li, H. J. Jiang, Z. Y. Yu and C. Hu, Dynamical analysis of rumor spreading model in homogeneous complex networks, Applied Mathematics and Computation, 359 (2019), 374-385.  doi: 10.1016/j.amc.2019.04.076.  Google Scholar

[15]

X. Liang, L. Zhang and X. Q. Zhao, Basic reproduction ratios for periodic abstract functional differential equations (with application to a spatial model for lyme disease), Journal of Dynamic and Differential Equations, 31 (2019), 1247-1278.  doi: 10.1007/s10884-017-9601-7.  Google Scholar

[16]

Y. J. Lou and X. Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, Journal of Mathematical Biology, 62 (2011), 543-568.  doi: 10.1007/s00285-010-0346-8.  Google Scholar

[17]

Y. T. Luo, L. Zhang, T. T. Zheng and Z. D. Teng, Analysis of a diffusive virus infection model with humoral immunity, Cell-to-cell Transmission and Nonlinear Incidence. Physica A, 535 (2019), 122415, 20pp. doi: 10.1016/j.physa.2019.122415.  Google Scholar

[18]

D. Maki and M. Thomson, Mathematical Models and Applications, Prentice-Hall, Englewood Cliffs, 1973.  Google Scholar

[19]

P. Miao, Z. D. Zhang, C. W. Lim and X. D. Wang, Hopf bifurcation and hybrid control of a delayed ecoepidemiological model with nonlinear incidence rate and Holling type Ⅱ functional response, Mathematical Problems in Engineering, 2018 (2018), 6052503, 12pp. doi: 10.1155/2018/6052503.  Google Scholar

[20]

R. Peng and X. Q. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471.  doi: 10.1088/0951-7715/25/5/1451.  Google Scholar

[21]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Pren-tice Hall, Englewood Cliffs, 1967.  Google Scholar

[22]

X. Ren, Y. Tian, L. Liu and X. Liu, A reaction-diffusion within-host HIV model with cell-to-cell transmission, Journal of Mathematical Biology, 76 (2018), 1831-1872.  doi: 10.1007/s00285-017-1202-x.  Google Scholar

[23]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, in: Math. Surveys Monger. vol. 41, American Mathematical Society, Providence, RI, 1995.  Google Scholar

[24]

H. L. Smith and X. Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Analysis: Theory Methods & Applications, 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2.  Google Scholar

[25]

S. T. Tang, Z. D. Teng and H. Miao, Global dynamics of a reaction-diffusion virus infection model with humoral immunity and nonlinear incidence, Computers and Mathematics with Applications, 78 (2019), 786-806.  doi: 10.1016/j.camwa.2019.03.004.  Google Scholar

[26]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM Journal on Applied Mathematics, 70 (2009), 188-211.  doi: 10.1137/080732870.  Google Scholar

[27]

H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, Journal of Mathematical Biology, 30 (1992), 755-763.  doi: 10.1007/BF00173267.  Google Scholar

[28]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[29]

N. K. Vaidya, F. B. Wang and X. F. Zou, Avian influenza dynamics in wild birds with bird mobility and spatial heterogeneous environment, Discrete and Continuous Dynamical Systems, 17 (2012), 2829-2848.  doi: 10.3934/dcdsb.2012.17.2829.  Google Scholar

[30]

J. Wang, L. Zhao and R. Huang, SIRaRu rumor spreading model in complex networks, Physica A, 398 (2014), 43-55.  doi: 10.1016/j.physa.2013.12.004.  Google Scholar

[31]

W. Wang and X. Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM Journal on Applied Dynamical Systems, 11 (2012), 1652-1673.  doi: 10.1137/120872942.  Google Scholar

[32]

W. Wang and X. Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM Journal on Applied Mathematics, 71 (2011), 147-168.  doi: 10.1137/090775890.  Google Scholar

[33]

J. L. Wang, F. L. Xie and T. Kuniya, Analysis of a reaction-diffusion cholera epidemic model in a spatially heterogeneous environment, Communications in Nonlinear Science and Numerical Simulation, 80 (2020), 104951, 20pp. doi: 10.1016/j.cnsns.2019.104951.  Google Scholar

[34]

W. Wang, W. B. Ma and X. L. Lai, Repulsion effect on superinfecting virions by infected cells for virus infection dynamic model with absorption effect and chemotaxis, Nonlinear Analysis: Real World Applications, 33 (2017), 253-283.  doi: 10.1016/j.nonrwa.2016.04.013.  Google Scholar

[35]

R. Wu and X. Q. Zhao, A reaction-diffusion model of vector-borne disease with periodic delays, Journal of Nonlinear Science, 29 (2019), 29-64.  doi: 10.1007/s00332-018-9475-9.  Google Scholar

[36]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar

[37]

D. M. Xiao and S. G. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Mathematical Biosciences, 208 (2007), 419-429.  doi: 10.1016/j.mbs.2006.09.025.  Google Scholar

[38]

Y. Yu, Z. D. Zhang and Q. S. Bi, Multistability and fast-slow analysis for van der Pol-Duffing oscillator with varying exponential delay feedback factor, Applied Mathematical Modelling, 57 (2018), 448-458.  doi: 10.1016/j.apm.2018.01.010.  Google Scholar

[39]

R. Zhang, Y. Wang, Z. D. Zhang and Q. S. Bi, Nonlinear behaviors as well as the bifurcation mechanism in switched dynamical systems, Nonlinear Dynamics, 79 (2015), 465-471.   Google Scholar

[40]

C. Zhang, J. G. Gao, H. Q. Sun and J. L. Wang, Dynamics of a reaction-diffusion SVIR model in a spatial heterogeneous environment, Physica A, 533 (2019), 122049, 15pp. doi: 10.1016/j.physa.2019.122049.  Google Scholar

[41]

X. Q. Zhao, Basic reproduction ratios for periodic compartmental models with time delay, Journal of Dynamic and Differential Equations, 29 (2017), 67-82.  doi: 10.1007/s10884-015-9425-2.  Google Scholar

[42]

X. Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

[43]

L. H. Zhu, G. Guan and Y. M. Li, Nonlinear dynamical analysis and control strategies of a network-based SIS epidemic model with time delay, Applied Mathematical Modelling, 70 (2019), 512-531.  doi: 10.1016/j.apm.2019.01.037.  Google Scholar

[44]

L. H. Zhu, W. S. Liu and Z. D. Zhang, Delay differential equations modeling of rumor propagation in both homogeneous and heterogeneous networks with a forced silence function, Applied Mathematics and Computation, 370 (2020), 124925, 22pp. doi: 10.1016/j.amc.2019.124925.  Google Scholar

[45]

L. H. Zhu and X. Y. Huang, SIRaRu rumor spreading model in complex networks, Communications in Theoretical Physics, 72 (2020), 015002. Google Scholar

[46]

L. H. Zhu, M. X. Liu and Y. M. Li, The dynamics analysis of a rumor propagation model in online social networks, Physica A, 520 (2019), 118-137.  doi: 10.1016/j.physa.2019.01.013.  Google Scholar

[47]

L. H. Zhu, H. Y. Zhao and H. Y. Wang, Partial differential equation modeling of rumor propagation in complex networks with higher order of organization, Chaos, 29 (2019), 053106, 23pp. doi: 10.1063/1.5090268.  Google Scholar

[48]

L. H. Zhu, X. Zhou, Y. M. Li and Y. X. Zhu, Stability and bifurcation analysis on a delayed epidemic model with information dependent vaccination, Physica Scripta, 94 (2019), 125202. doi: 10.1088/1402-4896/ab2f04.  Google Scholar

[49]

L. H. Zhu, H. Y. Zhao and H. Y. Wang, Stability and spatial patterns of an epidemi-like rumor propagation model with diffusions, Physica Scripta, 94 (2019), 085007. Google Scholar

[50]

M. Zhu and Y. Xu, A time-periodic dengue fever model in a heterogeneous environment, Mathematics and Computers in Simulation, 155 (2019), 115-129.  doi: 10.1016/j.matcom.2017.12.008.  Google Scholar

Figure 1.  The asymptotic behavior of the solution of system (4)
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Figure 2.  The uniform persistence of rumor propagation
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Figure 3.  Projection diagram in the $ tx $-plane
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Figure 4.  Distribution of rumor collectors and rumor-infective users at $ t = 0.5 $ for different diffusion coefficient $ D = 0.001,1,5 $
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Figure 5.  Two incidence functions
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Figure 6.  Contour surfaces of $ \mathcal{R}^0 $ with consideration of $ \beta,\theta,A\in[0,1] $
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Figure 7.  (a) The density of susceptible users. (b) The density of collectors. (c) The density of infective users. (d) The rumor-free equilibrium point $ E_0 $ is globally asymptotically stable
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Figure 8.  (a) The density of susceptible users. (b) The density of collectors. (c) The density of infective users. (d) The rumor-prevailing equilibrium point $ E^\star $ is locally asymptotically stable
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