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Stable transition layers in an unbalanced bistable equation
A theoretical approach to understanding rumor propagation dynamics in a spatially heterogeneous environment
1. | School of Mathematical Sciences, Jiangsu University, Zhenjiang, 212013, China |
2. | School of Mathematical Sciences, Nanjing Normal University Nanjing, 210023, China |
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.
References:
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doi: 10.1007/978-3-663-11336-2_1. |
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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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, , American Mathematical Society, Providence, RI, 1988. |
[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. |
[12] |
H. W. Hethcote,
The mathematical of infectious diseases, SIAM Review, 42 (2000), 599-653.
doi: 10.1137/S0036144500371907. |
[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. |
[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. |
[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. |
[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. |
[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. |
[18] |
D. Maki and M. Thomson, Mathematical Models and Applications, Prentice-Hall, Englewood Cliffs, 1973. |
[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. |
[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. |
[21] |
M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Pren-tice Hall, Englewood Cliffs, 1967. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[36] |
J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996.
doi: 10.1007/978-1-4612-4050-1. |
[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. |
[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. |
[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. |
[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. |
[42] |
X. Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003.
doi: 10.1007/978-0-387-21761-1. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, , American Mathematical Society, Providence, RI, 1988. |
[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. |
[12] |
H. W. Hethcote,
The mathematical of infectious diseases, SIAM Review, 42 (2000), 599-653.
doi: 10.1137/S0036144500371907. |
[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. |
[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. |
[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. |
[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. |
[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. |
[18] |
D. Maki and M. Thomson, Mathematical Models and Applications, Prentice-Hall, Englewood Cliffs, 1973. |
[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. |
[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. |
[21] |
M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Pren-tice Hall, Englewood Cliffs, 1967. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[36] |
J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996.
doi: 10.1007/978-1-4612-4050-1. |
[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. |
[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. |
[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. |
[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. |
[42] |
X. Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003.
doi: 10.1007/978-0-387-21761-1. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |







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