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On some model problem for the propagation of interacting species in a special environment
doi: 10.3934/dcds.2020342

## Spatial dynamics and optimization method for a network propagation model in a shifting environment

 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  April 2020 Revised  August 2020 Published  October 2020

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

In this paper, a reaction-diffusion $ISCT$ rumor propagation model with general incidence rate is proposed in a spatially heterogeneous environment. We first summarize the well-posedness of global solutions. Then the basic reproduction number $\mathcal{R}_0$ is introduced for the model which contains the spatial homogeneity as a special case. The threshold-type dynamics are also established in terms of $\mathcal{R}_0$, including the global asymptotic stability of the rumor-free steady state and the uniform persistence of all positive solutions. Furthermore, by applying a controller to this model, we investigate the optimal control problem. Employing the operator semigroup theory, we prove the existence, uniqueness and some estimates of the positive strong solution to the controlled system. Subsequently, the existence of the optimal control strategy is established with the aid of minimal sequence techniques and the first order necessary optimality conditions for the optimal control is deduced. Finally, some numerical simulations are performed to validate the main analysis. The results of our study can theoretically promote the regulation of rumor propagation on the Internet.

Citation: Linhe Zhu, Wenshan Liu. Spatial dynamics and optimization method for a network propagation model in a shifting environment. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020342
##### References:
 [1] H. Amann, Dynamical theory of quasilinear parabolic equations III: Global existence, Mathematische Zeitschrift, 202 (1989), 219-250.  doi: 10.1007/BF01215256.  Google Scholar [2] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in Function Spaces, Differential Operators and Nonlinear Analysis (Friedrichroda, 1992), Teubner-Texte zur Mathematik, Vol. 133, Teubner, Stuttgart, 1993, 9–126. doi: 10.1007/978-3-663-11336-2_1.  Google Scholar [3] N. C. Apreutesei, Necessary optimality conditions for three species reaction-diffusion system, Applied Mathematics Letters, 24 (2011), 293-297.  doi: 10.1016/j.aml.2010.10.008.  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] J. Cui, Y. Sun and H. Zhu, The impact of media on the control of infectious diseases, Journal of Dynamic and Differential Equations, 20 (2008), 31-53.  doi: 10.1007/s10884-007-9075-0.  Google Scholar [6] D. J. Daley and D. G. Kendall, Epidemic and rumors, Nature, 204 (1964), 1118. Google Scholar [7] 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 [8] J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025.  Google Scholar [9] 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 [10] A. Laaroussi, M. Rachik and M. Elhia, An optimal control problem for a spatiotemporal SIR model, International Journal of Dynamics and Control, 6 (2018), 384-397.  doi: 10.1007/s40435-016-0283-5.  Google Scholar [11] W. Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, Journal of Mathematical Biology, 23 (1986), 187-204.  doi: 10.1007/BF00276956.  Google Scholar [12] X. D. Liu, T. Li, H. Xu and W. J. Liu, Spreading dynamics of an online social information model on scale-free networks, Physica A, 514 (2019), 497-510.  doi: 10.1016/j.physa.2018.09.085.  Google Scholar [13] 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 [14] 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. doi: 10.1016/j.physa.2019.122415.  Google Scholar [15] D. Maki and M. Thomson, Mathematical Models and Applications, Prentice-Hall, Englewood Cliffs, 1973.  Google Scholar [16] R. H. Martin and H. L. Smith, Abstract functional-differnential equations and reaction-diffusion systems, Transactions of The American Mathematical Society, 321 (1990), 1-44.  doi: 10.2307/2001590.  Google Scholar [17] 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, 6052503. doi: 10.1155/2018/6052503.  Google Scholar [18] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice Hall, Englewood Cliffs, 1967.  Google Scholar [19] J. P. Raymond and F. Tr$\ddot{o}$ltzsch, Second order sufficient optimality conditions for nonlinear parabolic control problems with state contraints, Discrete and Continous Dynamic Systems, 6 (2000), 431-450.  doi: 10.3934/dcds.2000.6.431.  Google Scholar [20] 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 [21] 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 [22] 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 [23] 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 [24] 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 [25] 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 [26] 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 [27] 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 [28] 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 [29] T. Wang, J. He and X. Wang, An information spreading model based on online social networks, Physica A, 490 (2018), 488-496.  doi: 10.1016/j.physa.2017.08.078.  Google Scholar [30] J. H. Wu, Theory and Applications of Partial Functional- Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar [31] H. L. Xiang and B. Liu, Solving the inverse problem of an SIS epidemic reaction-diffusion model by optimal control methods, Computers and Mathematics with Applications, 70 (2015), 805-819.  doi: 10.1016/j.camwa.2015.05.025.  Google Scholar [32] H. L. Xiang, B. Liu and Z. Fang, Optimal control strategies for a new ecosystem governed by reaction-diffusion equations, Journal of Mathmatical Analysis and Applications, 467 (2018), 270-291.  doi: 10.1016/j.jmaa.2018.07.001.  Google Scholar [33] D. Xu, X. Xu, Y. Xie and C. Yang, Optimal control of an SIVRS epidemic spreading model with virus variation based on complex networks, Communications in Nonlinear Science and Numerical Simulation, 48 (2017), 200-210.  doi: 10.1016/j.cnsns.2016.12.025.  Google Scholar [34] X. Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar [35] M. Zhou, H. L. Xiang and Z. X. Li, Optimal control strategies for a reaction-diffusion epidemic system, Nonlinear Analysis: Real World Applications, 46 (2019), 446-464.  doi: 10.1016/j.nonrwa.2018.09.023.  Google Scholar [36] 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 [37] 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 [38] 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 [39] 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. doi: 10.1063/1.5090268.  Google Scholar [40] 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. doi: 10.1016/j.amc.2019.124925.  Google Scholar

show all references

##### References:
 [1] H. Amann, Dynamical theory of quasilinear parabolic equations III: Global existence, Mathematische Zeitschrift, 202 (1989), 219-250.  doi: 10.1007/BF01215256.  Google Scholar [2] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in Function Spaces, Differential Operators and Nonlinear Analysis (Friedrichroda, 1992), Teubner-Texte zur Mathematik, Vol. 133, Teubner, Stuttgart, 1993, 9–126. doi: 10.1007/978-3-663-11336-2_1.  Google Scholar [3] N. C. Apreutesei, Necessary optimality conditions for three species reaction-diffusion system, Applied Mathematics Letters, 24 (2011), 293-297.  doi: 10.1016/j.aml.2010.10.008.  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] J. Cui, Y. Sun and H. Zhu, The impact of media on the control of infectious diseases, Journal of Dynamic and Differential Equations, 20 (2008), 31-53.  doi: 10.1007/s10884-007-9075-0.  Google Scholar [6] D. J. Daley and D. G. Kendall, Epidemic and rumors, Nature, 204 (1964), 1118. Google Scholar [7] 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 [8] J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025.  Google Scholar [9] 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 [10] A. Laaroussi, M. Rachik and M. Elhia, An optimal control problem for a spatiotemporal SIR model, International Journal of Dynamics and Control, 6 (2018), 384-397.  doi: 10.1007/s40435-016-0283-5.  Google Scholar [11] W. Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, Journal of Mathematical Biology, 23 (1986), 187-204.  doi: 10.1007/BF00276956.  Google Scholar [12] X. D. Liu, T. Li, H. Xu and W. J. Liu, Spreading dynamics of an online social information model on scale-free networks, Physica A, 514 (2019), 497-510.  doi: 10.1016/j.physa.2018.09.085.  Google Scholar [13] 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 [14] 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. doi: 10.1016/j.physa.2019.122415.  Google Scholar [15] D. Maki and M. Thomson, Mathematical Models and Applications, Prentice-Hall, Englewood Cliffs, 1973.  Google Scholar [16] R. H. Martin and H. L. Smith, Abstract functional-differnential equations and reaction-diffusion systems, Transactions of The American Mathematical Society, 321 (1990), 1-44.  doi: 10.2307/2001590.  Google Scholar [17] 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, 6052503. doi: 10.1155/2018/6052503.  Google Scholar [18] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice Hall, Englewood Cliffs, 1967.  Google Scholar [19] J. P. Raymond and F. Tr$\ddot{o}$ltzsch, Second order sufficient optimality conditions for nonlinear parabolic control problems with state contraints, Discrete and Continous Dynamic Systems, 6 (2000), 431-450.  doi: 10.3934/dcds.2000.6.431.  Google Scholar [20] 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 [21] 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 [22] 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 [23] 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 [24] 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 [25] 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 [26] 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 [27] 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 [28] 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 [29] T. Wang, J. He and X. Wang, An information spreading model based on online social networks, Physica A, 490 (2018), 488-496.  doi: 10.1016/j.physa.2017.08.078.  Google Scholar [30] J. H. Wu, Theory and Applications of Partial Functional- Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar [31] H. L. Xiang and B. Liu, Solving the inverse problem of an SIS epidemic reaction-diffusion model by optimal control methods, Computers and Mathematics with Applications, 70 (2015), 805-819.  doi: 10.1016/j.camwa.2015.05.025.  Google Scholar [32] H. L. Xiang, B. Liu and Z. Fang, Optimal control strategies for a new ecosystem governed by reaction-diffusion equations, Journal of Mathmatical Analysis and Applications, 467 (2018), 270-291.  doi: 10.1016/j.jmaa.2018.07.001.  Google Scholar [33] D. Xu, X. Xu, Y. Xie and C. Yang, Optimal control of an SIVRS epidemic spreading model with virus variation based on complex networks, Communications in Nonlinear Science and Numerical Simulation, 48 (2017), 200-210.  doi: 10.1016/j.cnsns.2016.12.025.  Google Scholar [34] X. Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar [35] M. Zhou, H. L. Xiang and Z. X. Li, Optimal control strategies for a reaction-diffusion epidemic system, Nonlinear Analysis: Real World Applications, 46 (2019), 446-464.  doi: 10.1016/j.nonrwa.2018.09.023.  Google Scholar [36] 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 [37] 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 [38] 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 [39] 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. doi: 10.1063/1.5090268.  Google Scholar [40] 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. doi: 10.1016/j.amc.2019.124925.  Google Scholar
The transmission sketch of the ISCT model
The rumor-free steady state $E_0(x)$ is globally asymptotically stable
The positive solutions of system (7)-(9) are uniformly persistent
Projection diagram in the $tx$-plane
The solution surface to state equations and adjoint equations on $\Omega_T$
The optimal control $u(x,t)$ on $\Omega_T$
Control effect comparison diagram of $S(x,t)$ at $t = 1$
Control effect comparison diagram of $T(x,t)$ at $t = 1$
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