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

Linhe Zhu; Wenshan Liu

doi:10.3934/dcds.2020342

Article Contents

2021, Volume 41, Issue 4: 1843-1874. Doi: 10.3934/dcds.2020342

This issue Previous Article Anomalous time-scaling of extreme events in infinite systems and Birkhoff sums of infinite observables Next Article Reversible perturbations of conservative Hénon-like maps

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

Linhe Zhu^1, , and
Wenshan Liu^1,2,

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
^* Corresponding author: Linhe Zhu

Received: April 2020

Revised: August 2020

Early access: October 2020

Published: April 2021

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)

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Abstract

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.

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Mathematics Subject Classification: Primary: 35K57; Secondary: 92D25.

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Figure 1. The transmission sketch of the ISCT model

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Figure 2. The rumor-free steady state $ E_0(x) $ is globally asymptotically stable

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Figure 3. The positive solutions of system (7)-(9) are uniformly persistent

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Figure 4. Projection diagram in the $ tx $-plane

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Figure 5. The solution surface to state equations and adjoint equations on $ \Omega_T $

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Figure 6. The optimal control $ u(x,t) $ on $ \Omega_T $

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Figure 7. Control effect comparison diagram of $ S(x,t) $ at $ t = 1 $

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Figure 8. Control effect comparison diagram of $ T(x,t) $ at $ t = 1 $

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