Dynamical Analysis and Optimal Control for a SEIR Model Based on Virus Mutation in WSNs
Abstract
:1. Introduction
1.1. Research Background
1.2. Related Work
1.3. Contributions
- An SEIR model based on virus mutation is established to describe the propagation process of mutated virus in WSNs.
- Calculating the basic reproduction number R0 of the improved model by the next generation matrix method. Besides, the local and global stability of the two equilibria are proved and simulated by the Routh criterion and the Lyapunov stability method. Moreover, the influence of the repair rate γ1 and γ2 on the basic reproduction number is also revealed in the simulation part.
- Based on the Pontryagin maximum principle, the optimal control variable pairs of the repair ratio of infected nodes and the repair ratio of mutated infected nodes are obtained. The simulation results show that the optimal control strategy ensures the security of wireless sensor networks and minimizes the maintenance cost.
2. Modeling
2.1. Dynamic Equation
2.2. Calculation of the Equilibrium Point and the Basic Reproduction Number
3. Dynamic Analysis and Optimal Strategy
3.1. Subs Stability Analysis of P0
3.2. Stability Analysis of P*
3.3. Optimal Strategy
4. Simulation
4.1. Stable Analysis of Disease-Free Equilibrium
4.2. Stable Analysis of Epidemic Equilibrium
4.3. Optimal Control
4.3.1. Evolution of Sensor Nodes
4.3.2. Total Cost and Control Variables
4.3.3. Influence of Control Variables on the Basic Reproduction Number R0
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Symbol | Description |
---|---|
b | Birth rate or Death rate |
λ1 | Transmission rate of I1 nodes |
λ2 | Transmission rate of I2 nodes |
λ3 | Transmission rate of E nodes |
ε | Probability at which E nodes are converted to I1 nodes |
μ | Probability of virus mutation |
γ1 | Repair rate of I1 nodes |
γ2 | Repair rate of I2 nodes |
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Liu, G.; Chen, J.; Liang, Z.; Peng, Z.; Li, J. Dynamical Analysis and Optimal Control for a SEIR Model Based on Virus Mutation in WSNs. Mathematics 2021, 9, 929. https://doi.org/10.3390/math9090929
Liu G, Chen J, Liang Z, Peng Z, Li J. Dynamical Analysis and Optimal Control for a SEIR Model Based on Virus Mutation in WSNs. Mathematics. 2021; 9(9):929. https://doi.org/10.3390/math9090929
Chicago/Turabian StyleLiu, Guiyun, Jieyong Chen, Zhongwei Liang, Zhimin Peng, and Junqiang Li. 2021. "Dynamical Analysis and Optimal Control for a SEIR Model Based on Virus Mutation in WSNs" Mathematics 9, no. 9: 929. https://doi.org/10.3390/math9090929