Propagation Dynamics of an Epidemic Model with Heterogeneous Control Strategies on Complex Networks
Abstract
:1. Introduction
2. Description and Formation of Epidemic Models
- (1)
- Birth and death: Each vacant node i in the network randomly selects a neighbor. If the neighbor is a vacant node, the state of i remains unchanged. If the neighbor is a nonvacant node, the vacant node i will be activated to generate a new susceptible node at the birth rate b. Each nonvacant node becomes a vacant node at a natural death rate d per unit time. We assume that each nonvacant node has the same birth contact ability A (where ) due to physiological constraints.
- (2)
- Immunization and quarantine (): At each time step, susceptible individuals with degree k are immunized at the immune rate . The infected nodes with degree k will be quarantined at rate . The quarantined individuals will recover to a susceptible node at rate . Nodes with the same degree have identical quarantine and immunization strategies, while those with different degrees have different strategies.
- (3)
- SIS epidemic framework: Infection : At the initial moment, some nodes are randomly selected as infected nodes. At each time step, the possibility that each infected node i will connect to its neighboring nodes is , where represents the infectivity of infected nodes with degree k, and [38,39,40], = A [41], = [42], = [43]. If an infected node i interacts with a susceptible node j along a connecting edge, node j has a possibility of being infected by i at a transmission rate . For a node with degree k, the overall transmission rate is .
3. Equilibria and Basic Reproduction Number
4. Stability Analysis for SIQS Model
4.1. Stability Analysis of Disease-Free Equilibrium
- Case 1
- If then we obtain
- Case 2
- If , since the sum of all eigenvalues is equal to the trace of the matrix, when , , and , .
4.2. Global Stability of Endemic Equilibrium
5. The Optimal Control for the SIQS Model
6. Simulations
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Symbols | Description |
---|---|
Proportion of nodes with degree k. | |
Average degree . | |
n | Maximum degree. |
b | Birth rate. |
d | Natural death rate. |
Fertile contact probability between a node with degree k and its neighbors. | |
Transmission rate of infected nodes with degree k. | |
Vaccination rate of susceptible nodes with degree k. | |
Quarantine rate of infected nodes with degree k. | |
Recovery rate of infected nodes. | |
Recovery rate of quarantined nodes. |
Basic reproduction number | 0.1734 | 0.6590 | 0.9538 | 5.3671 | 7.1561 | 8.9451 |
Final infection scale | 0 | 0 | 0 | 0.0449 | 0.0597 | 0.0724 |
Quarantining Control Strategies | Final Infection Scale | Epidemic Threshold |
---|---|---|
No quarantine | 0.0713 | 0.0468 |
Acquaintance quarantine | 0.0568 | 0.0724 |
Proportional quarantine | 0.0424 | 0.0798 |
Target quarantine | 0.0182 | 0.1483 |
Optimal Control | Max Control | No Control |
---|---|---|
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Wang, Y.; Chen, S.; Yu, D.; Liu, L.; Shang, K.-K. Propagation Dynamics of an Epidemic Model with Heterogeneous Control Strategies on Complex Networks. Symmetry 2024, 16, 166. https://doi.org/10.3390/sym16020166
Wang Y, Chen S, Yu D, Liu L, Shang K-K. Propagation Dynamics of an Epidemic Model with Heterogeneous Control Strategies on Complex Networks. Symmetry. 2024; 16(2):166. https://doi.org/10.3390/sym16020166
Chicago/Turabian StyleWang, Yan, Shanshan Chen, Dingguo Yu, Lixiang Liu, and Ke-Ke Shang. 2024. "Propagation Dynamics of an Epidemic Model with Heterogeneous Control Strategies on Complex Networks" Symmetry 16, no. 2: 166. https://doi.org/10.3390/sym16020166