| Citation: |
Wenjie Gu, Qianqian Zheng, Jianwei Shen. INSTABILITY-DRIVEN PATTERN FORMATION IN A NETWORK SIR MODEL WITH INDIRECT TRANSMISSION AND QUASI-LAPLACIAN DIFFUSION[J]. Journal of Applied Analysis & Computation, 2026, 16(4): 1860-1883. doi: 10.11948/20250348
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INSTABILITY-DRIVEN PATTERN FORMATION IN A NETWORK SIR MODEL WITH INDIRECT TRANSMISSION AND QUASI-LAPLACIAN DIFFUSION
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Abstract
This study investigates how indirect transmission and diffusion asymmetry shape epidemic dynamics in a network-organized SIR model. Using linear stability analysis and eigenmode decomposition, we derive explicit conditions for Hopf bifurcation, Turing instability, and their interaction. The results show that indirect transmission significantly shifts epidemic thresholds, while asymmetric diffusion across network nodes promotes the activation of additional eigenmodes and the emergence of spatially heterogeneous infection patterns. Numerical simulations on random and quasi-Laplacian networks reveal transitions among stable equilibria, periodic outbreaks, and mixed Hopf-Turing regimes, with the specific pattern determined jointly by biological parameters and network topology. To validate the theory, the model was calibrated using real influenza surveillance data from 44 countries. The observed periodicity and spatial clustering closely match the model predictions, demonstrating that instability-driven mechanisms can explain real-world influenza oscillations and heterogeneity. These findings provide a unified theoretical and data-supported framework for understanding epidemic pattern formation and designing interventions that target indirect transmission and mobility-induced spatial instabilities.
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References
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Wenjie Gu, Qianqian Zheng, Jianwei Shen. INSTABILITY-DRIVEN PATTERN FORMATION IN A NETWORK SIR MODEL WITH INDIRECT TRANSMISSION AND QUASI-LAPLACIAN DIFFUSION[J]. Journal of Applied Analysis & Computation, 2026, 16(4): 1860-1883. doi: 10.11948/20250348
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Figure 1.
The stability of system (2.1) without network. (a) Hopf bifurcation occurs and system is periodic when
$ \beta=0.01 $ $ \beta=0.05 $ $ \beta $ -
Figure 2.
The bifurcation of system (2.1) without network. (a) The bifurcation about
$ \alpha $ $ \gamma $ $ \delta $ -
Figure 3.
The instability condition of system (2.1). (a) The minimum with
$ \sigma=\frac{d_S}{d_I} $ $ F_i(\beta) $ $ \Lambda $ $ d_S=3,d_I=0.1 $ $ F_i(\beta) $ $ \Lambda $ $ d_S=2,d_I=0.1 $ -
Figure 4.
Turing instability and pattern formation of system (2.1) when
$ d_I=0.1 $ $ \Lambda $ $ d_S=3,p=0.01 $ $ d_S=3,p=0.01 $ $ \Lambda $ $ d_S=2,p=0.2 $ $ d_S=2,p=0.2 $ -
Figure 5.
The instability and pattern formation of system (2.1) when
$ \beta=0.02,d_S=2,d_I=0.1 $ $ \Lambda $ $ p=0.2 $ $ p=0.2 $ $ \Lambda $ $ p=0.02 $ $ p=0.02 $ -
Figure 6.
The distribution of
$ \Lambda $ $ \Lambda $ $ \varepsilon=1 $ $ \Lambda $ $ \varepsilon=0.8 $ $ \Lambda $ $ \beta_c $ -
Figure 7.
The pattern formation and the distribution of
$ \phi^{(1)}_i $ $ \varepsilon=1,\beta=0.0449 $ $ Re(c_1(\beta))=-0.0070<0 $ $ \varepsilon=1,\beta=0.035<\beta_c $ $ Re(c_1(\beta))=-0.0038<0 $ $ \phi^{(1)}_i $ $ \varepsilon=1 $ -
Figure 8.
The pattern formation and the distribution of
$ \phi^{(1)}_i $ $ \varepsilon=0.8,\beta=0.035 $ $ Re(c_1(\beta)) = 0.0017>0 $ $ \varepsilon=0.8,\beta=0.01 $ $ Re(c_1(\beta))=0.0137>0 $ $ \phi^{(1)}_i $ $ \varepsilon=0.8 $ -
Figure 9.
The new confirmed cases about influenza in Africa, Eastern Mediterranean and Western Pacific regions from 11.14.2021 to 10.12.2025. Data source: FluNet (
https://www.who.int/tools/flunet ).