Curvature-Weighted Contact Networks: Spectral Reduction and Global Stability in a Markovian SIR Model
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| Format: | Preprint |
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2025
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| _version_ | 1866914219788599296 |
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| author | Santos, Marcilio Ferreira dos |
| author_facet | Santos, Marcilio Ferreira dos |
| contents | We propose a new network-based SIR epidemic model in which transmission is modulated by a curvature-weighted contact matrix that encodes structural and geometric features of the underlying graph. The formulation encompasses both adjacency-driven and Markovian mixing, allowing heterogeneous interactions to be shaped by curvature-sensitive topological properties. We prove that the basic reproduction number satisfies \[ R_0=\fracβγλ_{\max}(M), \] where $M$ is the curvature-weighted transmission operator. Using Perron--Frobenius theory together with linear and nonlinear Lyapunov functionals, we establish: (i) global asymptotic stability of the disease-free equilibrium when $R_0<1$, and (ii) existence and global asymptotic stability of a unique endemic equilibrium when $R_0>1$. Our results show that curvature acts as a geometric regularizer of connectivity, lowering spectral radii, raising effective epidemic thresholds, and organizing the long-term dynamics through monotone contraction toward the endemic state. This framework generalizes classical network epidemiology by integrating geometric information directly into transmission operators, providing a rigorous foundation for epidemic dynamics on structurally heterogeneous networks. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_10331 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Curvature-Weighted Contact Networks: Spectral Reduction and Global Stability in a Markovian SIR Model Santos, Marcilio Ferreira dos Dynamical Systems Probability Populations and Evolution We propose a new network-based SIR epidemic model in which transmission is modulated by a curvature-weighted contact matrix that encodes structural and geometric features of the underlying graph. The formulation encompasses both adjacency-driven and Markovian mixing, allowing heterogeneous interactions to be shaped by curvature-sensitive topological properties. We prove that the basic reproduction number satisfies \[ R_0=\fracβγλ_{\max}(M), \] where $M$ is the curvature-weighted transmission operator. Using Perron--Frobenius theory together with linear and nonlinear Lyapunov functionals, we establish: (i) global asymptotic stability of the disease-free equilibrium when $R_0<1$, and (ii) existence and global asymptotic stability of a unique endemic equilibrium when $R_0>1$. Our results show that curvature acts as a geometric regularizer of connectivity, lowering spectral radii, raising effective epidemic thresholds, and organizing the long-term dynamics through monotone contraction toward the endemic state. This framework generalizes classical network epidemiology by integrating geometric information directly into transmission operators, providing a rigorous foundation for epidemic dynamics on structurally heterogeneous networks. |
| title | Curvature-Weighted Contact Networks: Spectral Reduction and Global Stability in a Markovian SIR Model |
| topic | Dynamical Systems Probability Populations and Evolution |
| url | https://arxiv.org/abs/2512.10331 |