分数布朗运动躁声驱动的SIS流行病模型解的存在唯一性
Existence and Uniqueness of Solutions for an SIS Epidemic Model Driven by Fractional Brownian Motion
摘要: 现实中，传染病的传播过程往往受到具有长程相关特征的环境躁声影响。本文研究了一类由Hurst 指数H ∈ (1/2, 1) 的分数布朗运动驱动的随机SIS 流行病模型。 首先，通过压缩映射原理，证明了 该方程局部解的存在唯一性。 其次，针对分数Malliavin 导数矩估计中的相关技术难题，本文通 过合适的变量变换与不等式放缩技巧，证明了状态比值过程的有界性。 最后，通过构造Lyapunov 函数并结合反证法，严格证明了该随机微分方程存在唯一的全局正解，且方程轨道以概率1 始终保 持在生物学合理区间(0, N ) 内。
Abstract: Real-world epidemic transmission is frequently subject to environmental noise with long-range dependence. This paper investigates a stochastic SIS model driven by frac- tional Brownian motion with Hurst parameter H ∈ (1/2, 1). First, the local existence and uniqueness of the solution are established using the contraction mapping prin- ciple. To address moment estimation challenges for fractional Malliavin derivatives, the boundedness of the state ratio process is proved via variable transformations and inequality scaling. Finally, by constructing a Lyapunov function and applying proof by contradiction, the existence and uniqueness of a global positive solution is rigor- ously demonstrated, with trajectories almost surely remaining within the biologically feasible interval (0, N ).
文章引用：俞乐梵, 田琳琳, 闫理坦. 分数布朗运动躁声驱动的SIS流行病模型解的存在唯一性[J]. 应用数学进展, 2026, 15(5): 175-187. https://doi.org/10.12677/AAM.2026.155218

参考文献

[1] Capasso, V. and Serio, G. (1978) A Generalization of the Kermack-Mckendrick Deterministic Epidemic Model. Mathematical Biosciences, 42, 43-61. [Google Scholar] [CrossRef] 
[2] Hethcote, H.W. and Yorke, J.A. (1984) Gonorrhea Transmission Dynamics and Control. In: Lecture Notes in Biomathematics, Vol. 56, Springer-Verlag.
[3] Brauer, F., Allen, L.J.S., van den Driessche, P. and Wu, J. (2008) Mathematical Epidemiology. In: Lecture Notes in Mathematics, Springer-Verlag.
[4] Lajmanovich, A. and Yorke, J.A. (1976) A Deterministic Model for Gonorrhea in a Nonhomo- geneous Population. Mathematical Biosciences, 28, 221-236. [Google Scholar] [CrossRef] 
[5] Nold, A. (1980) Heterogeneity in Disease-Transmission Modeling. Mathematical Biosciences, 52, 227-240. [Google Scholar] [CrossRef] 
[6] Feng, Z., Huang, W. and Castillo-Chavez, C. (2005) Global Behavior of a Multi-Group SIS Epidemic Model with Age Structure. Journal of Differential Equations, 218, 292-324. [Google Scholar] [CrossRef] 
[7] Lamb, K.E., Greenhalgh, D. and Robertson, C. (2011) A Simple Mathematical Model for Genetic Effects in Pneumococcal Carriage and Transmission. Journal of Computational and Applied Mathematics, 235, 1812-1818. [Google Scholar] [CrossRef] 
[8] Gray, A., Greenhalgh, D., Hu, L., Mao, X. and Pan, J. (2011) A Stochastic Differential Equa- tion SIS Epidemic Model. SIAM Journal on Applied Mathematics, 71, 876-902. [Google Scholar] [CrossRef] 
[9] Zhao, Y., Jiang, D. and O’Regan, D. (2013) The Extinction and Persistence of the Stochastic SIS Epidemic Model with Vaccination. Physica A: Statistical Mechanics and its Applications, 392, 4916-4927. [Google Scholar] [CrossRef] 
[10] Greenhalgh, D., Liang, Y. and Mao, X. (2015) Demographic Stochasticity in the SDE SIS Epidemic Model. Discrete and Continuous Dynamical Systems—B, 20, 2859-2884. [Google Scholar] [CrossRef] 
[11] Greenhalgh, D., Liang, Y. and Mao, X. (2016) SDE SIS Epidemic Model with Demographic Stochasticity and Varying Population Size. Applied Mathematics and Computation, 276, 218- 238. [Google Scholar] [CrossRef] 
[12] Hu, G., Liu, M. and Wang, K. (2012) The Asymptotic Behaviours of an Epidemic Model with Two Correlated Stochastic Perturbations. Applied Mathematics and Computation, 218, 10520-10532. [Google Scholar] [CrossRef] 
[13] Liang, Y., Greenhalgh, D. and Mao, X. (2016) A Stochastic Differential Equation Model for the Spread of HIV amongst People Who Inject Drugs. Computational and Mathematical Methods in Medicine, 2016, 1-14. [Google Scholar] [CrossRef] 
[14] Caraballo, T. and Keraani, S. (2018) Analysis of a Stochastic SIR Model with Fractional Brownian Motion. Stochastic Analysis and Applications, 36, 895-908. [Google Scholar] [CrossRef] 
[15] Cai, S., Cai, Y. and Mao, X. (2019) A Stochastic Differential Equation SIS Epidemic Model with Two Independent Brownian Motions. Journal of Mathematical Analysis and Applications, 474, 1536-1550. [Google Scholar] [CrossRef] 
[16] Duncan, T.E., Hu, Y. and Pasik-Duncan, B. (2000) Stochastic Calculus for Fractional Brow- nian Motion I. Theory. SIAM Journal on Control and Optimization, 38, 582-612. [Google Scholar] [CrossRef] 
[17] Hu, Y. (2005) Integral Transformations and Anticipative Calculus for Fractional Brownian Motions. In: Memoirs of the American Mathematical Society, AMS. [Google Scholar] [CrossRef] 
[18] Mishura, Y. and Shevchenko, G. (2012) Mixed Stochastic Differential Equations with Long- Range Dependence: Existence, Uniqueness and Convergence of Solutions. Computers & Math- ematics with Applications, 64, 3217-3227. [Google Scholar] [CrossRef] 
[19] Biagini, F. (2008) Stochastic Calculus for Fractional Brownian Motion and Applications, Prob- ability and Its Applications. Springer.
[20] Hu, Y. and Øksendal, B. (1996) Wick Approximation of Quasilinear Stochastic Differen- tial Equations. In: K¨orezlio˘glu, H., et al., Eds., Stochastic Analysis and Related Topics V, Birkha¨user, 203-231. 10 [Google Scholar] [CrossRef] 
[21] Nualart, D. (2006) The Malliavin Calculus and Related Topics. Springer.
[22] Hu, Y. (2018) Itˆo Type Stochastic Differential Equations Driven by Fractional Brownian Mo- tions of Hurst Parameter. Stochastics, 90, 720-761. [Google Scholar] [CrossRef] 
[23] Arnold, L. (1972) Stochastic Differential Equations: Theory and Applications. Wiley.
[24] Hasminskii, R.Z. (1980) Stochastic Stability of Differential Equations. Sijthoff & Noordhoff. [25] Friedman, A. (1976) Stochastic Differential Equations and Their Applications. Academic Press.

为你推荐