Barbera Elvira, Consolo Giancarlo, Valenti Giovanna
Department of Mathematics and Computer Science, University of Messina, V. le F. D'Alcontres 31, I-98166 Messina, Italy.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Nov;88(5):052719. doi: 10.1103/PhysRevE.88.052719. Epub 2013 Nov 25.
A one-dimensional hyperbolic reaction-diffusion model of epidemics is developed to describe the dynamics of diseases spread occurring in an environment where three kinds of individuals mutually interact: the susceptibles, the infectives, and the removed. It is assumed that the disease is transmitted from the infected population to the susceptible one according to a nonlinear convex incidence rate. The model, based upon the framework of extended thermodynamics, removes the unphysical feature of instantaneous diffusive effects, which is typical of parabolic models. Linear stability analyses are performed to study the nature of the equilibrium states against uniform and nonuniform perturbations. Emphasis is given to the occurrence of Hopf and Turing bifurcations, which break the temporal and the spatial symmetry of the system, respectively. The existence of traveling wave solutions connecting two steady states is also discussed. The governing equations are also integrated numerically to validate the analytical results and to characterize the spatiotemporal evolution of diseases.
建立了一个一维双曲型传染病反应扩散模型，以描述在三种个体相互作用的环境中疾病传播的动态：易感者、感染者和康复者。假设疾病根据非线性凸发生率从感染人群传播到易感人群。该模型基于扩展热力学框架，消除了抛物型模型中典型的非物理瞬时扩散效应特征。进行线性稳定性分析以研究平衡态对均匀和非均匀扰动的性质。重点关注霍普夫分岔和图灵分岔的出现，它们分别打破了系统的时间和空间对称性。还讨论了连接两个稳态的行波解的存在性。对控制方程进行了数值积分，以验证分析结果并刻画疾病的时空演化。
