Department of Applied Mathematics, Yuncheng University, Yuncheng 044000, Shanxi, China.
Math Biosci Eng. 2014 Jun;11(3):449-69. doi: 10.3934/mbe.2014.11.449.
Infection age is an important factor affecting the transmission of infectious diseases. In this paper, we consider an SIRS model with infection age, which is described by a mixed system of ordinary differential equations and partial differential equations. The expression of the basic reproduction number R0 is obtained. If R0≤1 then the model only has the disease-free equilibrium, while if R0>1 then besides the disease-free equilibrium the model also has an endemic equilibrium. Moreover, if R0<1 then the disease-free equilibrium is globally asymptotically stable otherwise it is unstable; if R0>1 then the endemic equilibrium is globally asymptotically stable under additional conditions. The local stability is established through linearization. The global stability of the disease-free equilibrium is shown by applying the fluctuation lemma and that of the endemic equilibrium is proved by employing Lyapunov functionals. The theoretical results are illustrated with numerical simulations.
感染年龄是影响传染病传播的一个重要因素。在本文中，我们考虑了一个带有感染年龄的 SIRS 模型，该模型由常微分方程和偏微分方程的混合系统来描述。得到了基本再生数 R0 的表达式。如果 R0≤1，则模型只有无病平衡点，而如果 R0>1，则模型除了无病平衡点外，还有一个地方病平衡点。此外，如果 R0<1，则无病平衡点是全局渐近稳定的，否则是不稳定的；如果 R0>1，则在附加条件下地方病平衡点是全局渐近稳定的。局部稳定性通过线性化来建立。通过应用波动引理证明了无病平衡点的全局稳定性，通过使用李雅普诺夫泛函证明了地方病平衡点的全局稳定性。理论结果通过数值模拟进行了说明。
