In this work, we study the global dynamics of a new SIRI epidemic model with demographics, graded cure and relapse in a complex heterogeneous network. First, we analytically make out the epidemic threshold ${\mathcal{R}}_0$ which strictly depends on the topology of the underlying network and the model parameters. Second, we show that ${\mathcal{R}}_0$ plays the role of a necessary and sufficient condition between extinction and permanence of the disease. More specifically, by using new Lyapunov functions, we establish that the disease free-equilibrium state E⁰ is globally asymptotically stable when ${\mathcal{R}}_0\le 1,$ otherwise we proved the existence and uniqueness of the endemic state E*. Then, we show that E* is globally asymptotically stable. Finally, we present a series of numerical simulations to confirm the correctness of the established analytical results.

在这项工作中，我们研究了一个新的SIRI流行病模型的全球动力学，该模型具有人口统计学，分级治愈和复发的复杂异构网络。首先，我们分析确定流行阈${\mathcal{[R}}_0$这严格取决于基础网络的拓扑和模型参数。其次，我们证明${\mathcal{[R}}_0$在疾病的灭绝和持久性之间起着必要和充分条件的作用。更具体地说，通过使用新的Lyapunov函数，我们确定当以下情况时，疾病的自由平衡状态E ⁰是全局渐近稳定的${\mathcal{[R}}_0\le \mathrm{1个}，$否则，我们证明了流行状态E *的存在和唯一性。然后，我们证明E *是全局渐近稳定的。最后，我们提出了一系列数值模拟，以确认所建立分析结果的正确性。