Backward bifurcation has significant implications for disease control. Deriving necessary and sufficient conditions for backward bifurcation is of paramount importance to understand the reasons for its occurrence and devise effective control strategies. In this paper, we review the methods that lead to necessary and sufficient conditions for backward bifurcation in infectious disease models. We review separately the methods that apply to ODEs and methods that apply to PDEs. We further propose a new method, applicable to both ODEs and PDEs. We illustrate the methods on three examples: a novel ODE model of cholera with vaccination, a PDE version of the cholera model with vaccination, and on an eight equation model of dengue, taken from the literature.

向后分叉对疾病控制具有重要意义。弄清楚反向分叉的必要和充分条件对于理解其发生的原因并制定有效的控制策略至关重要。在本文中，我们回顾了导致传染病模型中反向分叉的必要条件和充分条件的方法。我们分别回顾了适用于ODE的方法和适用于PDE的方法。我们进一步提出了一种适用于ODE和PDE的新方法。我们在三个示例中说明了这些方法：从文献中提取的霍乱疫苗的新型ODE模型，带有疫苗接种的霍乱模型的PDE版本以及登革热的八个方程模型。