The study of the dynamics of human infectious disease using deterministic models is typically carried out under the assumption that a critical mass of individuals is available and involved in the transmission process. However, in the study of animal disease dynamics where demographic considerations often play a significant role, this assumption must be weakened. Models of the dynamics of animal populations often naturally assume that the presence of a minimal number of individuals is essential to avoid extinction. In the ecological literature, this a priori requirement is commonly incorporated as an Allee effect. The focus here is on the study disease dynamics under the assumption that a critical mass of susceptible individuals is required to guarantee the population's survival. Specifically, the emphasis is on the study of the role of an Allee effect on a Susceptible-Infectious (SI) model where the possibility that susceptible and infected individuals reproduce, with the S-class the best fit. It is further assumed that infected individuals loose some of their ability to compete for resources, the cost imposed by the disease. These features are set in motion in as simple model as possible. They turn out to lead to a rich set of dynamical outcomes. This toy model supports the possibility of multi-stability (hysteresis), saddle node and Hopf bifurcations, and catastrophic events (disease-induced extinction). The analyses provide a full picture of the system under disease-free dynamics including disease-induced extinction and proceed to identify required conditions for disease persistence. We conclude that increases in (i) the maximum birth rate of a species, or (ii) in the relative reproductive ability of infected individuals, or (iii) in the competitive ability of a infected individuals at low density levels, or in (iv) the per-capita death rate (including disease-induced) of infected individuals, can stabilize the system (resulting in disease persistence). We further conclude that increases in (a) the Allee effect threshold, or (b) in disease transmission rates, or in (c) the competitive ability of infected individuals at high density levels, can destabilize the system, possibly leading to the eventual collapse of the population. The results obtained from the analyses of this toy model highlight the significant role that factors like an Allee effect may play on the survival and persistence of animal populations. Scientists involved in biological conservation and pest management or interested in finding sustainability solutions, may find these results of this study compelling enough to suggest additional focused research on the role of disease in the regulation and persistence of animal populations. The risk faced by endangered species may turn out to be a lot higher than initially thought.

中文翻译：

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具有 Allee 效应和疾病修正适应性的野外种群的简单流行病学模型。

使用确定性模型研究人类传染病的动态通常是在假设有足够数量的个体可用并参与传播过程的情况下进行的。然而，在人口统计因素通常起着重要作用的动物疾病动态研究中，必须削弱这一假设。动物种群动态模型通常自然而然地假设最少数量的个体的存在对于避免灭绝至关重要。在生态文献中，这种先验要求通常被纳入为 Allee 效应。这里的重点是在假设需要大量易感个体来保证种群生存的情况下研究疾病动态。具体来说，重点是研究 Allee 效应对易感感染 (SI) 模型的作用，其中易感和受感染个体繁殖的可能性，S 级最适合。进一步假设受感染的个体失去了一些竞争资源的能力，即疾病带来的成本。这些功能在尽可能简单的模型中设置。结果证明，它们会导致一系列丰富的动态结果。该玩具模型支持多重稳定性（滞后）、鞍点和 Hopf 分叉以及灾难性事件（疾病引起的灭绝）的可能性。这些分析提供了无病动态下系统的全貌，包括疾病引起的灭绝，并继续确定疾病持续存在所需的条件。我们得出的结论是，(i) 物种的最大出生率，或 (ii) 受感染个体的相对繁殖能力，或 (iii) 受感染个体在低密度水平下的竞争能力增加，或 (iv) ) 受感染个体的人均死亡率（包括疾病引起的）可以稳定系统（导致疾病持续存在）。我们进一步得出结论，(a) Allee 效应阈值，或 (b) 疾病传播率，或 (c) 高密度感染个体的竞争能力的增加会破坏系统的稳定性，可能导致最终的崩溃的人口。从该玩具模型的分析中获得的结果突出了 Allee 效应等因素可能对动物种群的生存和持久性发挥的重要作用。参与生物保护和害虫管理或对寻找可持续性解决方案感兴趣的科学家可能会发现，这项研究的这些结果足以令人信服，建议对疾病在动物种群调节和持久性中的作用进行额外的重点研究。濒危物种面临的风险可能比最初想象的要高得多。