We investigate an SIRS epidemic model with spatial diffusion and nonlinear incidence rates. We show that for small diffusion rate of the infected class [Formula: see text], the infected population tends to be highly localized at certain points inside the domain, forming K spikes. We then study three distinct destabilization mechanisms, as well as a transition from localized spikes to plateau solutions. Two of the instabilities are due to coarsening (spike death) and self-replication (spike birth), and have well-known analogues in other reaction-diffusion systems such as the Schnakenberg model. The third transition is when a single spike becomes unstable and moves to the boundary. This happens when the diffusion of the recovered class, [Formula: see text] becomes sufficiently small. In all cases, the stability thresholds are computed asymptotically and are verified by numerical experiments. We also show that the spike solution can transit into an plateau-type solution when the diffusion rates of recovered and susceptible class are sufficiently small. Implications for disease spread and control through quarantine are discussed.

中文翻译：

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具有扩散的SIR模型中的局部爆发。

我们调查具有空间扩散和非线性发生率的SIRS流行病模型。我们表明，对于受感染类别的低扩散率[公式：参见文本]，受感染人群倾向于高度集中在域内的某些点，形成K峰值。然后，我们研究了三种不同的去稳定机制，以及从局部峰值到平稳解的过渡。其中两个不稳定性是由于变大（尖峰死亡）和自我复制（尖峰出生）造成的，并且在其他反应扩散系统（例如Schnakenberg模型）中具有众所周知的类似物。第三个过渡是单个尖峰变得不稳定并移至边界时。当恢复的类的扩散程度变得足够小时，就会发生这种情况。在所有情况下，渐近计算了稳定性阈值，并通过数值实验对其进行了验证。我们还表明，当回收和易感类别的扩散速率足够小时，加标溶液可以转变为平稳型溶液。讨论了通过隔离对疾病传播和控制的影响。