This paper presents a nonlinear family of stochastic SEIRS models for diseases such as malaria in a highly random environment with noises from the disease transmission and natural death rates, and also from the random delays of the incubation and immunity periods. Improved analytical methods and local martingale characterizations are applied to find conditions for the disease to persist near an endemic steady state, and also for the disease to remain permanently in the system over time. Moreover, the ergodic stationary distribution for the stochastic process describing the disease dynamics is defined, and the statistical characteristics of the distribution are given numerically. The results of this study show that the disease will persist and become permanent in the system, regardless of (1) whether the noises are from the disease transmission rate and/or from the natural death rates or (2) whether the delays in the system are constant or random for individuals in the system. Furthermore, it is shown that “weak” noise is associated with the existence of an endemic stationary distribution for the disease, while “strong” noise is associated with extinction of the population over time. Numerical simulation examples for Plasmodium vivax malaria are given.

中文翻译：

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一类疾病模型的平稳分布和随机持续性：疟疾案例研究

本文提出了一个非线性随机 SEIRS 模型家族，用于高度随机环境中的疟疾等疾病，其中包含疾病传播和自然死亡率以及潜伏期和免疫期的随机延迟的噪声。改进的分析方法和局部鞅特征被应用于寻找疾病在地方性稳定状态附近持续存在的条件，以及疾病随着时间的推移永久保留在系统中的条件。此外，定义了描述疾病动态的随机过程的遍历平稳分布，并以数值方式给出了分布的统计特征。这项研究的结果表明，这种疾病会在系统中持续存在并成为永久性的，无论 (1) 噪声是否来自疾病传播率和/或自然死亡率，或者 (2) 系统中的延迟对于系统中的个体而言是恒定的还是随机的。此外，研究表明，“弱”噪声与该疾病的地方性平稳分布的存在有关，而“强”噪声与随着时间的推移种群灭绝有关。给出了间日疟原虫疟疾的数值模拟实例。