We derive the time-dependent probability distribution for the number of infected individuals at a given time in a stochastic Susceptible-Infected-Susceptible (SIS) epidemic model. The mean, variance, skewness, and kurtosis of the distribution are obtained as a function of time. We study the effect of noise intensity on the distribution and later derive and analyze the effect of changes in the transmission and recovery rates of the disease. Our analysis reveals that the time-dependent probability density function exists if the basic reproduction number is greater than one. It converges to the Dirac delta function on the long run (entirely concentrated on zero) as the basic reproduction number tends to one from above. The result is applied using published COVID-19 parameters and also applied to analyze the probability distribution of the aggregate number of COVID-19 cases in the United States for the period: January 22, 2020-March 23, 2021. Findings show that the distribution shifts concentration to the right until it concentrates entirely on the carrying infection capacity as the infection growth rate increases or the recovery rate reduces. The disease eradication and disease persistence thresholds are calculated.

我们在随机的易感性-易感性（SIS）流行病模型中得出给定时间的受感染个体数的时间依赖性概率分布。随时间获得分布的均值，方差，偏度和峰度。我们研究了噪声强度对分布的影响，然后推导并分析了疾病的传播和恢复率变化的影响。我们的分析表明，如果基本复制数大于1，则存在与时间有关的概率密度函数。从长远来看，它会收敛到狄拉克增量函数（完全集中在零上），因为基本复制数从上趋于一。使用已发布的COVID-19参数应用该结果，还可以用于分析美国2020年1月22日至2021年3月23日这段时间内COVID-19病例总数的概率分布。随着感染增长率的提高或恢复率的降低，浓度会向右移动，直到完全集中在携带感染的能力上。计算出疾病根除和疾病持久性阈值。