我們引入了一個簡單的模型，該模型既捕捉了有症狀的感染者的隔離，也捕捉了應對遏制政策或行為變化的人群範圍內的隔離做法，並表明該模型準確地捕捉了觀察到的生長行為

Effective containment explains subexponential growth in recent confirmed COVID-19 cases in China

• Published In SCIENCE Volume 368 | Issue 6492 15 May 2020
• Total Citations ： 267
• Date: 15 May 2020
• DOI: 10.1126/science.abb4557

Abstract:

The recent outbreak of coronavirus disease 2019 (COVID-19) in mainland China was characterized by a distinctive subexponential increase of confirmed cases during the early phase of the epidemic, contrasting with an initial exponential growth expected for an unconstrained outbreak. We show that this effect can be explained as a direct consequence of containment policies that effectively deplete the susceptible population. To this end, we introduce a parsimonious model that captures both quarantine of symptomatic infected individuals, as well as population-wide isolation practices in response to containment policies or behavioral changes, and show that the model captures the observed growth behavior accurately.

The insights provided here may aid the careful implementation of containment strategies for ongoing secondary outbreaks of COVID-19 or similar future outbreaks of other emergent infectious diseases.
中國大陸最近爆發的2019冠狀病毒病(COVID-19)的特點是，在疫情初期，確診病例明顯呈亞指數增長，而不是預期的無約束爆發時的指數增長.我們表明，這種影響可以解釋為有效地耗盡易感人群的遏制政策的直接後果。為此，我們引入了一個簡單的模型，該模型既捕捉了有症狀的感染者的隔離，也捕捉了應對遏制政策或行為變化的人群範圍內的隔離做法，並表明該模型準確地捕捉了觀察到的生長行為， 本文提供的見解可能有助於仔細實施遏制戰略，以應對正在發生的COVID-19二次暴發或未來類似的其他突發傳染病暴發。

Model

To test the hypothesis that the observed growth behavior can be caused by mitigation policies that apply to both infected and susceptible individuals.
we extend the SIR model by two additional mechanisms, one of which can be
interpreted as a process of removing susceptibles from the transmission process: First, we assume that general public containment efforts or individual behavioral changes in response to the epidemic effectively remove individuals from the interaction dynamics or significantly reduce their participation in the transmission dynamics. We will refer to this mechanism as “containment” in the following.
Second, we account for the removal of symptomatic infected individuals, which we will refer to as “quarantine” procedures.
檢驗觀察到的生長行為可能由適用於受感染和易感個體的緩解政策引起的假設。
我們通過兩個附加機制擴充套件了SIR模型，其中一個機制可以是被解釋為從傳播過程中消除易感因素的過程：首先，我們假設一般的公共遏制措施或個人行為改變有效地將個人從互動動態中移除，或顯著減少他們在傳播動態中的參與。在下文中，我們將這種機制稱為“遏制”。
第二，我們考慮清除有症狀的感染者，我們稱之為“隔離”程式。

a generalization of the standard SIR model, henceforth referred to as the **SIR-X ** model.標準SIR模型的一般化，此後稱為SIR- x模型。

The rate parameters α and β quantify the transmission rate and the recovery rate of the standard SIR model, respectively. Additionally,the impact of containment efforts is captured by the terms proportional to the containment rate k0 that is effective in both I and S populations, because measures such as social distancing and curfews affect the whole population alike. Infected individuals are removed at rate k corresponding to quarantine measures that only affect symptomatic infecteds.
The new compartment X quantifies symptomatic, quarantined infecteds. Here we assume that the fraction X(t) is proportional to the empirically confirmed and reported cases C(t) and that the time period between sampling and test results can be neglected. The case k0 = 0 corresponds to a scenario in which the general population is unaffected by policies or does not commit behavioral changes in response to an epidemic. The case k0 = 0 corresponds to a scenario in which symptomatic infecteds are not isolated specifically.

速率引數 α and β 分別量化了標準 SIR 模型的傳播速率和恢復速率。此外，遏制努力的影響可以用與遏制率 k0成比例的術語來表示，k0對 i 型和 s 型人口都有效，因為社會距離和宵禁等措施對整個人口都有同樣的影響。感染者被清除的速度為 k，相當於隻影響有症狀感染者的隔離措施。
新的隔離室 x 量化了有症狀的隔離感染者。在這裡，我們假設分數 x (t)與經驗確認和報告的案例 c (t)成正比，並且可以忽略抽樣和檢驗結果之間的時間間隔。K0 = 0的情況對應於一種情況，在這種情況下，一般人口不受政策的影響，或者不對流行病作出行為變化的反應。K0 = 0的情況與有症狀的感染者不是特別孤立的情況相對應。

In the basic SIR model that captures unconstrained, free spread of the disease, the basicreproduction number R0 is related to transmission and recovery rate by
becauseis the average time an infected individual remains infectious before
recovery or removal. Here, the time period
that an infected individual remains infectious issuch that theeffective, or “observed,” reproduction numberis smaller than R0;free because both k0 > 0 and k > 0.

The key mechanism at work in the model defined by Eqs. 1 to 4 is the exponentially fast depletion of susceptibles in addition to isolation of infecteds. This effect is sufficient to account for the observed scaling law in the number of confirmed cases for a plausible range of model parameters as discussed below
在方程式1至4所定義的模型中起作用的關鍵機制是除隔離受感染者外，易感者的指數快速耗竭。這種效應足以解釋如下所述的模型引數的合理範圍內，在已確認的案例數中觀察到的標度律

Conclusion

The model reproduces the empirical case counts in all provinces well for plausible parameter values. The quality of the reproduction of the case counts in all 29 affected provinces can be used to estimate the peak time of the number of asymptomatic or oligosymptomatic infected individuals in the population, which is the key quantity for estimating the time when an outbreak will wane.
該模型較好地再現了各省份的經驗案例數，得到了合理的引數值。在所有29個受影響省份中，病例計數的複製質量可用於估計人口中無症狀或少症狀感染者人數的高峰時間，這是估計疫情消退時間的關鍵數量

We stress that our model describes the general effects of containment mechanisms, effectively averaged over many applied strategies or individual changes of behavior. Our analysis therefore cannot identify the efficacy of specific actions. As the implementation of drastic measures such as mandatory curfews can have severe consequences for both individuals as well as a country’s society and economy, decisions about their application should not be made lightly.
我們強調，我們的模型描述了遏制機制的一般影響，有效地平均了許多應用策略或行為的個人變化。因此，我們的分析不能確定具體行動的有效性。由於強制宵禁等嚴厲措施的實施可能對個人以及一個國家的社會和經濟產生嚴重後果，因此不應輕率地決定是否實施這些措施。