難以發現的無症狀感染者是 COVID-19 的其中一個著名特點。因此，實際受到感
染的人數可能會遠大於官方所報告的人數。受此啟發，我們開始研究如何去估計社
交圖中真實的感染人數。在本文中，我們假設疾病的傳播是使用independent cascade model。當疾病傳播結束時，我們會在社交圖中觀察到受到感染並被檢測出來的點集合，我們將此情況稱作為一個事件 (Event)。
假設在疾病傳播開始時只存在單一病源，我們想（i）找到觀察到的事件發生的機率
（ii）估計給定事件的預期感染人數。為了解決這兩個問題，我們運用了隨機圖方法。
對於線圖，我們推導出了這兩個問題的閉合解。對於樹圖，我們提出了線性時間演算
法來計算這兩個值。對於一般圖，我們設計了基於最小Steiner樹的重要性採樣方法來估計這兩個值。最後模擬結果表示出，我們的重要性抽樣方法比標準模擬方法的相對
誤差要小得多。

One of the prominent features of COVID-19 is that asymptomatic infections are difficult to detect. As such, the actual number of infections might be much larger than the officially reported number. Motivated by this, we consider the problem of estimating the number
of infections with undetectable infections in a social graph. We use the independent cascade model for disease propagation in a graph. When the disease propagation ends, we observe an event that consists of a set of reported cases in the social graph. Assuming that there is a single source in the graph at the beginning of the disease propagation, we are interested in (i) finding the probability of the observed event and (ii) estimating the expected number of infections given that event. To address these two questions, we adopt a random graph approach. For line graphs, we derive closed-form expressions of
these two quantities. For trees, we propose linear-time algorithms to compute these two quantities. For general graphs, we devise efficient importance sampling methods based on Steiner minimal trees to estimate these two quantities. Our simulation results show that our importance sampling methods achieve much smaller relative errors than those of the standard simulation methods.

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