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During COVID-19, we've heard many comparisons between SARS-CoV-2 and the Flu, especially in terms of the case fatality rate (CFR). Currently, the Flu is thought to be around 0.1%, and according to the latest CDC estimates (as of 2020-05-28), COVID-19 is around 0.4% (or 0.26% when including asymptomatic cases).

I haven't been able to find any information about the Common Cold's CFR. Do we have any estimates for where that sits?

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SARS-Cov-2 is a distinct virus that causes a distinct disease: COVID-19.

Virus influenzae is a distinct virus (or a term for a small group of similar viruses) that causes a distinct disease: seasonal influenza or "flu."

This is why you can compare death rates of SARS-Cov-2 and influenza virus, or COVID-19 and flu.

"Common cold" refers to a group of various viral infections of the nose and throat, which can be caused by at least 200 different viruses. This means you can't compare death rates of a diverse group of diseases called common cold and a single disease, such as COVID-19 or flu.

When a common cold extends to the lower respiratory tract, it is, by definition, no longer called common cold, but laryngitis, bronchitis or pneumonia, for example.

When a person gets a common cold that becomes complicated as pneumonia and dies from it, the cause of death is recorded as pneumonia not as a common cold.

If you stick with a definition of common cold linked above:

Common cold is a self-limited contagious disease that can be caused by a number of different types of viruses. The common cold is medically referred to as a viral upper respiratory tract infection.

...then you can't even say that common cold is deadly.

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I had trouble finding any specific estimates for your question after various searches, but given the common cold is much milder than the flu, one can presume it is significantly less than the CFR of the flu.

The issue with the common cold isn't the virus itself, but instead the complications that patients can get after the virus, which can rarely but potentially become dangerous. For instance, a bacterial sinus infection can develop -- in children, 5-13% of bacterial sinus infections have a preceding viral infection. Serious but rare neurological diseases, like Guillain-Barre Syndrome, can also occur after a common cold.

The common cold can also exacerbate the symptoms one has from other health conditions, such as COPD, which is a common cause for hospitalization among adults with COPD.

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