This may not strike most as a bioinformatics, but getting the key clinical outcome is essential in understanding the molecular basis of pathogenicity.
I think the mortality rate is over-reported. This is not to say the situation of 2019-nCov is not serious - it is very serious.
The two essential factors missing in your equation for 2019-nCov are:
- Age. Mortality is likely skewed towards the old and very young
- Seroprevalence. To identify the number of asymptomatic patients.
Old people have a weakened immune system (e.g. reduced T-cell response). The very young have an underdeveloped immune system. The problem is hospital cases are skewed towards those with severe infection, which is likely the old and very young. In 2019-nCov the first infecteds were middle aged men, but likely to be restricted to those exposed to the "Wuhan meat market".
Asymptomatic "spreaders" are known in influenza and are suspected in 2019-nCov. Asymptomatics will inflate the mortality rate (because we're not detecting them).
- One solution is to conduct seroprevalence surveys. Serological surveys screening for antibodies (IgG) against 2019-nCov will identify asymptomatic patients, using the first formula you stated. However, would you seriously go around the general population in a 2019-nCov epidemic taking blood samples? Errhhh no.
- The bio-statistics approach is to use a catalytic model and critical to this is the age of the patient. Catalytic models are a priori models that incorporates the age-prevelance profile of infection. By introducing the incidence and age of hospitalisation of e.g. the old and the young, the rest of the distribution, i.e. asymptomatics can be calculated by regression. This estimates the total population of infecteds and again you use equation 1 to calculate the mortality rate recorded by the hospital.
- Mathematical models based on transmission dynamics. The problem here is the rate of transmission is unknown.
I would assume the hospital is the place virus-induced mortality is reasonably accurate.
The second formula you described works very well for Ebolavirus, no-one has ever heard of asymptomatic Ebolavirus. It would also work well for another betacoronavirus Middle Eastern respiratory syndrome. The key problem with 2019-nCov is we don't know the number of asymptomatic patients or patients with such mild infections they never report illness.
In summary, I think the mortality is being over-reported if it is based on your equation 2. Equation 1, however, will also over-report the mortality rate.