How to interpret expected death count?

That looks OK.

The "trick" is to keep track of persons and time.

Working with:

    CMR = 0.16 / 10,000 / day
    population = 5 million

The proportion of the population that dies each day is:

    p = 0.16 / 10,000 = 0.000016

So the number of deaths in the 5 million population is:

    Daily deaths = p * 5,000,000 = 0.000016 * 5,000,000 = 80

There are about 365 days in a year (leap years have 366) so the number of deaths in a year is:

    Annual deaths = 80 * 365 = 29,200

which is what you get.

A few things to consider ...

You have to bear in mind that your estimate of CMR may not be very accurate. This is often the case with mortality estimates made from surveys.

Since we are dealing with estimates we should account for uncertainty. An easy way of doing this is to do the calculation for the lower 95% confidence limit, the point estimate, and the upper 95% confidence limit for CMR.

Populations change over time. We have a start population (5 million). If population is growing at 2.41% per year (that's the 2017 estimate for 2017 for Rwanda) then after a year that 5 million becomes:

    5,000,000 * (1 + 2.41 / 100) = 5,120,500

You can use the mid-point:

    (5,000,000 + 5,120,500) / 2 = 5,060,250

Using this gives:

    Annual deaths = 0.000016 * 5,060,250 * 365 = 29552

Assuming (your data will likely be different) the 95% CI for the CMR is [0.09,0.25] we have:

    95% LCL = 0.000009 * 5,060,250 * 365 = 16,623
    95% LCL = 0.000025 * 5,060,250 * 365 = 46,175

To be able to "safely say" I think you should (if possible) check your CMR, use 95% CIs to account for imprecision in the CMR estimate, and correct the population for growth. When reporting results you should use words like "estimated" and "predicted" and present the estimate with a 95% confidence interval. With what I have above I'd say something like " an estimated 29,552 (95% CI = 16,623 - 46,175) deaths predicted to occur over the year".

I hope this is of some use.

It gets a bit confusing as person-times vary. Often we have "per 1000 per day" or "per 100,000 per year" or even or "per 1000 live births" (with time unspecified). With emergencies we have "per 10,000 per day". I find that keeping track of the persons and the times and checking all conversions and calculations helps.

You numbers are large because you have a large population (5 million) and a long period (365 days). I find it useful to familiarise myself with magnitudes so I can spot when something is way off. Try, for example, going to WolframAlpha and specifying "mortality Burundi". This give 115,050 deaths per year on a 10.9 million population. This is much larger then your 29,200. If we standardise to 5 million it is:

    115050 * (5 / 10.9) = 52,775

Your number does not seem so very large.

I hope this helps.

We need to assume that the mortality rate for the rural area is from a sample that is representative of the rural population and the mortality rate for the urban area is representative of the urban population. In that case we can combine the two estimates.

A simple average (as you have above) is not appropriate unless the two populations are of equal size. What is needed is a weighted average:

               r1 * w1 + r2 * w2 + ... + rn * rn
  Pooled rate = ---------------------------------
                       w1 + w2 + wn

where:

  r1 = rate for area 1
  r2 = rate for area 2
  .
  . and so-on
  .
  w1 = population in area 1
  w2 = population in area 2
  .
  . and so-on
  .

Assuming:

  Area    Population    Rate
  ----    ----------   -----
  Rural        98578    0.35
  Urban        21422    0.17
  ----    ----------   -----

We get:

               0.35 * 98578 + 0.17 * 21422
  Pooled rate = -------------------------- = 0.32 / 1000 / year
                      98578 + 21422

The pooled rate is closer to the rural rate because, in this example, most people live in the rural area (urbanisation is at c. 18%).

We should always present a confident interval (or similar) when we make an estimate. This is not so easy. We need to pool variances. Survey samples are complex and the variance is influenced by the proportion, the sample size, and the survey design effect. We can get over this by calculating the standard error (SE) from survey results from each survey:

       Upper Confidence Limit - Lower Confidence Limit
  SE = -----------------------------------------------
                         2 * 1.96

With the example results we have:

             
  Rural SE = (0.41 - 0.30) / (2 * 1.96) = 0.0281
  Urban SE = (0.20 - 0.11) / (2 * 1.96) = 0.0230

The pooled SE is:

                  ( SE1^2 * w1 + SE2^2 * w2 + ...  SEn^2 * wn )
  Pooled SE = sqrt( ----------------------------------------- )
                  (               w1 + w2 + wn                )

where:

  SE1 = SE for survey 1
  SE2 = SE for survey 2
  .
  . and so-on
  .
  w1  = population in area for survey 1
  w2  = population in area for survey 3
  .
  . and so-on
  .

With the example results we have:

  Pooled SE = sqrt((0.0281^2 * 98578 + 0.0230^2 * 21422) / (98578 + 21422))
            = 0.0273

The pooled estimate is:

  Point estimate = 0.32
  95% LCL = 0.32 - 1.96 * 0.0273 = 0.27
  95% UCL = 0.32 + 1.96 * 0.0273 = 0.38

Notes ...

(a) When you do these sorts of calculation you should do them to the full precision throughout and only round at the end. I did not do this above so there will be some accumulated rounding error in the final result above.

(b) This is an approximate method based on estimates that were probably made using approximate methods. The 95% CI is only "nominal" and is unlikely to have 95% coverage. Theses sorts of methods are usually good enough.

(c) The original results are very different from each other (i.e. 0.35 vs. 0.17 with 95$ CIs that do not overlap). When this is the case then you should present the results separately as well as the pooled results.

(d) Check my arithmetic.

I hope this is of some use.

I would not do this. I would try to find statistics on urbanisation for the country you are working in and use that. This will likely be better tha assuming 50% urbanisation. If you have to make a guess then it is best to make an informed guess.

You can find useful data from UN DESA / Population Division. The "Data in Excel Files" section is useful.