fill in blanks with exponential estimates

Sure your result is not exponential, you are using a function na.approx() to impute the values using linear interpolation. The zoo package you are using offers to interpolate using cubic spline interpolation using na.spline() function, but this function does not produce exponential curve either.

x <- expand.grid(X2009H1N1 = "0-17 years",
                 type = "Cases",
                 month = seq(as.Date("2009-04-12"), 
                             to = as.Date("2010-03-12"), 
                             by = "month")) %>% 
  bind_cols(data.frame(MidLevelRange = c(0,NA,NA,NA,NA,NA,8000000,16000000,18000000,19000000,19000000,19000000),
                       lowEst = c(0,NA,NA,NA,NA,NA,5000000,12000000,12000000,13000000,14000000,14000000)))

x %>% arrange(month, X2009H1N1) %>% 
  group_by(X2009H1N1, type) %>% 
  mutate(aprox_MidLevelRange = zoo::na.spline(MidLevelRange))

The problem with cubic spline interpolation is that your lowest values will be interpolated as negative, depends whether this is a behavior you are looking for or not:

# A tibble: 8 x 6
# Groups:   X2009H1N1, type [1]
  X2009H1N1  type  month      MidLevelRange   lowEst aprox_MidLevelRange
  <fct>      <fct> <date>             <dbl>    <dbl>               <dbl>
1 0-17 years Cases 2009-04-12             0        0                  0 
2 0-17 years Cases 2009-05-12            NA       NA          -18568160.
3 0-17 years Cases 2009-06-12            NA       NA          -25223342.
4 0-17 years Cases 2009-07-12            NA       NA          -22929832.
5 0-17 years Cases 2009-08-12            NA       NA          -14651914.
6 0-17 years Cases 2009-09-12            NA       NA           -3353875.
7 0-17 years Cases 2009-10-12       8000000  5000000            8000000.

Have a look at the imputeTS package. It offers plenty of imputation functions for time series. Take a look at this paper to get a good overview of all offered options

In your case using Stineman interpolation ( imputeTS::na_interpolation(x, option ="stine") could maybe be a suitable option.

Here for the example you provided:

x <- expand.grid(
  X2009H1N1 = "0-17 years",
  type = "Cases",
  month = seq(as.Date("2009-04-12"),
    to = as.Date("2010-03-12"),
    by = "month"
  )
) %>%
  bind_cols(data.frame(
    MidLevelRange = c(0, NA, NA, NA, NA, NA, 8000000, 16000000, 18000000, 19000000, 19000000, 19000000),
    lowEst = c(0, NA, NA, NA, NA, NA, 5000000, 12000000, 12000000, 13000000, 14000000, 14000000)
  ))

x %>%
  arrange(month, X2009H1N1) %>%
  group_by(X2009H1N1, type) %>%
  mutate(aprox_MidLevelRange = imputeTS::na_interpolation(MidLevelRange, option = "stine"))

This gives you:

# A tibble: 12 x 6
# Groups:   X2009H1N1, type [1]
   X2009H1N1  type  month      MidLevelRange   lowEst aprox_MidLevelRange
   <fct>      <fct> <date>             <dbl>    <dbl>               <dbl>
 1 0-17 years Cases 2009-04-12             0        0                  0 
 2 0-17 years Cases 2009-05-12            NA       NA             593718.
 3 0-17 years Cases 2009-06-12            NA       NA            1335612.
 4 0-17 years Cases 2009-07-12            NA       NA            2289061.
 5 0-17 years Cases 2009-08-12            NA       NA            3559604.
 6 0-17 years Cases 2009-09-12            NA       NA            5336975.
 7 0-17 years Cases 2009-10-12       8000000  5000000            8000000 
 8 0-17 years Cases 2009-11-12      16000000 12000000           16000000 
 9 0-17 years Cases 2009-12-12      18000000 12000000           18000000 
10 0-17 years Cases 2010-01-12      19000000 13000000           19000000 
11 0-17 years Cases 2010-02-12      19000000 14000000           19000000 
12 0-17 years Cases 2010-03-12      19000000 14000000           19000000 

So just comparing interpolation functions I guess this could be the best option.

Just plot yourself the different interpolation options, to see the differences. In general this are the interpolation options:

imputeTS::na_interpolation(x, option ="linear")
imputeTS::na_interpolation(x, option ="spline")
imputeTS::na_interpolation(x, option ="stine")

linear / spline options from imputeTS are the same as zoo::approx()/ zoo::spline(). stine does not exist in zoo.