I have a data as following. I want to add two random time points, one for birth one for death.

 

       data={{1, 10}, {1.06117, 11}, {1.21241, 12}, {1.33893, 13}, {1.41358, 14}, {2, 14}};

    ListLinePlot[data, Mesh -> Full, MeshStyle -> {Red, PointSize[0.01]},
      Frame -> True, PlotTheme -> "Detailed", 
     FrameLabel -> {"Time", "Population"}, ImageSize -> Medium, 
     InterpolationOrder -> 0]
[![enter image description here][1]][1]

Here is one desired output 

 

       data2={{1, 10}, {1.06117, 11}, {1.10169, 11}, {1.21241, 12}, {1.33893, 
      13}, {1.41358, 14}, {1.51463, 14}, {2, 14}};

    ListLinePlot[{data, data2}, Mesh -> Full, 
     MeshStyle -> {Red, PointSize[0.01]}, Frame -> True, 
     PlotTheme -> "Detailed", FrameLabel -> {"Time", "Population"}, 
     ImageSize -> Medium, InterpolationOrder -> 0]

[![enter image description here][2]][2]



birth is added at `{1.10169, 11}` and death is added at `{1.51463, 14}`. I wanna keep both path, i.e. blue and yellow paths. Second path (yellow path ) must also start at `{1,10}` and finish at `{2,14}`. Population size can increase or decrease only by 1. Any idea?

**Edit:** Let me generalized the algorithm.

Given interval `[i,i+1]` ,in this case `[1,2]`, we have 3 types of move (equal probability, 1/3 each ).

Choose one of the move.


 1.  Type 1 move: Add 1 birth and 1 death randomly on the interval `[1,2]` which is done successfully by @Carl and @kglr.
 2.  Type 2 move: Select 1 birth and 1 death from interval and remove them. If there is no birth remove only 1 death, and similarly if there is no death remove only 1 birth.
 3. Type 3 move: Select 1 reaction time, birth OR death, and shifted  somewhere else between `[1,2]`




  [1]: https://i.sstatic.net/N2Wcc.png
  [2]: https://i.sstatic.net/RpXgA.png