Simulating correlated lifetimes at pre-specified correlation level

I am trying to simulate remission times for 100 patients from an exponential distribution with mean 1 year. I also want to simulate after-remission times for these 100 patients. But it is very much likely in the real life scenario that those who remit quickly are likely to have comparatively longer after-remission times and those who remit slowly are likely to have comparatively shorter after-remission times. So the after-remission time is likely to be correlated negatively to the remission time.

I want to test the performance of an estimator at certain levels of correlation (say, r=-0.5, r=-0.8 etc.). Could you please suggest how should I simulate the remission times and the after-remission times imposing the condition of certain levels of correlations?

Should I consider the relationship to be linear or non-linear?

I know from the real life data that the average after-remission time is about 1.5 years. How can I reflect this information in my simulated data, imposing the level of correlation?

I am using R for this simulation study.