Here is an implementation in R. We'll use Poisson regression, though this may not be appropriate for your data. You'll want to do a deviance goodness of fit test on your model once it is fit. If you reject the null of this test, you should then try a negative binomial regression, or if there are a lot of zeros, a zero-inflated Poisson. I can add details as necessary.
First, we'll simulate some data. Shown below is the fake data I made
Since these are counts, we'll hypothesize that the distribution of the data on any given data is Poisson with mean $\lambda(t)$. So imagine that the data above are the data you have. Fit the Poisson regression to that data and continue. You don't seem to be interested in inference, so I will skip right to the simulation bit.
With the model I fit, I can now predict the mean of the Poisson distribution for any day (remember, the data for any given day is Poisson distributed). What I'll do is predict those means, then simulate data from Poisson distributions with those means. Here is the result
The red line is the mean of the distributions, the blue dots are the simulated data. The intensity of the blue color is a proxy for how frequent the simulated points are. For instance, at $t = 0$ points are very frequent at $y=10$ but not so frequent at $y = 20$.
Here is the code to make this example.
#Simulate Data for regression
times = c(0,sort(sample(0:365, size = 7)))
B = c(log(10),-0.01)
lin.pred = model.matrix(~times)%*%B
y = map(lin.pred, ~rpois(20,exp(.x)))
d = data_frame(times = times, y = y) %>%
#Fit the poisson regression
model = glm(y~times, data = d, family = 'poisson')
#Make some simulated data
predict.at = seq(0,365,30)
means = exp(predict(model, newdata = list(times = predict.at)))
simulated_data = map(means, ~rpois(500,.x))
simd = data_frame(times = predict.at,
y = simulated_data) %>% unnest()
lined = data_frame(times = predict.at, y = means)
#Plot the simulated data
geom_point(color = 'blue', alpha = 0.15)+
geom_line(data = lined, color = 'red')
Obviously, this should just be a starting point, not an all out solution. I can't say anything of substance about your data without having it in my hands. Is this what you were thinking of?