The mathematics of computing confidence and prediction bands of curves fit by nonlinear regression are explained in this Cross-Validated page. It shows that the bands are not always/usually symmetrical.
And here is an explanation with more words and less math:
First, let's define G|x, which is the gradient of the parameters at a particular value of X and using all the best-fit values of the parameters. The result is a vector, with one element per parameter. For each parameter, it is defined as dY/dP, where Y is the Y value of the curve given the particular value of X and all the best-fit parameter values, and P is one of the parameters.)
G'|x is that gradient vector transposed, so it is a column rather than a row of values.
Cov is the covariance matrix (inversed Hessian from last iteration). It is a square matrix with the number of rows and columns equal to the number of parameters. Each item in the matrix is the covariance between two parameters. We use Cov to refer to the normalized covariance matrix, where each value is between -1 and 1.
c = G'|x * Cov * G|x.
The result is a single number for any value of X.
The confidence and prediction bands are centered on the best fit curve, and extend above and below the curve an equal amount.
The confidence bands extend above and below the curve by:
= sqrt(c)*sqrt(SS/DF)*CriticalT(Confidence%, DF)
The prediction bands extend a further distance above and below the curve, equal to:
In both these equations, the value of c (defined above) depends on the value of X, so the confidence and prediction bands are not a constant distance from the curve. The value of SS is the sum-of-squares for the fit, and DF is the number of degrees of freedom (number of data points minus number of parameters). CriticalT is a constant from the t distribution based on the confidence level you want (traditionally 95%) and the number of degrees of freedom. For 95% limits, and a fairly large df, this value is close to 1.96. If DF is small, this value is higher.