We have a sigmoid with the form
$$y=\frac{e^{x+a}}{b+ce^{x+d}}$$
where $a,b,c,d$ are real constants.
We have known points $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4)$ all satisfying the equation of the sigmoid, and we need to find the constants $a,b,c,d$.
So is it possible to solve the following system for those constants:
$$\left\{\begin{matrix} y_1=\frac{e^{x_1+a}}{b+ce^{x_1+d}}\\ {}\\ y_2=\frac{e^{x_2+a}}{b+ce^{x_2+d}}\\ {}\\ y_3=\frac{e^{x_3+a}}{b+ce^{x_3+d}}\\ {}\\ y_4=\frac{e^{x_4+a}}{b+ce^{x_4+d}} \end{matrix}\right.$$
In other words, we need to express $a,b,c,d$ in terms of $x_1,x_2,x_3,x_4,y_1,y_2,y_3,y_4$.
Wolfram Alpha did not solve this problem as required. What I think, but not sure, that there exist infinitely many combinations of $a,b,c,d$ so that the given points lie on the sigmoid.
If that is the case, I need only one of those combinations.
Here is a clarification graph
Any help would be really appreciated. Thanks in advance!
$$y=\frac{e^{x+a}}{b+ce^{x+d}}$$ $$e^{x+a}=e^ae^x\quad\text{and}\quad e^{x+d}=e^de^x$$ $$y=\frac{e^ae^{x}}{b+ce^de^{x}}=\frac{e^{x}}{be^{-a}+ce^{d-a}e^{x}}$$ Let $\quad B=be^{-a}\quad$ and $\quad C=ce^{d-a}$ $$y=\frac{e^{x}}{B+Ce^{x}}$$ Thus you have not four unknows $a,b,c,d$ but only two unknowns $B,C$. This means that if a couple $B,C$ is solution they are an infinity of solutions for $a,b,c,d$ since $b=Be^a$ and $c=Ce^{a-d}$ . You can take arbitrary $a,d$.
This shows that the question is put on the wrong way.
With four given points $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4)$ leading to four equations but only two unknows, in general the system has no exact solution.
Only two points are sufficient to find $B,C$ from $\quad\begin{cases} y_1=\frac{e^{x_1}}{B+Ce^{x_1}}\\ y_2=\frac{e^{x_2}}{B+Ce^{x_2}} \end{cases}$
I suppose that you can solve the linear system $\quad\begin{cases} B+e^{x_1}C=\frac{e^{x_1}}{y_1}\\ B+e^{x_2}C=\frac{e^{x_2}}{y_1} \end{cases}$
If ANY four points are given, the curve cannot fit exactly. One have to proceed with a regression method which will give be best approximates for $B,C$ according o some criteria of fitting. This can be easily carried out.
Please, reedit your question without ambiguity so that one can give you more help to correctly solve the problem.
In addition :
If the question specifies that four points $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4)$ are given and that the function $y(x)$ must exactly fit, then four independant parameters $a,b,c,d$ are necessary.
An infinity of functions are likely to be convenient. For exemple : $$y=a+\frac{e^{c\,x}}{b+d\,e^{c\,x}}$$ But I bet that this form of equation will not fully satisfy you. In the wording of the question they are not enough specifications about what is expected to chose an equation fully satisfying.