How to plot and animate numerically calculated Poisson integral

I want to plot the expression formed from numerically calculated Poisson integrals (aka fundamental solutions of heat equation). I can only get numerical values. This question arises from my previous one.

ODE system. We extract the solutions.

s = NDSolve[{u'[x] == -3 W[x] + x, W'[x] == u[x] - W[x]^3, u[0] == -1,
    W[0] == 1}, {u, W}, {x, 0, 200}]
G = First[u /. s]
g = First[W /. s]

They will serve as initial conditions in Poisson integrals.
Now we choose parameter, some x and t and integration limits.

[Epsilon] = 1/10
T = -1/2
X = 10
p1 = -200
p2 = 200

Now we consctruct the expression.

Q1 = 1/( 2 Sqrt[Pi *((T) + 1)*([Epsilon])^(2)]) NIntegrate[
   Exp[-(Abs[X - [Xi]])^2/(4*((T) + 
          1)*([Epsilon])^(2))] g[[Xi]] G[[Xi]] 
(-1/(2*([Epsilon])^2)), {[Xi], p1, p2}]
Q2 = 1/( 2 Sqrt[Pi *((T) + 1)*([Epsilon])^(2)]) NIntegrate[
   Exp[-(Abs[X - [Xi]])^2/(4*((T) + 
          1)*([Epsilon])^(2))] g[[Xi]], {[Xi], p1, p2}]
q = (-2 ([Epsilon])^2 )*(Q1/Q2)

I need to plot and animate q for any x and t intervals. By the code above I can only get numerical values.

Tried to get the tables of values and then animate it like this…

plots = Table[
   Plot[(1/( 2 Sqrt[Pi *((t) + 1)*([Epsilon])^(2)]) NIntegrate[
        Exp[-(Abs[x - [Xi]])^2/(4*((t) + 
               1)*([Epsilon])^(2))] g[[Xi]] G[[Xi]] (-1/(2*(
[Epsilon])^2)), {[Xi], p1, p2}])/(1/( 
         2 Sqrt[Pi *((t) + 1)*([Epsilon])^(2)]) NIntegrate[
        Exp[-(Abs[x - [Xi]])^2/(4*((t) + 
               1)*([Epsilon])^(2))] g[[Xi]], {[Xi], p1, p2}]), {x, 
     0, 2}, PlotRange -> {-10, 10}], {t, -2, 0, .25}];
ListAnimate[plots]

But Mathematica is just running and I receive the error that one number "is too small to represent as a normalized machine number".

I think it should be very simple but I am a newbie in Wolfram Mathematica so I’m sorry if the question is too trivial. Hope to get help.