Passing ParametricNDSolve through NIntegrate?

Question

I am trying to integrate (function2) an integrand (function1) that is a function of a ParametricNDSolve output (solution1). This somehow works if I use Integrate for function1, as opposed to using NIntegrate. But, if I use NIntegrate, I receive a ParametricNDSolve error saying "500 cannot be used as a parameter." Is there a solution to this? Any help would be greatly appreciated!

Alex

a = 0.03; b = d = 1; c = 0.5; K = HMax = 2000; v = 10^(-7);

solution1[g_, t0_] := H[t] /. ParametricNDSolve[{X'[t] == 
  X[t]*b*H[t]/HMax - X[t]*d*(X[t] + Y[t])/K, 
 Y'[t] == 
  Y[t]*b*(1 - c*g)*H[t]/HMax + 
   Y[t]*b*(1 - c*g)*g*(1 - H[t]/HMax) - Y[t]*d*(X[t] + Y[t])/K, 
 H'[t] == Y[t]*b*(1 - c*g)*(1 - H[t]/HMax)*g - a*H[t], 
 X[0] == ((b E^((b E^(-a t0) K t0)/HMax) K^2)/(d E^(a t0) (-1 + 
         E^((b E^(-a t0) K t0)/HMax)) HMax + b K)), Y[0] == 1, 
 H[0] == Exp[-a*t0]*HMax}, {X[t], Y[t], H[t]}, {t, 0, 
 10000}, {t0}] // First;

function1[g_, t0_] := NIntegrate[t*solution1[g, t0]/K, {t, 0, 200}]

function2[g_] :=Block[{},1 -Exp[-v*NIntegrate[function1[g, T], {T, 0, 500},Method -> {Automatic, "SymbolicProcessing" -> 0}]]]

function2[0.4]

Answer 1

We can set g as parameter in ParametricNDSolve

a = 0.03; b = d = 1; c = 0.5; K = HMax = 2000; v = 10^(-7);
 sol = ParametricNDSolve[{X'[t] == 
     X[t]*b*H[t]/HMax - X[t]*d*(X[t] + Y[t])/K, 
    Y'[t] == 
     Y[t]*b*(1 - c*g)*H[t]/HMax + Y[t]*b*(1 - c*g)*g*(1 - H[t]/HMax) -
       Y[t]*d*(X[t] + Y[t])/K, 
    H'[t] == Y[t]*b*(1 - c*g)*(1 - H[t]/HMax)*g - a*H[t], 
    X[0] == ((b E^((b E^(-a t0) K t0)/HMax) K^2)/(d E^(a t0) (-1 + 
            E^((b E^(-a t0) K t0)/HMax)) HMax + b K)), Y[0] == 1, 
    H[0] == Exp[-a*t0]*HMax}, {X, Y, H}, {t, 0, 10000}, {g, t0}];

function1[g_, t0_] := 
  NIntegrate[t*H[g, t0][t]/K /. sol, {t, 0, 200}];
function1[10, 1]

1059.44

function2[g_] := 
 Block[{}, 
  1 - Exp[-v*
     NIntegrate[function1[g, T], {T, 0, 500}, 
      Method -> {Automatic, "SymbolicProcessing" -> 0}]]]

function2[0.4]

0.468472

But the last code get some warning message.