Trying to plot the norm of the solutions to NDsolve

• plot the norm of the solutions of a differential equation system versus time.

Plot[Evaluate@(Norm[Through[sol[## & @@ #][t]]] & /@ set), {t, 0, tmax}, 
 PlotRange -> All, AspectRatio -> 1, ImageSize -> Large, 
 PlotLegends -> Placed[ToString /@ set, Top], 
 PlotLabel -> (Norm[{P[t], B[t], i[t]}])]

• plot one component of the solutions of a differential equation system versus time.

Row[ParametricPlot[Evaluate@Thread[{t, (Through[sol[## & @@ #][t]] & /@ set)[[All, #]]}], 
    {t, 0, tmax}, PlotRange -> All, 
    PlotLegends -> Placed[set[[All, #]], Top] , AspectRatio -> 1, 
    ImageSize -> 300, PlotLabel -> ({P[t], B[t], i[t]}[[#]])] & /@ {1,
    2, 3}, Spacer[5]]

Alternatively, you can use Plot:

Row[Plot[Evaluate@(Through[sol[## & @@ #][t]] & /@ set)[[All, #]], {t, 0, tmax}, 
    PlotRange -> All, AspectRatio -> 1, ImageSize -> 300, 
    PlotLegends -> Placed[set[[All, #]], Top] , 
    PlotLabel -> ({P[t], B[t], i[t]}[[#]])] & /@ {1, 2, 3}, Spacer[5]]

sol = ParametricNDSolveValue[{ODEsys, {P[0] == init1, B[0] == init2,i[0] == init0}}, {P[t], B[t], i[t]}, {t, 0, tmax}, {init1, init2,init0}]

plot of solutions:

Plot[Table[# &[ Apply[sol, set[[i]]]]  , {i, 1, Length[set]}] , {t, 0,tmax}, PlotRange -> {0, Automatic}]

plot of euclidean norm:

Plot[Table[Sqrt[ #.#] &[ Apply[sol, set[[i]]] ], {i, 1, Length[set]}] , {t, 0,tmax}, PlotRange -> {0, Automatic}]