Components of (Subscript[R, 0])

Manipulate[
Plot[((βm v)/(βh+v)),{v,0,10},Frame->True,FrameLabel->{Style[v,FontSize->Large],Style["β(v)",FontSize->Large]},PlotRange->{0,5}],"Transmission rate as function of virulence (\!\(\*FractionBox[\(\*SubscriptBox[\(β\), \(max\)]\\ v\), \(\*SubscriptBox[\(β\), \(h\)] + v\)]\))",{{βm,4,"(Subscript[β, max])"},0,5},{{βh,1,"(Subscript[β, h])"},0,2}]

Manipulate[
Plot[(n/(v+m+c)),{v,0,10},Frame->True,FrameLabel->{Style[v,FontSize->Large],Style["(n/(v + m + c))",FontSize->Large]},PlotRange->{0,3}],"Everything else as function of virulence",{{n,2.5,"n"},0,5},{{c,0.5,"c"},0,3},{{m,0.5,"m"},0,3}]

Put it all together to plot (Subscript[R, 0])

Manipulate[
Plot[((βm v)/(βh+v))(n/(v+m+c)),{v,0,4},Frame->True,FrameLabel->{Style[v,FontSize->Large],Style["(Subscript[R, 0])",FontSize->Large]},PlotRange->{0,5},Epilog->{Opacity[If[showMax,0.5,0]],EdgeForm[Opacity[If[showMax,0.7,0]]],Red,Disk[{(Sqrt[c βh+m βh]),((n (Sqrt[(c+m) βh]) βm)/((c+m+(Sqrt[(c+m) βh])) (βh+(Sqrt[(c+m) βh]))))},{0.07,0.07GoldenRatio}]}],"Fitness of virus",{{n,2.5,"n"},0,5},{{c,0.5,"c"},0.001,3},{{m,0.5,"m"},0.001,3},{{βm,4,"(Subscript[β, max])"},0,5},{{βh,1,"(Subscript[β, h])"},0,2},{{showMax,False,"Show Maximum"},{True,False}}]

Answer: Find function for maximum

(Subscript[R, 0])==((β[v] n)/(v+m+c));

Substitute expression for β[v]

((βm v)/(βh+v))(n/(v+m+c));

Take the derivative with respect to v

derv=D[((βm v)/(βh+v))(n/(v+m+c)),v]

Set that equal to 0

derv==0

Solve for v.  Note, only what positive v.

Solve[derv==0,v]