Is this expression positive or negative given the assumptions?

Question

Consider:

Expre78 = ( b βa (μ (-1 + ν) - ξ) + μ (μ + ξ + ρ) (γa + μ + σ))/(μ (μ + ξ + ρ) σ);

Assuming[
   {b > 0, βa > 0, μ > 0, 0 < ν < 1, βi > 0, ξ > 0, ρ > 0, γa > 0, σ > 0}, 
   Reduce[$Assumptions \[Implies] Expre78 >= 0, {}, Reals]
]

Is this expression positive or negative given the assumptions?

Edit:

if r0<1 then i want to check whether Expre78 <0

r0=(b (\[Mu] - \[Mu] \[Nu] + \[Xi]) (\[Beta]a (\[Alpha] + \[Gamma]i + \
    \[Eta] + \[Mu]) + \[Beta]i \[Sigma]))/(\[Mu] (\[Alpha] + \[Gamma]i + \
    \[Eta] + \[Mu]) (\[Mu] + \[Xi] + \[Rho]) (\[Gamma]a + \[Mu] + \
    \[Sigma]));

        Expre78 = (b \[Beta]a (\[Mu] (-1 + \[Nu]) - \[Xi]) + \[Mu] (\[Mu] + \
\[Xi] + \[Rho]) (\[Gamma]a + \[Mu] + \[Sigma]))/(\[Mu] (\[Mu] + \[Xi] \
+ \[Rho]) \[Sigma]);

Assuming[{b > 0, \[Beta]a > 0, \[Beta]i > 0, \[Mu] > 0, 
  0 < \[Nu] < 1, \[Gamma]a > 0, \[Gamma]i > 0, \[Xi] > 0, \[Rho] > 
   0, \[Eta] > 0, \[Alpha] > 0, \[Sigma] > 0, r0 < 1}, 
 Reduce[$Assumptions \[Implies] Expre78 <= 0, {}, Reals]]

Answer 1

Clear["Global`*"]

Expre78 = (b βa (μ (-1 + ν) - ξ) + μ (μ + ξ + ρ) (γa + μ + σ))/
   (μ (μ + ξ + ρ) σ);

It can be either.

To be nonnegative

Assuming[{b > 0, βa > 0, μ > 0, 
  0 < ν <= 1, ξ > 0, ρ > 0, γa > 0, σ > 0}, 
 Reduce[Expre78 >= 0] // Simplify]

(* b βa (μ - μ ν + ξ) <= μ (μ + ξ + ρ) (\
γa + μ + σ) *)

vars = Variables[Level[Expre78, {-1}]];

Nonnegative values

Expre78 >= 0 /. 
 FindInstance[{Expre78 >= 0, b > 0, βa > 0, μ > 0, 
   0 < ν <= 1, ξ > 0, ρ > 0, γa > 0, σ > 0}, vars, 5]

(* {True, True, True, True, True} *)

Negative values

Expre78 < 0 /. 
 FindInstance[{Expre78 < 0, b > 0, βa > 0, μ > 0, 
   0 < ν <= 1, ξ > 0, ρ > 0, γa > 0, σ > 0}, vars, 5]

(* {True, True, True, True, True} *)

EDIT: For the revised question

Clear["Global`*"]

r0 = (b (μ - μ ν + ξ) (βa (α + γi + η + μ) + βi σ))/
  (μ (α + γi + η + μ) (μ + ξ + ρ) (γa + μ + σ));

Expre78 = (b βa (μ (-1 + ν) - ξ) + μ (μ + ξ + ρ) * 
  (γa + μ + σ))/(μ (μ + ξ + ρ) σ);

vars = Variables[Level[Expre78 && r0, {-1}]]

{b, α, βa, βi, γa, γi, η, μ, ν, ξ, ρ, σ}

There are no nonpositive cases

FindInstance[{Expre78 <= 0, b > 0, βa > 0, βi > 0, μ > 0, 
  0 < ν < 1, γa > 0, γi > 0, ξ > 0, ρ > 0, η >
    0, α > 0, σ > 0, r0 < 1}, vars]

(* {} *)

or

Reduce[{Expre78 <= 0, b > 0, βa > 0, βi > 0, μ > 0, 
  0 < ν < 1, γa > 0, γi > 0, ξ > 0, ρ > 0, η >
    0, α > 0, σ > 0, r0 < 1}]

(* False *)

There is at least one positive case

inst = FindInstance[{Expre78 > 0, 
   b > 0, βa > 0, βi > 0, μ > 0, 
   0 < ν < 1, γa > 0, γi > 0, ξ > 0, ρ > 0,
   η > 0, α > 0, σ > 0, r0 < 1}, vars]

(* {{b -> 7/2, α -> 1, βa -> 1, βi -> 3/2, γa -> 1,
   γi -> 1, η -> 1, μ -> 1, ν -> 1/2, ξ -> 1,
   ρ -> 1, σ -> 1}} *)

Expre78 /. inst[[1]]

5/4