Consider the following paper:

"A note on global stability for a tuberculosis model" by Gao and Huang: https://doi.org/10.1016/j.aml.2017.05.004

The methodology is understood in this paper apart from some technical things; How did the author arrive at constants $m_i, i=1,...,5$? They achieved this by killing the variable terms ($xu, xv, z, u, v, yu, yv$), however when I try solving the system, it outputs all $m_i$'s equalling zero. Can someone show a detailed answer how they arrived at these constants and how they got their final form of the Lyapunov function (page 168)?

Code:

```
sol = Solve[{-m1 \[Beta] is + m3 l \[Beta] ss is/ls +
m4 (1 - l) \[Beta] ss ==
0, -m1 \[Rho]1 \[Beta] ts + m3 \[Rho]1 l \[Beta] ss ts/ls +
m4 (1 - l) \[Beta] \[Rho]1 ss ts/is ==
0, -(\[Mu] + \[Delta]) m3 + m4 \[Delta] ls/is == 0,
m1 \[Beta] is + m2 \[Rho]2 \[Beta] is -
m4 (\[Mu] + \[Alpha] + \[Gamma]) + m5 (\[Mu] + \[Rho]) == 0,
m1 \[Rho]1 \[Beta] ts + m2 \[Rho]1 \[Rho]2 \[Beta] ts +
m3 \[Rho] ts/ls - m5 (\[Mu] + \[Rho]) ==
0, -m2 \[Rho]2 \[Beta] is + m3 \[Rho]2 \[Beta] vs is/ls ==
0, -m2 \[Rho]1 \[Rho]2 \[Beta] ts +
m3 \[Rho]1 \[Rho]2 \[Beta] vs ts/ls == 0 }, {m1, m2, m3, m4, m5}]
```

EDIT:

Trying with the authors co-efficients, I managed to get:

```
m1 = ss;
m2 = (\[Delta] vs )/(((1 - l) (\[Mu] + \[Delta]) + \[Delta] l));
m3 = (\[Delta] ls )/(((1 - l) (\[Mu] + \[Delta]) + \[Delta] l));
m4 = ((\[Mu] + \[Delta]) is )/(((1 -
l) (\[Mu] + \[Delta]) + \[Delta] l));
m5 = (1/(\[Mu] + \[Rho])) (\[Rho]1 \[Beta] ss ts + \[Rho]1 \[Rho]2 \
\[Beta] ((\[Delta] vs ts)/((1 -
l) (\[Mu] + \[Delta]) + \[Delta] l)) + ((\[Rho] \[Delta] \
ts)/((1 - l) (\[Mu] + \[Delta]) + \[Delta] l)));
```

$xu$ term:

```
FullSimplify[-m1 \[Beta] is + m3 l \[Beta] ss is /ls +
m4 (1 - l) \[Beta] ss]
(* 0 *)
```

$xv$ term:

```
FullSimplify[-m1 \[Rho]1 \[Beta] ts + m3 \[Rho]1 l \[Beta] ss ts/ls +
m4 (1 - l) \[Beta] \[Rho]1 ss ts/is]
(* 0 *)
```

$z$ term:

```
FullSimplify[-(\[Mu] + \[Delta]) m3 + m4 \[Delta] ls/is]
(* 0 *)
```

$u$ term:

```
FullSimplify[
m1 \[Beta] is + m2 \[Rho]2 \[Beta] is -
m4 (\[Mu] + \[Alpha] + \[Gamma]) + m5 (\[Mu] + \[Rho])]
(* (-is \[Alpha] (\[Delta] + \[Mu]) -
is (\[Gamma] + \[Mu]) (\[Delta] + \[Mu]) +
is ss \[Beta] (\[Delta] + \[Mu] - l \[Mu]) - (-1 +
l) ss ts \[Beta] \[Mu] \[Rho]1 + is vs \[Beta] \[Delta] \[Rho]2 +
ts \[Delta] (\[Rho] + \[Beta] \[Rho]1 (ss +
vs \[Rho]2)))/(\[Delta] + \[Mu] - l \[Mu]) *)
```

$v$ term:

```
FullSimplify[
m1 \[Rho]1 \[Beta] ts + \[Rho]2 \[Beta] \[Rho]1 ts m2 +
m3 \[Rho] ts/ls - m5 (\[Mu] + \[Rho])]
(* 0 *)
```

$yu$ term:

```
FullSimplify[m3 \[Rho]2 \[Beta] vs is/ls - \[Rho]2 \[Beta] is m2]
(* 0 *)
```

$yv$ term:

```
FullSimplify[
m3 \[Rho]2 \[Rho]1 \[Beta] vs ts/ls - \[Rho]2 \[Rho]1 \[Beta] ts m2]
(* 0 *)
```

The $u$ term is causing some problems here, why?

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