How does this expression follow algebraically from the last one?

Question

I was reading this paper:

Global stability for an HIV/AIDS epidemic model with different latent stages and treatment

Everything is understood apart from on page 7 of the pdf (page 1486 in the document). How does the author algebraically go from the second line to the last line for the equation of $\dot V$?

I don't understand how he "generates" more terms in the last line compared to the one preceding it.

EDIT: For those who cannot access the paper

The system: \begin{align*} \dot S &= \Lambda - (\beta_1 S I_2 +\beta_2 S J )-\mu S \\ \dot I_1 &= p\beta_1 S I_2 +q\beta_2 S J +\xi_1 J -b_1 I_1\\ \dot I_2 &= (1-P)\beta_1 S I_2 +(1-q)\beta_2 S J +\epsilon I_1 +\xi_2 J -b_2 I_2\\ \dot J &= p_1 I_2 -b_3 J\\ \dot A &= p_2 J - b_4 A \end{align*} Equilibrium point: \begin{align*} S^* &= \frac{\Lambda}{\mu \mathcal{R}_0}\\ I_1^* &= \frac{1}{b_1}\left[ p \beta_1 \frac{\Lambda b_3 }{\left(\beta_1 b_3 +\beta_2 p_1 \right)J^* +\mu p_1}\right. +\left. q \beta_2 \frac{\Lambda p_1}{\left(\beta_1 b_3 +\beta_2 p_1 \right)J^* +\mu p_1}+\xi_1\right]J^*\\ I_2^* &= \frac{b_3}{p_1}J^*\\ J^*&= \frac{\mu p_1 }{\beta_1 b_3 +\beta_2 p_1}\left(\mathcal{R}_0-1\right)\\ A^*&=\frac{p_2}{b_4}J^* \end{align*}

Theorem: If $p=q$ and $\mathcal{R}_0 >1$ then the above equilibrium point is globally stable.

Proof:

Define Lyapunov function: $$V = S-S^* \ln S+ B( I_1-{I_1}^* \ln I_1) +C(I_2-{I_2}^* \ln I_2) + D( J -J^* \ln J)$$

Derivative is given by:

$$\dot V =\left(1-\frac{S^*}{S}\right) \dot S + B\left(1-\frac{{I_1}^*}{I_1} \right)\dot I_1 + C\left(1-\frac{{I_2}^*}{I_2}\right)\dot I_2+D\left(1-\frac{J^*}{J} \right)\dot J$$

The author then does substitutions i.e replaces $\Lambda$, $b_1$, $b_2$, $b_3$ by making the original system equal to $0$. He finds the constants $B$, $C$ and $D$ by killing the variable co-efficients, giving:
\begin{align*} B &= \frac{\epsilon}{\epsilon p +b_1(1-p)}\\ C&= \frac{b_1}{\epsilon p +b_1(1-p)}\\ D &= \frac{b1 b_2}{p_1[\epsilon p +b_1(1-p)]} -\frac{\beta_1 S^*}{p_1} \end{align*}

Next he does another substitution $ x=\dfrac{S}{S^*}$, $y=\dfrac{I_1}{I_1^*}$, $z=\dfrac{I_2}{I_2^*}$ and $u=\dfrac{J}{J^*}$ to which he arrives at: \begin{align*} \dot V &= -\mu S^* \frac{\left(1-x\right)^2}{x}+ \left[ \beta_1 S^* {I_2}^*+\beta_2 S^* J^*+B p\beta_1 S^* {I_2}^* + B q \beta_2 S^* J^*\right.\\ &+ B \xi_1 J^*+ C(1-p)\beta_1 S^* {I_2}^*+C(1-q)\beta_2 S^* J^*\\ &+\left. C \epsilon {I_1}^*+C \xi_2 J^*+D p_1 {I_2}^*\right]-x\left[ C(1-p)\beta_1 S^* {I_2}^* \right] -\frac{xz}{y}B p\beta_1 S^* {I_2}^*\\ & -\frac{xu}{y}B q \beta_2 S^* J^* -\frac{u}{y}B \xi_1 J^* - \frac{xu}{z}C(1-q)\beta_2 S^* J^*\\ &- \frac{y}{z}C \epsilon {I_1}^* - \frac{u}{z}C \xi_2 J^*-\frac{z}{u}D p_1 {I_2}^*\\ &- \frac{1}{x}\left[\beta_1 S^* {I_2}^*+ \beta_2 S^* J^*\right]\\ &= -\mu S^* \frac{\left(1-x\right)^2}{x} + \frac{b_1}{\epsilon p +b_1(1-p)}(1-p)\beta_1 S^* I_2^* \left(2-x-\frac{1}{x}\right)\\ &+\frac{b_1}{\epsilon p +b_1(1-p)}\xi_2 J^*\left(2-\frac{u}{z}-\frac{z}{u}\right)\\ &+\frac{b_1}{\epsilon p +b_1(1-p)}(1-q)\beta_2 S^* J^* \left(3-\frac{1}{x}-\frac{xu}{z}-\frac{z}{u}\right)\\ &+\frac{\epsilon}{\epsilon p +b_1(1-p)}p \beta_1 S^* I_2^*\left(3-\frac{1}{x}-\frac{xz}{y}-\frac{y}{z}\right)\\ &+\frac{\epsilon}{\epsilon p +b_1(1-p)} q\beta_2 S^* J^*\left(4-\frac{1}{x}-\frac{y}{z}-\frac{z}{u}-\frac{xu}{y} \right)\\ &+ \frac{\epsilon}{\epsilon p +b_1(1-p)}\xi_1 J^*\left(3-\frac{y}{z}-\frac{z}{u}-\frac{u}{y} \right) \end{align*}

I don't understand how he "generates" more terms in the last line compared to the one preceding it.

Answer 1

Posting Mathematica code, as requested ...

(Some notational conventions: I write bb for $\beta$, ee for $\epsilon$, xx for $\xi$, LL for $\Lambda$, mm for $\mu$; a trailing s corresponds to a superscript $*$, as in Ss for $S^*$; and I tend to double my uppercase variables ---BB, CC, DD for $B$, $C$, $D$--- to avoid conflicting with Mathematica's predefined C and D operators. Also, I don't bother with index subscripts; eg, bb1 is $\beta_1$ and I2s is $I_2^*$. )

The first pass at calculating the difference of the left- and right-hand sides of the target equality. I did some preliminary grouping, ignored the common first term on each side, and also temporarily re-introduced $B$ and $C$ for the right-hand side to reduce clutter):

Factor[(
  Ss (bb1 I2s + bb2 Js) (1 - 1/x)
+ BB ( p bb1 I2s Ss (1 - x z/y)
     + q bb2 Js Ss (1 - x u/y)
     + xx1 Js (1 - u/y) )
+ CC ( (1 - p) bb1 I2s Ss (1 - x) 
     + (1 - q) bb2 Js Ss (1 - x u/z) 
     + ee I1s (1 - y/z)
     + xx2 Js (1 - u/z) )
+ DD p1 I2s (1 - z/u) 
) - (
  BB ( p bb1 I2s Ss (3 - 1/x - x z/y - y/z)
     + q bb2 Js Ss (4 - 1/x - y/z - z/u - x u/y)
     + xx1 Js (3 - y/z - z/u - u/y) )
+ CC ( (1 - p) bb1 I2s Ss (2 - x - 1/x)
     + xx2 Js (2 - u/z - z/u)
     + (1 - q) bb2 Js Ss (3 - 1/x - x u/z - z/u) )
)]

Terms from the $B$ and $C$ group cancel, but there's still a bit of a mess left.

Second pass, substituting the $B$, $C$, $D$ definitions:

Factor[% /. {
         DD -> b1 b2/p1/(ee p + b1 (1 - p)) - bb1 Ss/p1,
         BB -> ee/(ee p + b1 (1 - p)),
         CC -> b1/(ee p + b1 (1 - p)) }] 

The mess expands.

Third pass, substituting the "equilibrium point" expressions, and $q\to p$:

Factor[% /. {
  Ss  -> LL/mm/R0,
  I1s -> Js/b1 ((p bb1 LL b3 + q bb2 LL p1)/((bb1 b3 + bb2 p1) Js + mm p1) + xx1),
  I2s -> b3/p1 Js } /. {
  Js  -> p1 mm/(bb1 b3 + bb2 p1) (R0 - 1),
  q   -> p}]

This gives the following result. (I'm omitting the denominator, and factors R0-1 (which the article assumes is non-zero in the given context) and u-z.)

  LL ( ee p + b1 (1-p) ) (b3 bb1 + bb2 p1)
- R0 mm ( b1 b2 b3 - p1 (ee xx1 + b1 xx2) )

The article (after equation (3.13)) calculates that, for the situation under consideration, $$R_0 = \frac{(\epsilon p + b_1 (1-p))(b_3 \beta_1 + \beta_2 p_1)\frac{\Lambda}{\mu}}{b_1b_2b_3-p_1(\epsilon \xi_1+b_1\xi_2)}$$ which is precisely what is needed to make the above expression vanish. $\square$

There may be a possibility of getting to zero without explicitly unravelling every substitution; exploring that possibility is left as an exercise to the reader.