# How do we solve this system?

consider: \begin{align} \dot S &= A-\beta S(I+\rho_1 T)-(\mu+p)S\\[1ex] \dot V &=pS-\rho_2 \beta V(I+\rho_1 T)-\mu V\\[1ex] \dot L &=l \beta S(I+\rho_1 T)+\rho_2 \beta V(I+\rho_1 T)-(\mu+\delta)L +\rho T\\[1ex] \dot I &=(1-l)\beta S(I+\rho_1 T)+\delta L-(\mu+\alpha+\gamma)I\\[1ex] \dot T &=\gamma I-(\mu+\rho)T \end{align} with $$N=S+V+L+I+T$$.

to find the equilibriums, we set the above system to $$0$$: \begin{align} A-\beta S(I+\rho_1 T)-(\mu+p)S&=0\\[1ex] pS-\rho_2 \beta V(I+\rho_1 T)-\mu V&=0\\[1ex] l \beta S(I+\rho_1 T)+\rho_2 \beta V(I+\rho_1 T)-(\mu+\delta)L +\rho T&=0\\[1ex] (1-l)\beta S(I+\rho_1 T)+\delta L-(\mu+\alpha+\gamma)I&=0\\[1ex] \gamma I-(\mu+\rho)T&=0 \end{align}

How we solve this for $$S,V,L,T$$ in terms of $$I$$?

For example we see: $$T^*=\frac{\gamma}{\mu+\rho}I^*$$

I tried using solve but this isn't working. This is the code I used:

Solve[A - \[Beta] S (i + \[Rho]1 T) - (\[Mu] + p) S == 0,   p S - \[Rho]2 \[Beta] V (i + \[Rho]1 T) - \[Mu] V == 0,   l \[Beta] S (i + \[Rho]1 T) + \[Rho]2 \[Beta] V (i + \[Rho]1 T) - (\[Mu] + \[Delta]) L + \[Rho] T ==    0, (1 - l) \[Beta] S (i + \[Rho]1 T) + \[Delta] L - (\[Mu] + \[Alpha] + \[Gamma]) i == 0, \[Gamma] i - (\[Mu] + \[Rho]) T == 0, {S,    V, L, T}]

• "I tried using solve but this isn't working." Please post the code you tried. Jan 24 at 14:52
• @RohitNamjoshi Solve[A - \[Beta] S (i + \[Rho]1 T) - (\[Mu] + p) S == 0, p S - \[Rho]2 \[Beta] V (i + \[Rho]1 T) - \[Mu] V == 0, l \[Beta] S (i + \[Rho]1 T) + \[Rho]2 \[Beta] V (i + \[Rho]1 T) - (\ \[Mu] + \[Delta]) L + \[Rho] T == 0, (1 - l) \[Beta] S (i + \[Rho]1 T) + \[Delta] L - (\[Mu] + \ \[Alpha] + \[Gamma]) i == 0, \[Gamma] i - (\[Mu] + \[Rho]) T == 0, {S, V, L, T}]
– Math
Jan 24 at 15:02
• This is not a linear system. There are terms involving products like S * I, both of which are solution variables. Jan 24 at 15:21
• @SjoerdSmit My bad, I rushed the title. Fixed now.
– Math
Jan 24 at 15:23
• added the code from the comment to the OP
– kcr
Jan 24 at 15:52

## 2 Answers

Clear["Global*"]

You have five equations with only four unknowns. Use the option MaxExtraConditions. The use of FullSimplify slows this down considerably.

sol = Solve[{
A - β S (i + ρ1 T) - (μ + p) S == 0,
p S - ρ2 β V (i + ρ1 T) - μ V == 0,
l β S (i + ρ1 T) + ρ2 β V (i + ρ1 T) - (μ + δ) L + ρ T == 0,
(1 - l) β S (i + ρ1 T) + δ L - (μ + α + γ) i == 0,
γ i - (μ + ρ) T == 0},
{S, V, L, T}, MaxExtraConditions -> 1] // FullSimplify


• How long did it take to run this on your computer?
– Math
Jan 24 at 17:02
• Using version 13.0.0 for Mac OS X x86 (64-bit) (December 3, 2021) the AbsoluteTiming is 2.41174 Jan 24 at 17:05

Use lower case names, otherwise there is danger of a conflict with built in names. You use "I" what in MMA is the imaginary unit. I therefore replaced it by W. In addition using the hint from Bob Halon:

sol = Solve[{A - β S (W + ρ1 T) - (μ + p) S == 0,
p S - ρ2 β V (W + ρ1 T) - μ V == 0,
l β S (W + ρ1 T) + ρ2 β V (W + ρ1 T) - \
(μ + δ) L + ρ T ==
0, (1 - l) β S (W + ρ1 T) + δ L - (μ + \
α + γ) W == 0, γ W - (μ + ρ) T ==
0}, {S, V, L, T}, MaxExtraConditions -> 1] // FullSimplify


Update

To bring this into more readable form:

Grid[Transpose[{{"S= ", "V= ", "L= ", "T= "}, {S, V, L, T} /.
sol[[1]]}] , Alignment -> Left]
`

• both solutions are nice, however, can we have it in the form like in the paper?
– Math
Jan 25 at 13:12
• See the update. Jan 25 at 13:32
• what I meant was, if you can access the paper, you'll notice his solutions are in a "nice form" using one another, do you know how to do this?
– Math
Jan 25 at 14:56