Question

Help: Stability of Solutions in Mathematical Systems

Answer 1

whats the question?

Answer 2

sorry...

i was asked to use stability analysis to determineif equilibrium point is stable

Answer 3

i didn't really understand how the Stability Analysis works, would you guys please explain it to me. the process as well

Answer 4

:( i'm not sure if i can be of any help
are you talking about control systems, routh tables,root locus and such?

Answer 5

i was given a mathematical model for HIV spread in the community... ifound the equilibrium points already, I just don't know how to determine if the points are stable or not

Answer 6

ahh...ok :)
can you type the differential equation/solution ?

Answer 7

@darkprince14

Answer 8

wait
sorry.net.lag

Answer 9

example of a differential equation-
y'-y=0

:)

Answer 10

thank u qwerty :)

Answer 11

np (;

Answer 12

\[dS=\Pi S - c \beta\frac{ I }{ N }S - \mu S\]\[dI = c\beta \frac{ S }{ N }I - \mu I - \upsilon I\]

Answer 13

@baru

Answer 14

π - immigration rate of males
c - number of partners/male
β - rate of infection per contact
μ - natural death rate or emigration rate
υ - death rate due to HIV

Answer 15

whut on earth is that xD, makes no sense to me
i'll just tag some smart people
@ganeshie8

Answer 16

anyway, how did you find equilibrium points?

Answer 17

Notice that the function is not changing at both those points.
So they are called equilibrium points.

A tiny disturbance at red point ends the equilibrium, so the red point is called unstable euqilibrium point.

However, after small disturbances at green point, the ball always comes back to its equilibrium state, so the green point is called stable euilibrium.

The same terminology/concept should apply to differential equations too...

Answer 18

\[dS=\Pi S - c \beta\frac{ I }{ N }S - \mu S\]\[dI = c\beta \frac{ S }{ N }I - \mu I - \upsilon I\]

I see that S and I are dependent variables
whats the independent variable ?

Answer 19

well N = S+I. so I guess every variable used in the equation except S, I and N is independent

Answer 20

what do N,S,I stand for?

Answer 21

I am not talking about constants
with respect to what variable are you differentiating ?

Answer 22

S is for the susceptible, I stands for Infected, and N is the total population

Answer 23

I don't know, sorry.
I started the stability analysis and did Jacobian and then plugged in equilibrium poins. don't know if I'm heading in the right direction

Answer 24

I may not be useful here,
let me tag @SithsAndGiggles

Answer 25

\[\begin{cases}S'=\pi S-c\beta\dfrac{IS}{S+I}-\mu S\\[1ex]
I'=c\beta\dfrac{IS}{S+I}-\mu I-\upsilon I\end{cases}\]If this problem is about what I think it is then I'm pretty sure you need to find the \(S\)- and \(I\)-nullclines first. These occur for \(S'=0\) and \(I'=0\), respectively.
\[\pi S-c\beta\frac{IS}{S+I}-\mu S=0~~\implies~~S=0,\,\frac{\pi-\mu}{c\beta-\pi+\mu}S=I\]
\[c\beta\frac{IS}{S+I}-\mu I-\upsilon I=0~~\implies~~I=0,\,I=\left(\frac{c\beta}{\mu+\upsilon}-1\right)S\]
The fixed/equilibrium points occur at the intersections of these nullclines, and the only one I'm seeing is \((S,I)=(0,0)\)... which poses quite a problem in the next step.

Consider the Jacobian:
\[J=\begin{pmatrix}\pi-c\beta\dfrac{I^2}{(S+I)^2}-\mu&-c\beta\dfrac{S^2}{(S+I)^2}\\[1ex]c\beta\dfrac{I^2}{(S+I)^2}&c\beta\dfrac{S^2}{(S+I)^2}-\mu-\upsilon\end{pmatrix}\]Around any fixed point \((x,y)\), the nonlinear system behaves like the linear system
\[\begin{cases}S'=J_{1,1}(x,y)S+J_{1,2}(x,y)I\\[1ex]
I'=J_{2,1}(x,y)S+J_{2,2}(x,y)I\end{cases}\]but the Jacobian's components don't exist for \((0,0)\), hence the aforementioned problem...