Exercise: SIR MODEL

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Exercise SIR MODEL
(Infected individuals do not move, they stay at
home) What is the effect of diffusion? How is
the behavior affected by the diffusion
coefficient D? What if you have two nests of
infection?
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File-gtopen?mathmodel?satarupa?SIR_NEW_MODEL Save
this .

• create a math Model? Spatial for BOX geometry.
• Copy Paste the Constants, VolumeVariable and
  Functions. Add diffusionRate
• as constant.
• 2.Cut Initial concentration for infected
  population. We want to set infected
• population in a particular place. So we will
  declare it as Function.
• 3. We have no Flux BC.
• 4. Infected people do not move, so no diffusion
  for Infectected population, i.e.
• ODE .

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Part-1
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Part-2
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What happens to Healthy Population
Healthy people move arround and if they come near
infected people, who are in the middle, they get
sick !!
Time plot
Line plot
S_init9.0,D 1.0
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Infected population stays at the middle , see how
the concentration changes as you increase the
time.
Line plot, t 10
Line plot, t .3
Time plot
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Recovered Population
Line plot
Time plot
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Now consider two nests of infection- that is
infection in two places
Save this SIR model with a new name to modify it
.
Only change ? Function I_init
((((x-5)2 y2) lt 1 ) (((x-5)2 (y-10)2)
lt 1 )) 0.2
It specifies two two places of infected
population with the concentration 0.2
Thats all !!!
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Susceptible (D1)
Line plot
Time plot
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Infected
Line plot
Time plot
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Recovered
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When diffusion rate 0
Infection becomes epidemic in the infected region
If healthy people dont move.
Nothing happens outside the infected region
Recovered
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Fitzhugh-Nagumo system with voltage (ions)
spreading along the axon
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Create 2D analytic geometry. Set size x1, Y
0.5, origin at (0.0). Save it with a name .
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Go file? new? math Model?Spatial? click the
geometry you just created file? open? math
Model?satarupa?FHN_model (ode model) Save this.
New conditions for our new system
1.Copy the constants from the old F-N model (ODE
model) and paste, cut Constant V_init, because
V is now a sptial variable, i.e. a Function 2.
Constant V_diffusionRate 0.0003 3. Copy
paste VolumeVariable and Function.Add new
function for V_init.
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We will set PDE and ODE here
CompartmentSubDomain subVolume1 Priority 0
BoundaryXm Flux BoundaryXp Flux
PdeEquation V
BoundaryXm 0.0
BoundaryXp 0.0
Rate J1 Diffusion
V_diffusionRate Initial
V_init OdeEquation C RateJ2 Initial
C_init
We have 1 ODE for C
Click Apply changes.
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The code looks like -
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Click equation viewer --
Close this window and click simulation
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Run simulation for t100, I0, 0.05, 0.2
can you increase parameter I and get periodic
firing?
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For I0.0
V at t0.0
C at t0.0
Time plot C
Time plot V
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Time plot for V with I 0.05
Time plot for V with I 0.2
Time plot for C with I 0.2
Time plot for C with I 0.05
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Time plot for I0.2, t 1000 sec
C
V
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Reaction-Diffusion system of the
activator-inhibitor type that appears to account
for many important types of pattern formation
and morphogenesis observed in development .
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Pattern Formation
The development of a higher organism out of a
single fertilised egg is one of the most
fascinating aspects of biology. A central
question is how the cells, which carry identical
genetic code, become different from each
other. Spontaneous pattern formation in
initially almost homogeneous systems is also
common in inorganic systems. Large sand dunes
are formed despite the fact that the wind
permanently redistributes the sand. Sharply
contoured and branching river systems (which are
in fact quite similar to the branching patterns
of a nerve) are formed due to erosion despite the
fact that the rain falls more or less
homogeneously over the ground
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(No Transcript)
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Model equations
When activator spreads much more slowly than the
inhibitor, periodically spaced peaks of
activator evolve
Exercise 3 on a 2D domain, would you have
stripes or spots?
In my model I have taken ee, deltad
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For this model we will take retangular geometry.
File? Open?Geometry?Satarupa? rectangle save
it. File? New?MathModel?Spatial? click the
geometry you just saved.
Constants, VolumeVariables and Functions
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PDEs
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Click Equation viewer to see the equations
Run simulation for t10, and use line tool to see
kymograph results.
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For e.9,d5, D1.0, t10
i
a
For e.9,d15, D1.0, t10
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Rectangualr geometry
For e.9,d15, D1.0, t10
a
i
Stripes and other patterns can be produced by
reaction diffusion mechanism in 2D domain under
variety of initial conditions and chemical
interactions.
Change initial conditions and different constant
parameters for various pattern.
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Exercise (Fisher equation)
Animals or bacteria grow until the local
environment cannot handle the population and
then spread by diffusion. The result is the
invasion and colonization. Mathematically, you
see the traveling wave solution. How does it
look in 2D? How does the wave speed depend on
the value of parameter D?
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For this model we will take retangular geometry.
File? Open?Geometry?Satarupa? rectangle save
it. File? New?MathModel?Spatial? click the
geometry you just saved.
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Now we have to declare two Functions. 1st
Function is simple Function J P(1-P)
Second Function ? initial concentration of
P Function P_init (1.0 / (1.0 exp((2.0 k
(-1.0 x)))))
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The code looks like---
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Run simulation for t20, and use line tool to see
kymograph results.
t2
t0
t5.7
t14.2
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See the kymograph