2 2 getting sick m oving around getting sick recovering 2 2 recovering m oving around S S SI D t x I SI I t R R I D t x 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?
37
Embed
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.
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
2
2
getting sickmoving around
getting sick recovering
2
2
recoveringmoving around
S SSI D
t x
ISI I
t
R RI D
t x
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?
create a math Model Spatial for BOX geometry.
1. 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 .
File->openmathmodelsatarupaSIR_NEW_MODELSave this .
Part-1
Part-2
Healthy people move arround and if they come near infected people, who are in the middle, they get sick !!
What happens to Healthy Population:
Time plot
Line plot
S_init=9.0,D= 1.0
Infected population stays at the middle , see how the concentration changes as you increase the time.
Line plot, t= .3
Time plot
Line plot, t= 10
Recovered Population:
Time plot Line plot
Now consider two nests of infection- that is infection in two places:
Save this SIR model with a new name to modify it .
can you increase parameter I and get periodic firing?
For I=0.0V at t=0.0 C at t=0.0
Time plot CTime plot V
Time plot for V with I= 0.05 Time plot for V with I= 0.2
Time plot for C with I= 0.2Time plot for C with I= 0.05
Time plot for I=0.2, t= 1000 sec
V C
Reaction-Diffusion system of the activator-inhibitor type that appears to account for many important types of pattern formation and morphogenesis observed in development .
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
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?
Model equations:
In my model I have taken e=e, delta=d
For this model we will take retangular geometry.
File OpenGeometrySatarupa rectangle save it.
File NewMathModelSpatial click the geometry you just saved.
Constants, VolumeVariables and Functions
PDEs :
Click Equation viewer to see the equations:
Run simulation for t=10, and use line tool to see kymograph results.
For e=.9,d=5, D=1.0, t=10
a
For e=.9,d=15, D=1.0, t=10
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.
For e=.9,d=15, D=1.0, t=10
Rectangualr geometry:
a i
2
21
1, 11, , 0
0, 1
P PP P D
t xx
D P x tx
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?
For this model we will take retangular geometry.
File OpenGeometrySatarupa rectangle save it.
File NewMathModelSpatial click the geometry you just saved.
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))))) ;
The code looks like---
Run simulation for t=20, and use line tool to see kymograph results.