SIR Model

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We will create Mathematical codes in Vcell for

• SIR Model
• Fitzhugh-Nagumo Model

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SIR Model
The population can be subdivided into a set of
distinct classes, dependent upon their
experience with respect to the disease. These are
Susceptible, Infectious or Recovered. Thus comes
the term the SIR model.
Rate of transmission--
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Modeling
Assumptions 1. Non-lethal epidemic (i.e.
SIR Nconstant) 2. Once a person recovered,
can not get infected
rate of change of suceptible - (infection
rate)S, Infection rate depends on number of
infectious individuals (I) and the Probabily to
catch the infection (b). The equation for
susceptible is-
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Rate of change of infectious population-
, g is the recovery rate
Equation for infectious population -
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Rate of change of recovered population-
Initial condition S(0)N-I0 I(0)I0 R0. SIR
N constant
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Start Math Model in Vcell
File?New?MathModel?Non-spatial
Start writing code here
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First declare the variable parameters for
example S_init, I_init, R_init ( s0.9,
I0.1 R0) Then constants, b and g (2.0
and 1.0 )
Declare VolumeVariable S, I and R
Declare 3 Function Susceptible, Infectious,
Recovered populations
Write 3 ODEs as before
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Finally the Model will be like this.
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Run simulation and see the results
S(0)0.995 I(0)0.005, b2.0, g1.0 Time20 sec
dI/dt bSI - gI I (bS - g) S(0)gt(g/b),
dI/dt gt0 epidemics S(0)lt(g/b), dI/dtlt0 no
epidemics
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S(0)9.0,I(0)0.1,bg1.0
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S(0)0.9, I(0)0.1, bg1.0
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Fitzhugh-Nagumo Model for Neural Impulses This
Model is a simplification of the Hodgkin-Huxley
Model.

• Propagation of nerve signal is electrical in
  nature.
• The signal propagates down the length of the axon
  to synapses, which are connected to the
  neighboring neurons.
• Propagated signal is called action potential
• Neuronal signals travel along the cell membrane
  of the axon in the form of local voltage
  difference accross the membrane.

Reference http//www.scholarpedia.org/article/Fit
zHugh-Nagumo_model
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V Excitability of the system (Membrane
potential in the axon) C openness of ion
channels, representing combined forces that tend
to return the state of the axonal membrane to
rest I externally applied voltage or
stimulus that leads to an excitation
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Start writing MathModel in Vcell
Constant declaration V_init C_init I
Try, V_init0.1 C_init0 I0
VolumeVariable V VolumeVariable C
Function J1 I V (0.2-V) (V-1) C
Function J2 0 .002(V-C)
ODEs as before
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Simple MathModel
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Run simulation Try initially, I0 C0 and V
0.1 t_end 200 sec
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I0 C0 and V 0.3 t_end 800 sec
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Exercise 1 play with this Fitzhugh-Nagumo
model in VC keep all parameters parameters the
same as in the previous slide start increasing
I to 0.05, then 0.1, then 0.15, then 0.2 Think
about what is going on! Try to plot V(C) and
think some more
Run the simulation for t_end 2000 sec
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C-V curve for different values of I.,
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Spatial diagram C and V w.r.t. time for
different values of I.
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Exercise 2
Consider SIR model in more detail.
B20 alpha1 beta0.1 gamma1 lambda1
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Exercise1
This is my math model.
In my model Betab Alpha a Gammag Lambdal
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S(0)10, I(0)1, b 0.1
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For b0.2
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Seasonal getting sick with period 1 year (for
example Flu )
Take parameters-- B20 alpha1 beta0.1
gamma1 lambda1
Try to play with this model in VC vary
parameters think about what is going on
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Something new--
Here beta is not constant, rather it fluctuates
periodically. beta beta_0 (1cos t)
So we have to declare this periodic function.
HOW ??? Write, Function K cos (t) Use this
function to write other functions Rest of the
code is same as before !!!!!
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My MathModel looks like this...
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For small b0.1, almost no infection
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For b0.2
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S(0)10 I(0)1 b0.3, lga1.0