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These lecture slides are delivered at The LNM Institute of Information Technology by Dr. Sham Thakur for subject of Mathematical Modeling and Simulation. Its main points are: Continuous, Mathematical, Models, First, Order, Analytical, Constant, Coefficient, Case, Population
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where f is linear in y. Examples include equations with constant coefficients,
or equations with variable coefficients:
or, more conventionally,
f( t,y) dt
dy
y ayb
First Order Linear Models in Biophysics
Example 1: A Model from Biology
rN t
N
N Nt
rN dt
dN ^ Mathematical model
Where, r is proportionality constant. This will be the parameter of the system
Example 1: A Model from Biology
Example 1: A Model from Biology
When time is zero, we say N = N(0);
Then C = N(0). rN; dt
dN
This value of r is in # per hour. This is termed as growth constant.
N 0 0. 1 N 0 N 0 expr 1
N N 0 exp rt
When time t = 1 hour, the N icreases by 10%. Let us put this in the model.
This model is showing very important law of exponential growth.
Example 1: A Model from Biology
Let us consider N(0)=1;
rN; dt
dN
r = 0.0953 is in # per hour.
N N 0 exp rt
Let us model this law of exponential growth by MATLAB.
simout To Workspace
Scope 1/s Integrator
Gain
(^00 2 4 6 8 )
50
100
150
200
250
300
Number of organisms
time(hours)
using ode45 in matlab
Population gain rate:
gain rate N; population^ gain rate aN
Where, a is proportionality constant.
Population loss rate:
loss rate N ; 2 population loss rateb N^2
Where, b is proportionality constant.
Example 2: Population of Fruit flies
Balance Equation:
minus loss rates.
Where, a and b are proportionality constants.
Let us consider both of them as positive numbers.
Example 2: Population of Fruit flies
Example 1: A Model from Biology
N(0)=100;
aN bN^2 dt
dN
a = 0.1 in # per day. b = 0.0005 in # per day.
0 20 40 60 80 100
0
50
100
150
200
250
Number of flies
time(days)
using ode45 in matlab simout To Workspace
Scope
u^2 Math Function
1/s Integrator
-0. Gain
Gain
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Variable Coefficient Case:
Method of Integrating Factors
Major features/properties of the following mathematical model:
Dependent and independent variable: y and t,
Order: one;
Linearity: It is a linear equation as it has no product term:
Homogeneity: It is non-homogeneous equation with a force term ( exp( t/2 ).
Conditions: Initial conditions can be given.
Coefficients: There are constant coefficients.
Driving term type: It has analytical term as opposed to a tabular form.
Model Equation Type: It is a single ordinary differential equation based model. It
is time dependent first order deterministic model.
Example 3: Population of germs
/ 2 2 t y y e
Example 3: Analytical solution
Let us find analytical solution using Integrating Factor
Consider the following, and recall product rule:
/ 2 2 t y y e
y
Example 3: General Solution
The red lines are direction field and the blue lines are solutions with different initial conditions.
Example 4: Population of viruses
Consider another type of viruses in experiments showing following loss
properties:
population at that time and proportionality constant is about 1/5.
y t y
Rate of change of population = rate of growth – rate of loss
Let us call population at any time t as y(t). Then
Rate of growth = 0.0 ; first rate of loss = 5 – t.
Second Rate of loss = y/.
y ^ t^ y^ Mathematical model