Continuous Mathematical Models 1-Mathematical Modeling and Simulation-Lecture Slides, Slides of Mathematical Modeling and Simulation

Download Continuous Mathematical Models 1-Mathematical Modeling and Simulation-Lecture Slides and more Slides Mathematical Modeling and Simulation in PDF only on Docsity!

Lecture Slides

on

Modeling and Simulation

Continuous Mathematical Models

• A linear first order model based on ODE has the following general form:

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 ' a(t)yb(t)

y ayb

p( t)y g(t)

dt

dy

First Order Linear Models in Biophysics

• Consider a colony of micro-organisms reproducing

through simple cell division under ideal conditions of

unlimited food supply and total absence of any predators.

• It is observed that the population increases q (10% every

hour).

• Find a mathematical model that will also produce the

same results.

Example 1: A Model from Biology

• Consider N to be number of organisms present at any time t.
• The after Δt time they will have increased by amount ΔN
• This change ΔN is directly proportional to N and the time interval Δt.

rN t

N  



N  Nt

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 expr  1 

N  N  0 exp rt

When time t = 1 hour, the N icreases by 10%. Let us put this in the model.

1. 1 exp r  r ln 1. 1   0. 0953

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:

• Consider N to be number of organisms present at any time t.
• The after Δt time they will have increased by amount ΔN
• This change ΔN is directly proportional to N and the time interval Δt.

gain rate N; population^ gain rate aN

Where, a is proportionality constant.

Population loss rate:

• It is propotional to the square of populational at that time:

loss rate N ; 2  population loss rateb N^2

Where, b is proportionality constant.

Example 2: Population of Fruit flies

Balance Equation:

• The net rate of change of population will be equal to gain

minus loss rates.

aN bN^2

dt

dN

  Mathematical model

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

Docsity.com

Variable Coefficient Case:

• A model with linear first order ODEs and variable

coefficients may look like this:

• The method of integrating factors is used to find

solution.

• It involves multiplying this equation by a function ( t ),

chosen so that the resulting equation is easily

integrated.

p( t)y g(t)

dt

dy

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.

e y

dt

dy t

 /^2  2

Example 3: Population of germs

/ 2 2 t  y  y  e

Example 3: Analytical solution

Let us find analytical solution using Integrating Factor

Multiplying both sides by ( t ), we obtain

We will choose ( t ) so that left side is derivative of known quantity.

Consider the following, and recall product rule:

Choose ( t ) so that

/ 2 2 t y  y  e

  y

dt

d t

dt

dy

t y t

dt

d ( )

(t ) 2 (t)  (t ) e^2 t

( ) 2 (t)y e /^2 (t)

dt

dy

 t    t 

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:

1. They have rate of growth equal to zero.
2. Their first loss rate is t – 5; where t is in seconds.
3. They have another way of loss and its rate is proportional to the

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