Download What Mathematical modelling then to do with

By Prof. J. B. Shukla
International Internet University for Research in Science and
Technology, HQ Kanpur
Website : www.iiurst.org
Formerly, IIT Kanpur
Email: [email protected]
Introduction
From where shall I begin.
Let us begin from the very beginning
What is research?
 Research is to discover some new characteristics about a
system/situations which has not been explored before.
“The greatest force on earth is the human soul (mind) on
fire”
…Forsyth
“People see things and ask why. I see thinks that are not and
ask why not”
…George Bernard Shaw
 Researchers therefore must have that kind of vision and
mind for quality research
Introduction contd.
CHIT JAL PAWAK GAGAN SAMIRA
PANCH TATVA SE BANA SHARIRA
...Tulsidas
 The survival of all living beings depends upon the following FIVE Tatvas
(Resources)
 Air
 Water
 Earth
 Fire (Energy)
 Space(Environment in the Universe)
 Therefore, our research must be directed towards conservation and preservation of
these resources by using modern science and technology
 The role of research should therefore be to preserve these resources provided by
nature and not used them to the extend so that they do not remain available for
future generation
Mathematical Model
“An equation for me has no meaning unless it
expresses the thought of God.”
S. Ramanujan
Thought of God  Nature
Nature includes any thing and every thing in this
universe including planet earth.
As mentioned before research must be focussed on
nature and society.
Mathematical Model contd.
 Mathematics has permeated in all sciences.
 Physical Science, Environmental, Biological Sciences,
Social Sciences.
Systems
Math Models
•
Unknown and known variables
Mathematical relations among the variables
and parameters
All systems are governed by some laws, principles and
hypothesis.
Mathematical Model Cont.
 Models predicts the future.
 The future can not be experimented upon.
 Model predicts the system’s future behaviour on the basis of what





is known today.
Mathematical models are complex and non linear O.D.Es/PDEJ
etc.
Solution are difficult, as no. of unknown variables are large.
They can be analysed using both quantitative techniques as well as
qualitative methods.
Stability analysis
Validations is needed.
Math. Model
Solution
System
validation
interpretation
Infection Diseases
These are diseases which are transmitted by
infection.
Factors affecting the spread of such diseases:
•
•
•
•
•
•
•
•
•
No. of susceptibles
No. of infectives
No of exposed persons
No. of removed persons
Rate of contact
Incubation period
Rates of removal
Genetic and immunological factors
Nature of diseases
Other Factors
i.
ii.
iii.
iv.
v.
vi.
vii.
viii.
Demographic (Human)
Environmental
Ecological
Societal/Economic
Growth of Vector, Carrier, Bacteria population.
Diffusion of population in the habitat
Convective effect caused by motion of the medium
Migration between two habitats
Not better
Better
Some More Factors
a) Isolation
b) Nutrition
c) Sanitation
d) Medication




Religion, Caste Creed.
Social interaction
Professional Interactions
Interactions at work place.
Modes of Transmission
X
Susceptible
Y
Infective
Direct Contact/Transmission
• Direct contact may be with the infective or with the
disease agent.
ii. Indirect Contact/Transmission
 Disease agent are passed on the susceptible through
the medium of some kinds.
i.
Direct Transmission
Type
Pathway for
transmission
Diseases
Direct Contact
Skin to skin mucosa to
mucosa, sexual
STD, AIDS, Leprosy, Eye
infection
Droplet infection
Spray of droplets of
Common old, T.B.
saliva, other secretions by
coughing, sneezing,
spitting etc
Contact with soil, etc.
Direct contact of
susceptible with disease
agent in soil etc.
Hookworm
Larvae, tetanus
Indirect Transmission
Type
Pathway for Transmission
Diseases
Vector, Carrier, borne
Vectors transmit infection by
biting or by depositing
infective material on food
skin etc.
Malaria, Dysentery
,Typhoid fever by House
fly, Cholera
Medium Borne
Through the agency of food,
water, ice, blood, serum,
tissues, injection, cloths, soil,
etc.
Typhoid fever (TB)
Air borne bacteria,
viruses, Droplets,
Infected Dust
Small droplets in the air may
evaporate leave behind
viruses bacterial which
remain in the air and can get
to susceptibles, Sneezing and
spitting on the ground wall
etc.
Chicken Pox, Measles,
Influenza, H1N1, TB, etc
Aims/ Objectives of Study
1) How fast a disease is spreading?
2) How much of the total population has been
infected?
3) How of much of the remaining will be infected at a
particular time/ location
4) What are the causes and factors for the spread of a
particular disease
5) What is the effect of control measures, if any other
the spread etc.
A General Epidemic model
 X: Susceptible Y = Infective
dX
 A   XY  dX   Y   Z  k1 X
dt
dY
  XY  dX  Y   Y  1Y
dt
dZ
  1Y  dZ   Z
dt
dZ1
 k1 X  dZ1
dt
N  X  Y  Z  Z1
dN
 A  dN   Y
dt
Z = Removed Z1 = Isolation
Some Ideas
1. Non mixing models for the spread of infections
diseases
1.
Effects of sanitation
II. Effect of nutrition
III. Effects of medication
IV. Effects of awareness
I.
Effects of environmental and ecological factors on the
spread of infections diseases
3. Effect of diffusion (Movement of population)
4. Effects of movement of convection effects on the spread
of infection diseases.
2.
Non mixing model
 β-Interaction coefficient, rate of contracts.
 It should dependent upon the following
i. Space
ii. Social factors such as going together to Churches,
Temples, Dargahs, Malls, Fairs, Feats And Parties
iii. Office factors such as working in the same office,
department, building etc sitting on the same table.
iv. Professional factors such as doing research together
v. Travel factors such as going to together in same vehicle
vocation doing the office vacation
vi. Awareness of the disease
Non Mixing Models contd…
 Effect of Space

X
 2Y
 A   X Y  d 2
t
s


2 X
  dX  d1 2
s


Y
 2Y 
 2Y
  X  Y  d 2   dY  d 2 2
t
s 
s

Effect of Sanitation
 S = Sanitation variable
 B = Bacteria
dX
 A   ( S )Y  dX
dt
dY
  ( S )Y  dX
dt
dB
  Y   0 B  1 BS
dt
dS
  B  0 ( S  S0 )  1 BS
dt
Effect of Medication
 M= Variable for Medical support
dX
 A   ( M )Y  dX
dt
dY
  ( M )Y  dX
dt
dM
  Y  0 ( M  M 0 )  1 BM
dt
Where B(M) to be defined suitably
Effect of Nutrition
 H= Healthy Population Nu = Variable for nutrition
dH
 A0  dH  1 XN
dt
dX
 A   ( Nu )Y  dX  1 XN
dt
dY
  ( Nu )Y  dX   Y
dt
dNu
  Y  0 ( Nu  Nu0 )
dt
N  X Y  H
dN
 A0  A  dN   Y
dt
Effect of Media Awareness
Me=Variable for Awareness
dX
 A   ( M e ) XY  dX
dt
dY
  ( M e ) XY  dX
dt
dM e
  Y  0 ( M e  M e 0 )
dt
How to Consider
Environmental/Ecological Effects
 First we consider the changes in environmental/ecological
factors
 Then their effect on e human / agent (vector, bacteria
carriers etc).

dE
E 
  E 1 
   0 E  1 EN
dT
 Em 
dT
 Q(N)  Q0 T
dt
Q( N )  Q0  QN
N – Is the total human population
E – Ecological density (biomass density)
T – Density (Environmental factor)
How to consider spatial effects
A simple case with convection and diffusion

X
X
 2Y 
2 X
U
 A   x  Y  d 2   dX  d 2
t
s
s 
s


Y
Y
 2Y 
 2Y
U
  x  Y  d 2   dY  d 2 2
t
s
s 
s
