What Mathematical modelling then to do with
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]
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
“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
CHIT JAL PAWAK GAGAN SAMIRA
PANCH TATVA SE BANA SHARIRA
The survival of all living beings depends upon the following FIVE Tatvas
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
“An equation for me has no meaning unless it
expresses the thought of God.”
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,
Unknown and known variables
Mathematical relations among the variables
All systems are governed by some laws, principles and
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
Solution are difficult, as no. of unknown variables are large.
They can be analysed using both quantitative techniques as well as
Validations is needed.
These are diseases which are transmitted by
Factors affecting the spread of such diseases:
No. of susceptibles
No. of infectives
No of exposed persons
No. of removed persons
Rate of contact
Rates of removal
Genetic and immunological factors
Nature of diseases
Growth of Vector, Carrier, Bacteria population.
Diffusion of population in the habitat
Convective effect caused by motion of the medium
Migration between two habitats
Some More Factors
Religion, Caste Creed.
Interactions at work place.
Modes of Transmission
• Direct contact may be with the infective or with the
ii. Indirect Contact/Transmission
Disease agent are passed on the susceptible through
the medium of some kinds.
Skin to skin mucosa to
STD, AIDS, Leprosy, Eye
Spray of droplets of
Common old, T.B.
saliva, other secretions by
Contact with soil, etc.
Direct contact of
susceptible with disease
agent in soil etc.
Pathway for Transmission
Vector, Carrier, borne
Vectors transmit infection by
biting or by depositing
infective material on food
,Typhoid fever by House
Through the agency of food,
water, ice, blood, serum,
tissues, injection, cloths, soil,
Typhoid fever (TB)
Air borne bacteria,
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
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
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
5) What is the effect of control measures, if any other
the spread etc.
A General Epidemic model
X: Susceptible Y = Infective
A XY dX Y Z k1 X
XY dX Y Y 1Y
1Y dZ Z
k1 X dZ1
N X Y Z Z1
A dN Y
Z = Removed Z1 = Isolation
1. Non mixing models for the spread of infections
Effects of sanitation
II. Effect of nutrition
III. Effects of medication
IV. Effects of awareness
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.
Non mixing model
β-Interaction coefficient, rate of contracts.
It should dependent upon the following
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
A X Y d 2
dX d1 2
X Y d 2 dY d 2 2
Effect of Sanitation
S = Sanitation variable
B = Bacteria
A ( S )Y dX
( S )Y dX
Y 0 B 1 BS
B 0 ( S S0 ) 1 BS
Effect of Medication
M= Variable for Medical support
A ( M )Y dX
( M )Y dX
Y 0 ( M M 0 ) 1 BM
Where B(M) to be defined suitably
Effect of Nutrition
H= Healthy Population Nu = Variable for nutrition
A0 dH 1 XN
A ( Nu )Y dX 1 XN
( Nu )Y dX Y
Y 0 ( Nu Nu0 )
N X Y H
A0 A dN Y
Effect of Media Awareness
Me=Variable for Awareness
A ( M e ) XY dX
( M e ) XY dX
Y 0 ( M e M e 0 )
How to Consider
First we consider the changes in environmental/ecological
Then their effect on e human / agent (vector, bacteria
0 E 1 EN
Q(N) Q0 T
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
A x Y d 2 dX d 2
x Y d 2 dY d 2 2