What Mathematical modelling has to do with infectious diseases

By Prof. J. B. ShuklaInternational Internet University for Research in Science and

Technology, HQ KanpurWebsite : www.iiurst.org

Formerly, IIT KanpurEmail: jbs@iitk.ac.in

IntroductionFrom where shall I begin.Let us begin from the very beginningWhat 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. RamanujanThought of God NatureNature 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.

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 analysisValidations is needed.

System

Math. Model Solution

interpretationvalidation

Mathematical Model Cont.

Infection DiseasesThese 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 Factorsi. Demographic (Human)ii. Environmentaliii. Ecologicaliv. Societal/Economicv. Growth of Vector, Carrier, Bacteria population.vi. Diffusion of population in the habitatvii. Convective effect caused by motion of the mediumviii.Migration between two habitats

Not better Better

Some More Factorsa) Isolationb) Nutritionc) Sanitationd) Medication Religion, Caste Creed. Social interaction Professional Interactions Interactions at work place.

Modes of Transmission

i. 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.

XSusceptible

YInfective

Direct TransmissionType Pathway for

transmissionDiseases

Direct Contact Skin to skin mucosa to mucosa, sexual

STD, AIDS, Leprosy, Eye infection

Droplet infection Spray of droplets of saliva, other secretions by coughing, sneezing, spitting etc

Common old, T.B.

Contact with soil, etc. Direct contact of susceptible with disease agent in soil etc.

HookwormLarvae, tetanus

Indirect TransmissionType Pathway for

TransmissionDiseases

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 Study1) 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/ location4) What are the causes and factors for the

spread of a particular disease5) What is the effect of control measures, if

any other the spread etc.

A General Epidemic modelX: Susceptible Y = Infective Z = Removed

Z1 = Isolation1

1

1

11 1

1

dXA XY dX Y Z k X

dtdY

XY dX Y Y YdtdZ

Y dZ ZdtdZ

k X dZdtN X Y Z Z

dNA dN Y

dt

Some Ideas1. 1. Non mixing models for the spread of

infections diseasesI. Effects of sanitationII. Effect of nutritionIII. Effects of medicationIV. Effects of awareness

2. 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 modelβ-Interaction coefficient, rate of contracts.It should dependent upon the followingi. Spaceii. Social factors such as going together to Churches,

Temples, Dargahs, Malls, Fairs, Feats And Partiesiii. Office factors such as working in the same office,

department, building etc sitting on the same table.iv. Professional factors such as doing research

togetherv. Travel factors such as going to together in same

vehicle vocation doing the office vacationvi. Awareness of the disease

Non Mixing Models contd…Effect of Space

2 2

12 2

2 2

22 2

X Y XA X Y d dX d

t s s

Y Y YX Y d dY d

t s s

Effect of SanitationS = Sanitation variableB = Bacteria

0 1

0 0 1

( )

( )

( )

dXA S Y dX

dtdY

S Y dXdtdB

Y B BSdtdS

B S S BSdt

Effect of MedicationM= Variable for Medical support

Where B(M) to be defined suitably

0 0 1

( )

( )

( )

dXA M Y dX

dtdY

M Y dXdtdM

Y M M BMdt

Effect of NutritionH= Healthy Population Nu = Variable for

nutrition0 1

1

0 0

0

( )

( )

( )

dHA dH XN

dtdX

A Nu Y dX XNdtdY

Nu Y dX YdtdNu

Y Nu NudtN X Y H

dNA A dN Y

dt

Effect of Media AwarenessMe=Variable for Awareness

0 0

( )XY

( )

( )

e

e

ee e

dXA M dX

dtdY

M XY dXdtdM

Y M Mdt

How to Consider Environmental/Ecological EffectsFirst we consider the changes in

environmental/ecological factorsThen their effect on e human / agent (vector, bacteria

carriers etc).

N – Is the total human population E – Ecological density (biomass density)T – Density (Environmental factor)

0 1

0

0

1

(N) Q

( )

m

dE EE E EN

dT E

dTQ T

dtQ N Q QN

How to consider spatial effectsA simple case with convection and

diffusion 2 2

2 2

2 2

22 2

X X Y XU A x Y d dX d

t s s s

Y Y Y YU x Y d dY d

t s s s

Thanks