What Mathematical modelling has to do with infectious diseases

1. What Mathematical modelling has to do with infectious diseases By Prof. J. B. Shukla International Internet University for Research in Science and Technology, HQ Kanpur Website : www.iiurst.org Formerly, IIT Kanpur Email: jbs@iitk.ac.in

2. 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

3. 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

4. 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.

5. Mathematical Model contd. • Mathematics has permeated in all sciences. • Physical Science, Environmental, Biological Sciences, Social Sciences. Systems Unknown and known variables Math Models Mathematical relations among the variables and parameters • All systems are governed by some laws, principles and hypothesis.

6. 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. Solution Math. Model System interpretation validation

7. 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

8. Other Factors • 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

9. Some More Factors • Isolation • Nutrition • Sanitation • Medication • Religion, Caste Creed. • Social interaction • Professional Interactions • Interactions at work place.

10. Modes of Transmission X Susceptible Y Infective • Direct Contact/Transmission • Direct contact may be with the infective or with the disease agent. • Indirect Contact/Transmission • Disease agent are passed on the susceptible through the medium of some kinds.

11. Direct Transmission

12. Indirect Transmission

13. Aims/ Objectives of Study • How fast a disease is spreading? • How much of the total population has been infected? • How of much of the remaining will be infected at a particular time/ location • What are the causes and factors for the spread of a particular disease • What is the effect of control measures, if any other the spread etc.

14. A General Epidemic model • X: Susceptible Y = Infective Z = Removed Z1 = Isolation

15. Some Ideas • 1. Non mixing models for the spread of infections diseases • Effects of sanitation • Effect of nutrition • Effects of medication • Effects of awareness • Effects of environmental and ecological factors on the spread of infections diseases • Effect of diffusion (Movement of population) • Effects of movement of convection effects on the spread of infection diseases.

16. Non mixing model • β-Interaction coefficient, rate of contracts. • It should dependent upon the following • Space • Social factors such as going together to Churches, Temples, Dargahs, Malls, Fairs, Feats And Parties • Office factors such as working in the same office, department, building etc sitting on the same table. • Professional factors such as doing research together • Travel factors such as going to together in same vehicle vocation doing the office vacation • Awareness of the disease

17. Non Mixing Models contd… • Effect of Space

18. Effect of Sanitation • S = Sanitation variable • B = Bacteria

19. Effect of Medication • M= Variable for Medical support Where B(M) to be defined suitably

20. Effect of Nutrition • H= Healthy Population Nu = Variable for nutrition

21. Effect of Media Awareness Me=Variable for Awareness

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

23. How to consider spatial effects A simple case with convection and diffusion

24. Thanks