Researchable or impossible choice of topic: Insights from summer schools by

1. Researchable or impossible choice of topic: Insights from summer schoolsby J.Y.T. Mugisha Biomathematics Modelling Group Makerere University Lecture presented at DIMACS Advanced Study Institute, Makerere University, 20 – 31 July, 2009 Department of Mathematics, www.math.ac.ug

2. Introduction • The choice of topic to research on during summer school is supposed to be stimulated from the rather few, very fast presentation from guest lecturers • Guest lecturers often assume an audience that is both biologically and mathematically stable • there is little time for a summer school student to read and internalise the lecture • Then, time comes and the students are put in groups; of varying background • by the end of the day there is a problem to formulate. The question is, when does the topic look researchable or impossible? Department of Mathematics, www.math.ac.ug

3. CLUES from Other Summer Schools • In the following slides, I give a few lessons from other summer schools on the choice of subject to research on and highlight the interesting patterns of topic choice Department of Mathematics, www.math.ac.ug

4. Nature of models • Many epidemics take the usual dynamics of being SI, SIS, SIR, SIRS, SIRZ, SEI,SEIS, SEIR, SEIRS, with others getting new names after identifying their development pattern such as SVI • Then the nature of modelling involved depends on the individual mathematician expertise and the characteristics of the epidemic: Deterministic ODES, Stochastic Models, Delay ODES, Discrete Models such as in patched space, PDEs as in most vaccination models (Measles, Pertussis, Malaria, HIV/AIDS) and structured models (Age, size, shape) • Also the nature of model analysis depends on the orientation of one's mathematical skills: You will find a pure mathematician tackle the model analysis from pure mathematics angle, a dynamical systems person goes straight to finding strange attractors and whether there will be chaos in the system, an applied mathematician formulating and analyzing towards that direction and an algebraist must bring in algebraic approach. Department of Mathematics, www.math.ac.ug

5. But looking for what? • In all, there are basic fundamentals that we need the model to bring out on an epidemic • what is the epidemic curve (describing the intensity of the outbreak)? • what is the death impact? • economic impact? • what is the level of disease incidence (no. of new occurences) and prevalence • what is the basic reproductive number • any equilibrium (steady) states and what is the nature and conditions (for) of their stability? • Through these questions and of course, others coming out of the "system study" (literature and on-spot data about the disease) results regarding your model start to take shape. • BUT are these known concepts before a student embarks on the battle of topic choice Department of Mathematics, www.math.ac.ug

6. Model • The model you would like to formulate is to eradicate the disease completely (disease-free population), • or to have it to a contained situation (stable endemic state) • or to have a control measure to hold its devastation in cases of re-emerging infections (vaccination). • In cases of endemic situation, you want to suggest best treatment regimes and protocols to enable effective patient administration, cost-effective drug administration and less drug resistance. • If it is a re-emerging disease, there would be literature on the way it attacks, forms and patterns of outbreak and how existing models were formulated and what they are targeting. • This gives you a clue on designing a new model to address the failures of the previous one. This also may call for new parameter definition and estimation. Therefore, the borrowing from the existing models of the same diseases would help one not to waste much research time identifying a problem, formulating, solving, validating and improving the model. Department of Mathematics, www.math.ac.ug

7. In case of emerging epidemics, they are in may cases new outbreaks. These come as a surprise and every person (health worker, technocrat and beaurocrat) in the process of scampering around suggests a possible control measure. To a mathematician, study the situation (do not take too long studying the situation however!). Find out the the epidemiological and demographic patterns of the outbreak, then may be you will be able to characterize the epidemic as an SI, SIS, SIR, SIRS, SEIR and build from there. Of course not all will be based on the mentioned types of models. You will find yourself baptizing yours after a while of formulation efforts like MSEIR and MSEIRS in Hethcote (2000).Maybe the characteristics of a new and outbreak are similar to the one that is existing, or that was eradicated. This is where literature comes in to play an important role to address such a situation in building a model. Department of Mathematics, www.math.ac.ug

8. What is it that is modelled Below, I present a couple of examples of models developed from simple to complex, giving a reflection that would answer questions as to: Do mathematical models exist for any disease or epidemic? This exposition should bring excitement to the reader to dig even far back in literature, the genesis of use of mathematical modelling in epidemiology. Department of Mathematics, www.math.ac.ug

9. Human-Mosquito Interaction model for mosquito control Department of Mathematics, www.math.ac.ug

10. Mosquito model Department of Mathematics, www.math.ac.ug

11. What do the terms mean Department of Mathematics, www.math.ac.ug

12. Are assumptions to this model realistic and where do you get them from? Department of Mathematics, www.math.ac.ug

13. … Department of Mathematics, www.math.ac.ug

14. Tuberculosis Model with slow and fast progression Department of Mathematics, www.math.ac.ug

15. Improved dynamics? Department of Mathematics, www.math.ac.ug

16. Simple model for dynamics of streptococci infections (MTBI 1999) Department of Mathematics, www.math.ac.ug

17. Herpes Zoster Department of Mathematics, www.math.ac.ug

18. And Foot-and Mouth Disease Model base on county metapopulation • FMD spreads very fast in a given region once not checked. • An affected region is made up of counties. • In the county are farms or call them patches assumed to homogeneous mixing of animals. • But the habitats or patches are heterogeneously distributed in activity and characteristics. • FMD is checked by Vaccination (mass or ring), Isolation (Quarantine) and/or mass destruction of the affected animals. • Let us take a ring vaccination strategy (effective in reducing high risk, movement). • What is the effective ring size to adopt to balance two goals: prompt vaccination and vaccinating over as large an area as possible without animal movement. Department of Mathematics, www.math.ac.ug

19. Simple Model for FMD Department of Mathematics, www.math.ac.ug

20. What does this model want to do? • We need to now carry out a vigorous protection after detection of an infected farm. We apply a reactive response approach with ring vaccination. • reactive in that control measures are implemented only after an outbreak has been reported • responsive in that we target vaccination according to which farms have been diagnosed with FMD • Ring vaccination means vaccinating in a ring with certain radius around the diseased counties. Department of Mathematics, www.math.ac.ug

21. Anything achieved? • The idea behind ring vaccination is that farms that have close contact with an infected one are at higher risk of becoming infected and hence must be protected • To have a bigger coverage of the counties this model can be refined to incorporate spatial dynamics. In such a way, we would address the transmission for inter-county and intra-county .[Further reading see Keeling \& Woolhouse, 2003, Chowell, G, et al., 2005; Diaz, E. et al 2006) Department of Mathematics, www.math.ac.ug

22. The Crude HIV/AIDS Model: Where are we building from? (1986 model) • Assume recruitment by births Department of Mathematics, www.math.ac.ug

23. Trends in HIVAIDS Modelling • From this model, one was able to changes in parameters when treatment is introduced, when one takes good diet, when one used a condom, when one practices zero-grazing (being faithful to one sexual partner) etc. Since then, excellent models have been built on this and mathematical modellers are adding on new improvements with advances other efforts to contain the epidemic. Department of Mathematics, www.math.ac.ug

24. Discussion… Many other models available and improvements needed • Cholera • Sleeping Sickness (Trypanosomiasis) • Bilharzia (Schistosomiasis) and helminths through snails in lakes (Anderson and May (1991)) • River Blindness (Onchocerciasis) • Tumor and cancer growth and therapy • Meningitis • Chagas (Busenberg and Vargas), Pertussis (Hethcote) • Polio, Measles, and their vaccination strategies • Many of the STDs (Gonorrhea, syphilis, Candida etc) • Influenza • Diabetes mellitus (Mahan • West Nile Virus, baboonic fever, rift valley virus • new emerging infections such as SARS and Bird flu • Zoonotic diseases: Tick-borne diseases as East Coast fever, Nagana • Plant diseases: Coffee and banana wilt, cassava mosaic • dispersal models • and modeling pathogenesis and treatment • Criss-cross Models • Mathematical Epidemiology ideas to social models • Within Host models and co-infection models Department of Mathematics, www.math.ac.ug

25. END.. THANK YOU and special thanks to DIMACS Department of Mathematics, www.math.ac.ug