Modelling Infectious Diseases with discontinuities

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With the recent Ebola outbreak, the need for mathematical models to understand transmission between and within countries worldwide is critical. Although we have had some success in reducing disease prevalence in the Western world, many diseases continue to threaten developing nations, with over 50% of the world's population at risk for one or more transmissable infections. Emerging infections such as West Nile Virus, SARS and swine flu have threatened the West in recent years, with potential pandemics from avian influenza, MERS and others looming. Tools at our disposal include vaccines, drugs, quarantine and education.***Many of the effects of such interventions occur in short bursts or shocks. For example, National Immunization Days (NIDs) for measles, polio etc occur in many developing countries over 1-2 days, twice a year. During this time, millions of children are vaccinated at once. India vaccinates 174,000,000 children in a single NID. On a smaller scale, antiretroviral treatment for HIV involves taking pills, whose duration of action is about 20 minutes long, significantly shorter than the period of hours between pills. Quarantine often involves relocation of infected individuals occuring on a short timescale, whereas education can involve bursts of media information happening quickly.******To analyse these effects, we will develop mathematical models using impulsive differential equations, impulse extension equations and Filippov systems. These methods are used to analyse short, sharp shocks, either in the state variables or their derivatives. By harnessing the power of short-burst modelling, a great many problems can be analysed using novel mathematical techniques. This will be useful in an applied context when dealing with biological, physical or other real-word models where precision of the results are important.************