Modelling infectious diseases with stochastic discontinuities

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Infectious diseases are a global problem worldwide. 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 transmissible 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. 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. 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. Impulsive differential equations are founded upon the assumption that it is often natural to assume that sufficiently short perturbations in the system occur instantaneously, since their length is negligible in comparison with the duration of the process. Impulse extension equations address the question of the validity of the assumption that the duration of short bursts can be ignored by extending the impulsive differential equation to include its continuous analogue, in order to compare the two. Filippov systems are dynamical systems with discontinuities in the derivatives. Filippov systems lend themselves to imposing an economic threshold: when the cost of a disease is sufficiently high, action will be instigated. We will use these formulations to analyse the effects of stochastic variations on the pivot points. Much work has been done on stochastic differential equations in the past; however, very little has been done on the effects of stochasticity on discontinuous approximations. E.g., for impulsive differential equations, the timing of the impulse may vary, as well as the strength of the jump. The location of Filippov thresholds may be subject to variation, which may have implications for both real and virtual equilibria that are located near the threshold. By harnessing the power of short-burst modelling, a great many problems can be analysed using novel mathematical techniques. By investigating the effect of stochastic variations on the threshold, we can develop an interface between mathematics and human behaviour. This will be useful in an applied context when dealing with biological, physical or other real-world models where thresholds are important, but the actions of humans may reduce the predictability of the outcome.