Dmitry Savostyanov (University of Essex)
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Mathematical modelling of infectious disease is an important area of applied mathematics. The Kermack--McKendrick compartmental SIR model is quite simple but also quite powerful --- it describes the epidemics with a system of ordinary differential equations (ODEs), which can be easily solved using a suitable numerical method, and predicts the behaviour of outbreaks very similar to that observed in many recorded epidemics. Even though compartmental models are almost hundred years old now, they are still widely used not only in a classroom, but also to predict the development of dangerous diseases and to inform Government strategies in case of emergency. The quality of a mathematical model, and our understanding of its assumptions and applicability in a particular scenario, is therefore crucial to make correct decisions to protect public health and respond to epidemics effectively when they occur.
The fundamental assumption of a compartmental model is that the population is well-mixed: there is no firm boundary between susceptible, infected and recovered individuals. Everyone interacts with everyone at once, similar to chemical molecules in a mixture. Although this assumption may be appropriate on a later stages of epidemic, it clearly limits the model's capability to accurately describe and predict the early stages, when the infection is largely localised in one location and is carried to other locations through a network of transport and/or social and community links.
If we consider how a disease progresses through a network, only neighbouring nodes can participate in transmission --- the network is not well-mixed. Hence, the compartmental model is no longer fit for purpose, and has to be replaced with a probabilistic model, where we estimate the probability for each node to be in susceptible, infected or recovered state at a given time. Importantly, the states of the neighbours are not independent --- quite the opposite! --- a susceptible person in direct contact with an infected person is likely to become infected soon. This means that instead of considering individual probabilities, we have to describe the evolution of the joint probability distribution, accounting for the states of all nodes at once. This high--dimensional problem struggles from the curse of dimensionality --- the number of unknowns grows exponentially with the number of nodes, and traditional ODE solvers can't cope with he growing complexity when the number of nodes exceeds several tens. For this reason, the problem is typically solved using Stochastic Simulation Algorithms (SSA), such as Monte Carlo and its variants.
Using our experience with high--dimensional problems, such as Fokker--Planck, Chemical Master Equation and Quantum Spin Dynamics, we consider applying tensor product algorithms to solve this high--dimensional ODE with high accuracy, and hence obtain a full probabilistic picture of the disease transfer through the network. In preliminary experiments we find tensor product approach to be successful in principle. In particular, it can accurately estimate the probabilities of rare events, as well as higher moments of the observed quantities, where SSA often struggles.
This is a work in progress! The presented results are in preparation for publication. We will appreciate all feedback and suggestions regarding this work.
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