Epidemics have shaped, sometimes more than wars and natural disasters, demo graphic aspects of human populations around the world, their health habits and their economies. Ebola and the Middle East Respiratory Syndrome (MERS) are clear and current examples of potential hazards at planetary scale.
During the spread of an epidemic disease, there are phenomena, like the sudden extinction of the epidemic, that can not be captured by deterministic models. As a consequence, stochastic models have been proposed during the last decades. A typical forward problem in the stochastic setting could be the approximation of the expected number of infected individuals found in one month from now. On the other hand, a typical inverse problem could be, given a discretely observed set of epidemiological data, infer the transmission rate of the epidemic or its basic reproduction number.
Markovian epidemic models are stochastic models belonging to a wide class of pure jump processes known as Stochastic Reaction Networks (SRNs), that are intended to describe the time evolution of interacting particle systems where one particle interacts with the others through a finite set of reaction channels. SRNs have been mainly developed to model biochemical reactions but they also have applications in neural networks, virus kinetics, and dynamics of social networks, among others.
4
This PhD thesis is focused on novel fast simulation algorithms and statistical
inference methods for SRNs.
Our novel Multilevel Monte Carlo (MLMC) hybrid simulation algorithms provide
accurate estimates of expected values of a given observable of SRNs at a prescribed final time. They are designed to control the global approximation error up to a userselected accuracy and up to a certain confidence level, and with near optimal computational work. We also present novel dualweighted residual expansions for fast estimation of weak and strong errors arising from the MLMC methodology.
Regarding the statistical inference aspect, we first mention an innovative multi scale approach, where we introduce a deterministic systematic way of using upscaled likelihoods for parameter estimation while the statistical fittings are done in the base model through the use of the Master Equation. In a diāµerent approach, we derive a new forwardreverse representation for simulating stochastic bridges between con secutive observations. This allows us to use the wellknown EM Algorithm to infer the reaction rates. The forwardreverse methodology is boosted by an initial phase where, using multiscale approximation techniques, we provide initial values for the EM Algorithm.
Date of Award  Jan 2015 

Original language  English 

Awarding Institution   Computer, Electrical and Mathematical Science and Engineering


Supervisor  Raul Tempone (Supervisor) 

 stochastic numerics
 stochastic processes
 multilevel monte carlo
 stochastic reaction networks
 Simulation
 statistical inference