Bayesian inference for indirectly observed stochastic processes, applications to epidemic modelling

Stochastic processes are mathematical objects that offer a probabilistic representation of how some quantities evolve in time. In this thesis we focus on estimating the trajectory and parameters of dynamical systems in cases where only indirect observations of the driving stochastic process are available. We have first explored means to use weekly recorded numbers of cases of Influenza to capture how the frequency and nature of contacts made with infected individuals evolved in time. The latter was modelled with diffusions and can be used to quantify the impact of varying drivers of epidemics as holidays, climate, or prevention interventions. Following this idea, we have estimated how the frequency of condom use has evolved during the intervention of the Gates Foundation against HIV in India. In this setting, the available estimates of the proportion of individuals infected with HIV were not only indirect but also very scarce observations, leading to specific difficulties. At last, we developed a methodology for fractional Brownian motions (fBM), here a fractional stochastic volatility model, indirectly observed through market prices. The intractability of the likelihood function, requiring augmentation of the parameter space with the diffusion path, is ubiquitous in this thesis. We aimed for inference methods robust to refinements in time discretisations, made necessary to enforce accuracy of Euler schemes. The particle Marginal Metropolis Hastings (PMMH) algorithm exhibits this mesh free property. We propose the use of fast approximate filters as a pre-exploration tool to estimate the shape of the target density, for a quicker and more robust adaptation phase of the asymptotically exact algorithm. The fBM problem could not be treated with the PMMH, which required an alternative methodology based on reparameterisation and advanced Hamiltonian Monte Carlo techniques on the diffusion pathspace, that would also be applicable in the Markovian setting

Modeling the role of public health intervention measures in halting the transmission of monkeypox virus

Monkeypox (mpox), a zoonotic viral disease caused by the monkeypox virus (mpoxv), is endemic in many countries in West Africa and is sometimes exported to other parts of the world. The recent outbreak of mpoxv in humans, in endemic and non-endemic countries, has created substantial public health concern worldwide. This research uses a mechanistic model to study the transmission dynamics of mpoxv epidemics in the USA. Our model describes the interaction between different categories of individuals represent various infection phases and hospitalization processes. The model also takes into account the extent of compliance with non-pharmaceutical intervention strategies (NPIs), such as using condoms during sexual contact, quarantine and avoiding large gatherings. The model's equilibria are analyzed, and results on asymptotic stability are obtained. Moreover, the basic reproductive number and other threshold quantities are used to establish the conditions for a forward or backward bifurcation. Our model accurately captures the incidence curves from mpox surveillance data for the USA, indicating that it can be used to explain mpoxv transmission and suggest some effective ways to enhance control efforts. In addition, numerical simulations are carried out to examine the influence of some parameters on the overall dynamics of the model. A partial rank correlation coefficient is adopted for the sensitivity analysis to determine the model most important parameters, which require close attention for effective mpoxv prevention and control. We conclude that it is especially important to ensure that NPIs are properly followed to mitigate mpoxv outbreaks effectively

Modelling vector-borne diseases: epidemic and inter-epidemic activities with application to Rift Valley fever

A Thesis submitted to the Faculty of Science in ful lment of the requirements for the degree of Doctor of Philosophy, School of Computer Science and Applied Mathematics. Johannesburg, 2016.In this thesis in order to study the complex dynamics of Rift Valley fever (RVF) we combine two modelling approaches: equation-based and simulation-based modelling. In the first approach we first formulate a deterministic model that includes two vector populations, Aedes and Culex mosquitoes with one host population (livestock), while considering both horizontal and vertical transmissions. An easy applicable expression of the basic reproduction number, R0 is derived for both periodic and non-periodic environment. Both time invariant and time varying uncertainty and sensitivity analysis of the model is carried out for quantifying the attribution of model output variations to input parameters over time and novel relationships between R0 and vertical transmission are determined providing important information useful for improving disease management. Then, we analytically derive conditions for stability of both disease-free and endemic equilibria. Using techniques of numerical simulations we perform bifurcation and chaos analysis of the model under periodic environment for evaluating the effects of climatic conditions on the characteristic pattern of disease outbreaks. Moreover, extending this model including vectors other than mosquitoes (such as ticks) we evaluate the possible role of ticks in the spread and persistence of the disease pointing out relevant model parameters that require further attention from experimental ecologists to further determine the actual role of ticks and other biting insects on the dynamics of RVF. Additionally, a novel host-vector stochastic model with vertical transmission is used to analytically determine the dominant period of disease outbreaks with respect to vertical transmission efficiency. Then, novel relationships among vertical transmission, invasion and extinction probabilities and R0 are determined. In the second approach a novel individual-based model (IBM) of complete mosquito life cycle built under daily temperature and rainfall data sets is designed and simulated. The model is applied for determining correlation between abundance of mosquito populations and rainfall regimes and is then used for studying disease inter-epidemic activities. We find that indeed rainfall is responsible for creating intra- and inter-annual variations observed in the abundance of adult mosquitoes and the length of gonotrophic cycle, number of eggs laid per blood meal, adults age-dependent survival and fight behaviour are among the most important features of the mosquito life cycle with great epidemiological impacts in the dynamics of RVF transmission. These indicators could be of great epidemiological significance by allowing disease control program managers to focus their e orts on specific features of vector life cycle including vertical transmission ability and diapause. We argue that our IBM model is an ideal extendible framework useful for further investigations of other relevant host-vector ecological and epidemiological questions for providing additional knowledge important for improving the length and quality of life of humans and domestic animals.LG201