Bayesian model assessment for stochastic epidemic models

Acrucial practical advantage of infectious diseases modelling as a public health tool lies in its application to evaluate various disease-control policies. However, such evaluation is of limited use, unless a sufficiently accurate epidemic model is applied. If the model provides an adequate fit, it is possible to interpret parameter estimates, compare disease epidemics and implement control procedures. Methods to assess and compare stochastic epidemic models in a Bayesian framework are not well-established, particularly in epidemic settings with missing data. In this thesis, we develop novel methods for both model adequacy and model choice for stochastic epidemic models. We work with continuous time epidemic models and assume that only case detection times of infected individuals are available, corresponding to removal times. Throughout, we illustrate our methods using both simulated outbreak data and real disease data. Data augmented Markov Chain Monte Carlo (MCMC) algorithms are employed to make inference for unobserved infection times and model parameters. Under a Bayesian framework, we first conduct a systematic investigation of three different but natural methods of model adequacy for SIR (Susceptible-Infective-Removed) epidemic models. We proceed to develop a new two-stage method for assessing the adequacy of epidemic models. In this two stage method, two predictive distributions are examined, namely the predictive distribution of the final size of the epidemic and the predictive distribution of the removal times. The idea is based onlooking explicitly at the discrepancy between the observed and predicted removal times using the posterior predictive model checking approach in which the notion of Bayesian residuals and the and the posterior predictive p−value are utilized. This approach differs, most importantly, from classical likelihood-based approaches by taking into account uncertainty in both model stochasticity and model parameters. The two-stage method explores how SIR models with different infection mechanisms, infectious periods and population structures can be assessed and distinguished given only a set of removal times. In the last part of this thesis, we consider Bayesian model choice methods for epidemic models. We derive explicit forms for Bayes factors in two different epidemic settings, given complete epidemic data. Additionally, in the setting where the available data are partially observed, we extend the existing power posterior method for estimating Bayes factors to models incorporating missing data and successfully apply our missing-data extension of the power posterior method to various epidemic settings. We further consider the performance of the deviance information criterion (DIC) method to select between epidemic models

Cubic autocatalysis in a reaction-diffusion annulus: semi-analytical solutions

Semi-analytical solutions for cubic autocatalytic reactions are considered in a circularly symmetric reaction-diffusion annulus. The Galerkin method is used to approximate the spatial structure of the reactant and autocatalyst concentrations. Ordinary differential equations are then obtained as an approximation to the governing partial differential equations and analyzed to obtain semi-analytical results for this novel geometry. Singularity theory is used to determine the regions of parameter space in which the different types of steady-state diagram occur. The region of parameter space, in which Hopf bifurcations can occur, is found using a degenerate Hopf bifurcation analysis. A novel feature of this geometry is the effect, of varying the width of the annulus, on the static and dynamic multiplicity. The results show that for a thicker annulus, Hopf bifurcations and multiple steady-state solutions occur in a larger portion of parameter space. The usefulness and accuracy of the semi-analytical results are confirmed by comparison with numerical solutions of the governing partial differential equations

Bayes Factors for Partially Observed Stochastic Epidemic Models

We consider the problem of model choice for stochastic epidemic models given partial observation of a disease outbreak through time. Our main focus is on the use of Bayes factors. Although Bayes factors have appeared in the epidemic modelling literature before, they can be hard to compute and little attention has been given to fundamental questions concerning their utility. In this paper we derive analytic expressions for Bayes factors given complete observation through time, which suggest practical guidelines for model choice problems. We adapt the power posterior method for computing Bayes factors so as to account for missing data and apply this approach to partially observed epidemics. For comparison, wealso explore the use of a deviance information criterion for missing data scenarios. The methods are illustrated via examples involving both simulated and real data

Prevalence of Breast Tumors and Methods of Prevention: A Cross-sectional Study

Background: Breast cancer is one of the most prevalent forms of cancer in women and one of the most severe and significant public health concerns in developing nations. This study aimed to determine the prevalence of breast tumors and women’s preventive behavior. Methods: A descriptive, correlational cross-sectional design was employed for this study. The study was conducted at (jeddah). Participants were selected during the period from September to November 2022. Population of this study were adult women (Aged >18 years) at KSA. Study instruments consisted of the following domains sociodemographic data, anthropometric measurements, information related to menstrual cycle and pregnancy, obstetric history, family history, practices of breast self-examination, procedures of early detection and knowledge, attitude and practice assessment for methods of prevention. Results: The study included 420 women of different ages. Breast cancer was found among 82 women (19.5%). The mean age among all study participants was 33.96 + 14.79 years with median age of 28 years. More than half of study participants had normal BMI (n= 220, 52.4%) while third participants were overweight (n= 136, 32.4%). Among participants, 18.1% had a history of post-partum complications, 38.3% had undergone previous surgery, 1.4% had experienced vascular moles, 18.6% had a history of fibroid uterus, 6% had cervical polyps, and 5% had endometriosis. Table 3 presents obstetric history among study participants. More than half of study participants underwent previous hysteroscopy (n= 235, 56%). On the other hand, 81 women had a family history of breast cancer (19.3%). Most of women in this study perform self-examination of the breast (n= 300, 71.4%) and 102 women underwent fine needle aspiration procedure (FNA) (24.3%). The FNA result was positive among 81 women. Furthermore, 124 women underwent mammography (29.5%) and the result was positive among 67 participants. Breast cancer is found among 82 women (19.5%). Women in this study agreed that they should have clinical breast examination at any time (n= 191, 45.5%) while other women believed they should have this examination in certain circumstances such as mastodynia (n= 61, 14.5%), history of benign breast tumors (n= 38, 9%), obesity (n= 37, 8.8%) and family history of breast cancer (n= 32, 7.6%). Conclusion: Breast cancer prevalence was 19.5%. Urban residency was predominant, with varying educational levels. Marital status, income, family size, and work differed among participants. Chronic conditions and diverse anthropometric measurements were observed. Obstetric history showed early marriage and delivery ages, limited abortions, and varied complications. Family history indicated links to chronic diseases and cancers. Participants exhibited awareness about breast cancer risk factors and methods for early detection

Bayesian model assessment for stochastic epidemic models

Acrucial practical advantage of infectious diseases modelling as a public health tool lies in its application to evaluate various disease-control policies. However, such evaluation is of limited use, unless a sufficiently accurate epidemic model is applied. If the model provides an adequate fit, it is possible to interpret parameter estimates, compare disease epidemics and implement control procedures. Methods to assess and compare stochastic epidemic models in a Bayesian framework are not well-established, particularly in epidemic settings with missing data. In this thesis, we develop novel methods for both model adequacy and model choice for stochastic epidemic models. We work with continuous time epidemic models and assume that only case detection times of infected individuals are available, corresponding to removal times. Throughout, we illustrate our methods using both simulated outbreak data and real disease data. Data augmented Markov Chain Monte Carlo (MCMC) algorithms are employed to make inference for unobserved infection times and model parameters. Under a Bayesian framework, we first conduct a systematic investigation of three different but natural methods of model adequacy for SIR (Susceptible-Infective-Removed) epidemic models. We proceed to develop a new two-stage method for assessing the adequacy of epidemic models. In this two stage method, two predictive distributions are examined, namely the predictive distribution of the final size of the epidemic and the predictive distribution of the removal times. The idea is based onlooking explicitly at the discrepancy between the observed and predicted removal times using the posterior predictive model checking approach in which the notion of Bayesian residuals and the and the posterior predictive p−value are utilized. This approach differs, most importantly, from classical likelihood-based approaches by taking into account uncertainty in both model stochasticity and model parameters. The two-stage method explores how SIR models with different infection mechanisms, infectious periods and population structures can be assessed and distinguished given only a set of removal times. In the last part of this thesis, we consider Bayesian model choice methods for epidemic models. We derive explicit forms for Bayes factors in two different epidemic settings, given complete epidemic data. Additionally, in the setting where the available data are partially observed, we extend the existing power posterior method for estimating Bayes factors to models incorporating missing data and successfully apply our missing-data extension of the power posterior method to various epidemic settings. We further consider the performance of the deviance information criterion (DIC) method to select between epidemic models

Computational Methods for Estimating the Evidence and Bayes Factor in SEIR Stochastic Infectious Diseases Models Featuring Asymmetrical Dynamics of Transmission

Stochastic epidemic models may offer a vitally essential public health tool for comprehending and regulating disease progression. The best illustration of their importance and usefulness is perhaps the substantial influence that these models have had on the global COVID-19 epidemic. Nonetheless, these models are of limited practical use unless they provide an adequate fit to real-life epidemic outbreaks. In this work, we consider the problem of model selection for epidemic models given temporal observation of a disease outbreak through time. The epidemic models are stochastic individual-based transmission models of the Susceptible–Exposed–Infective–Removed (SEIR) type. The main focus is on the use of model evidence (or marginal likelihood), and hence the Bayes factor is a gold-standard measure of merit for comparing the fits of models to data. Even though the Bayes factor has been discussed in the epidemic modeling literature, little focus has been given to the fundamental issues surrounding its utility and computation. Based on various asymmetrical infection mechanism assumptions, we derive analytical expressions for Bayes factors which offer helpful suggestions for model selection problems. We also explore theoretical aspects that highlight the need for caution when utilizing the Bayes factor as a model selection technique, such as when the within-model prior distributions become more asymmetrical (diffuse or informative). Three computational methods for estimating the marginal likelihood and hence Bayes factor are discussed, which are the arithmetic mean estimator, the harmonic mean estimator, and the power posterior estimator. The theory and methods are illustrated using artificial data

Semi-analytical solutions for autocatalytic reaction-diffusion equations

In this thesis autocatalytic reactions in a diffusion cell are considered. Cubic autocatalytic reactions have proved a useful test-bed for examining static and dynamic stability of chemical systems. Various extensions and aspects of this model are considered such as mixed quadratic-cubic reactions, the effect of a precursor chemical, a circularly symmetric cell and delay feedback control. For all of these models, the Galerkin method is used to approximate the spatial structure of the reactant and autocatalyst concentrations. Ordinary differential equations are then obtained as an approximation to the governing partial differential equations and analyzed to obtain semi-analytical results for the reaction-diffusion cell. Singularity theory is used to determine the regions of parameter space in which the different types of steady-state diagram occur. The region of parameter space in which Hopf bifurcations can occur is found by a local stability analysis of the semi-analytical model. In chapter 2 mixed quadratic and cubic reactions are considered. The effect of varying the relative importance of the quadratic and cubic reaction terms and the diffusion coefficient of the two species are examined in detail with much additional complexity, involving bifurcation patterns and stability, found. In chapter 3 the effect of a precursor chemical is examined. The reactant is supplied by two mechanisms, diffusion via the cell boundaries and decay of an abundant precursor chemical present in the reactor. The effect of varying the relative importance of the precursor chemical is examined in detail. In chapter 4 a circularly symmetric reaction diffusion cell or annulus is considered. This allows geometric effects, such as varying the width of the annulus, to be examined. This is one of the first studies to consider reactions in a circularly symmetric annulus. In chapter 5 feedback control with delay is examined by varying the boundary reservoir concentrations in response to the concentrations in the centre of the reactor. The effect of varying the strength of the feedback and the delay are both considered. This study illustrates that feedback control can modify the stability of the system, in terms of stabilizing and destabilizing regions of parameter space. A new technique for finding the regions of parameter space is used in the case of delay, as expressions for the degenerate Hopf points can not be obtained. For each of the models considered steady-state profiles, bifurcation diagrams, parameter space stability maps and limit-cycle dynamics are drawn. The numerical solutions of the governing partial differential equations are also obtained for comparison, and show the usefulness and accuracy of the semi-analytical results. The methods used here have a general applicability and can be used to develop semi-analytical solutions to other chemical systems for which diffusion is important

Mixed quadratic-cubic autocatalytic reaction-diffusion equations: Semi-analytical solutions

Semi-analytical solutions for autocatalytic reactions with mixed quadratic and cubic terms are considered. The kinetic model is combined with diffusion and considered in a onedimensional reactor. The spatial structure of the reactant and autocatalyst concentrations are approximated by trial functions and averaging is used to obtain a lower-order ordinary differential equation model, as an approximation to the governing partial differential equations. This allows semi-analytical results to be obtained for the reaction–diffusion cell, using theoretical methods developed for ordinary differential equations. Singularity theory is used to investigate the static multiplicity of the system and obtain a parameter map, in which the different types of steady-state bifurcation diagrams occur. Hopf bifurcations are also found by a local stability analysis of the semi-analytical model. The transitions in the number and types of bifurcation diagrams and the changes to the parameter regions, in which Hopf bifurcations occur, as the relative importance of the cubic and quadratic terms vary, is explored in great detail. A key outcome of the study is that the static and dynamic stability of the mixed system exhibits more complexity than either the cubic or quadratic autocatalytic systems alone. In addition it is found that varying the diffusivity ratio, of the reactant and autocatalyst, causes dramatic changes to the dynamic stability. The semi-analytical results are show to be highly accurate, in comparison to numerical solutions of the governing partial differential equations

Semi-analytical solutions for cubic autocatalytic reaction-diffusion equations; the effect of a precursor chemical

Semi-analytical solutions for a cubic autocatalytic reaction, with linear decay and a precursor chemical, are considered. The model is coupled with diffusion and considered in a one-dimensional reactor. In this model the reactant is supplied by two mechanisms, diffusion via the cell boundaries and decay of an abundant precursor chemical present in the reactor. The Galerkin method is used to approximate the spatial structure of the reactant and autocatalyst concentrations in the reactor. Ordinary differential equations are then obtained as an approximation to the governing partial differential equations and analyzed to obtain semi-analytical results for the reaction-diffusion cell. Singularity theory and a local stability analysis are used to determine the regions of parameter space in which the different types of bifurcation diagrams and Hopf bifurcations occur. The effect of the precursor chemical concentration is examined in detail and some novel behaviours are identified