On the Fokker-Planck approximation to the Boltzmann collision operator

The Boltzmann equation (BE) is a mesoscopic model that provides a description of how gases undergoing a binary collision process evolve in time, however there is no general analytical approach for finding its solutions and direct numerical treatment using quadrature methods is prohibitively expensive due to the dimensions of the problem. For this reason, models that are able to capture the behaviour of solutions to the BE, but which are simpler to treat numerically and analytically are highly desirable. The Fokker-Planck collision operator is one such collision model, which is suited well to numerical solutions using stochastic particle methods, and is the subject of this thesis. The stochastic numerical solutions of the Fokker-Planck model suffer heavily from noise when the speed of the flow is low. We develop two methods that are able to reduced the variance of the estimators of the particle method. The first is a common random number method, which produces a correlated equilibrium solution where thermodynamic fields are known. The second is a importance sampling method, where weights are attached to the particles. This means that particles close to equilibrium do not contribute to the noise of the estimators. We also develop a randomised quasi-Monte Carlo scheme for solving the diffusion equation, which has a faster rate of convergence than simple Monte Carlo methods. The relative simplicity of the functional form of the Fokker-Planck collision operator makes it possible to find analytic solutions in simple cases. We consider a spatially homogeneous, isotropic gas with elastic collisions in the presence of forcing and dissipation and derive self-consistent non-equilibrium steady-state solutions. Previous numerical evidence exists that suggest such forcing and dissipation mechanisms, widely separated, give rise to steady-states of the BE that are close to Maxwellian, with a direct energy cascade and an inverse particle cascade. Using our analytic solutions, we are able to investigate the dependence of such solutions on the forcing and dissipation scales, and find that in the inertial range, the interaction is non-local. We then show that the “extreme driving” mechanism, responsible for a family of non-universal power-law solutions for inelastic granular gases, where the flux of energy is towards lower scales, is also able to produce inverse energy cascades for the elastic system

SCHISTOX: An individual based model for the epidemiology and control of schistosomiasis.

A stochastic individual based model, SCHISTOX, has been developed for the study of schistosome transmission dynamics and the impact of control by mass drug administration. More novel aspects that can be investigated include individual level adherence and access to treatment, multiple communities, human sex population dynamics, and implementation of a potential vaccine. Many of the model parameters have been estimated within previous studies and have been shown to vary between communities, such as the age-specific contact rates governing the age profiles of infection. However, uncertainty remains as there are wide ranges for certain parameter values and a few remain relatively unknown. We analyse the model dynamics by parameterizing it with published parameter values. We also discuss the development of SCHISTOX in the form of a publicly available open-source GitHub repository. The next key development stage involves validating the model by calibrating to epidemiological data

How Does Treatment Coverage and Proportion Never Treated Influence the Success of Schistosoma mansoni Elimination as a Public Health Problem by 2030?

Background: The 2030 target for schistosomiasis is elimination as a public health problem (EPHP), achieved when the prevalence of heavy-intensity infection among school-Aged children (SAC) reduces to &lt;1%. To achieve this, the new World Health Organization guidelines recommend a broader target of population to include pre-SAC and adults. However, the probability of achieving EPHP should be expected to depend on patterns in repeated uptake of mass drug administration by individuals. Methods: We employed 2 individual-based stochastic models to evaluate the impact of school-based and community-wide treatment and calculated the number of rounds required to achieve EPHP for Schistosoma mansoni by considering various levels of the population never treated (NT). We also considered 2 age-intensity profiles, corresponding to a low and high burden of infection in adults. Results: The number of rounds needed to achieve this target depends on the baseline prevalence and the coverage used. For low-and moderate-Transmission areas, EPHP can be achieved within 7 years if NT ≤10% and NT &lt;5%, respectively. In high-Transmission areas, community-wide treatment with NT &lt;1% is required to achieve EPHP. Conclusions: The higher the intensity of transmission, and the lower the treatment coverage, the lower the acceptable value of NT becomes. Using more efficacious treatment regimens would permit NT values to be marginally higher. A balance between target treatment coverage and NT values may be an adequate treatment strategy depending on the epidemiological setting, but striving to increase coverage and/or minimize NT can shorten program duration.</p

How Does the Proportion of Never Treatment Influence the Success of Mass Drug Administration Programs for the Elimination of Lymphatic Filariasis?

Background: Mass drug administration (MDA) is the cornerstone for the elimination of lymphatic filariasis (LF). The proportion of the population that is never treated (NT) is a crucial determinant of whether this goal is achieved within reasonable time frames. Methods: Using 2 individual-based stochastic LF transmission models, we assess the maximum permissible level of NT for which the 1% microfilaremia (mf) prevalence threshold can be achieved (with 90% probability) within 10 years under different scenarios of annual MDA coverage, drug combination and transmission setting. Results: For Anopheles-Transmission settings, we find that treating 80% of the eligible population annually with ivermectin + albendazole (IA) can achieve the 1% mf prevalence threshold within 10 years of annual treatment when baseline mf prevalence is 10%, as long as NT &lt;10%. Higher proportions of NT are acceptable when more efficacious treatment regimens are used. For Culex-Transmission settings with a low (5%) baseline mf prevalence and diethylcarbamazine + albendazole (DA) or ivermectin + diethylcarbamazine + albendazole (IDA) treatment, elimination can be reached if treatment coverage among eligibles is 80% or higher. For 10% baseline mf prevalence, the target can be achieved when the annual coverage is 80% and NT ≤15%. Higher infection prevalence or levels of NT would make achieving the target more difficult. Conclusions: The proportion of people never treated in MDA programmes for LF can strongly influence the achievement of elimination and the impact of NT is greater in high transmission areas. This study provides a starting point for further development of criteria for the evaluation of NT.</p

Data type and definition used to classify 57 compliance papers included in the analysis.

Longitudinal compliance data was reported by both longitudinal studies (cohort, randomised controlled trials) and cross-sectional studies. Abbreviations: CDD–community drug distributor, DOT–directly observed treatment.</p

Summary overview of the 89 studies included in the analysis split by the epidemiological classification of cross-sectional or longitudinal data.

The full data extraction of the 165 studies is provided in S2 Data. Frequency and percentage of studies falling into each category is shown as “n (%)”. Note for ‘Sample Population’, each stratification totals the number of studies, whilst the remaining rows of the table represents a single stratum. Abbreviations: LF–lymphatic filariasis, N–number of studies, OV–onchocerciasis, preSAC–preschool-aged children, SAC–school-aged children, SCH–schistosomiasis, STH–soil-transmitted helminths.</p

Workflow of screening and inclusion of studies as defined by the PRISMA guidelines.

Workflow of screening and inclusion of studies as defined by the PRISMA guidelines.</p

The search strategy employed to generate the review.

The search strategy employed to generate the review.</p

PRISMA guidelines for full text of review and abstract.

PRISMA guidelines for full text of review and abstract.</p

Choropleth map of studies published globally from 2016 to 2022 (Papers).

Countries where no papers were published are shown in grey. Four studies are represented as 11 datapoints where studies reported data for more than one country, totalling 96 datapoints from 89 studies. (LF, OV, STH, SCH, Trachoma) Maps showing total population of each country requiring treatment for each of the five (PC)-NTDs. Note: STH map shows total preSAC and SAC population only requiring treatment. All maps have been focused to latitude less than 50 degrees. Data taken from WHO global observatory PCT databank [28]. Abbreviations: LF–lymphatic filariasis, OV–Onchocerciasis, SCH–schistosomiasis, STH–soil-transmitted helminths. Map data acquired from the open source tidyverse package using the map_data() function [29].</p