Epidemiological cellular automata: a case study involving AIDS

The spread of disease is a major health concern in many parts of the world. In the absence of vaccines and treatments, the only method to stop the spread of disease is to control population movements. Human mobility is one of the causes of the geographical spread of emergent human infectious diseases and plays a key role in human-mediated bio-invasion, the dominant factor in the global biodiversity crisis. One of the most serious emergent infectious diseases in the last 30 years or so is AIDS (acquired immunodeficiency syndrome), where multiple pathogen species infect a human body. HIV/AIDS is now considered much more commonplace than previously thought. AIDS leads to interaction effects between the pathogens that may alter previously understood patterns of disease spread. There has been longstanding interest in how to model population movements in order to find optimal control strategies for a particular disease. The simulation models proposed here use cellular automata based on sound mathematical principles and epidemiological theory to model HIV/AIDS to provide a suitable framework to study the spatial spread of disease in different scenarios. This work investigates how probabilistic parameters affect the model in terms of time, location, gender, age and subgroups of the population. The cellular automaton modelling approach is used to forecast numbers of cases in different subgroups. An approach using wavelet transforms analysis is illustrated to understand the impact of delay on the spread of infectious disease. The results confirm that the higher the frequency, then the slower the spread of disease and vice versa. The thesis concludes with showing how co-infection can be modelled in future work on a theoretical base

Investigating disease persistence in host vector systems: dengue as a case study

The investigation of pathogen persistence in vector-borne diseases is important in different ecological and epidemiological contexts. In this thesis, I have developed deterministic and stochastic models to help investigating the pathogen persistence in host-vector systems by using efficient modelling paradigms. A general introduction with aims and objectives of the studies conducted in the thesis are provided in Chapter 1. The mathematical treatment of models used in the thesis is provided in Chapter 2 where the models are found locally asymptotically stable. The models used in the rest of the thesis are based on either the same or similar mathematical structure studied in this chapter. After that, there are three different experiments that are conducted in this thesis to study the pathogen persistence. In Chapter 3, I characterize pathogen persistence in terms of the Critical Community Size (CCS) and find its relationship with the model parameters. In this study, the stochastic versions of two epidemiologically different host-vector models are used for estimating CCS. I note that the model parameters and their algebraic combination, in addition to the seroprevalence level of the host population, can be used to quantify CCS. The study undertaken in Chapter 4 is used to estimate pathogen persistence using both deterministic and stochastic versions of a model with seasonal birth rate of the vectors. Through stochastic simulations we investigate the pattern of epidemics after the introduction of an infectious individual at different times of the year. The results show that the disease dynamics are altered by the seasonal variation. The higher levels of pre-existing seroprevalence reduces the probability of invasion of dengue. In Chapter 5, I considered two alternate ways to represent the dynamics of a host-vector model. Both of the approximate models are investigated for the parameter regions where the approximation fails to hold. Moreover, three metrics are used to compare them with the Full model. In addition to the computational benefits, these approximations are used to investigate to what degree the inclusion of the vector population in the dynamics of the system is important. Finally, in Chapter 6, I present the summary of studies undertaken and possible extensions for the future work

Noise induced changes to dynamic behaviour of stochastic delay differential equations

This thesis is concerned with changes in the behaviour of solutions to parameter-dependent stochastic delay differential equations

Mathematics of Viral Infections: A Review of Modeling Approaches and A Case-Study for Dengue Dynamics

In this thesis we use mathematical models to study the mechanisms by which diseases spread. Transmission dynamics is modelled by the class of SIR models, where the abbreviation stands for susceptible (S), infected (I) and recovered (R). These models are also called compartmental models, and they serve as the basic mathematical framework for understanding the complex dynamics of infectious diseases. Theory developed for the SIR framework can be applied the real-world dynamics, for instance to the spread of the dengue virus. We look at how parameters such as the as basic reproduction number, R0, drive epidemics by allowing transitions from a disease-free equilibrium (DFE) when R0 1. A case study was carried out to investigate dengue transmission dynamics in a single serotype model by using a vector-to-human compartmental model. Here the approach is to explore the underlying dynamical structures, as well as looking at the projected impact of possible interventions such as vaccines and vector-control measures

Kinetic Modelling of Epidemic Dynamics: Social Contacts, Control with Uncertain Data, and Multiscale Spatial Dynamics

In this survey we report some recent results in the mathematical modelling of epidemic phenomena through the use of kinetic equations. We initially consider models of interaction between agents in which social characteristics play a key role in the spread of an epidemic, such as the age of individuals, the number of social contacts, and their economic wealth. Subsequently, for such models, we discuss the possibility of containing the epidemic through an appropriate optimal control formulation based on the policy maker’s perception of the progress of the epidemic. The role of uncertainty in the data is also discussed and addressed. Finally, the kinetic modelling is extended to spatially dependent settings using multiscale transport models that can characterize the impact of movement dynamics on epidemic advancement on both one-dimensional networks and realistic two-dimensional geographic settings

Mathematics of Climate Change and Mosquito-borne Disease Dynamics

abstract: The role of climate change, as measured in terms of changes in the climatology of geophysical variables (such as temperature and rainfall), on the global distribution and burden of vector-borne diseases (VBDs) remains a subject of considerable debate. This dissertation attempts to contribute to this debate via the use of mathematical (compartmental) modeling and statistical data analysis. In particular, the objective is to find suitable values and/or ranges of the climate variables considered (typically temperature and rainfall) for maximum vector abundance and consequently, maximum transmission intensity of the disease(s) they cause. Motivated by the fact that understanding the dynamics of disease vector is crucial to understanding the transmission and control of the VBDs they cause, a novel weather-driven deterministic model for the population biology of the mosquito is formulated and rigorously analyzed. Numerical simulations, using relevant weather and entomological data for Anopheles mosquito (the vector for malaria), show that maximum mosquito abundance occurs when temperature and rainfall values lie in the range [20-25]C and [105-115] mm, respectively. The Anopheles mosquito ecology model is extended to incorporate human dynamics. The resulting weather-driven malaria transmission model, which includes many of the key aspects of malaria (such as disease transmission by asymptomatically-infectious humans, and enhanced malaria immunity due to repeated exposure), was rigorously analyzed. The model which also incorporates the effect of diurnal temperature range (DTR) on malaria transmission dynamics shows that increasing DTR shifts the peak temperature value for malaria transmission from 29C (when DTR is 0C) to about 25C (when DTR is 15C). Finally, the malaria model is adapted and used to study the transmission dynamics of chikungunya, dengue and Zika, three diseases co-circulating in the Americas caused by the same vector (Aedes aegypti). The resulting model, which is fitted using data from Mexico, is used to assess a few hypotheses (such as those associated with the possible impact the newly-released dengue vaccine will have on Zika) and the impact of variability in climate variables on the dynamics of the three diseases. Suitable temperature and rainfall ranges for the maximum transmission intensity of the three diseases are obtained.Dissertation/ThesisDoctoral Dissertation Applied Mathematics 201

The Role of Vector Trait Variation in Vector-Borne Disease Dynamics

Many important endemic and emerging diseases are transmitted by vectors that are biting arthropods. The functional traits of vectors can affect pathogen transmission rates directly and also through their effect on vector population dynamics. Increasing empirical evidence shows that vector traits vary significantly across individuals, populations, and environmental conditions, and at time scales relevant to disease transmission dynamics. Here, we review empirical evidence for variation in vector traits and how this trait variation is currently incorporated into mathematical models of vector-borne disease transmission. We argue that mechanistically incorporating trait variation into these models, by explicitly capturing its effects on vector fitness and abundance, can improve the reliability of their predictions in a changing world. We provide a conceptual framework for incorporating trait variation into vector-borne disease transmission models, and highlight key empirical and theoretical challenges. This framework provides a means to conceptualize how traits can be incorporated in vector borne disease systems, and identifies key areas in which trait variation can be explored. Determining when and to what extent it is important to incorporate trait variation into vector borne disease models remains an important, outstanding question

Mathematical modelling of epidemic systems influenced by maternal antibodies and public health intervention

The general subject area of research considered in this thesis is population level epidemic modelling of infectious diseases, with specific application to the problems of model indeterminacy and systems that include processes associated with maternally acquired immunity. The work presents the derivation and analysis of a lumped systems model framework to study the influence of maternal antibodies on the population dynamics of infection among neonate and young infant age classes. The proposed models are defined by sets of ordinary and partial differential equations that describe the variation of distinct states in the natural history of infection with respect to time and/or age. The model framework is extended to explore the potential population level outcomes and consequences of mass maternal immunisation: an emerging targeted vaccine strategy that utilises the active transfer of neutralising antibodies during pregnancy in order to supplement neonatal immunity during the first few months of life. A qualitative analysis of these models has highlighted the importance of interaction with early childhood targeted vaccination campaigns, the potential to invoke transient epidemic behaviour and the prospective advantages of seasonal administration. The work considers the implications of structural identifiability, indistinguishability and formal sensitivity analyses on a number of fundamental model structures within the proposed framework. These methods are used to establish whether a postulated model structure, or the individual parameters within a known structure, are uniquely determinable from a given set of empirical observations. The main epidemiological measures available for the validation of epidemic models are inherently based on records of clinical disease or age serological surveys, which are not explicitly representative of infection and provide a very limited observation of the full system state. The analyses suggest that these issues give rise to problems of indeterminacy even in the most simple models, such that certain system characteristics cannot be uniquely estimated from available data

The Effects of Spatio-Temporal Heterogeneities on the Emergence and Spread of Dengue Virus

The dengue virus (DENV) remains a considerable global public health concern. The interactions between the virus, its mosquito vectors and the human host are complex and only partially understood. Dependencies of vector ecology on environmental attributes, such as temperature and rainfall, together with host population density, introduce strong spatiotemporal heterogeneities, resulting in irregular epidemic outbreaks and asynchronous oscillations in serotype prevalence. Human movements across different spatial scales have also been implicated as important drivers of dengue epidemiology across space and time, and further create the conditions for the geographic expansion of dengue into new habitats. Previously proposed transmission models often relied on strong, unrealistic assumptions regarding key epidemiological and ecological interactions to elucidate the effects of these spatio-temporal heterogeneities on the emergence, spread and persistence of dengue. Furthermore, the computational limitations of individual based models have hindered the development of more detailed descriptions of the influence of vector ecology, environment and human mobility on dengue epidemiology. In order to address these shortcomings, the main aim of this thesis was to rigorously quantify the effects of ecological drivers on dengue epidemiology within a robust and computational efficient framework. The individual based model presented included an explicit spatial structure, vector and human movement, spatio-temporal heterogeneity in population densities, and climate effects. The flexibility of the framework allowed robust assessment of the implications of classical modelling assumptions on the basic reproduction number, R₀, demonstrating that traditional approaches grossly inflate R₀ estimates. The model's more realistic meta-population formulation was then exploited to elucidate the effects of ecological heterogeneities on dengue incidence which showed that sufficient levels of community connectivity are required for the spread and persistence of dengue virus. By fitting the individual based model to empirical data, the influence of climate and on dengue was quantified, revealing the strong benefits that cross-sectional serological data could bring to more precisely inferring ecological drivers of arboviral epidemiology. Overall, the findings presented here demonstrate the wide epidemiological landscape which ecological drivers induce, forewarning against the strong implications of generalising interpretations from one particular setting across wider spatial contexts. These findings will prove invaluable for the assessment of vector-borne control strategies, such as mosquito elimination or vaccination deployment programs