Dissertations / Theses on the topic 'Mathematical modelling - Epidemiology'

The central theme of this dissertation is the development of a new approach to conceptualize and quantify dependence structures of capture-recapture data for closed populations, with specific emphasis on epidemiological applications. We introduce a measure of source dependence: the Coefficient of Incremental Dependence (CID). Properties of this and the related Coefficient of Source Dependence (CSD) of Vandal, Walker, and Pearson (2005) are presented, in particular their relationships to the conditional independence structures that can be modelled by hierarchical joint log-linear models (HJLLM). From these measures, we develop a new class of marginal log-linear models (MLLM), which we compare and contrast to HJLLMs.
We demonstrate that MLLMs serve to extend the universe of dependence structures of capture-recapture data that can be modelled and easily interpreted. Furthermore, the CIDs and CSDs enable us to meaningfully interpret the parameters of joint log-linear models previously excluded from the analysis of capture-recapture data for reasons of non-interpretability of model parameters.
In order to explore the challenges and features of MLLMs, we show how to produce inference from them under both a maximum likelihood and a Bayesian paradigm. The proposed modelling approach performs well and provides new insight into the fundamental nature of epidemiological capture-recapture data.

In this thesis I developed and analysed several mathematical models that describe the dynamics of infectious diseases spreading in a population simultaneously with people becoming aware of the presence of the disease and thus modifying their behaviour. This is achieved using compartmental models, with further extensions to models with time delays and the administration of vaccines. Resulting mathematical models were analysed using the techniques of dynamical systems and bifurcations theory, complemented by direct numerical simulations. Design of optimal strategies maximising the reduction of infection rates subject to logistical constraints were studied within the new modelling framework and with a view to be used in realistic contexts. Of particular interest is the design and analysis of the impact of local and global awareness campaigns, as well as the administration of vaccines to minimise the spread of infections.

This thesis explores aspects of Streptococcus pneumoniae (pneumococcus) epidemiology and control, in view of the possible widespread introduction of conjugate vaccines in England and Wales. A review and analysis of a range of different epidemiological data sources showed that the burden of pneumococcal disease in England and Wales is high and remains mostly a condition of the very young and the elderly. A meta-analysis demonstrated the effectiveness of the polysaccharide vaccine against invasive pneumococcal disease among healthy elderly, to whom vaccination was not recommended at the start of this work. Using this result, a costeffectiveness analysis assessed the economic acceptability of such a programme, from the public health perspective. A better understanding of pneumococcal carriage and transmission is required to assess the effectiveness and cost-effectiveness of mass vaccination strategies with the pneumococcal conjugate vaccine. A novel model framework was developed and fitted to a longitudinal dataset of carriage in UK families. The results demonstrated an inverse relationship between duration of carriage and age and highlighted the importance of both family size and composition for persistence in a household. Great dissimilarities were estimated among the specific serotypes in terms of transmissibility, duration of carriage and level of competition. Realistic age structured dynamic models were developed and used to investigate the impact of a range of vaccine strategies. The importance of serotype replacement, as a consequence of vaccination, was demonstrated. The economic acceptability of alternative interventions with the conjugate vaccine depended on the magnitude of its indirect effects. Herd immunity had a considerable impact on the overall cost-effectiveness of the programmes since it may substantially reduce the burden of disease in older age groups. However, serotype replacement may counterbalance this reduction and lead to a non cost-effective result.

A large proportion of chlamydial and gonococcal infections are asymptomatic. In lower- and middle-income countries like South Africa, where syndromic management is practiced, it is likely that a large proportion of curable STIs go untreated, as screening for asymptomatic STIs is rarely conducted. Due to the lack of empirical data on the efficacy of STI screening programs, dynamic mathematical modelling has been used to assess the impact of screening, but most previous modelling studies have focused on high-income settings. Here we utilize dynamic mathematical modelling to evaluate the potential impact of opportunistic STI screening programs on the incidence and prevalence of Chlamydia trachomatis and Neisseria gonorrhea in South Africa. We extended an existing agent-based model of heterosexual HIV and STI transmission in South Africa to investigate the impact of targeted screening strategies directed at high risk groups including youth, female sex workers, pregnant women and patients in HIV care. All four screening strategies resulted in reductions in general and key population STI transmission. Opportunistic STI screening of youth and ART patients were shown to be most effective and represent viable interventions for reducing STI transmission in the South African population. Additionally, we compared the modelled impact of a standardized screening program to results obtained from other published mathematical models of chlamydia screening. Differences between models could be attributed to differences in the modelled heterogeneity in sexual behaviour as well as differences in assumptions about immunity following chlamydia recovery.

Thesis (MSc (Mathematics))--University of Stellenbosch, 2009.
ENGLISH ABSTRACT: In this thesis, a mathematical model describing the interaction between HIV and TB in the presence of TB superinfection is presented. The model takes into account two strains of Mycobacterium tuberculosis (MTB), where one strain is drug-sensitive and the other is resistant to at least one of the first-line anti-tuberculosis drugs. The impact of TB superinfection on the incidence and prevalence of TB in HIV-negative and HIVTB coinfected individuals is evaluated. Various control measures such as condom use, antiretroviral therapy, isoniazid preventive therapy and increased TB detection are studied using this model. Numerical results show that TB superinfection increases the prevalence and incidence of TB and its impact is more in HIV-negative than HIV-TB coinfected individuals. The results also show that TB superinfection promotes strain coexistence and increases the associated HIV mortality. Increased condom use was found to have a high positive impact towards the control of the two epidemics. Antiretroviral therapy decreases the TB notification rate and its impact on HIV prevalence increases with the coverage and efficacy. Isoniazid preventive therapy has a clear effect on the TB prevalence. Finally, increased TB detection was found to have a less impact on the TB incidence in HIV-TB coinfected individuals
AFRIKAANSE OPSOMMING: In hierdie verhandeling word ´n wiskundige model vir die interaksie tussen MIV en TB, in ´n situasie met TB superinfeksie voorgelˆe. Die model neem twee variante van TB in ag. Een van die variante is sensitief vir MTB behandeling, terwyl die ander weerstandig is vir ten minste een van die eerste-linie TB behandenings. Die impak van TB superinfeksie op die insidensie and prevalensie van TB in MIV negatiewe en MIV-TB ko-ge˜ınfekteerde individu word ondersoek. Veskeie beheer maatreels soos kondoom gebruik, anti-retrovirale behandeling (vir MIV) en isonazid voorkomende behandeling en verhoodge TB deteksie (vir TB) word ondersoek. Numeriese resultate wys TB superinfeksie verhoog die prevalense en insidensie van TB en dat dit ´n groter bydrae maak by MIV negatief as by MIV-TB ko-geinfekteerde individu. Die resultate wys veder TB superinfeksie promofeer variant kohabitasie en verhoog MIV verwante mortalitieit. Verhoogde kondoom gebruik is gevind om ´n positiewe bydrae te maak tot die beheer van beide epidemies. Anti-retrovirale terapie verlaag die TB aanmeldings koers en die impak van ART verhoog saam met ´n verhoging in die dekking en effektiwiteit daarvan. Voorkomende behandeling het ´n beduidende impak op TB prevalensie. Ons vind dat TB deteksie ´n beperkte impak maak op TB insidensie by MIV-TB ko-geinfekteerde individu

This thesis is concerned with the mathematical modelling for human immunodeficiency virus (HIV) and hepatitis C virus (HCV) epidemics in Australia. There are two parts to this thesis. Part I is aimed at modelling the transmission of HIV and HCV via needle sharing among injecting drug users (IDUs). The dynamical model of an epidemic through needle sharing among IDUs is derived. This model reveals the correlation between needle sharing and the epidemic prevalence among IDUs. The simulations of HIV and HCV prevalence and incidence among IDUs in Australia are made with this model. The comparison of simulated results with literature estimates shows that the modelled results are consistent with the literature estimates. The effects of needle sharing and cleaning on HIV and HCV prevalence and incidence among IDUs in Australia are evaluated. Part II is devoted to modelling the spread of HIV in the general community in Australia. A mathematical model is formulated to assess the epidemiological consequences of injecting drug use and sexual transmission in Australia. The effects of highly active antiretroviral therapies (HAART) on the HIV epidemic are included. The modelled results are in broad agreement with the literature estimates and observed data. The long-term effects of HAART are also discussed.

Antibiotic resistant pathogens, such as methicillin-resistant Staphylococcus aureus (MRSA), and vancomycin-resistant enterococci (VRE), are an increasing burden on healthcare systems. Hospital acquired infections with these organisms leads to higher morbidity and mortality compared with the sensitive strains of the same species and both VRE and MRSA are on the rise worldwide including in Australian hospitals. Emerging community infectious diseases are also having an impact on hospitals. The Severe Acute Respiratory Syndrome virus (SARS Co-V) was noted for its propensity to spread throughout hospitals, and was contained largely through social distancing interventions including hospital isolation. A detailed understanding of the transmission of these and other emerging pathogens is crucial for their containment. The statistical inference and mathematical models used in this thesis aim to improve understanding of pathogen transmission by estimating the transmission rates of contagions and predicting the impact of interventions. Datasets used for these studies come from the Princess Alexandra Hospital in Brisbane, Australia and Shanxi province, mainland China. Epidemiological data on infection outbreaks are challenging to analyse due to the censored nature of infection transmission events. Most datasets record the time on symptom onset, but the transmission time is not observable. There are many ways of managing censored data, in this study we use Bayesian inference, with transmission times incorporated into the augmented dataset as latent variables. Hospital infection surveillance data is often much less detailed that data collected for epidemiological studies, often consisting of serial incidence or prevalence of patient colonisation with a resistant pathogen without individual patient event histories. Despite the lack of detailed data, transmission characteristics can be inferred from such a dataset using structured HiddenMarkovModels (HMMs). Each new transmission in an epidemic increases the infection pressure on those remaining susceptible, hence infection outbreak data are serially dependent. Statistical methods that assume independence of infection events are misleading and prone to over-estimating the impact of infection control interventions. Structured mathematical models that include transmission pressure are essential. Mathematical models can also give insights into the potential impact of interventions. The complex interaction of different infection control strategies, and their likely impact on transmission can be predicted using mathematical models. This dissertation uses modified or novel mathematical models that are specific to the pathogen and dataset being analysed. The first study estimates MRSA transmission in an Intensive Care Unit, using a structured four compartment model, Bayesian inference and a piecewise hazard methods. The model predicts the impact of interventions, such as changes to staff/patient ratios, ward size and decolonisation. A comparison of results of the stochastic and deterministic model is made and reason for differences given. The second study constructs a Hidden Markov Model to describe longitudinal data on weekly VRE prevalence. Transmission is assumed to be either from patient to patient cross-transmission or sporadic (independent of cross-transmission) and parameters for each mode of acquisition are estimated from the data. The third study develops a new model with a compartment representing an environmental reservoir. Parameters for the model are gathered from literature sources and the implications of the environmental reservoir are explored. The fourth study uses a modified Susceptible-Exposed-Infectious-Removed (SEIR) model to analyse data from a SARS outbreak in Shanxi province, China. Infectivity is determined before and after interventions as well as separately for hospitalised and community symptomatic SARS cases. Model diagnostics including sensitivity analysis, model comparison and bootstrapping are implemented.

Thesis (PhD)--Stellenbosch University, 2014.
ENGLISH ABSTRACT: Water-borne infections have been a menace in many countries around the globe, claiming millions of lives. Cholera in particular has spread to all continents and now on its seventh epidemic. Although control measures have been continually developed through sanitation, vaccination and rehydration, the infection still devastates populations whenever there is an outbreak. In this research work, mathematical models for cholera transmission dynamics with focus on the impact of sanitation and hygiene, metapopulation spread, optimal control and biological control using a bacteriophage specific for pathogenic Vibrio cholerae are constructed and analysed. Vital analyses for the models are precisely given as well as numerical results depicting long term behaviour and the evolution of populations over time. The results of our analysis indicate that; improved sanitation and hand-hygiene are vital in reducing cholera infections; the spread of disease across metapopulations characterised by exchange of individuals and no cross community infection is associated with synchronous fluctuation of populations in both adjacent communities; during control of cholera, the control measures/efforts ought to be optimal especially at the beginning of the epidemic where the outbreak is often explosive in nature; and biological control if well implemented would avert many potential infections by lowering the concentration of pathogenic vibrios in the aquatic environment to values lower than the infectious dose.
AFRIKAANSE OPSOMMING: Water-infeksies is ’n bedreiging in baie lande regoor die wêreld en eis miljoene lewens. Cholera in die besonder, het op sy sewende epidemie na alle kontinente versprei. Hoewel beheermaatreëls voortdurend ontwikkel word deur middel van higiëne, inentings en rehidrasie, vernietig die infeksie steeds bevolkings wanneer daar ’n uitbraak voorkom. In hierdie navorsingswerk, word wiskundige modelle vir cholera-oordrag dinamika met die fokus op die impak van higiëne, metabevolking verspreiding, optimale beheer en biologiese beheer met behulp van ’n bakteriofaag spesifiek vir patogene Vibrio cholerae gebou en ontleed. Noodsaaklike ontledings vir die modelle is gegee sowel as numeriese resultate wat die langtermyn gedrag uitbeeld en die ontwikkeling van die bevolking oor tyd. Die resultate van ons ontleding dui daarop dat; verbeterde higiëne is noodsaaklik in die vermindering van cholera infeksies; die verspreiding van die siekte oor metapopulaties gekenmerk deur die uitruil van individue en geen kruis gemeenskap infeksie wat verband houmet sinchrone skommeling van bevolkings in beide aangrensende gemeenskappe; tydens die beheer van cholera,behoort die beheermaatreëls/pogings optimaal te wees veral aan die begin van die epidemie waar die uitbreking dikwels plofbaar in die natuur is; en biologiese beheer, indien dit goed geïmplementeer word, kan baie potensiële infeksies voorkom deur ’n vermindering in die konsentrasie van patogene vibrio in die water tot waardes laer as die aansteeklike dosis.

Orientador: Rodney Carlos Bassanezi
Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
Made available in DSpace on 2018-08-13T04:06:10Z (GMT). No. of bitstreams: 1 SaleseSilva_JoseGilson_M.pdf: 1291910 bytes, checksum: b6f441a8e24badc2915c8f5301ca28ce (MD5) Previous issue date: 2009
Resumo: Apresentamos uma abordagem resumida da epidemiologia matemática, dos modelos epidemiológicos tipo SI, SIS e SIR, bem como um embasamento sobre as equações de diferenças finitas. Utilizamos um modelo matemático determinístico em tempo discreto tipo SI, simplificado, para interpreta¸c¿ao e análise da epidemia de hanseníase do município de Imperatriz-MA, com base em dados relativos ao número de contatos registrados e casos notificados no per'iodo de 1994 a 2007. Apresentamos também uma validação do modelo e histórico da epidemia em Imperatriz.
Abstract: We present an approach summarized from epidemiology mathematics , from the models epidemiological type Oneself SI SS e SIR , as well as a basement on the subject of the equations of differences finite. Uses a model mathematical deterministic in time discreet type SI about to interpretation e analysis from epidemics from hansen'iase of the county of Imperatriz-MA , based on dice relative the number of contacts registered e cases notifying into the period of 1994 the 2007. We present also a validation of the model e historical from epidemics in the Imperatriz.
Mestrado
Biomatematica
Mestre em Matemática

The British cattle population hosts a diverse community of endemic pathogens that impact the sustainability of beef and dairy production. As such, there has been a tremendous amount of ongoing research to develop more cost-effective strategies for controlling disease at the industry level. Cattle movements have come under particular scrutiny over the past decade both because of their role in spreading many economically important diseases and because the movements of individual cattle in Great Britain have been explicitly recorded in a centralized electronic database since 1998. Numerous studies have shown that these cattle movements organize into complex networks with key structural and temporal features that influence transmission dynamics. Building on previous work, this thesis used a variety of epidemiological and statistical models to highlight limitations in the current approaches to controlling disease as well as opportunities for reducing endemic disease prevalence through targeted interventions. Empirical disease data from the national bovine tuberculosis (bTB) control programme and from two seroprevalence studies of bovine viral diarrhoea virus (BVDV) in Scottish cattle herds were used in conjunction with movement data from the Cattle Tracing System (CTS) database. Endemic diseases are often challenging to control due to lack of affordable and accurate diagnostic tests as well as the presence of subclinically infected carriers that can easily escape detection. There was evidence that combined issues with the sensitivity and specificity of routine surveillance methods for bTB were contributing to a low level of disease transmission within and between Scottish cattle herds from 2002 to 2009. For BVDV, herds that purchased pregnant beef dams, beef dams with a calf at foot, and open dairy heifers were significantly more likely to be seropositive even though these movements were responsible for only a small number of network contacts. In both cases, targeting the subset of high risk movements with disease specific biosecurity measures may be a more cost-effective use of limited national disease control resources. Other researchers have suggested that control strategies should target multiple diseases simultaneously to reduce trade-offs in resource allocation. Using key indicators of herd reproductive performance derived from the CTS database, it was shown that improving the reproductive management of herds operating below industry standards could reduce endemic disease prevalence by reducing the movements of replacement breeding cattle. A series of network generation algorithms were also developed to study the effects of restricting contact formation based on key demographic and network characteristics of actively trading cattle farms. Strategies that increased network fragmentation either by forcing highly connected farms to form contacts with other highly connected farms or preventing the formation of movements with a high predicted betweenness centrality were found to be particularly effective in limiting disease transmission. For these models to be useful in guiding future policy decisions, it is important to incorporate financial and behavioural drivers of dynamic network change. Following the introduction of pre- and post-movement testing requirements for cattle imported into Scotland from endemic bTB regions, there was a significant decline in cross-border movements, which has likely contributed to the decreasing risk of bTB outbreaks as much as testing itself. Many endemic cattle diseases such as BVDV also spread through local transmission mechanisms, which may undermine the success of disease control programmes that exclusively target cattle movements. There was also evidence that in the absence of national animal legislation, few farmers were likely to adopt biosecurity measures against BVDV. This may be related to the perceived inefficacy of recommendations as well as general unawareness of farm disease status due to the non-specific clinical signs of BVDV outbreaks. Although the CTS database was originally intended for use in slaughter traceback investigations, results from this thesis show how the basic records of births, deaths, and movements can be used to generate valuable insights into the epidemiology of endemic cattle diseases. The findings also emphasize that the management decisions of individual herds can have a substantial impact on industry level transmission dynamics, which offers unique opportunities to develop novel and more cost-effective disease control programmes.

L’Influenza Aviaire Hautement Pathogène H5N1 (IAHP à H5N1) a été déclaré en Thaïlande lors de plusieurs vagues épidémiques puis lors de cas sporadiques entre 2004 et 2008, et le risque persiste encore en Thaïlande. La plupart des foyers confirmés d’IAHP à H5N1 en Thaïlande ont eu lieu dans les populations de volailles de basse-cour. Les poulets de basse-cour sont élevés pour de nombreuses raisons, y compris pour un revenu supplémentaire. Ce commerce est géré informellement par les collecteurs de volailles qui peuvent être classés en collecteur-abatteur (TS), sous-collecteur (HT) et collecteur de collecteurs (TT). Ces collecteurs se déplacent entre villages avec les mêmes véhicules et matériel non nettoyés. Nous avons mesuré leurs activités commerciales dans l'espace et le temps. Nous avons développé un modèle spatial dynamique compartimental et stochastique de la filière de poulet de basse-cour dans une province. Nos résultats indiquent que ces échanges commerciaux peuvent contribuer à la propagation d’IAHP à H5N1 grâce au chevauchement des zones de collecte et aux distances parcourues. Des variations temporelles ont été observées lors de certaines grandes fêtes rituelles tel le Nouvel An chinois. Nous avons développé un modèle SIR couplé au modèle de la filière de poulets. Nous avons supposé que la maladie peut se propager par deux moyens : localement entre villages voisins et par les échanges commerciaux. Nous avons utilisé ce modèle infectieux de base pour tester plusieurs mesures de contrôle connexes. Nos résultats suggèrent que lors d’un foyer une interdiction totale et rapide de tous les échanges de volailles devrait être mise en oeuvre ainsi qu’une campagne de désinfection
Highly pathogenic avian influenza H5N1 (HPAI H5N1) was recognized in Thailand by multiple epidemic waves and some sporadic cases between 2004 and 2008 but the risk of disease remerging in Thailand still remains up to present. Most of HPAI H5N1 confirmed outbreaks in Thailand occurred in backyard chicken populations. Backyard chickens are reared for many purposes including for additional cash income. Backyard chicken trade is informally managed by poultry traders which can be categorized into trader–slaughterhouse (TS), household trader (HT) and trader of trader (TT). These traders roam around different villages with the same unclean vehicle and facilities. Thus, their trade patterns in space and time are necessary to be elaborately studied. In our study, we developed a spatial compartmental stochastic dynamic model of backyard chicken trade network in a province of Thailand. Our model results indicated that the structure of poultry networks may contribute to HPAI H5N1 spread through overlapped catchment areas and long distance trades. Also, temporal variations of live poultry movements were observed during some major ritual festivals especially Chinese New Year. Subsequently, we developed an SIR model upon the dynamic model of backyard chicken trade network. It was assumed in this study that the disease can spread by two means: local spread to neighboring uninfected villages and spread by poultry traders. Then, we used this baseline infectious model to test multiple related control measures. Our ultimate results suggested that a complete ban of all poultry traders should be promptly implemented with poultry area disinfection campaign once the outbreak occurs

The human immunodeficiency virus (HIV) epidemic is far from over. Antiretroviral therapy (ART) is effective at suppressing viral replication within a patient but it must be taken daily and is life-long. Therefore, the development of a therapy that could induce drug-free remission or constitute a functional cure is a key focus of HIV research. In this thesis I explore three mechanisms which could lead to more individuals being able to control their viraemia in the absence of ART: (1) T-cell immunity, (2) early initiation of ART, and (3) viral evolution. Firstly, a strong HIV-specific T-cell response has been linked to rare cases of spontaneous viral control, but the extent to which this arm of the immune response contributes to viral control is debated. Several types of data are used to answer this question, including the rate at which the virus evolves to escape the CD8+ T-cell response. I study the frequency of incident immune escape in the largest cohort used for this purpose to date. Secondly, some patients, with characteristics dissimilar to spontaneous HIV controllers, are able to control the virus for years after the interruption of ART that was initiated early in infection. I use mathematical models to investigate a new hypothesis for the differing outcomes of early- and late- initiated ART. Thirdly, since HIV is a relatively new infection of humans it is still adapting to its new host. Recent studies suggest that the virus could be evolving towards decreased virulence at the population level. I study whether the widespread administration of ART has the potential to alter the course of virulence evolution and might result in a further attenuated virus. I conclude by discussing the implications of these results for viral control at the individual level and also for population level epidemic control.

The recent emergence of highly virulent strains of the pathogen causing wheat stem rust has been acknowledged as a threat to global food security. In infected wheat fields, vast amounts of pathogenic fungal spores are produced that can be carried away by wind. For targeted disease surveillance and control it is important to estimate when, where and how many fungal spores are dispersed from infected to susceptible wheat fields. In this study, high-performance computational resources are used to investigate long-distance dispersal revealing atmospheric pathways that connect entire continents. Mechanistic simulations of turbulent atmospheric spore dispersal are conducted. The analyses bring together a variety of data, including international field disease surveys and finely resolved meteorological model data. The UK Met Office's Langrangian stochastic particle dispersion model, NAME, is applied, extended and coupled to other models in a set of case studies. In the first case study, spore dispersal is analysed across Southern/East Africa, the Middle East, and Central/South Asia by simulating billions of stochastic trajectories of fungal spores over dynamically changing host and environmental landscapes. The circumstances under which virulent strains, such as Ug99, pose a risk to globally important wheat producing areas are identified. Simulation results indicate a negligible risk for dispersal from key wheat producing countries on the East African continent (Ethiopia, Kenya) directly to India and Pakistan. However, there is a considerable risk for atmospheric transport from the Arabian Peninsula to South Asia. Spore dispersal trends are quantified between all countries in the domain providing estimates which can be used to improve targeted sampling and control. In the second case study, dispersal from southern Africa to Australia is analysed. Simulation results, as well as data from phenotypic and genotypic analyses, support the hypothesis that extremely long-distance airborne dispersal across the Indian Ocean is possible, albeit rare. This indicates that the pathogen populations on the two continents are connected and underlines the importance of sharing surveillance intelligence between continents. The third case study focusses on Ethiopia, determining likely origins of strain TKTTF that recently caused severe epidemics in East Africa's largest wheat producing country. The analyses suggest inflow into Ethiopia from the Middle East via Yemen, consistent with field survey data. The risk for inflow of pathogens into Ethiopia from key neighbouring countries is ranked for different months of the wheat season. In the last results chapter a pilot study is summarized testing the feasibility of an automated short-term forecasting system for spore dispersal from the latest field disease detection sites. Whilst the functionality and practical relevance of the forecasting system is demonstrated, considerable challenges remain for testing the forecasts. The predictive simulation framework described in this thesis can be applied to any wheat producing area worldwide to assess dispersal risks. The research has broader relevance because long-distance dispersal is a key mechanism for the transmission of several crop and livestock diseases, and also plays an important role in other areas of ecology.

Gyrodactylus salaris Malmberg, 1957, is a notifiable freshwater ecto-parasite that infects both wild and farmed populations of Atlantic salmon (Salmo salar, L.). It has caused catastrophic damage to wild salmon stocks in Norway since its accidental introduction in 1975, reducing salmon density in some rivers by 98% over a period of five years. It is estimated that G. salaris has cost the Norwegian salmon industry more than 500 million EUR. Currently the UK has G. salaris free status under EU law, however, it is believed that if G. salaris emerged in the UK the impact would be similar to that witnessed in Norway. The aim of this thesis is to develop mathematical models that describe the salmon-G. salaris system in order to gain a greater understanding of the possible long-term impact the parasite may have on wild populations of Atlantic salmon in G. salaris-free territories such as the UK. Mathematical models, including deterministic, Leslie matrix and individual based models, were used to investigate the impact of G. salaris on Atlantic salmon at the individual and population level. It is known that the Atlantic strain of Atlantic salmon, examples of which occur naturally in Norway and the UK, does not have any resistance to G. salaris infections and the parasite population is able to quickly grow to epidemic levels. In contrast, the Baltic strain of Atlantic salmon, examples of which occur naturally in Sweden and Russia, exhibits some form of resistance and the parasite is unable to persist. Thus, baseline models were extended to include immunity to infection, a trade-off on salmon reproductive rate, and finally, to consider interactions between populations of G. salaris and multiple strains of salmon exhibiting varying levels of immunity from fully susceptible to resistant. The models proposed predict that in the absence of host resistance or an immune response infections by G. salaris will result in an epidemic followed by the extinction of the salmon host population. Models also predict that if salmon are able to increase their resistance to G. salaris infections through mutations, salmon population recovery after the epidemic is indeed possible within 10-15 years post introduction with low level parasite coexistence. Finally, models also highlight areas where additional information is needed in order to improve predictions and enable the estimation of important parameter values. Model predictions will ultimately be used to assist in future contingency planning against G. salaris outbreaks in the UK and possibly as a basis for future models describing other fish/ecto-parasite systems.

Thesis (MSc)--Stellenbosch University, 2014.
ENGLISH ABSTRACT: Clinical results have indicated that abuse of multiple drugs/substances has devastating health and social consequences. The combined abuse of alcohol and the highly addictive methamphetamine has worsened the drug epidemic in South Africa, especially in the Western Cape Province. Using non-linear ordinary differential equations, we formulate a deterministic mathematical model for alcohol-methamphetamine coabuse epidemic. We prove that the growth of the co-abuse epidemic is dependent on the threshold parameters of the individual substances of abuse. The substance with the maximum reproduction number dominates the epidemic. We also prove that the equilibria points of the co-abuse sub-models are locally and globally asymptotically stable when the sub-model threshold parameters are less than unity. Using parameters values derived from the sub-model fittings to data, a population estimate of co-users of alcohol and methamphetamine under treatment is estimated with a prevalence of about 1%. Although the results show of a small proportion of co-users of alcohol and methamphetamine in the province, the prevalence curve is indicative of a persistent problem. Numerical simulation results reveal that co-abuse epidemic would persists when both reproduction numbers are greater than one. Results from sensitivity analysis shows that the individual substance transmission rates between users of methamphetamine and/or alcohol and the general susceptible population are the most vital parameters in the co-abuse epidemic. This suggests the need to emphasise on preventive measures through educational campaigns and social programs that ensure minimal recruitment into alcohol or methamphetamine abuse. Model analysis using the time-dependent controls (policies) emphasizes the need to allocate even more resources on educational campaigns against substance abuse and on effective treatment services that minimizes or eliminates rampant cases of relapse into substance abuse.
AFRIKAANSE OPSOMMING: Kliniese resultate toon dat die misbruik van meer as een dwelmmiddel verwoestende gesondheids-en sosiale gevolge het. Die gekombineerde misbruik van alkohol en die hoogsverslawende methamphetamine het die dwelm-epidemie in Suid-Afrika vererger, veral in die Wes-Kaapse provinsie. Deur van nie-lineere gewone diffensiaalvergelykings gebruik te maak, formuleer ons ’n deterministiese wiskundige model vir epidemie van die gesamentlike misbruik van alkohol en methamphetamine. Ons toon aan dat die groei van die sogenaamde mede-misbruik epidemie afhanklik is van die drumpelparameters van die individuele middels wat misbruik word. Die middels met die grootste voortbringende syfer domineer die epidemie. Ons bewys ook dat die ekwilibriumpunte van die mede-misbruik submodelle plaaslik en globaal asimptoties stabiel is wanneer die sub-model drumpelparameters kleiner as een is. Deur die submodelle op werklike data te pas word waardes vir die drumpelparameters afgelei en word daar beraam dat daar ongeveer 1% van die populasie mede-misbruikers van alkohol en methamphetamine onder behandeling is. Alhoewel die data ’n klein persentasie van mede-misbruikers van alkohol en methamphetamine in die provinsie toon, dui die voorkomskurwe op ’n groeiende endemie en voortdurende probleem. Resultate uit numeriese simulasie toon dat die mede-misbruik epidemie sal voortduur indien beide reproduserende syfers groter as een sal wees. Resultate van sensitiwiteitsanalise toon dat die individuele middeloordragkoerse tussen gebruikers van methamphetamine en/of alkohol en die gewone vatbare populasie die mees noodsaaklike parameters in die mede-misbruik epidemie is. Dit stel voor dat daar meer klem gelê moet word op voorkomingsmaatreëls deur opvoedkundige veldtogte en sosiale programme om te verseker dat minder alkohol en/of methamphetamine misbruik sal word. Model-analise wat gebruik maak van tyd-afhanklike kontroles (beleide) lê verder klem op die feit dat selfs meer hulpbronne aan opvoedkundige veldtogte teen dwelmmisbruik toegewy moet word, asook die effektiewe behandeling wat gevalle van terugval in dwelmmisbruik sal minimeer of elimineer.

Introduction: Antimalarial resistance, particularly artemisinin resistance, is a major threat to P. falciparum malaria elimination efforts worldwide. Urgent intervention is required to tackle artemisinin resistance but field data on which to base planning of strategies are limited. The aims were to collect available field data and develop population level mathematical models of P. falciparum malaria treatment and artemisinin resistance in order to determine the optimal strategies for elimination of artemisinin resistant malaria in Cambodia and treatment of pre-hospital and severe malaria in Cambodia and Bangladesh. Methods: Malaria incidence and parasite clearance data from Cambodia and Bangladesh were collected and analysed and modelling parameters derived. Population dynamic mathematical models of P. falciparum malaria were produced. Results: The modelling demonstrated that elimination of artemisinin resistant P. falciparum malaria would be achievable in Cambodia in the context of artemisinin resistance using high coverages with ACT treatment, ideally combined with LLITNs and adjunctive single dose primaquine. Sustained efforts would be necessary to achieve elimination and effective surveillance is essential, both to identify the baseline malaria burden and to monitor parasite prevalence as interventions are implemented. A modelled policy change to rectal and intravenous artesunate in the context of pre-existing artemisinin resistance would not compromise the efficacy of ACT for malaria elimination. Conclusions: By being developed rapidly in response to specific questions the models presented here are helping to inform planning efforts to combat artemisinin resistance. As further field data become available, their planned on-going development will produce increasingly realistic and informative models which can be expected to play a central role in planning efforts for years to come.

Thesis (DTech (Mechanical Engineering))--Cape Peninsula University of Technology, 2012
In this thesis, we investigate certain key aspects of mathematical modelling to explain the epidemiology of HIV/AIDS at the workplace and to assess the potential benefits of proposed control strategies. Deterministic models to investigate the effects of the transmission dynamics of HIV/AIDS on labour force productivity are formulated. The population is divided into mutually exclusive but exhaustive compartments and a system of differential equations is derived to describe the spread of the epidemic. The qualitative features of their equilibria are analyzed and conditions under which they are stable are provided. Sensitivity analysis of the reproductive number is carried out to determine the relative importance of model parameters to initial disease transmission. Results suggest that optimal control theory in conjunction with standard numerical procedures and cost effective analysis can be used to determine the best intervention strategies to curtail the burden HIV/AIDS is imposing on the human population, in particular to the global economy through infection of the most productive individuals. We utilise Pontryagin’s Maximum Principle to derive and then analyze numerically the conditions for optimal control of the disease with effective use of condoms, enlightenment/educational programs, treatment regime and screening of infectives. We study the potential impact on productivity of combinations of these conventional control measures against HIV. Our numerical results suggest that increased access to antiretroviral therapy (ART) could decrease not only the HIV prevalence but also increase productivity of the infected especially when coupled with prevention, enlightenment and screening efforts.

Includes bibliographical references (leaves 115-120) Establishes a Bayesian Belief Network (BBN) to model uncertainty of aerosols released from cooling towers and Geographic Information Systems (GIS) to create a wind dispersal model and identify potential cooling towers as the source of infection. Demonstrates the use of GIS and BBN in environmental epidemiology and the power of spatial information in the area of health.

Ce travail de thèse traite de la modélisation et du contrôle des maladies infectieuses à l’aide des automates cellulaires. Nous nous sommes d’abord focalisés sur l’étude d’un modèle de type SEIR. Nous avons pu monter d’une part qu’un voisinage fixe pouvait entrainer une sous-évaluation de l’incidence et de la prévalence et d’autre part que sa structure a un impact direct sur la structure de la distribution de la maladie. Nous nous sommes intéressés également la propagation des maladies vectorielles à travers un modèle de type SIRS-SI multi-hôtes dans un environnement hétérogène.Les hôtes y étaient caractérisés par leur niveau de compétence et l’environnement par la variation du taux de reproduction et de mortalité. Son application à la maladie de Chagas, nous a permis de montrer que l’hétérogénéité de l’habitat et la diversité des hôtes contribuaient à faire baisser l’infection. Cependant l’un des principaux résultats de notre travail à été la formulation du nombre de reproduction spatiale grâce à deux matrices qui représentent les coefficients d’interactions entre les différentes cellules du réseau
This PhD thesis considers the general problem of epidemiological modelling and control using cellular automata approach.We first focused on the study of the SEIR model. On the one hand, we have shown that the traditionnal neighborhood contribute to underestimate the incidence and prevalence of infection disease. On the other hand, it appeared that the spatial distribution of the cells in the lattice have a real impact on the disease spreading. The second study concerns the transmission of the vector-borne disease in heterogeneous landscape with host community. We considered a SIRS-SI with various level of competence at witch the environnment heterogeneity has been characterized by the variation of the birth flow and the death rate. We simulated the Chagas disease spreading and shown that the heterogeneity of habitat and host diversity contribute to decrease the infection. One of the most important results of our work, was the proposition of the spatial reproduction number expression based on two matrices that represent the interaction factors between the cells in the lattice

L'évaluation économique de nouveaux vaccins exige de modéliser la transmission infectieuse au sein de la population, et donc des hypothèses sur la structure et la répartition des contacts. Les matrices de contact obtenues à partir d’enquête de population ont été déterminées pour 8 pays européens. Mais aucune donnée de ce type n'existe à ce jour pour la France. L’étude ComesF (Contact Matrix Estimation – France) vise à combler cette lacune.MéthodologieL'enquête s’est effectuée sur 3 périodes (Février-Mars, Avril, Mai-Avril) avec 278 participants communs à la première et dernière période. Les participants devaient rapporter tous leurs contacts au cours de 2 jours consécutifs dans un journal, avec l'âge, le sexe, l'endroit, la fréquence, le type et la durée du contact.En combinant des enquêtes sérologiques de 2009 et 2013 et les données de couverture vaccinales, nous avons modélisé la séroprévalence de la rougeole, des oreillons et de la rubéole; puis extrapolé la susceptibilité selon l’âge par département à l'année d'intérêt (2016) ; enfin le potentiel épidémique et l'incidence relative selon l’âge d'une future épidémie ont été estimés.Nous avons analysé l'influence de conditions météorologiques sur les variations temporelles des matrices de contact. La population de l'étude a été analysée selon le jour et la météorologie pour estimer le nombre moyen de contacts et le potentiel de transmission estimée avec le R0. Nous avons effectué une revue systématique de la littérature sur les différences selon le genre pour la grippe, la rougeole, les oreillons et la rubéole, puis exploré l'impact du genre sur les matrices de contact et la modélisation des maladies infectieuses.Résultats2033 participants ont rapporté 38881 contacts (médiane pondérée [premier quartile-troisième quartile] : 8 [5–14] par jour) et 54378 contacts avec les contacts professionnels supplémentaires (9 [5–17]). Contrairement à l'âge, le genre, la taille du foyer, les vacances scolaires, le week-end et l'activité professionnelle, la période de l'année influait peu le nombre de contacts ou les schémas de contact. Les schémas de contact étaient influencés par l'âge indépendamment du lieu de contact, et par le genre, les femmes ayant 8 % plus de contacts que les hommes. La plupart des contacts avaient lieu à la maison et à l'école, mais l'ajout des contacts professionnels modifiait la structure des schémas de contact. Les vacances scolaires et les week-ends réduisaient le nombre de contacts, et le R0 de 33 % et de 28 %, respectivement. Le risque pour les Oreillons et la Rubéole concerne surtout le Sud Est et le Centre de la France, alors que le risque pour la rougeole est plus dispersé. Le risque varie avec le genre pour la Rougeole et la Rubéole. Outre les bébés < 1 an, l’épidémie toucherait surtout les adolescents et les jeunes adultes.Les conditions météorologiques influençaient les schémas de contact différemment entre les jours de semaine ou les weekends. La correction pour analyse répétée limitait le nombre de résultats significatif, mais la tendance pour un effet de la météorologie variant entre les jours de semaine et le week-end restait.Les différences de genre dans le schéma de contact pourraient expliquer en partie les différences de genre dans l'épidémiologie des maladies infectieuses. L'utilisation de données spécifiques par genre avait un impact significatif sur le résultat de la modélisation du risque d’une épidémie.Les matrices de contact françaises partageaient de nombreux points communs avec les autres matrices européennes, notamment avec un impact substantiel des fermetures d’école en cas d’épidémie sur la progression de l’épidémie. Le risque d'une nouvelle épidémie de rougeole persiste, mais prédomine pour les oreillons. L'effet des conditions climatiques sur les schémas de contact était modeste, voire négligeable. L’utilisation des données spécifiques par genre est à considérer en modélisation
The economic evaluation of new vaccines requires the modeling of infectious disease transmission within a population, which in turn requires some assumption of specific mixing patterns. Matrixes generated from social contact studies were determined for 8 European countries. To date, no such data exist for France. The ComesF study (Contact Matrix Estimation – France) aimed to fill this gap.MethodologyThe survey was carried out over 3 different periods (Feb-Mar, Apr, Apr-May) with 278 participants common to the first and the last periods. Participants had to list all their contacts for 2 consecutive days in a diary, with the age, sex, location, frequency, type and duration of the contact, from which we estimated French contact matrixes.Combining cross-sectional serological surveys from 2009 and 2013 and vaccine coverage information, we have determined an optimal model for the serology of measles, mumps and rubella for the year of the data collection; age-dependent susceptibility by department was then derived to the year of interest (2016), and effective reproduction number and age-dependent relative incidence of a potential outbreak were estimated using the French contact matrixes.We analysed the influence of meteorological conditions on the temporal variations in mixing patterns. The population of the study was split according to the day and the weather at the time when the diary was filled in. The mean number of contacts and the potential for transmission summarized with R0 were calculated for type and location of contact under different weather conditions.We conducted a systematic review on gender differences in infection focusing on influenza, measles, mumps and rubella. Finally, we provided an exploration of the impact of gender on mixing patterns, and eventually the potential implication for modelling.ResultsThe 2033 participants reported 38 881 contacts (weighted median [first quartile-third quartile]: 8[5–14] per day), and 54 378 contacts with supplementary professional contacts (9[5–17]). Contrary to age, gender, household size, holidays, weekend and occupation, the period of the year had little influence on the number of contacts or the mixing patterns. Contact patterns were highly assortative with age, irrespective of the location of the contact, and gender, with women having 8% more contacts than men. Although most contacts occurred at home and school, the inclusion of professional contacts modified the structure of the mixing patterns. Holidays and weekends reduced the number of contacts dramatically, and as proxies for school closure, reduced R0 by 33% and 28%, respectively.The risk for Mumps and Rubella mainly concerned southeastern and south central France, while the risk for measles was more scattered over the country. Risk differed with gender for Measles and Rubella. Besides infants under 1, the highest share of participation would concern teenagers and young adults.The weather had a differential effect on social mixing according to the type of day, notably weekdays and weekend. But correction for repeated analysis made some results no more significant, although the trend for a differential effect between weekdays and weekend remained.Gender differences in social mixing might explain some gender differences in infectious disease epidemiology. Using gender-specific susceptibility and gender-specific contact matrixes had a significant impact on the result of the modeling. Despite the differences, French contact matrixes shared many aspects with those of other European countries. Notably, school closures were likely to have a substantial impact on the spread of close contact infections in France. While the risk of a new Measles outbreak persists, it predominates for Mumps. The effect of weather on social mixing was mild, if not negligible. Gender differences in modelling should be emphasized

Vaccination has been used to control the spread of infectious diseases for centuries with widespread success. Deterministic models studying the spread of infectious disease often use the assumption of mass vaccination; however, these models do not allow for the inclusion of human behaviour. Since current vaccination campaigns are voluntary in nature, it is important to extend the study of infectious disease models to include the effects of human behaviour. To model the effects of vaccination behaviour on the spread of influenza, we examine a series of models in which individuals vaccinate according to memory or individual decision-making processes based upon self-interest. Allowing individuals to vaccinate proportionally to an exponentially decaying memory function of disease prevalence, we demonstrate the existence of a Hopf bifurcation for short memory spans. Using a game-theoretic influenza model, we determine that lowering the perceived vaccine risk may be insufficient to increase coverage to established target levels. Utilizing evolutionary game theory, we examine models with imitation dynamics both with and without a decaying memory function and show that, under certain conditions, periodic dynamics occur without seasonal forcing. Our results suggest that maintaining diseases at low prevalence with voluntary vaccination campaigns could lead to subsequent epidemics following the free-rider dilemma and that future research in disease control reliant on individual-based decision-making need to include the effects of human behaviour.

Pathogen evolution poses a significant challenge to public health, as efforts to control the spread of infectious diseases struggle to keep up with a shifting target. To better understand this adaptive process, we turn to mathematical modelling. Specifically, we use multi-type branching processes to describe a pathogen's stochastic spread among members of a host population or growth within a single host. In each case, there is potential for new pathogen strains with different characteristics to arise through mutation. We first develop a specific model to study the emergence of a newly introduced infectious disease, where the pathogen must adapt to its new host or face extinction in this population. In an extension of previous models, we separate the processes of host-to-host contacts and disease transmission, in order to consider each of their contributions in isolation. We also allow for an arbitrary distribution of host contacts and arbitrary mutational pathways/rates among strains. This framework enables us to assess the impact of these various factors on the chance that the process develops into a large-scale epidemic. We obtain some intriguing results when interpreted in a biological context. Secondly, motivated by a desire to investigate the time course of pathogen evolutionary processes more closely, we derive some novel theoretical results for multi-type branching processes. Specifically, we obtain equations for: (1) the distribution of waiting time for a particular type to arise; and (2) the distribution of population numbers over time, conditioned on a particular type not having yet appeared. A few numerical examples scratch the surface of potential applications for these results, which we hope to develop further.
Thesis (Master, Mathematics & Statistics) -- Queen's University, 2010-07-28 11:43:22.984

Some Epidemic models with fractional derivatives were proved to be well-defined, well-posed and more accurate [34, 51, 116], compared to models with the conventional derivative. An Ebola epidemic model with non-linear transmission is fully analyzed. The model is expressed with the conventional time derivative with a new parameter included, which happens to be fractional (that derivative is called the 􀀀derivative). We proved that the model is well-de ned and well-posed. Moreover, conditions for boundedness and dissipativity of the trajectories are established. Exploiting the generalized Routh-Hurwitz Criteria, existence and stability analysis of equilibrium points for the Ebola model are performed to show that they are strongly dependent on the non-linear transmission. In particular, conditions for existence and stability of a unique endemic equilibrium to the Ebola system are given. Numerical simulations are provided for particular expressions of the non-linear transmission, with model's parameters taking di erent values. The resulting simulations are in concordance with the usual threshold behavior. The results obtained here may be signi cant for the ght and prevention against Ebola haemorrhagic fever that has so far exterminated hundreds of families and is still a ecting many people in West-Africa and other parts of the world. The full comprehension and handling of the phenomenon of shattering, sometime happening during the process of polymer chain degradation [129, 142], remains unsolved when using the traditional evolution equations describing the degradation. This traditional model has been proved to be very hard to handle as it involves evolution of two intertwined quantities. Moreover, the explicit form of its solution is, in general, impossible to obtain. We explore the possibility of generalizing evolution equation modeling the polymer chain degradation and analyze the model with the conventional time derivative with a new parameter. We consider the general case where the breakup rate depends on the size of the chain breaking up. In the process, the alternative version of Sumudu integral transform is used to provide an explicit form of the general solution representing the evolution of polymer sizes distribution. In particular, we show that this evolution exhibits existence of complex periodic properties due to the presence of cosine and sine functions governing the solutions. Numerical simulations are performed for some particular cases and prove that the system describing the polymer chain degradation contains complex and simple harmonic poles whose e ects are given by these functions or a combination of them. This result may be crucial in the ongoing research to better handle and explain the phenomenon of shattering. Lastly, it has become a conjecture that power series like Mittag-Le er functions and their variants naturally govern solutions to most of generalized fractional evolution models such as kinetic, di usion or relaxation equations. The question is to say whether or not this is always true! Whence, three generalized evolution equations with an additional fractional parameter are solved analytically with conventional techniques. These are processes related to stationary state system, relaxation and di usion. In the analysis, we exploit the Sumudu transform to show that investigation on the stationary state system leads to results of invariability. However, unlike other models, the generalized di usion and relaxation models are proven not to be governed by Mittag-Le er functions or any of their variants, but rather by a parameterized exponential function, new in the literature, more accurate and easier to handle. Graphical representations are performed and also show how that parameter, called ; can be used to control the stationarity of such generalized models.
Mathematical Sciences
Ph. D. (Applied Mathematics)

This thesis presents mathematical models for the dynamics of vaccine preventable diseases, specifically looking at the New Zealand situation. Through the use of integral and differential equations, we develop models and compare the results of these to known data. Using game theory analysis we determine and compare the proportion of the population that needs to be vaccinated in order to minimise the expected costs to the individuals in the population and to the community. Two different scenarios and methods are considered, where the effects of vaccination last only one epidemic cycle (using an integral equation method) and where vaccination is effective over an entire lifetime (using a differential equation method). For both scenarios, we find that the minimum cost for the individuals is reached when a lower proportion of the population is vaccinated than needed for the minimum cost to the community. We then elaborate on the integral equation method to produce a model for repeated epidemics of measles in a population, where a discrete mapping is used to include the year to year demographics of the population. The results of this model show a different epidemic pattern then that produced from a differential equation model, with numerical problems encountered. From here on, we use differential equation models in our analysis. A critique and extension to an existing model for the dynamics of the hepatitis B virus is presented, with discussion on the appropriateness of the model’s construct for predicting the incidence of infection. Alternative differential equation models for hepatitis B virus and immunisation that include splitting the population into age groups with nonhomogeneous mixing are presented. The results of these models are compared with the known data on incidence of infection and carriage in New Zealand, showing how affective different immunisation schedules may have been. Differential equation models are then presented for meningococcal B virus epidemiology in New Zealand, with the models incorporating different features of the virus until the best model is found that fits the New Zealand data. Each model is compared with the known incidence of infection, with the population being either treated as a whole or split into age groups with non-homogeneous mixing. The effect of vaccination is included in this model so that we can explore the future of the infection in the population, and how best to tackle any future epidemics. The model shows that the current vaccination campaign was the best solution for controlling the epidemic, but there will be epidemics in the future that will need subsequent vaccination campaigns to limit the number of infections.

Tese de mestrado, Bioinformática e Biologia computacional (Biologia computacional),Universidade de Lisboa, Faculdade de Ciências, 2015
In recent years our understanding of infectious-disease epidemiology has been greatly increased through mathematical modelling. The major goal of any mathematical study in epidemiology is to develop understanding of the interplay between the variables that determine the course of infection within an individual, and the variables that control the pattern of infections within communities of people. The epidemiology of multi-type pathogen systems, such as dengue, malaria and pneumococcus are notoriously challenging. Direct and indirect interactions between multiple strains shape pathogen population processes, both at the level of a single host and at the population level. Quantifying these interactions is crucial, and the new technologies that are now available to detect multiple infections with different pathogen types are opening new avenues in this endeavour. In this thesis, motivated by the pneumococcus system, we study the colonization dynamics by a multi-type pathogen and focus particularly on co-colonization phenomena, which reflects the simultaneous colonization/infection (terms used in this thesis interchangeably) by two antigenic types of the same pathogen. We pretend to introduce strain ratios, first quantified by Brugger et al. (2010), when modelling the co-colonization phenomena. Therefore, a mathematical epidemiological model is constructed using ordinary differential equations to examine the prevalence and distribution of the co-colonization in the population. Interestingly, we find one scenario where the infection can still persist despite the basic reproduction number R0 being below 1. The phenomena of backward bifurcation is also observed. Moreover, the proportion of each double infected class, at equilibrium, is independent of the size of susceptible or single infected class. Based on a static epidemiological point of view, we also develop an within-host model to study the distribution of co-colonization in an average host. Both models show a clear equal abundance ratio (1:1) prevalence and this seems to be robust despite varying the parameters.
A Epidemiologia é uma ciência que estuda quantitativamente a distribuição dos fenómenos de saúde/doença, e seus factores condicionantes e determinantes, nas populações humanas. Esta permite ainda avaliar a eficácia das intervenções realizadas no âmbito da saúde pública. O fundador da teoria epidemiológica moderna é Ronald Ross cujo estudo no ciclo de vida da malária concedeu-lhe o Nobel em 1902. Este utilizou a modelação matemática para investigar a eficácia das intervenções na prevenção desta doença. No entanto, foi só no final do século XX que a modelação matemática se tornou mais popular. Nos últimos anos o nosso conhecimento relativo à epidemiologia das doenças infecciosas desenvolveu-se bastante devido à modelação matemática. O principal objectivo de qualquer estudo matemático em epidemiologia é melhorar o nosso entendimento relativo às relações das variáveis que determinam o curso de uma infecção quer ao nível do indivíduo como ao nível das comunidades. No entanto, devemos ter sempre em conta que os modelos são sempre abstracções/simplificações dos fenómenos em estudo e os resultados obtidos aproximações do sistema real. A modelação têm sido aplicada para o estudo de diversas doenças infecciosas tal como a sarampo, HIV ou a dengue. Estes modelos revelam-se ferramentas essenciais para compreender a dinâmica das doenças infecciosas e auxiliar no planeamento e controlo das mesmas. Nesta tese, estou interessada em estudar as dinâmicas das doenças infecciosas, mas mais precisamente, explorar através da modelação matemática o fenómeno de co-colonização ou também designado por múltipla colonização. Esta significa a colonização simultânea do hospedeiro por vários microorganismos (da mesma espécie ou diferente). É sabido desde há décadas que a co-colonização é um fenómeno comum na natureza e com importantes consequências para o hospedeiro e parasita. Para o hospedeiro, representa um desafio extra para o seu sistema imunitário. Para o parasita, conduz a interacções directas e indirectas entre as diversas estirpes alterando a sua dinâmica e transmissão. Geralmente este fenómeno agrava o estado de saúde do individuo em comparação com as infecções simples, ou seja, quando o individuo é unicamente colonizado por um parasita. Quantificar a interacção entre as diversas estirpes envolvidas revela-se por isso fundamental, e as novas tecnologias que estão hoje em dia disponíveis para detectar os diferentes patogénios envolvidos, estão a abrir caminho nesta área. Recentemente, Brugger et al. (2010) revelou com os seus estudos na bactéria Streptococcus pneumoniae, também conhecida por pneumoccocus, que a co-colonização tem uma prevalência de 7:9%. Aparentemente, é também mais comum para o hospedeiro apresentar sensivelmente a mesma proporção, usualmente designada por 1:1, entre as duas estirpes da bactéria. Esta prevalência foi também observada independentemente por Valente et al. (2012), mas desta vez em indivíduos saudáveis. Este padrão parece ser, por isso, independente do estado de saúde do indivíduo. O pneumococcus é uma bactéria gram-positiva que normalmente vive assimptomáticamente na nasofaringe e cuja prevalência está aumentada nos primeiros cinco anos de vida de um indivíduo. Ocasionalmente, esta pode migrar para outras regiões do corpo e potencialmente causar uma série de doenças, desde infecções respiratórias ligeiras (otites, etc.) até doenças mais invasivas (pneumonia, septicémia, meningite, etc.). O fenómeno da co-colonização parece também ser um importante factor para a evolução desta espécie, uma vez que representa uma oportunidade para a transferência horizontal de genes. Incorporar esta informação sobre os rácios nos modelos é relevante, uma vez que pode auxiliar na compreensão da sua dinâmica de transmissão e potencialmente prever o impacto de políticas de intervenção, tal como a vacinação. Para um organismo tão diverso como o penumococcus, com mais de 90 estirpes diferentes identificadas, a compreensão da sua biologia está longe de estar completa, e formular modelos reais ainda representa um desafio. Nesta tese foi feito um estudo detalhado acerca do padrão de cocolonização na nasofaringe por múltiplas estirpes do pneumococcus. Mais precisamente, pretendo compreender os factores que justificam a sua prevalência na população e a distribuição dos rácios de cocolonização no caso do hospedeiro apresentar duas estirpes. O principal objectivo deste estudo foi desenhar um modelo matemático que representasse adequadamente a infecção pelo pneumococcus para que o seu output fosse suficientemente preciso para explicar as características da distribuição das estirpes no hospedeiro. Nesse sentido, usei duas abordagens diferentes (mas complementares) para modelar a co-colonização. Em primeiro lugar, usando equações diferenciais ordinárias, construí um modelo epidemiológico determinístico com estrutura nos tipos de co-colonização. Esta abordagem parte da dinâmica de uma população com vista a estudar a distribuição num único indivíduo. Portanto caracteriza-se como uma abordagem topdown. Numa segunda abordagem, criei um modelo probabilístico que a partir da dinâmica da infecção no indivíduo, permite observar a distribuição das estirpes na população. Esta abordagem caracteriza-se como bottom-up. Em ambos os modelos, os resultados que obtive evidenciaram os mecanismos imunitários e estocásticos responsáveis pela distribuição dos rácios de co-colonização. Foi observada uma clara predominância dos rácios 1:1 e este resultado parece ser robusto quando se variam os parâmetros dos modelos. Foram identificados os equilibrios do sistema (trivial e endémico) e avaliada a sua estabilidade. Curiosamente, no modelo epidemiológico, encontrei um cenário em que a infecção pode persistir apesar do número básico de reprodução R0 ser inferior a 1. Este fenómeno tem o nome de backward bifurcation e consiste numa alteração estrutural da estabilidade dos equilíbrios, que deve-se essencialmente ao facto do modelo desenvolvido estruturar os hospedeiros co-colonizados em classes. Estas em média apresentam um número básico de reprodução superior aos hospedeiros colonizados por uma única estirpe. Assim, contribuem em média para uma maior transmissão da infecção na população. Também a proporção de cada classe de hospedeiros duplamente infectados relativamente ao total de hospedeiros infectados, no equilíbrio, é independente da magnitude da classe dos susceptíveis ou dos infectados apenas por uma estirpe. Isto significa que quando o hospedeiro é infectado por uma segunda estirpe tem uma probabilidade fixa de apresentar um determinado rácio. Neste modelo epidemiológico foi também possível verificar, que o mecanismo responsável por desviar a distribuição em torno do rácio 1:1 baseia-se no pressuposto que cada classe de cocolonizados ter taxas de recuperação diferentes, onde umas classes recuperam mais rapidamente que outras. Este rácio traduz como os diferentes patogénios, como um "todo", estão expostos ao sistema imunitário do hospedeiro. Todas as simulações numéricas foram realizadas usando a linguagem de programação Python e o software cientifico Mathematica. Construir modelos epidemiológicos que reflictam o fenómeno de cocolonização é fundamental para melhor compreender determinadas doenças, mas também apresenta muitos desafios técnicos. Nomeadamente, quanto mais factores biológicos forem tidos em conta na modelação, no sentido de os tornar mais realistas, mais parâmetros serão introduzidos e mais complexa será a sua análise. No entanto, seria interessante no futuro incorporar factores como: a identidade das estirpes, a heterogeneidade dos hospedeiros e as variações na sua resposta imunitária. Para além disso, poderíamos ter ainda em conta o fenómeno de co-transmissão, ou seja, a infecção do hospedeiro por mais de um parasita durante o mesmo evento de transmissão. Com isto poderíamos, potencialmente, contribuir para o estudo da evolução da virulência destes patogénios. No entanto, é fundamental que hajam mais resultados experimentais para se fazer uma comparação e validação dos resultados teóricos com vista à criação de modelos biológicos mais representativos da realidade.