A stochastic SIRI epidemic model with relapse and media coverage

This work is devoted to investigate the existence and uniqueness of a global positive solution for a stochastic epidemic model with relapse and media coverage. We also study the dynamical properties of the solution around both disease-free and endemic equilibria points of the deterministic model. Furthermore, we show the existence of a stationary distribution. Numerical simulations are presented to confirm the theoretical results.Fondo Europeo de Desarrollo RegionalMinisterio de Economía y CompetitividadConsejería de Innovación, Ciencia y Empresa (Junta de Andalucía)Faculty of Sciences (Ibn Tofail University

Analysis of a stochastic distributed delay epidemic model with relapse and Gamma distribution kernel

In this work, we investigate a stochastic epidemic model with relapse and distributed delay. First, we prove that our model possesses and unique global positive solution. Next, by means of the Lyapunov method, we determine some sufficient criteria for the extinction of the disease and its persistence. In addition, we establish the existence of a unique stationary distribution to our model. Finally, we provide some numerical simulations for the stochastic model to assist and show the applicability and efficiency of our results.Ministerio de Ciencia, Innovación y Universidades (MICINN). EspañaEuropean Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER

A Probabilistic SIRI Epidemic Model Incorporating Incidence Capping and Logistic Population Expansion

This study presents a newly developed stochastic SIRI epidemic model, which combines logistic growth with a saturation incidence rate. This research mainly examines the presence and uniqueness of positive solutions within the formulated model. Furthermore, we aim to analyze the long-term performance of the system and provide valuable insights into disease extinction in a population. Our investigation delves into the conditions required for disease extinction, which are crucial in predicting and controlling the spread of deadly diseases. To substantiate our assertions, we have devised a stochastic Lyapunov function, which serves as a robust mathematical framework for demonstrating the presence of a discernible stationary ergodic distribution. This mathematical foundation significantly contributes to the understanding of model behavior. To complement our analytical findings, we conduct numerical simulations, which reinforce our results and provide a comprehensive understanding of the behavior of our proposed model, and open new avenues for future research in this area

A survey on Lyapunov functions for epidemic compartmental models

In this survey, we propose an overview on Lyapunov functions for a variety of compartmental models in epidemiology. We exhibit the most widely employed functions, and provide a commentary on their use. Our aim is to provide a comprehensive starting point to readers who are attempting to prove global stability of systems of ODEs. The focus is on mathematical epidemiology, however some of the functions and strategies presented in this paper can be adapted to a wider variety of models, such as prey–predator or rumor spreading

Modeling tuberculosis:a compromise between biological realism and mathematical tractability

Tese de doutoramento, Matemática (Análise Matemática), 2009, Universidade de Lisboa, Faculdade de CiênciasDisponível no document

Characterization of differentially culturable bacteria in exenic culture and from tuberculosis patients

A thesis submitted to the Faculty of Health Sciences, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the Degree of Doctor of Philosophy Johannesburg, March 2018.During tuberculosis (TB) disease, host-derived stresses and chemotherapy are thought to drive tubercle bacilli into differential growth states. This is evidenced by the presence of differentially culturable tubercle bacilli (DCTB) in the sputum of treatment naïve TB patients. These bacteria do not form colonies on solid media but can be cultured following supplementation of liquid media with culture filtrate as a source of growth stimulatory molecules. As DCTB are non-replicating and phenotypically drug tolerant, these organisms are thought to underpin the lengthy culture diagnosis and protracted treatment period required for TB disease. The purpose of this study was to investigate the use of culture filtrate in unmasking DCTB populations to: (1) quantify these populations in treatment naïve individuals, (2) assess the response of DCTB versus conventionally culturable bacteria to first-line treatment, (3) determine the relationship between DCTB cultured in the most probable number (MPN) assay with other TB culture methods and (4) to enhance currently employed culture diagnostic methods. The results from this study confirmed that treatment naïve individuals coinfected with HIV had significantly lower quanta of DCTB in their sputum compared to their HIV-negative counterparts. These findings implicate the host immune response in influencing the prevalence of DCTB in sputum. During treatment, four patterns of decline in DCTB were described. One quarter of the patient population accumulated DCTB during the first seven days of treatment, whilst approximately the same number of individuals displayed a rapid decline in DCTB during this period. The remaining individuals either displayed static or atypical patterns of DCTB over the first 14 days of treatment. Following treatment completion, residual DCTB was cultured in approximately two thirds of the patients analysed, suggesting that bacteriological sterilization of lungs was not achieved. These observations were confirmed using a novel fluorogenic probe specific for the detection of live Mycobacterium tuberculosis. DCTB cultured in the MPN assay was shown to directly correlate with current TB culture methods. These findings demonstrate a potential utility for the MPN assay in early bactericidal activity studies to assess the sterilising effect of new TB drugs on DCTB populations. Furthermore, the addition of culture filtrate to the BACTEC MGIT 960 assay reduced the rates of TB detection in smear-negative, HIV-positive individuals. Collectively, these observations demonstrate that the detection of DCTB in sputum can serve as a possible biomarker for treatment response. Further long term studies are required to determine if DCTB can use used to assess the risk of relapse disease and to test the efficacy of new drugs on persistent bacterial populations.LG201

Towards malaria prediction in Sri Lanka. modelling spatial and temporal variability of malaria case counts

This thesis was motivated by the need of the Anti Malaria Campaign (AMC) of Sri Lanka for malaria risk maps and malaria case number predictions to assist in the planning for malaria control. Despite a wealth of high resolution data collected over decades, a malaria forecasting system was not in place, and detailed island-wide maps of malaria incidence could permit the assessment of the malaria situation and its determinants. The overall aim of this thesis was to describe the spatial and seasonal distribution of malaria in Sri Lanka and associated factors, and to develop a malaria forecasting system. In this thesis, the spatial variation of malaria in Sri Lanka was described in relation to risk factors. Also, the risk and the impact of a tsunami natural disaster on malaria transmission and malaria control in Sri Lanka were evaluated. The relation in space between seasonality of malaria and seasonality of rainfall, and the relationship between monthly malaria case time series and monthly rainfall time series in Sri Lanka were quantified. A model for short term malaria prediction was developed and implemented in Sri Lanka for use by the AMC. This thesis also contributed a statistical methodology for analysing over dispersed temporal count data with non stationary and / or seasonal behaviour, such as observed in malaria case count time series in Sri Lanka. In Chapter 1, the stage was set by briefly describing malarial disease and the biology of malarial parasites and vectors relevant to Sri Lanka. The influence of weather on malaria transmission, and observed linkages between weather and malaria in terms of spatial and temporal patterns were introduced. Immunity was also briefly discussed, because it affects the translation of (unobserved) disease transmission patterns into patterns of observed malaria cases. A brief overview was given of the history of malaria and malaria control in Sri Lanka. Chapter 2 provided health professionals and the larger general public with the first island-wide incidence maps of Plasmodium vivax and Plasmodium falciparum malaria at sub district resolution. The distribution and seasonality of P. vivax and P. falciparum incidence was remarkably similar within each district, although they varied spatially. The annual malaria incidence changed over the 1995 – 2002 period, and the rate of change varied with the area, thus indicating the need for regular updates of the incidence maps. The spatial and temporal malaria distribution in the country was related to accessibility of areas for implementation of malaria control (in particular governed by the armed conflict and the peace process), and to socio economic and environmental factors. Also, the exposure of tourists to malaria infection was discussed. Chapter 3 provided a re-assessment of the malaria situation, including details on vector insecticide resistance, parasite drug resistance, and insights into the national policy for malaria diagnosis and treatment. The assessment and its publication were triggered by the tsunami that hit on 26 December 2004, and the ensuing international concern about possibilities of an increase of vector borne diseases. The likelihood of a widespread outbreak was estimated as limited. The public health system was deemed capable of dealing with the possible threat of a malaria outbreak. Concerns were expressed that the influx of foreign medical assistance, drugs, and insecticides could interfere with malaria surveillance, and the long term malaria control strategy of Sri Lanka, if not in accordance with government policy. Chapter 4 assessed the impact of the tsunami on the malaria situation and the national and international malaria control efforts in the year following the tsunami. Malaria incidence had decreased in most districts, including the ones that were hit hardest by the tsunami, and the whole-country malaria incidence time series did not deviate from the downward trend that started in 2000. The focus of national and international post tsunami malaria control efforts was supply of antimalarials, distribution of impregnated mosquito nets and increased monitoring in the affected area. Internationally donated antimalarials were either redundant or did not comply with national drug policy. There was no indication of increased malaria vector density. In Chapter 5, the spatial correlation between average seasonality of malaria and climatic seasonality of rainfall was studied. A simple index for seasonality was developed by making use of the characteristic of a varying degree of bimodality of seasonality present in both malaria and rainfall in Sri Lanka. The malaria seasonality index was significantly associated with the rainfall seasonality index in a regression taking spatial autocorrelation into account. This was in paradox with the negative correlation in space between annual rainfall and malaria endemicity (Chapter 2). Both rainfall and malaria may react independently to monsoonal periodicity, but given the fact that rainfall has an important impact on the availability and quality of breeding sites for malaria vectors, it is clear that rainfall seasonality is an important driver of malaria seasonality. In Chapter 6, the temporal correlation between monthly malaria case time series and monthly rainfall time series was explored for each district separately. For most districts, strong positive correlations were found for malaria time series lagging zero to three months behind rainfall. However, only for a few districts, weak positive (at lags zero and one) or weak negative (at lags two to six) correlations were found if autocorrelation and seasonality were removed from the series prior to crosscorrelation analysis, thus indicating that rainfall might have little potential use in a malaria forecasting system. These cross correlation analyses had the drawbacks that inter-annual effects were masked due to detrending of the data, and that potentially seasonally varying effects were not taken into account. Subsequent inter-annual analysis showed strong negative correlations between malaria and rainfall for a group of districts in the centre-west of the country. Seasonal inter-annual analysis showed that the effect of rainfall on malaria varied according to the season (and geography). Chapter 7 focused on the development of a malaria forecasting system for Sri Lanka, which could assist in the efficient allocation of resources for malaria control, especially when malaria is unstable and fluctuates in intensity both spatially and temporally. Several types of time series models were tested in their ability to predict the monthly number of malaria cases in districts one to four months ahead. Different districts required different prediction models, and the prediction accuracy varied with district and forecasting horizon. It was subsequently tested if rainfall or malaria patterns in neighbouring districts could improve prediction accuracy of the selected models. Only for a few districts, a modest improvement was made when rainfall was included in the models as a covariate. This modest improvement was not deemed sufficient to merit investing in a forecasting system for which rainfall data are routinely processed. The development and launch of a system for forecasting malaria by the AMC was described in addendum to Chapter 7. Throughout the statistical modelling in Chapter 7, it was assumed that logarithmically transformed malaria case data were approximately Gaussian distributed. However, such an approximation is less close when case numbers are low, as was the case at the time of writing. Therefore, in Chapter 8, a class of generalised multiplicative seasonal autoregressive integrated moving average models for the parsimonious and observation-driven modelling of non Gaussian, non stationary and / or seasonal time series data was developed. Chapter 9 provides a general discussion in which the contributions of this thesis are put into context, in which limitations of this thesis are discussed and directions for future research outlined