Mathematical modeling and optimal control of malaria

Malaria continues to infect millions and kill hundreds of thousands of people worldwide each year, despite over a century of research and attempts to control and eliminate this infectious disease. Challenges such as the development and spread of drug resistant malaria parasites, insecticide resistance to mosquitoes, climate change, the presence of individuals with subpatent malaria infections which normally are asymptomatic and behavioral plasticity in the mosquito hinder the prospects of malaria control and elimination. In this thesis, mathematical models of malaria transmission and control that address the role of drug resistance, immunity, iron supplementation and anemia, immigration and visitation, and the presence of asymptomatic carriers in malaria transmission are developed. A within-host mathematical model of severe Plasmodium falciparum malaria is also developed. First, a deterministic mathematical model for transmission of antimalarial drug resistance parasites with superinfection is developed and analyzed. The possibility of increase in the risk of superinfection due to iron supplementation and fortification in malaria endemic areas is discussed. The model results calls upon stakeholders to weigh the pros and cons of iron supplementation to individuals living in malaria endemic regions. Second, a deterministic model of transmission of drug resistant malaria parasites, including the inflow of infective immigrants, is presented and analyzed. The optimal control theory is applied to this model to study the impact of various malaria and vector control strategies, such as screening of immigrants, treatment of drug-sensitive infections, treatment of drug-resistant infections, and the use of insecticide-treated bed nets and indoor spraying of mosquitoes. The results of the model emphasize the importance of using a combination of all four controls tools for effective malaria intervention. Next, a two-age-class mathematical model for malaria transmission with asymptomatic carriers is developed and analyzed. In development of this model, four possible control measures are analyzed: the use of long-lasting treated mosquito nets, indoor residual spraying, screening and treatment of symptomatic, and screening and treatment of asymptomatic individuals. The numerical results show that a disease-free equilibrium can be attained if all four control measures are used. A common pitfall for most epidemiological models is the absence of real data; model-based conclusions have to be drawn based on uncertain parameter values. In this thesis, an approach to study the robustness of optimal control solutions under such parameter uncertainty is presented. Numerical analysis of the optimal control problem in the presence of parameter uncertainty demonstrate the robustness of the optimal control approach that: when a comprehensive control strategy is used the main conclusions of the optimal control remain unchanged, even if inevitable variability remains in the control profiles. The results provide a promising framework for the design of cost-effective strategies for disease control with multiple interventions, even under considerable uncertainty of model parameters. Finally, a separate work modeling the within-host Plasmodium falciparum infection in humans is presented. The developed model allows re-infection of already-infected red blood cells. The model hypothesizes that in severe malaria due to parasite quest for survival and rapid multiplication, the Plasmodium falciparum can be absorbed in the already-infected red blood cells which accelerates the rupture rate and consequently cause anemia. Analysis of the model and parameter identifiability using Markov chain Monte Carlo methods is presented.

Diagnostic investigation of ventilator-associated pneumonia using bronchoalveolar lavage: comparative study with a postmortem lung biopsy

The purpose of the present study was to validate the quantitative culture and cellularity of bronchoalveolar lavage (BAL) for the diagnosis of ventilator-associated pneumonia (VAP). A prospective validation test trial was carried out between 1992 and 1997 in a general adult intensive care unit of a teaching hospital. Thirty-seven patients on mechanical ventilation with suspected VAP who died at most three days after a BAL diagnostic procedure were submitted to a postmortem lung biopsy. BAL effluent was submitted to Gram staining, quantitative culture and cellularity count. Postmortem lung tissue quantitative culture and histopathological findings were considered to be the gold standard exams for VAP diagnosis. According to these criteria, 20 patients (54%) were diagnosed as having VAP and 17 (46%) as not having the condition. Quantitative culture of BAL effluent showed 90% sensitivity (18/20), 94.1% specificity (16/17), 94.7% positive predictive value and 88.8% negative predictive value. Fever and leukocytosis were useless for VAP diagnosis. Gram staining of BAL effluent was negative in 94.1% of the patients without VAP (16/17). Regarding the total cellularity of BAL, a cut-off point of 400,000 cells/ml showed a specificity of 94.1% (16/17), and a cut-off point of 50% of BAL neutrophils showed a sensitivity of 90% (19/20). In conclusion, BAL quantitative culture, Gram staining and cellularity might be useful in the diagnostic investigation of VAP.

A mathematical framework for group analysis of von Willebrand factor multimeric composition following luminography

The objective of the present study was to establish a method for quantitative analysis of von Willebrand factor (vWF) multimeric composition using a mathematical framework based on curve fitting. Plasma vWF multimers from 15 healthy subjects and 13 patients with advanced pulmonary vascular disease were analyzed by Western immunoblotting followed by luminography. Quantitative analysis of luminographs was carried out by calculating the relative densities of low, intermediate and high molecular weight fractions using laser densitometry. For each densitometric peak (representing a given fraction of vWF multimers) a mean area value was obtained using data from all group subjects (patients and normal individuals) and plotted against the distance between the peak and IgM (950 kDa). Curves were constructed for each group using nonlinear fitting. Results indicated that highly accurate curves could be obtained for healthy controls and patients, with respective coefficients of determination (r²) of 0.9898 and 0.9778. Differences were observed between patients and normal subjects regarding curve shape, coefficients and the region of highest protein concentration. We conclude that the method provides accurate quantitative information on the composition of vWF multimers and may be useful for comparisons between groups and possibly treatments.

Mathematical modeling of electro-rotation spectra of small particles in liquid solutions: Application to human erythrocyte aggregates

Electro-rotation can be used to determine the dielectric properties of cells, as well as to observe dynamic changes in both dielectric and morphological properties. Suspended biological cells and particles respond to alternating-field polarization by moving, deforming or rotating. While in linearly polarized alternating fields the particles are oriented along their axis of highest polarizability, in circularly polarized fields the axis of lowest polarizability aligns perpendicular to the plane of field rotation. Ellipsoidal models for cells are frequently applied, which include, beside sphere-shaped cells, also the limiting cases of rods and disks. Human erythrocyte cells, due to their particular shape, hardly resemble an ellipsoid. The additional effect of rouleaux formation with different numbers of aggregations suggests a model of circular cylinders of variable length. In the present study, the induced dipole moment of short cylinders was calculated and applied to rouleaux of human erythrocytes, which move freely in a suspending conductive medium under the effect of a rotating external field. Electro-rotation torque spectra are calculated for such aggregations of different length. Both the maximum rotation speeds and the peak frequencies of the torque are found to depend clearly on the size of the rouleaux. While the rotation speed grows with rouleaux length, the field frequency nup is lowest for the largest cell aggregations where the torque shows a maximum.

Comparison of anaerobic threshold determined by visual and mathematical methods in healthy women

Several methods are used to estimate anaerobic threshold (AT) during exercise. The aim of the present study was to compare AT obtained by a graphic visual method for the estimate of ventilatory and metabolic variables (gold standard), to a bi-segmental linear regression mathematical model of Hinkley's algorithm applied to heart rate (HR) and carbon dioxide output (VCO2) data. Thirteen young (24 ± 2.63 years old) and 16 postmenopausal (57 ± 4.79 years old) healthy and sedentary women were submitted to a continuous ergospirometric incremental test on an electromagnetic braking cycloergometer with 10 to 20 W/min increases until physical exhaustion. The ventilatory variables were recorded breath-to-breath and HR was obtained beat-to-beat over real time. Data were analyzed by the nonparametric Friedman test and Spearman correlation test with the level of significance set at 5%. Power output (W), HR (bpm), oxygen uptake (VO2; mL kg-1 min-1), VO2 (mL/min), VCO2 (mL/min), and minute ventilation (VE; L/min) data observed at the AT level were similar for both methods and groups studied (P > 0.05). The VO2 (mL kg-1 min-1) data showed significant correlation (P < 0.05) between the gold standard method and the mathematical model when applied to HR (r s = 0.75) and VCO2 (r s = 0.78) data for the subjects as a whole (N = 29). The proposed mathematical method for the detection of changes in response patterns of VCO2 and HR was adequate and promising for AT detection in young and middle-aged women, representing a semi-automatic, non-invasive and objective AT measurement.

Atomic Force Microscopy Investigation of NI Particles Deposited on Porous Silicon

In this thesis properties and influence of modification techniques of porous silicon were studied by Atomic Force Microscope (AFM). This device permits to visualize the surface topography and to study properties of the samples on atomic scale, which was necessary for recent investigation. Samples of porous silicon were obtained by electrochemical etching. Nickel particles were deposited by two methods: electrochemical deposition and extracting from NiCl2 ethanol solution. Sample growth was conducted in Saint-Petersburg State Electrotechnical University, LETI. Kelvin probe force microscopy (KPFM) and Magnetic force microscopy (MFM) were utilized for detailed information about surface properties of the samples. Measurements showed the difference in morphology correlating with initial growth conditions. Submicron size particles were clearly visible on surfaces of the treated samples. Although their nature was not clarified due to limitations of AFM technique. It is expected that surfaces were covered by nanometer scale Ni particles, which can be verified by implication of RAMAN device.

Mathematical modelling of the osmotic dehydration of cherry tomato (Lycopersicon esculentum var. cerasiforme)

Osmotic dehydration of cherry tomato as influenced by osmotic agent (sodium chloride and a mixed sodium chloride and sucrose solutions) and solution concentration (10 and 25% w/w) at room temperature (25°C) was studied. Kinetics of water loss and solids uptake were determined by a two parameter model, based on Fick's second law and applied to spherical geometry. The water apparent diffusivity coefficients obtained ranged from 2.17x10-10 to 11.69x10-10 m²/s.

Multicomponent diffusion during Prato cheese ripening: mathematical modeling using the finite element method

The partial replacement of NaCl by KCl is a promising alternative to produce a cheese with lower sodium content since KCl does not change the final quality of the cheese product. In order to assure proper salt proportions, mathematical models are employed to control the product process and simulate the multicomponent diffusion during the reduced salt cheese ripening period. The generalized Fick's Second Law is widely accepted as the primary mass transfer model within solid foods. The Finite Element Method (FEM) was used to solve the system of differential equations formed. Therefore, a NaCl and KCl multicomponent diffusion was simulated using a 20% (w/w) static brine with 70% NaCl and 30% KCl during Prato cheese (a Brazilian semi-hard cheese) salting and ripening. The theoretical results were compared with experimental data, and indicated that the deviation was 4.43% for NaCl and 4.72% for KCl validating the proposed model for the production of good quality, reduced-sodium cheeses.

Investigation of the effects of peppermint (Mentha piperita) on the biochemical and anthropometric profile of university students

The hypolipidemic effects of several medicinal plants have already been demonstrated, but many plants commonly used to treat diseases still need to be studied. Peppermint (Mentha piperita) is widely consumed by the population for different purposes, but not for the treatment of dyslipidemias. The objective of this study was to examine the effects of this plant on human biochemical and anthropometric profiles and blood pressure, based on the administration of peppermint juice twice daily for 30 days. Blood samples were collected before and after the treatment in order to determine the glycemic and lipid profiles, and the Body Mass Index (BMI) analysis was performed. Results indicated that 41.5% of the subjects showed a reduction in glycemia, 66.9% in total cholesterol levels, 58.5% in triacylglycerides, 52.3% in LDL-c (low-density lipoproteins) indices, 70% in GOT (glutamic-oxaloacetic transaminase) levels, 74.5% in GPT (glutamic-pyruvic transaminase) levels, and that 52% presented an increase in HDL-c (high-density lipoprotein cholesterol) indices. Also, 52.5% showed a decrease in blood pressure and 48.7% in BMI. The use of peppermint by humans can be considered beneficial in the prevention and treatment of risk factors of chronic degenerative diseases.

Application of mathematical models for the prediction of adsorption isotherms in solid mixture for mango powder refreshment

Solid mixtures for refreshment are already totally integrated to the Brazilian consumers' daily routine, because of their quick preparation method, yield and reasonable price - quite lower if compared to 'ready-to-drink' products or products for prompt consumption, what makes them economically more accessible to low-income populations. Within such a context, the aim of this work was to evaluate the physicochemical and mineral composition, as well as the hygroscopic behavior of four different brands of solid mixture for mango refreshment. The BET, GAB, Oswim and Henderson mathematical models were built through the adjustment of experimental data to the isotherms of adsorption. Results from the physiochemical evaluation showed that the solid mixtures for refreshments are considerable sources of ascorbic acid and reductor sugar; and regarding mineral compounds, they are significant sources of calcium, sodium and potassium. It was also verified that the solid mixtures for refreshments of the four studied brands are considered highly hygroscopic.

Mathematical modeling of dehydration of 'Fuji' and 'Gala' apples slices using infrared

The objective of this work was to determine and model the infrared dehydration curves of apple slices - Fuji and Gala varieties. The slices were dehydrated until constant mass, in a prototype dryer with infrared heating source. The applied temperatures ranged from 50 to 100 °C. Due to the physical characteristics of the product, the dehydration curve was divided in two periods, constant and falling, separated by the critical moisture content. A linear model was used to describe the constant dehydration period. Empirical models traditionally used to model the drying behavior of agricultural products were fitted to the experimental data of the falling dehydration period. Critical moisture contents of 2.811 and 3.103 kgw kgs-1 were observed for the Fuji and Gala varieties, respectively. Based on the results, it was concluded that the constant dehydration rates presented a direct relationship with the temperature; thus, it was possible to fit a model that describes the moisture content variation in function of time and temperature. Among the tested models, which describe the falling dehydration period, the model proposed by Midilli presented the best fit for all studied conditions.

Mathematical modeling of microbial growth in milk

A mathematical model to predict microbial growth in milk was developed and analyzed. The model consists of a system of two differential equations of first order. The equations are based on physical hypotheses of population growth. The model was applied to five different sets of data of microbial growth in dairy products selected from Combase, which is the most important database in the area with thousands of datasets from around the world, and the results showed a good fit. In addition, the model provides equations for the evaluation of the maximum specific growth rate and the duration of the lag phase which may provide useful information about microbial growth.

Investigation of cytotoxic and mutagenic effects of Malpighia glabra L. (barbados cherry) fruit pulp and vitamin C on plant and animal test systems

Fruits are important sources of nutrients in human diet, and Barbados Cherry (Malpighia glabra L.) is of particular interest due to its high content of antioxidants. Diets rich in fruits and vegetables protect individuals against diseases and cancer, but excessive intake of vitamins may act as pro-oxidant and generate changes in DNA. To evaluate the effect of different in natura (BAN) and frozen (BAF) Barbados Cherry pulp concentrations and synthetic vitamin C in liquid form (VC) on the chromosome level and the cell cycle division, root meristeme cells of Allium cepa L. and bone marrow cells of Wistar rats Rattus norvegicus, were used as test system. In Allium cepa L., BAN, at the highest concentration (0.4 mg.mL-1) and BAF, at the lowest concentration (0.2 mg.mL-1), inhibited cell division, and there was recovery of cell division after the recovery period in water only for BAN. In the Wistar rats, all treatments with Barbados Cherry, either acute or subchronic, were not cytotoxic or mutagenic; only the highest concentration of VC increased significantly the rate of chromosomal abnormalities. The data obtained are important to reinforce the use of Barbados Cherry fruit in the diet.

Investigation of glutathione peroxidase activity in chicken meat under different experimental conditions

Due to the fact that previous studies on the enzymatic activity of Glutathione peroxidase (GSH-Px) diverge widely in their methodology and results, this study aimed to investigate the influence of different analytical conditions on GSH-Px activity in chicken thighs from broilers that were fed different diets with different sources and concentrations of selenium. GSH-Px activity was evaluated six hours after slaughter and 120 days after frozen storage at -18 ºC. The different analytical conditions included time of pre-incubation (0, 10 and 30 minutes), reaction medium, types of substrate (H2O2 (0.72 mM, 7.2 mM, and 72 mM) and Terc-butil hydroperoxide 15 mM), and different buffer concentrations (buffer 1, potassium phosphate 50 mM pH 7.0 + EDTA 1 mM + mercaptoethanol 1 mM, and buffer 2, tris-HCl 50 mM pH 7.6 + EDTA 1 mM + mercapthanol 5 mM). The results show that the highest GSH-Px activity was observed when enzyme and substrate were in contact at 22 ºC without any pre-incubation, and that, when used at concentrations above 0.72 mM, hydrogen peroxide saturated the GSH-Px enzyme and inhibited its activity. The enzyme presented higher affinity to hydrogen peroxide when compared to terc-butil peroxide, and the addition of a buffer containing mercaptoethanol did not increase GSH-Px enzymatic activity. The activity of GSH-Px was not influenced by the source and concentration of selenium in the diet either. The obtained results allowed the determination of the best temperature of contact between the enzyme and substrate (22 ºC), the optimum concentration, and the type of substrate and buffer to be used. This information is extremely useful for future studies on GSH-Px activity in meat due to the divergence and little information found in the literature.

Evaluation of a new mathematical model to describe Clostridium perfringens growth during the cooling of cooked ground beef

A mathematical model previously developed to study microbial growth in food products under an isothermal environment was adapted to a time-varying temperature regime. The resulting model was applied to study the growth of Clostridium perfringens in meat products. This micro-organism is of particular relevance to public health and economy due to the loss of productivity caused by it. Results showed a similar performance of the model used compared to the Baranyi model under an isothermal situation and a slightly better performance under a non-isothermal temperature profile.