We introduce for the first time a within-dog model of Leishmania infection. The model illustrates the dynamics of the parasite on the skin and within the bone marrow, as well as the dynamics of a class of antibodies. Simulations suggest that within-host pathogen persists either in oscillatory regime or at an equilibrium. The model is fitted to within-dog data of these three dynamical variables. Furthermore, we develop a vector-host epidemiological model of population level dynamics of Leishmaniasis in dogs [1]. The within-dog and the population level models are linked through transmission and disease-induced mortality. The newly developed is mathematically well-posed. We compute the epidemiological reproduction number, which depends on the within-host parasite load and immune response. We find that the model has a disease-free equilibrium. We show that it has at least one endemic equilibrium if the reproduction number is above one. If the reproduction number is below one, the model may have no endemic equilibria or it may have even number of endemic equilibria, at least of which is locally stable. Thus, even if the reproduction number is below one, Leishmania can persist in dog populations, if it is introduced at sufficiently high level, meaning that even dog populations with sufficiently high immunity can sustain the disease. [1] M. Gilchrist, A. Sasaki, Modeling Host-–Parasite Coevolution: A Nested Approach Based on Mechanistic Models, Journal of Theoretical Biology 3, p.289--308, 2002
Genomic regulatory networks are examples of complex systems with distributed control and abundant feedback. The concept of genes that can constrain, or canalize, such a complex network to a specic behavior was first proposed by C. Waddington in 1942 [1]. Waddington stipulated the existence of genes that can produce reliable developmental effects against genetic mutations or environmental changes during evolution [2], [1]. Zhao et al. [3] made a clear distinction between master genes and canalizing genes. Both master and canalizing genes exert a strong control over many downstream gene pathways; however, canalizing genes have an additional ability of taking over the control and overriding other regulatory instructions. Canalizing genes produce adaptive and optimal reactions to environmental, stochastic and genetic perturbations and they are essential in a complex system, so it can achieve biological robustness and buffer itself from the eects of random alterations or operating errors. Our work suggests that the currently adopted denitions of canalizing and master genes should be modified to include a relative characterization of these properties in such a way that a particular gene does not have to be exclusively a master or a canalizing one.
A protein is commonly visualised as a discrete piecewise linear curve, its backbone geometry being characterised in terms of the extrinsically determined Ramachandran angles. However, in addition to the extrinsic geometry, the protein backbone has also two independent intrinsic geometric structures determined by the peptide planes, and the side chains and their relative orientations. We develop a novel methodology for analysis and 3D visualisation of protein structure, based on the construction of a series of orthonormal coordinate frames, along the protein side chains, and mapping the atoms positions onto a unit sphere. Thus, we obtain a consistent picture of side-chain covalent bonds spatial orientations, though from a different perspective --- that of an observer, who climbs up the side chains from one carbon atom to the next. We validate the method by studying the distribution of distal and proximal histidine, as well as of valine in the molecule of myoglobinin, on a statistical sample of all myoglobin entries from PDB (Protein Data Bank) with resolution better than 2.0 A. The obtained results are in a good agreement with the biological data. Going beyond traditional visualization schemes that step on laboratory frames, our novel 3D visualization method can be employed as a valuable visual tool for protein side chain construction as well as structure validation and refinement, complementary to widely used visualisation suits like VMD, Jmol, PyMOL and others.
The PCFH is an alternative technology to conventional disinfection methods used to control the microbial quality of water. To verify its effectiveness, it is necessary to generate new information related to inactivation kinetics of MS2 coliphage by PCFH. Therefore, in this research we studied the mathematical modeling of the virus inactivation by PCFH-Al/Fe-PILC in presence of a synthetic pattern of natural organic matter. The inactivation constant was subjected to two approximations, finding out that the experimental data were adjusted to the pseudo-first order Chick-Watson model with constant inactivation rate. With the PCFH technology it was possible to obtain the maximum inactivation constant $k=0.1648\, min^{-1}$ in the catalytic tests MS2-3 and MS2-7, kinetics indicates that a rapid inactivation occurs in the first minutes of the reaction, followed by a slow inactivation in the rest of the reaction time. This research suggests the potential of PCFH to improve the quality of drinking water
Conservation of forest resources is one of the most challenging problems in ecology and environmental science. For maintaining the ecological integrity of forest ecosystem and preserving biodiversity, knowing forest dynamics is of very importance. One of fundamental issues in these problems is to predict the variation of tree density caused by random factors. Antonovsky (1975) introduced a mono-species forest model with two age classes of trees. Kuznetsov et al. [J. Math. Biol. 32 (1994)] then introduced a mathematical model of mono-species forest with two age classes which takes into account the seed production and dispersion. Yagi ([3]) presented very interesting mathematical structures for that model including variation of tree and seed densities, and robustness of forest in deterministic environments. No random factor (noise) has been considered in these models. In the real world, noise including climate change, disaster (floods, hurricanes, tornadoes, earthquakes, tsunamis), and human behaviors always make effect on forest in various levels. In [1,2], we investigated the effects of noise on the mortality of old trees in the Antonovsky's model. In this talk, we present a clear picture showing the effects of noise on not only that parameter but also others in the model. We prove existence and uniqueness of global nonnegative solutions. We then investigate asymptotic behavior of solutions, showing limitations for sustainability or decline of forest against noise. Results are illustrated by numerical examples.
Protein folding is the dynamic physical process by which a protein structure assumes its functional shape or conformation and it is a consequence of the interactions through time between the amino acids of the primary structure. It should be treated as a dynamic process and emergent phenomenon, on the contrary to most of the previous research in this computational biology problem, where the intense research was focused on the computational prediction of the final folded structure. The folding process can be modeled with tools like cellular automata (CA) [1], which can provide and model its emergent and dynamic nature. We are working with CA and machine learning methods to automatically obtain a model of protein folding. CA were implemented with artificial neural networks (neural-CA), instead of classical CA transition rules, incorporating this way the generalization capabilities of connectionist models. Evolved CA decide the most appropriate moves between consecutive amino acids, to obtain the final folded (native) conformation that minimizes its Gibbs free energy (as in the real case). We have used neural-CA to model protein folding using the basic HP model (Hydrophobic-Polar), with the 2D and 3D lattices [2], more complex lattice models like the Face-Centered Cubic lattice model [3], and the low resolution atomic model of the Rosetta environment [4]. In all cases the importance relies on the automatic modeling of the process, using machine learning and only from known protein structures, on the contrary to the a priori (and non-exact) modeling of protein component interactions followed in molecular dynamic approaches.
We present a model to investigate the effects of of social learning and wealth in the dynamics of Renal Failure in Kenya. The model captures the progression of Renal failure from mild sickness, to chronic disease to total renal failure that requires a transplant for the patient. The progression from mild to chronic disease is a function of social learning while the availability of a transplant organ is a function of wealth. Mathematical results reveal the existence of the disease free and endemic equilibrium whose existence and stability depends on the control reproduction number, R c . The disease persist when the R c > 1 and dies out when R c < 1 . Control strategies like social learning are shown to be e ff ective tools that reduce the rate of progression to chronic kidney disease and total renal failure, while government subsidies and health insurance cover averts death due to the disease as a transplant will be available when there are resources to pay for the same. We show that a combination of social learning and financial ability to pay for a transplant drastically reduces the number of sick individuals progressing to renal failure and also prevent death due to the disease.
We present and analyze a vector-borne disease model with two gamma distributed delays representing the incubation periods of the disease in the vector and hosts. The model assumes a logistic growth for both the host and vector populations and includes a density-dependent biting rate. We start by highlighting the role of density dependency in the vector’s population by fitting the vector’ model to data on tsetse flies. Our fitting routine uses Bayesian based Markov Chain Monte Carlo methods and statistics comparison tests. Then we proceed with our mathematical analysis by investigating the impact of both (distributed) delays on the model’s equilibria and their stability properties. This leads to the derivation of an explicit conditions for the occurrence of backward bifurcation, showing in the process the role of nonlinearity in the bitting rate in shaping the model’s bifurcation behavior. Finally, a sensitivity analysis is performed by means of the forward sensitivity index of the basic reproductive number to compare the effect of the mean and shape parameters of the delay on the initial disease transmission.
The majority of disease transmission models in the literature contain systems of deterministic differential equations. However, it is known that some of the components in these models produce random behavior in real life. In this study, we use normally distributed random effects to investigate the random behavior of a model of Poliomyelitis transmission. We use random differential transformation method to obtain the approximate numerical characteristics of the model and compare these results with simulation results. It is seen that random modelling of transmission provides useful information for both the disease dynamics and the variations of the results for disease behavior.
Usually, when modelling population dynamics, e.g. predator-prey systems, the growth rate of a population is assumed to be proportional to the consumption. Its per-capita death rate is often assumed to be constant. Those assumption, however, are plausible from biological point of view in cases when there is sufficient food in the environment. If this is not the case, however, a generalization might be considered, taking into account that a minimal amount of energy intake is required in order for the population to reproduce and not to starve to death. A. Terry studied a predator-prey model with generic birth and death rates for the predator. In the present work, we consider two main questions, concerning the applicability of this idea. First, we show that introducing such rates might lead to qualitatively richer dynamics of the mathematical model. Also, we compare the classical and the generalized models in terms of their ability to fit experimental data. The latter is accomplished by studying examples of bacterial growth under inhibitory conditions.
The homogenization of some reaction-diffusion problems in a highly heterogeneous composite medium formed by two connected constituents separated by an imperfect interface is analyzed. The main feature of our setting is represented by the fact that, across this imperfect interface, both the solution and its flux are assumed to exhibit jumps. Several models arise at the limit. In particular, a modified bidomain model is obtained and compared to some existing models in the literature (see [1]-[4]). Our results can serve as a tool for biochemists interested in studying the complex mechanisms involved in the calcium dynamics in living cells.
This study aims to create a diagnostic system for early Keratoconus ( KTC ) identification. Keratoconus is a progressive ocular pathology that may lead to gradual corneal deformation, and might cause decreased quality of vision. KTC is easily identified at advanced stages by means of clinical parameters, measured by non-invasive devices. Meanwhile identification of early KTC is still a big challenge for practitioners. This work proposes a mathematical model of normal ( N ) and KTC eyes. Furthermore, a hybrid machine learning algorithm was implemented as a clinical support tool for ophthalmologists to reach a correct diagnosis. A gaussian-sphere mathematical model was considered to model both N and KTC corneas. 145 N and 312 KTC anterior and posterior corneal maps (layers), were collected at the Antwerp University Hospital (UZA, Belgium). The maps were fitted to the gaussian-spherical model in order to extract potential meaningful parameters to help identify early KTC . Moreover, extracted data were statistically analyzed and used to train a hybrid machine learning algorithm, which applies a probabilistic strategy to support vector machine ( SV M ) and multilayer perceptron ( MLP ) algorithms. Cross-validation techniques were used to validate and evaluate the accuracy of the diagnostic system. The mean squared error ( MSE ) of the gaussian-spherical model amounted to MSE ≤ o (10 - 3 ) mm , MSE 2 [0 : 002 ; 0 : 004] mm in the center, and MSE ~ 0 : 02 mm on corneal borders. The highest accuracy in classifying early KTC versus N eyes from the extracted parameters was 94% during the validation and 99% in the training step. The implemented diagnostic system results in an accurate tool for early KTC detection. Further work is needed to improve this system to model the progression of KTC .
This work analyze the dynamics of the electroencephalographic (EEG) signals of normal and epileptic patients. The Detrended Fluctuation Anal- ysis (DFA) and the Hurst exponent methods are used for estimation of the presence of long term correlations in physiological time series observed in healthy and unhealthy brains. The presence of long-range correlation in a biological time series is an usually response observed in healthy organisms. The complexity of the recorded signal can guarantee some adaptability of the organism to the situations of disturbances. By other hand, the absence of this correlation indicates the loss of this complexity. Non-parametric Wilcoxon test was used for both, namely healthy and unhealthy groups, in order to compare the mean values of the Hurst exponents. Comparison of the means values of the Hurst exponents of normal and epileptic patients, using the Wilcoxon test results, point to some signicant dierence between two groups [1].
A simplified diffusive predator-prey model of the Lotka-Volterra type is considered for annular habitat which is used for description of predator and prey coexistence at habitats surrounding lakes, mountains at particular heights,etc. The model is formulated as a system of two partial differential equations in which unknown populations of the predator and prey are described by functions depending on time and polar angle. A mixed problem is formulated so that the boundary conditions are $2\pi$-periodic and in the initial conditions is assumed that populations of the predator and prey are completely separated on the annular habitat. The problem is solved by the method of lines by means of which the original system of partial differential equation is converted to the system of several hundred nonlinear ordinary differential equations. It's shown that predator and prey start slowly propagating through the annular habitat and their interaction commences after a time interval in the course of which the population of the predator is decreasing and the population of the prey is increasing. An intensive interaction of the predator and prey occurs after their meeting. The dynamics of the transient populations of predator and prey and the tendency of their steady state is analyzed.
Using the center manifold theory for maps, we derive a theorem for the existence of backward bifurcation at bifurcation points of discrete dynamical systems. This theorem is an analogue of a theorem in [3] for continuous dynamical systems. We discuss applications to discrete dynamical systems in general [2], but more specifically those arising as dicretisations of continuous dynamical systems via the nonstandard finite difference method [1].
The Islets of Langerhans are mainly composed of insulin-secreting pancreatic β -cells, glucagon-secreting α -cells and somatostatin-secreting δ -cells[1]. At the cellular level, secretion of these hormones takes place through a common mech- anism involving glucose metabolism, electrical activity and Ca 2+ -handling[2]. In addition, pancreatic hormone secretion is regulated by intra-islet interactions including paracrine and autocrine signals, as well as electrical coupling mediated by gap junctions between β -cells[3]. Electrical coupling between β -cells has been previously studied both theo- retically and experimentally. In these studies, it was shown that β -cell coupling is essential for the synchronized release of insulin[1]. In addition, it was demon- strated that the lack of functional gap junctions leads to impaired pulsatile insulin secretion due to uncoordinated Ca 2+ oscillations[4]. In this work we used a computational model to assess the effect of morpho- logical and functional heterogeneities in the islet β -cells (including differences in cell sizes, β -cell interconnectivity and electrophysiological and Ca 2+ buffering properties) on the Ca 2+ signal produced in the cytosol, ultimately related to the secretory response of the islet β -cells.
The FitzHugh-Nagumo equation has various applications in the fields of flame propagation, logistic population growth, neurophysiology, autocatalytic chemical reaction and nuclear theory ...
Malaria is an infectious disease that is transmitted by female mosquitoes of the genus Anopheles [1]. Typical symptoms include fatigue, vomiting, headaches and fever with severe cases resulting in seizures, coma, yellow skin or death [1]. Prevention of mosquito bites and mosquito control measures can reduce the risk of disease. Preventative measures mainly consist of the use of insect repellents, mosquito nets, draining standing water and insecticides. A vector control intervention that resulted in a substantial reduction in the number of malaria notication cases is the use of indoor residual spraying (IRS). IRS is used for malaria control in the low altitude parts of the Limpopo Province in South Africa. Recent studies have pointed out various health risks to those with prolonged exposure to IRS. These include cancer, male infertility, miscarriage, developmental delay, nervous system and liver damage. We aim to assess the long-term eects of IRS on malaria spread in Limpopo along the matter of health risks posed by exposure to IRS. We propose an SEIR model for the transmission dynamics of malaria in the Limpopo province. The model consists of a system of ordinary dierential equations including the intervention IRS. We analyze this model to show its quantitative and qualitative behaviour. The model is further tted to data to estimate some key parameters related to malaria transmission and health risks.
