Showing papers in "Bellman Prize in Mathematical Biosciences in 2018"

A similarity-based method for prediction of drug side effects with heterogeneous information.

Abstract: Drugs can produce intended therapeutic effects to treat different diseases. However, they may also cause side effects at the same time. For an approved drug, it is best to detect all side effects it can produce. Otherwise, it may bring great risks for pharmaceuticals companies as well as be harmful to human body. It is urgent to design quick and reliable identification methods to detect the side effects for a given drug. In this study, a binary classification model was proposed to predict drug side effects. Different from most previous methods, our model termed the pair of drug and side effect as a sample and convert the original problem to a binary classification problem. Based on the similarity idea, each pair was represented by five features, each of which was derived from a type of drug property. The strong machine learning algorithm, random forest, was adopted as the prediction engine. The ten-fold cross-validation on five datasets with different negative samples indicated that the proposed model yielded a good performance of Matthews correlation coefficient around 0.550 and AUC around 0.8492. In addition, we also analyzed the contribution of each drug property for construction of the model. The results indicated that drug similarity in fingerprint was most related to the prediction of drug side effects and all drug properties gave less or more contributions.

The Bayesian adaptive lasso regression

Abstract: Classical adaptive lasso regression is known to possess the oracle properties; namely, it performs as well as if the correct submodel were known in advance. However, it requires consistent initial estimates of the regression coefficients, which are generally not available in high dimensional settings. In addition, none of the algorithms used to obtain the adaptive lasso estimators provide a valid measure of standard error. To overcome these drawbacks, some Bayesian approaches have been proposed to obtain the adaptive lasso and related estimators. In this paper, we consider a fully Bayesian treatment for the adaptive lasso that leads to a new Gibbs sampler with tractable full conditional posteriors. Through simulations and real data analyses, we compare the performance of the new Gibbs sampler with some of the existing Bayesian and non-Bayesian methods. Results show that the new approach performs well in comparison to the existing Bayesian and non-Bayesian approaches.

Optimal control strategies for dengue transmission in pakistan.

Abstract: This paper presents a deterministic model for dengue virus transmission The model is parameterized using data from the 2017 dengue outbreak in Pakistan We estimated the basic reproduction number (R0) without any interventions for the 2017 dengue outbreak in Peshawar district of Pakistan as R0≈264, the distribution of the reproduction number lies in the range R0∈[121,524] (with a mean R0≈264) Optimal control theory is then applied to investigate the optimal strategy for curtailing the spread of the disease using two time-dependent control variables determined from sensitivity analysis These control variables are insecticide use and vaccination The results show that the two controls avert the same number of infections in the district regardless of the weights on the costs this is due to the reciprocal relationship between the cost of insecticide use and vaccination A strong reciprocal relationship exists between the use of insecticide and vaccination; as the cost of insecticide increases the use of vaccination increases The use of insecticide on the other hand slightly increases when vaccination level decreases due to increase in cost

Use of the AHP methodology in system dynamics: Modelling and simulation for health technology assessments to determine the correct prosthesis choice for hernia diseases.

Abstract: Health technology assessments (HTAs) are often difficult to conduct because of the decisive procedures of the HTA algorithm, which are often complex and not easy to apply. Thus, their use is not always convenient or possible for the assessment of technical requests requiring a multidisciplinary approach. This paper aims to address this issue through a multi-criteria analysis focusing on the analytic hierarchy process (AHP). This methodology allows the decision maker to analyse and evaluate different alternatives and monitor their impact on different actors during the decision-making process. However, the multi-criteria analysis is implemented through a simulation model to overcome the limitations of the AHP methodology. Simulations help decision-makers to make an appropriate decision and avoid unnecessary and costly attempts. Finally, a decision problem regarding the evaluation of two health technologies, namely, the evaluation of two biological prostheses for incisional infected hernias, will be analysed to assess the effectiveness of the model.

Nonlocal strain gradient beam model for postbuckling and associated vibrational response of lipid supramolecular protein micro/nano-tubules.

Abstract: As a supramolecular construction, lipid protein micro/nano-tubules can be utilized in a variety of sustained biological delivery system. The high slenderness ratio of lipid tubules makes their hierarchical assembly into a desired architecture difficult. Therefore, an accurate prediction of mechanical behavior of lipid tubular is essential. The objective of this study is to capture size dependency in the postbuckling and vibrational response of the postbuckled lipid micro/nano-tubules more comprehensively. To this purpose, the nonlocal strain gradient elasticity theory is incorporated to the third-order shear deformation beam theory to develop an unconventional beam model. Hamilton's principle is put to use to establish the size-dependent governing differential equations of motion. After that, an improved perturbation technique in conjunction with Galerkin method is employed to obtain the nonlocal strain gradient load-frequency response and postbuckling stability curves of lipid micro/nano-tubules. It is revealed that by taking the nonlocal size effect into consideration, the influence of the type (geometrical parameters) of an axially compressed lipid micro/nano-tubule on its natural frequency in order decreases and increases within the prebuckling and postbuckling regimes. While the strain gradient size dependency plays an opposite role which causes that the influence of the type of lipid micro/nano-tubule on its natural frequency corresponding to the prebuckling and postbuckling domains increases and decreases, respectively.

Structural and practical identifiability analysis of outbreak models.

Abstract: Estimating the reproduction number of an emerging infectious disease from an epidemiological data is becoming more essential in evaluating the current status of an outbreak. However, these studies are lacking the fundamental prerequisite in parameter estimation problem, namely the structural identifiability of the epidemic model, which determines the possibility of uniquely determining the model parameters from the epidemic data. In this paper, we perform both structural and practical identifiability analysis to classical epidemic models such as SIR (Susceptible-Infected-Recovered), SEIR (Susceptible-Exposed-Infected-Recovered) and an epidemic model with the treatment class (SITR). We performed structural identifiability analysis on these epidemic models using a differential algebra approach to investigate the well-posedness of the parameter estimation problem. Parameters of these models are estimated from different data types, namely prevalence, cumulative incidences and treated individuals. Furthermore, we carried out practical identifiability analysis on these models using Monte Carlo simulations and Fisher's Information Matrix. Our study shows that the SIR model is both structurally and practically identifiable from the prevalence data. It is also structurally identifiable to cumulative incidence observations, but due to high correlations of the parameters, it is practically unidentifiable from the cumulative incidence data. Furthermore, we found that none of these simple epidemic models are practically identifiable from the cumulative incidence data which is the standard type of epidemiological data provided by CDC or WHO. Our analysis with simple SIR model suggest that the health agencies, if possible, should report prevalence rather than incidence data.

Practical identifiability and uncertainty quantification of a pulsatile cardiovascular model.

Abstract: Mathematical models are essential tools to study how the cardiovascular system maintains homeostasis. The utility of such models is limited by the accuracy of their predictions, which can be determined by uncertainty quantification (UQ). A challenge associated with the use of UQ is that many published methods assume that the underlying model is identifiable (e.g. that a one-to-one mapping exists from the parameter space to the model output). In this study we present a novel workflow to calibrate a lumped-parameter model to left ventricular pressure and volume time series data. Key steps include using (1) literature and available data to determine nominal parameter values; (2) sensitivity analysis and subset selection to determine a set of identifiable parameters; (3) optimization to find a point estimate for identifiable parameters; and (4) frequentist and Bayesian UQ calculations to assess the predictive capability of the model. Our results show that it is possible to determine 5 identifiable model parameters that can be estimated to our experimental data from three rats, and that computed UQ intervals capture the measurement and model error.

A simple SI-type model for HIV/AIDS with media and self-imposed psychological fear.

Abstract: Infectious diseases can have a large impact on society, as they cause morbidity, mortality, unemployment, inequality and other adverse effects. Mathematical models are invaluable tools in understanding and describing disease dynamics with preventive measures for controlling the disease. The roles of media coverage and behavioral changes due to externally imposed factors on the disease dynamics are well studied. However, the effect of self-imposed psychological fear on the disease transmission has not been considered in extant research, and this gap is addressed in the present investigation. We propose a simple SI-type model for HIV/AIDS to assess the effects of media and self-imposed psychological fear on the disease dynamics. Local and global dynamics of the system are studied. Global sensitivity analysis is performed to identify the most influential parameters that have significant impact on the basic reproduction number. After calibrating our model using HIV case data-sets for Uganda and Tanzania, we calculate the basic reproduction numbers in the study period using the estimated parameters. Furthermore, a comparison of the effects of awareness and self-imposed psychological fear effects reveals that awareness is more effective in eliminating the burden of HIV infection.

Mathematical modelling of pressure-driven micropolar biological flow due to metachronal wave propulsion of beating cilia.

Abstract: In this paper, we present an analytical study of pressure-driven flow of micropolar non-Newtonian physiological fluids through a channel comprising two parallel oscillating walls. The cilia are arranged at equal intervals and protrude normally from both walls of the infinitely long channel. A metachronal wave is generated due to natural beating of cilia and the direction of wave propagation is parallel to the direction of fluid flow. Appropriate expressions are presented for deformation via longitudinal and transverse velocity components induced by the ciliary beating phenomenon with cilia assumed to follow elliptic trajectories. The conservation equations for mass, longitudinal and transverse (linear) momentum and angular momentum are reduced in accordance with the long wavelength and creeping Stokesian flow approximations and then normalized with appropriate transformations. The resulting non-linear moving boundary value problem is solved analytically for constant micro-inertia density, subject to physically realistic boundary conditions. Closed-form expressions are derived for axial velocity, angular velocity, volumetric flow rate and pressure rise. The transport phenomena are shown to be dictated by several non-Newtonian parameters, including micropolar material parameter and Eringen coupling parameter, and also several geometric parameters, viz eccentricity parameter, wave number and cilia length. The influence of these parameters on streamline profiles (with a view to addressing trapping features via bolus formation and evolution), pressure gradient and other characteristics are evaluated graphically. Both axial and angular velocities are observed to be substantially modified with both micropolar rheological parameters and furthermore are significantly altered with increasing volumetric flow rate. Free pumping is also examined. An inverse relationship between pressure rise and flow rate is computed which is similar to that observed in Newtonian fluids. The study is relevant to hemodynamics in narrow capillaries and also bio-inspired micro-fluidic devices.

Who is the infector? Epidemic models with symptomatic and asymptomatic cases.

Abstract: What role do asymptomatically infected individuals play in the transmission dynamics? There are many diseases, such as norovirus and influenza, where some infected hosts show symptoms of the disease while others are asymptomatically infected, i.e. do not show any symptoms. The current paper considers a class of epidemic models following an SEIR (Susceptible → Exposed → Infectious → Recovered) structure that allows for both symptomatic and asymptomatic cases. The following question is addressed: what fraction ρ of those individuals getting infected are infected by symptomatic (asymptomatic) cases? This is a more complicated question than the related question for the beginning of the epidemic: what fraction of the expected number of secondary cases of a typical newly infected individual, i.e. what fraction of the basic reproduction number R0, is caused by symptomatic individuals? The latter fraction only depends on the type-specific reproduction numbers, while the former fraction ρ also depends on timing and hence on the probabilistic distributions of latent and infectious periods of the two types (not only their means). Bounds on ρ are derived for the situation where these distributions (and even their means) are unknown. Special attention is given to the class of Markov models and the class of continuous-time Reed–Frost models as two classes of distribution functions for latent and infectious periods. We show how these two classes of models can exhibit very different behaviour.

Final size of a multi-group SIR epidemic model: Irreducible and non-irreducible modes of transmission.

Abstract: A model of an epidemic outbreak incorporating multiple subgroups of susceptible and infected individuals is investigated. The asymptotic behavior of the model is analyzed and it is proved that the infected classes all converge to 0. A computational algorithm is developed for the cumulative final size of infected individuals over the course of the epidemic. The results are applied to the SARS epidemic in Singapore in 2003, where it is shown that the two-peak evolution of the infected population can be attributed to a two-group formulation of transmission.

Carrying capacity influence on the incomes of seiners exploiting marine species in the Atlantic coast of Morocco

Abstract: In this article, we seek to highlight that the increase of the carrying capacity of marine species does not always lead to an increase on the catch levels and on the incomes. To effectively support the theoretical outcomes, we take a bioeconomic model of several seiners exploiting Sardina pilchardus, Engraulis encrasicolus and Xiphias gladius marine species in the Atlantic coast of Morocco based on the parameters given by 'Institut National de Recherche Halieutique' INRH.

Stochastic hybrid models of gene regulatory networks - A PDE approach.

Abstract: A widely used approach to describe the dynamics of gene regulatory networks is based on the chemical master equation, which considers probability distributions over all possible combinations of molecular counts. The analysis of such models is extremely challenging due to their large discrete state space. We therefore propose a hybrid approximation approach based on a system of partial differential equations, where we assume a continuous-deterministic evolution for the protein counts. We discuss efficient analysis methods for both modeling approaches and compare their performance. We show that the hybrid approach yields accurate results for sufficiently large molecule counts, while reducing the computational effort from one ordinary differential equation for each state to one partial differential equation for each mode of the system. Furthermore, we give an analytical steady-state solution of the hybrid model for the case of a self-regulatory gene.

Coexistence and bistability of a competition model in open advective environments.

Abstract: The community composition in open advective environments, where individuals are exposed to unidirectional flow, is formed by the complex interplays of hydrological and biological factors. We investigate the coexistence mechanism of species by a reaction-diffusion-advection competition model proposed by Lutscher et al. in [19]. It turns out that the locations of two critical curves, which separate the stable region of the semi-trivial solutions from the unstable one, determines whether coexistence or bistability happens. Furthermore, the analytical and numerical results suggest a tradeoff driven coexistence mechanism. More precisely, there is a tradeoff between the dispersal strategy and growth competence which allows the transition of competition outcomes, including competition exclusion, coexistence and bistability. This shifting may have an effect on the community composition in aquatic habitat.

Patterns induced by super cross-diffusion in a predator-prey system with Michaelis-Menten type harvesting

Abstract: Turing instability and pattern formation in a super cross-diffusion predator-prey system with Michaelis-Menten type predator harvesting are investigated. Stability of equilibrium points is first explored with or without super cross-diffusion. It is found that cross-diffusion could induce instability of equilibria. To further derive the conditions of Turing instability, the linear stability analysis is carried out. From theoretical analysis, note that cross-diffusion is the key mechanism for the formation of spatial patterns. By taking cross-diffusion rate as bifurcation parameter, we derive amplitude equations near the Turing bifurcation point for the excited modes by means of weakly nonlinear theory. Dynamical analysis of the amplitude equations interprets the structural transitions and stability of various forms of Turing patterns. Furthermore, the theoretical results are illustrated via numerical simulations.

Michaelis-Menten equation for degradation of insoluble substrate.

Abstract: Kinetic studies of homogeneous enzyme reactions where both the substrate and enzyme are soluble have been well described by the Michaelis-Menten (MM) equation for more than a century. However, many reactions are taking place at the interface of a solid substrate and enzyme in solution. Such heterogeneous reactions are abundant both in vivo and in industrial application of enzymes but it is not clear whether traditional enzyme kinetic theory developed for homogeneous catalysis can be applied. Since the molar concentration of surface accessible sites (attack-sites) often is unknown for a solid substrate it is difficult to assess whether the requirement of the MM equation is met. In this paper we study a simple kinetic model, where removal of attack sites expose new ones which preserve the total accessible substrate, and denote this approach the substrate conserving model. The kinetic equations are solved in closed form, both steady states and progress curves, for any admissible values of initial conditions and rate constants. The model is shown to merge with the MM equation and the reverse MM equation when these are valid. The relation between available molar concentration of attack sites and mass load of substrate is analyzed and this introduces an extra parameter to the equations. Various experimental setups to practically and reliably estimate all parameters are discussed.

Canard phenomenon in a slow-fast modified Leslie-Gower model.

Abstract: Geometrical Singular Perturbation Theory has been successful to investigate a broad range of biological problems with different time scales. The aim of this paper is to apply this theory to a predator-prey model of modified Leslie-Gower type for which we consider that prey reproduces mush faster than predator. This naturally leads to introduce a small parameter ϵ which gives rise to a slow-fast system. This system has a special folded singularity which has not been analyzed in the classical work [15]. We use the blow-up technique to visualize the behavior near this fold point P. Outside of this region the dynamics are given by classical regular and singular perturbation theory. This allows to quantify geometrically the attractive limit-cycle with an error of O(ϵ) and shows that it exhibits the canard phenomenon while crossing P.

Influence of multiple delays in brain tumor and immune system interaction with T11 target structure as a potent stimulator.

Abstract: We present a mathematical model which describes the growth of malignant gliomas in presence of immune responses by considering the role of immunotherapeutic agent T11 target structure (T11TS). The model consider five populations, namely, glioma cells, macrophages, cytotoxic T-lymphocytes, TGF - β and IFN - γ. The model system has highly nonlinear terms with four discrete time lags, but remains tractable. The goal of this work is to better understand the effect of multiple delays on the interaction between gliomas and immune components in conjunction with an administration of T11 target structure. Analytically, we investigate the conditions for the asymptotic stability of equilibrium points, the existence of Hopf bifurcations and the maximum value of the delay to preserve the stability of limit cycle. For the set of parameter values estimated from experimental data, time delays have hardly any influence on the system behavior. Numerical simulations are carried out to investigate the dynamics of the model with different values for delays with and without administration of T11 target structure.

Modelling, singular perturbation and bifurcation analyses of bitrophic food chains.

Abstract: Two predator-prey model formulations are studied: the classical Rosenzweig–MacArthur (RM) model and the Mass Balance (MB) chemostat model. When the growth and loss rate of the predator is much smaller than that of the prey these models are slow–fast systems leading mathematically to a singular perturbation problem. In contradiction to the RM-model, the resource for the prey is modelled explicitly in the MB-model but this comes with additional parameters. These parameter values are chosen such that the two models become easy to compare. In both models the transcritical bifurcation, a threshold above which invasion of predator into prey-only system occurs, and the Hopf bifurcation where the interior equilibrium becomes unstable leading to a stable limit cycle occur. The slow-fast limit cycles are called relaxation oscillations which for increasing differences in time scales leads to the well known degenerated trajectories being concatenations of slow parts of the trajectory and fast parts of the trajectory. In the fast–slow version of the RM-model a canard explosion of the stable limit cycles occurs in the oscillatory region of the parameter space. To our knowledge this type of dynamics has not been observed for the RM-model and not even for more complex ecosystem models. When a bifurcation parameter crosses the Hopf bifurcation point the amplitude of the emerging stable limit cycles increases. However, depending of the perturbation parameter the shape of this limit cycle changes abruptly from one consisting of two concatenated slow and fast episodes with small amplitude of the limit cycle, to a shape with large amplitude of which the shape is similar to the relaxation oscillation, the well known degenerated phase trajectories consisting of four episodes (concatenation of two slow and two fast). The canard explosion point is accurately predicted by using an extended asymptotic expansion technique in the perturbation and bifurcation parameter simultaneously where the small amplitude stable limit cycles exist. The predicted dynamics of the MB-model is in a large part of the parameter space similar to that of the RM-model. However, the fast–slow version of MB-model does not predict a canard explosion phenomenon.

Gaussian process approximations for fast inference from infectious disease data.

Abstract: We present a flexible framework for deriving and quantifying the accuracy of Gaussian process approximations to non-linear stochastic individual-based models of epidemics. We develop this for the SIR and SEIR models, and we show how it can be used to perform quick maximum likelihood inference for the underlying parameters given population estimates of the number of infecteds or cases at given time points. We also show how the unobserved processes can be inferred at the same time as the underlying parameters.

Modelling mass diffusion for a multi-layer sphere immersed in a semi-infinite medium: application to drug delivery

Abstract: We present a general mechanistic model of mass diffusion for a composite sphere placed in a large ambient medium. The multi-layer problem is described by a system of diffusion equations coupled via interlayer boundary conditions such as those imposing a finite mass resistance at the external surface of the sphere. While the work is applicable to the generic problem of heat or mass transfer in a multi-layer sphere, the analysis and results are presented in the context of drug kinetics for desorbing and absorbing spherical microcapsules. We derive an analytical solution for the concentration in the sphere and in the surrounding medium that avoids any artificial truncation at a finite distance. The closed-form solution in each concentric layer is expressed in terms of a suitably-defined inverse Laplace transform that can be evaluated numerically. Concentration profiles and drug mass curves in the spherical layers and in the external environment are presented and the dependency of the solution on the mass transfer coefficient at the surface of the sphere analyzed.

Evolutionary stability of ideal free dispersal under spatial heterogeneity and time periodicity.

Abstract: Roughly speaking, a population is said to have an ideal free distribution on a spatial region if all of its members can and do locate themselves in a way that optimizes their fitness, allowing for the effects of crowding. Dispersal strategies that can lead to ideal free distributions of populations using them have been shown to exist and to be evolutionarily stable in a number of modeling contexts in the case of habitats that vary in space but not in time. Those modeling contexts include reaction-diffusion-advection models and the analogous models using discrete diffusion or nonlocal dispersal described by integro-differential equations. Furthermore, in the case of reaction-diffusion-advection models and their nonlocal analogues, there are strategies that allow populations to achieve an ideal free distribution by using only local information about environmental quality and/or gradients. We show that in the context of reaction-diffusion-advection models for time-periodic environments with spatially varying resource levels, where the total level of resources in an environment remains fixed but its location varies seasonally, there are strategies that allow populations to achieve an ideal free distribution. We also show that those strategies are evolutionarily stable. However, achieving an ideal free distribution in a time-periodic environment requires the use of nonlocal information about the environment such as might be derived from experience and memory, social learning, or genetic programming.

Modeling host-seeking behavior of African malaria vector mosquitoes in the presence of long-lasting insecticidal nets.

Abstract: The efficiency of spatial repellents and long-lasting insecticide-treated nets (LLINs) is a key research topic in malaria control. Insecticidal nets reduce the mosquito-human contact rate and simultaneously decrease mosquito populations. However, LLINs demonstrate dissimilar efficiency against different species of malaria mosquitoes. Various factors have been proposed as an explanation, including differences in insecticide-induced mortality, flight characteristics, or persistence of attack. Here we present a discrete agent-based approach that enables the efficiency of LLINs, baited traps and Insecticide Residual Sprays (IRS) to be examined. The model is calibrated with hut-level experimental data to compare the efficiency of protection against two mosquito species: Anopheles gambiae and Anopheles arabiensis. We show that while such data does not allow an unambiguous identification of the details of how LLINs alter the vector behavior, the model calibrations quantify the overall impact of LLINs for the two different mosquito species. The simulations are generalized to community-scale scenarios that systematically demonstrate the lower efficiency of the LLINs in control of An. arabiensis compared to An. gambiae.

Data mining of the cancer-related lncRNAs GO terms and KEGG pathways by using mRMR method.

Abstract: LncRNAs plays an important role in the regulation of gene expression. Identification of cancer-related lncRNAs GO terms and KEGG pathways is great helpful for revealing cancer-related functional biological processes. Therefore, in this study, we proposed a computational method to identify novel cancer-related lncRNAs GO terms and KEGG pathways. By using existing lncRNA database and Max-relevance Min-redundancy (mRMR) method, GO terms and KEGG pathways were evaluated based on their importance on distinguishing cancer-related and non-cancer-related lncRNAs. Finally, GO terms and KEGG pathways with high importance were presented and analyzed. Our literature reviewing showed that the top 10 ranked GO terms and pathways were really related to interpretable tumorigenesis according to recent publications.

Harvesting induced stability and instability in a tri-trophic food chain

Abstract: Non-equilibrium dynamics in the form of oscillations or chaos is often found to be a natural phenomenon in complex ecological systems. In this paper, we first analyze a tri-trophic food chain, which is an extension of the Rosenzweig-MacArthur di-trophic food chain. We then explore the impact of harvesting individual trophic levels to answer the following questions : a) when a non-equilibrium dynamics persists, b) whether it can locally be stabilized to a steady state, c) when the system switches from a stable steady state to a non-equilibrium dynamics and d) whether the Maximum Sustainable Yield (MSY) always exists when the top predator is harvested. It is shown that searching for a general theory to unify the harvesting induced stability must take into account the number of trophic levels and the degree of species enrichment, the outcomes that cannot be obtained from the earlier reports on prey-predator models. We also identify the situation where harvesting induces instability switching: the non-equilibrium state enters into a stable steady-state and then, upon more intensive harvesting, the steady-state again loses its stability. One of the new and important results is also that the MSY may not exist for harvesting the top predator. In general, our results contribute to biological conservation theory, fishery and ecosystem biodiversity management.

Coupled, multi-strain epidemic models of mutating pathogens.

Abstract: We introduce and analyze coupled, multi-strain epidemic models designed to simulate the emergence and dissemination of mutant (e.g. drug-resistant) pathogen strains. In particular, we investigate the mathematical and biological properties of a general class of multi-strain epidemic models in which the infectious compartments of each strain are coupled together in a general manner. We derive explicit expressions for the basic reproduction number of each strain and highlight their importance in regulating the system dynamics (e.g. the potential for an epidemic outbreak) and the existence of nonnegative endemic solutions. Importantly, we find that the basic reproduction number of each strain is independent of the mutation rates between the strains — even under quite general assumptions for the form of the infectious compartment coupling. Moreover, we verify that the coupling term promotes strain coexistence (as an extension of the competitive exclusion principle) and demonstrate that the strain with the greatest reproductive capacity is not necessarily the most prevalent. Finally, we briefly discuss the implications of our results for public health policy and planning.

On the intrinsic dynamics of bacteria in waterborne infections.

Abstract: The intrinsic dynamics of bacteria often play an important role in the transmission and spread of waterborne infectious diseases. In this paper, we construct mathematical models for waterborne infections and analyze two types of nontrivial bacterial dynamics: logistic growth, and growth with Allee effects. For the model with logistic growth, we find that regular threshold dynamics take place, and the basic reproduction number can be used to characterize disease extinction and persistence. In contrast, the model with Allee effects exhibits much more complex dynamics, including the existence of multiple endemic equilibria and the presence of backward bifurcation and forward hysteresis.

The operating diagram of a model of two competitors in a chemostat with an external inhibitor

Abstract: Understanding and exploiting the inhibition phenomenon, which promotes the stable coexistence of species, is a major challenge in the mathematical theory of the chemostat. Here, we study a model of two microbial species in a chemostat competing for a single resource in the presence of an external inhibitor. The model is a four-dimensional system of ordinary differential equations. Using general monotonic growth rate functions of the species and absorption rate of the inhibitor, we give a complete analysis for the existence and local stability of all steady states. We focus on the behavior of the system with respect of the three operating parameters represented by the dilution rate and the input concentrations of the substrate and the inhibitor. The operating diagram has the operating parameters as its coordinates and the various regions defined in it correspond to qualitatively different asymptotic behavior: washout, competitive exclusion of one species, coexistence of the species around a stable steady state and coexistence around a stable cycle. This bifurcation diagram which determines the effect of the operating parameters, is very useful to understand the model from both the mathematical and biological points of view, and is often constructed in the mathematical and biological literature.

A mathematical model for malaria transmission with asymptomatic carriers and two age groups in the human population.

Abstract: In this paper a conceptual mathematical model of malaria transmission proposed in a previous paper has been analyzed in a deeper detail. Among its key epidemiological features of this model, two-age-classes (child and adult) and asymptomatic carriers have been included. The extra mortality of mosquitoes due to the use of long-lasting treated mosquito nets (LLINs) and Indoor Residual Spraying (IRS) has been included too. By taking advantage of the natural double time scale of the parasite and the human populations, it has been possible to provide interesting threshold results. In particular it has been shown that key parameters can be identified such that below a threshold level, built on these parameters, the epidemic tends to extinction, while above another threshold level it tends to a nontrivial endemic state, for which an interval estimate has been provided. Numerical simulations confirm the analytical results.

Analysis of the peristaltic-ciliary flow of Johnson-Segalman fluid induced by peristalsis-cilia of the human fallopian tube.

Abstract: The present prospective theoretical analysis concerns with the peristalsis-cilia induced transport of a developing embryo from ampulla to intramural, in the human fallopian tube. A model of peristalsis-cilia induced flow of the Johnson–Segalman fluid within fallopian tubal fluid in a finite two dimensional narrow tube is developed. We solved highly non-linear PDE emerging from the modeling of proposed model using perturbation method. The series expressions for flow variables like axial and radial velocities, pressure gradient, stream function, volume flow rate and time mean volume flow rate are derived. The numerical integration is performed for appropriate residue time over tube length and pressure difference over wavelength. The analysis delineated that, involved parameters and constants have vice versa effects on axial velocity and appropriate residue time over tube length. Striking features of the pumping characteristics and the trapping phenomenon are discussed in detail. Furthermore, comparison of the peristaltic flow with the peristaltic-ciliary flow and Johnson–Segalman fluid with the linearly viscous fluid is made. It is revealed that appropriate residue time of the Johnson–Segalman fluid in the narrow tube is 3–4 days, which agreed with the time taken by the developing embryo from ampulla to intramural, in the human fallopian tube.