Singular perturbation analysis of cAMP signalling in Dictyostelium discoideum aggregates

In this paper, we use singular perturbation methods to study the structure of travelling waves for some reaction-diffusion models obtained from the Martiel-Goldbeter and Goldbeter-Segel's models of cAMP signalling in Dictyostelium discoideum. As a consequence, we derive analytic formulae for quantities like wave speed, maximum concentration and other magnitudes in terms of the different biochemical constants that appear in the model.     

Travelling-wave behaviour in a multiphase model of a population of cells in an artificial scaffold

This paper analyses travelling-wave behaviour in a recently-formulated multiphase model for the growth of biological tissue that comprises motile cells and water inside a porous scaffold. The model arises in the context of tissue engineering, and its purpose is to study how cells migrate and proliferate inside porous biomaterials. In suitable limits, tissue growth in the model is shown to occur in the form of travelling waves that can propagate either forwards or backwards, depending on the values of the parameters. In the case where the drag force between the scaffold and the cells is non-zero, the growth of the aggregate can be analysed in terms of the propagation of a constant-speed wavefront in a semi-infinite domain. A numerical (shooting) method is described for calculating the wave speed, and detailed results for how the speed varies with respect to the parameters are given. In the case where the drag force is zero, the size of the aggregate is shown either to grow or to shrink exponentially with time. These results may be of importance in determining the experimental factors that control tissue invasiveness in scaffolds thereby allowing greater control over engineered tissue growth.     

A mathematical model for indirectly transmitted diseases

We consider a mathematical model for the indirect transmission via a contaminated environment of a microparasite between two spatially distributed host populations having non-coincident spatial domains. The parasite is benign in a first population and lethal in the second one. Global existence results are given for the resulting reaction-diffusion system coupled with an ordinary differential equation. Then, invasion and persistence of the parasite are studied. A simplified model for the transmission of a hantavirus from bank vole to human populations is then analysed.     

Persistence in reaction diffusion models with weak allee effect

We study the positive steady state distributions and dynamical behavior of reaction-diffusion equation with weak Allee effect type growth, in which the growth rate per capita is not monotonic as in logistic type, and the habitat is assumed to be a heterogeneous bounded region. The existence of multiple steady states is shown, and the global bifurcation diagrams are obtained. Results are applied to a reaction-diffusion model with type II functional response, and also a model with density-dependent diffusion of animal aggregation. J. S. is partially supported by United States NSF grants DMS-0314736 and EF-0436318, College of William and Mary summer grants, and a grant from Science Council of Heilongjiang Province, China.     

An age-and-cyclin-structured cell population model for healthy and tumoral tissues

We present a nonlinear model of the dynamics of a cell population divided into proliferative and quiescent compartments. The proliferative phase represents the complete cell cycle (G ₁−S−G ₂−M) of a population committed to divide at its end. The model is structured by the time spent by a cell in the proliferative phase, and by the amount of Cyclin D/(CDK4 or 6) complexes. Cells can transit from one compartment to the other, following transition rules which differ according to the tissue state: healthy or tumoral. The asymptotic behaviour of solutions of the nonlinear model is analysed in two cases, exhibiting tissue homeostasis or tumour exponential growth. The model is simulated and its analytic predictions are confirmed numerically.     

Spreading speeds as slowest wave speeds for cooperative systems

It is well known that in many scalar models for the spread of a fitter phenotype or species into the territory of a less fit one, the asymptotic spreading speed can be characterized as the lowest speed of a suitable family of traveling waves of the model. Despite a general belief that multi-species (vector) models have the same property, we are unaware of any proof to support this belief. The present work establishes this result for a class of multi-species model of a kind studied by Lui [Biological growth and spread modeled by systems of recursions. I: Mathematical theory, Math. Biosci. 93 (1989) 269] and generalized by the authors [Weinberger et al., Analysis of the linear conjecture for spread in cooperative models, J. Math. Biol. 45 (2002) 183; Lewis et al., Spreading speeds and the linear conjecture for two-species competition models, J. Math. Biol. 45 (2002) 219]. Lui showed the existence of a single spreading speed c(*) for all species. For the systems in the two aforementioned studies by the authors, which include related continuous-time models such as reaction-diffusion systems, as well as some standard competition models, it sometimes happens that different species spread at different rates, so that there are a slowest speed c(*) and a fastest speed c(f)(*). It is shown here that, for a large class of such multi-species systems, the slowest spreading speed c(*) is always characterized as the slowest speed of a class of traveling wave solutions.     

A distributed parameter identification problem in neuronal cable theory models

Dendritic and axonal processes of nerve cells, along with the soma itself, have membranes with spatially distributed densities of ionic channels of various kinds. These ionic channels play a major role in characterizing the types of excitable responses expected of the cell type. These densities are usually represented as constant parameters in neural models because of the difficulty in experimentally estimating them. However, through microelectrode measurements and selective ion staining techniques, it is known that ion channels are non-uniformly spatially distributed. This paper presents a non-optimization approach to recovering a single spatially non-uniform ion density through use of temporal data that can be gotten from recording microelectrode measurements at the ends of a neural fiber segment of interest. The numerical approach is first applied to a linear cable model and a transformed version of the linear model that has closed-form solutions. Then the numerical method is shown to be applicable to non-linear nerve models by showing it can recover the potassium conductance in the Morris-Lecar model for barnacle muscle, and recover the spine density in a continuous dendritic spine model by Baer and Rinzel.     

A model of intracellular transport of particles in an axon

In this paper we develop a model of intracellular transport of cell organelles and vesicles along the axon of a nerve cell. These particles are moving alternately by processive motion along a microtubule with the aid of motor proteins, and by diffusion. The model involves a degenerate system of diffusion equations. We prove a maximum principle and establish existence and behavior of a unique solution. Numerical results show how the transportation of mass depends on the relevant parameters of the model.     

A structured-population model of Proteus mirabilis swarm-colony development

In this paper we present continuous age- and space-structured models and numerical computations of Proteus mirabilis swarm-colony development. We base the mathematical representation of the cell-cycle dynamics of Proteus mirabilis on those developed by Esipov and Shapiro, which are the best understood aspects of the system, and we make minimum assumptions about less-understood mechanisms, such as precise forms of the spatial diffusion. The models in this paper have explicit age-structure and, when solved numerically, display both the temporal and spatial regularity seen in experiments, whereas the Esipov and Shapiro model, when solved accurately, shows only the temporal regularity. The composite hyperbolic-parabolic partial differential equations used to model Proteus mirabilis swarm-colony development are relevant to other biological systems where the spatial dynamics depend on local physiological structure. We use computational methods designed for such systems, with known convergence properties, to obtain the numerical results presented in this paper.     

Anomalous spreading speeds of cooperative recursion systems

This work presents an example of a cooperative system of truncated linear recursions in which the interaction between species causes one of the species to have an anomalous spreading speed. By this we mean that this species spreads at a speed which is strictly greater than its spreading speed in isolation from the other species and the speeds at which all the other species actually spread. An ecological implication of this example is discussed in Sect. 5. Our example shows that the formula for the fastest spreading speed given in Lemma 2.3 of our paper (Weinberger et al. in J Math Biol 45:183–218, 2002) is incorrect. However, we find an extra hypothesis under which the formula for the faster spreading speed given in (Weinberger et al. in J Math Biol 45:183–218, 2002) is valid. We also show that the hypotheses of all but one of the theorems of (Weinberger et al. in J Math Biol 45:183–218, 2002) whose proofs rely on Lemma 2.3 imply this extra hypothesis, so that all but one of the theorems of (Weinberger et al. in J Math Biol 45:183–218, 2002) and all the examples given there are valid as they stand.     

Stability analysis for HIV infection delay model with protease inhibitor

In this article, we considered a model of HIV-1 infection with a protease inhibitor therapy and three delays. The frequency of the bifurcating periodic solution as well as the threshold value is approximated numerically using realistic parameter. The estimated threshold value is realistic and the frequency of the oscillations is consistent with that of the observed viral blips.     

On the stability of separable solutions of a sexual age-structured population dynamics model

The Sharpe-Lotka-McKendrick-von Foerster equations for non-dispersing age-sex-structured populations with a harmonic mean type mating law are considered and their separable solutions are analysed. For certain forms of the demographic rates the underlying evolution equations are reduced to systems of ODEs, the long time behavior of their solutions is studied, and the stability of separable solutions is discussed. It is found that for the constant death rates and constant sex ratio of newborns with stationary birth rates this model admits one one-parameter class of separable solutions, two such classes (repeated or different) or no such ones. In the case of special forms of age-dependent birth rates, solutions of one of these two different classes corresponding to the greater root of the characteristic equation are locally stable, solutions of the other one corresponding to the smaller root are unstable, and the population dies out if the model does not admit separable solutions or if initial densities of newborns are small enough in the case of the existence of separable solutions. In the case of constant vital rates, the model has no separable solutions or admits only one class of such ones that are globally stable.     

Plasmid-bearing,plasmid-free organisms competing for two complementary nutrients in a chemostat

A model of competition for two complementary nutrients between plasmid-bearing and plasmid-free organisms in a chemostat is proposed. A rigorous mathematical analysis of the global asymptotic behavior of the model is presented. The work extends the model of competition for a single-limited nutrient studied by Stephanopoulos and Lapidus [Chem. Engng. Sci. 443 (1988) 49] and Hsu, Waltman and Wolkowicz [J. Math. Biol. 32 (1994) 731].     

A stochastic model for prostate-specific antigen levels

We introduce a continuous stochastic model for the prostate-specific antigen (PSA) levels following radiotherapy and derive solutions for the associated partial differential (Kolmogorov-Chapman) equation. The solutions describe the evolution of the time-dependent density for PSA levels which take into account an absorbing condition along the boundary and various initial conditions. We include implications for single-dose and multi-dose radiation treatment regimens and discuss parameter estimation and sensitivity issues.     

Type II functional response for continuous, physiologically structured models

The goal of this work is to formulate a general Holling-type functional, or behavioral, response for continuous physiologically structured populations, where both the predator and the prey have physiological densities and certain rules apply to their interactions. The physiological variable can be, for example, a development stage, weight, age, or a characteristic length. The model leads to a Fredholm integral equation for the functional response, and, when inserted into population balance laws, it produces a coupled system of partial differential-integral equations for the two species, with a nonlocal integral term that arises from rules of interaction in the functional response. The general model is, typically, analytically intractable, but specialization to a structured prey-unstructured predator model leads to some analytic results that reveal interesting and unexpected dynamics caused by the presence of size-dependent handling times in the functional response. In this case, steady-states are shown to exist over long times, similar to the stable age-structure solutions for the McKendick-von Foerster model with exponential growth rates determined by the Euler-Lotka equation. But, for type II responses, there are early transient oscillations in the number of births that bifurcate in a few generations into either the decaying or growing steady-state. The bifurcation parameter is the initial level of prey. This special case is applied to a problem of the biological control of a structured pest population (e.g., aphids) by a predator (e.g., lady beetles).     

Sustainability and optimal control of an exploited prey predator system through provision of alternative food to predator

In the present paper, we develop a simple two species prey-predator model in which the predator is partially coupled with alternative prey. The aim is to study the consequences of providing additional food to the predator as well as the effects of harvesting efforts applied to both the species. It is observed that the provision of alternative food to predator is not always beneficial to the system. A complete picture of the long run dynamics of the system is discussed based on the effort pair as control parameters. Optimal augmentations of prey and predator biomass at final time have been investigated by optimal control theory. Also the short and large time effects of the application of optimal control have been discussed. Finally, some numerical illustrations are given to verify our analytical results with the help of different sets of parameters.     

A complex mathematical model of the human menstrual cycle

Despite the fact that more than 100 million women worldwide use birth control pills and that half of the world's population is concerned, the menstrual cycle has so far received comparatively little attention in the field of mathematical modeling. The term menstrual cycle comprises the processes of the control system in the female body that, under healthy circumstances, lead to ovulation at regular intervals, thus making reproduction possible. If this is not the case or ovulation is not desired, the question arises how this control system can be influenced, for example, by hormonal treatments. In order to be able to cover a vast range of external manipulations, the mathematical model must comprise the main components where the processes belonging to the menstrual cycle occur, as well as their interrelations. A system of differential equations serves as the mathematical model, describing the dynamics of hormones, enzymes, receptors, and follicular phases. Since the processes take place in different parts of the body and influence each other with a certain delay, passing over to delay differential equations is deemed a reasonable step. The pulsatile release of the gonadotropin-releasing hormone (GnRH) is controlled by a complex neural network. We choose to model the pulse time points of this GnRH pulse generator by a stochastic process. Focus in this paper is on the model development. This rather elaborate mathematical model is the basis for a detailed analysis and could be helpful for possible drug design.     

Optimal isolation control strategies and cost-effectiveness analysis of a two-strain avian influenza model

The most important and effective measures against disease outbreaks in the absence of valid medicines or vaccine are quarantine and isolation strategies. In this paper optimal control theory is applied to a system of ordinary differential equation describing a two-strain avian influenza transmission via the Pontryagin's Maximum Principle. To this end, a pair of control variables representing the isolation strategies for individuals with avian and mutant strains were incorporated into the transmission model. The infection averted ratio (IAR) and the incremental cost-effectiveness ratio (ICER) were calculated to investigate the cost-effectiveness of all possible combinations of the control strategies. The simulation results show that the implementation of the combination strategy during the epidemic is the most cost-effective strategy for avian influenza transmission. This is followed by the control strategy involving isolation of individuals with the mutant strain. Also observed was the fact that low mutating and more virulent virus results in an increased control effort of isolating individuals with the avian strain; and high mutating with more virulent virus results in increased efforts in isolating individuals with the mutant strain.     

Mathematical analysis of a cholera model with public health interventions

Cholera, an acute gastro-intestinal infection and a waterborne disease continues to emerge in developing countries and remains an important global health challenge. We formulate a mathematical model that captures some essential dynamics of cholera transmission to study the impact of public health educational campaigns, vaccination and treatment as control strategies in curtailing the disease. The education-induced, vaccination-induced and treatment-induced reproductive numbers R(E), R(V), R(T) respectively and the combined reproductive number R(C) are compared with the basic reproduction number R(0) to assess the possible community benefits of these control measures. A Lyapunov functional approach is also used to analyse the stability of the equilibrium points. We perform sensitivity analysis on the key parameters that drive the disease dynamics in order to determine their relative importance to disease transmission and prevalence. Graphical representations are provided to qualitatively support the analytical results.