Mathematical Medicine and Biology - recent issues

Pest management of a prey-predator model with sexual favoritism

Pei, Y., Yang, Y., Li, C., Chen, L. — 2009-06-02

Although sex of prey is an important factor for the risk of predating, few articles consider the consequences of sexual favoritism and the corresponding effects on the impulsive predator–prey dynamics and its utility in biological control. This paper investigates the pest management strategy of a prey–predator system model with sexual favoritism. An impulsive differential equation which models the process of periodically releasing natural enemies and spraying pesticides at different fixed time for pest control is proposed and investigated. It is proved that the pest-eradication periodic solution is globally asymptotically stable under the assumption that the release amount of the predator is greater than some critical value. Permanent conditions are established under the assumption that the release amount of the predator is less than another critical value. In particular, two single control strategies are proposed. Furthermore, we compare three pest control strategies and find that if we choose narrow-spectrum pesticides that targeted to a specific pest's life cycle to kill the pest, then the combined strategy is preferable. Finally, the corresponding system with no sexual favoritism is investigated. The results indicate that we can release fewer amount of the predators to eliminate the preys with sexual favoritism than without and any strong sexual favoritism will drive the pest towards extinction. In view of the biological meaning, the sexual favoritism plays a more active role in suppressing insect pests.

A bistable genetic switch which does not require high co-operativity at the promoter: a two-timescale model for the PU.1-GATA-1 interaction

Bokes, P., King, J. R., Loose, M. — 2009-06-02

The transcription factors PU.1 and GATA-1 antagonize each other in common myeloid progenitors and their relative abundance is thought to decide whether the cell follows the erythrocyte/megakaryocyte lineage or the granulocyte/macrophage lineage. We propose a kinetic model for the PU.1–GATA-1 interaction, analyse its phase space and interpret the results of our analysis. The conclusions have broader implications for the modelling of cell-fate selection.

Modelling the growth and stabilization of cerebral aneurysms

Watton, P. N., Ventikos, Y., Holzapfel, G. A. — 2009-06-02

Experimental and theoretical guidance is needed to understand how the collagen fabric evolves during the development of aneurysms. In this paper, we model the development of an aneurysm as a cylindrical/spherical membrane subject to 1D enlargement; these conceptual models reflect the development of fusiform and saccular cerebral aneurysms. The mechanical response is attributed to the elastin and collagen. We introduce variables which define the elastin and collagen fibre concentration; these evolve to simulate growth/atrophy of the constituents. A hypothetical aneurysm model is analysed: collagen stretch is constant, elastin degrades and collagen fibre concentration can adapt to maintain mechanical equilibrium. An analytic expression for the rate of evolution of the fibre concentration is derived. The functional form is dependent on (i) the current collagen fibre concentration, (ii) the deviations in the collagen fibre stretch from the attachment stretch, (iii) the rate of change of fibre stretch, (iv) the rate of loss of elastin and (v) the ratio of load borne by elastinous and collagenous constituents. Finally, numerical examples of aneurysm development are considered. Suitable candidates for the fibre concentration evolution equations are identified that yield stabilization of the aneurysm even when there is complete loss of elastin. This theoretical analysis provides the basis for the development of physiologically realistic models of aneurysm development.

A mathematical model for the deformation of the eyeball by an elastic band

Keeling, S. L., Propst, G., Stadler, G., Wackernagel, W. — 2009-06-02

In a certain kind of eye surgery, the human eyeball is deformed sustainably by the application of an elastic band. This article presents a mathematical model for the mechanics of the combined eye/band structure along with an algorithm to compute the model solutions. These predict the immediate and the lasting indentation of the eyeball. The model is derived from basic physical principles by minimizing a potential energy subject to a volume constraint. Assuming spherical symmetry, this leads to a two-point boundary-value problem for a non-linear second-order ordinary differential equation that describes the minimizing static equilibrium. By comparison with laboratory data, a preliminary validation of the model is given.

Flow dynamics in a stented ureter

Siggers, J. H., Waters, S., Wattis, J., Cummings, L. — 2009-02-26

Vesicorenal reflux is a major side effect associated with ureteric stent placement. In a stented upper urinary tract when the bladder pressure rises, such as during bladder spasms (due to irritation caused by the stent) or voiding of the bladder, it drives urine reflux up the ureter, which, in turn, may be a contributory factor for infections in the renal pelvis. We develop a mathematical model to examine urine flow in a stented ureter, assuming that it remains axisymmetric and treating the wall as a non-linear elastic membrane. The stent is modelled as a rigid, permeable, hollow, circular cylinder lying coaxially inside the ureter. The renal pelvis is treated as an elastic bag, whose volume increases in response to an increased internal pressure. Fluid enters the renal pelvis from the kidney with a prescribed flux. The stent, ureter and renal pelvis are filled with urine, and the bladder pressure is prescribed. We use the model to calculate the total volume of reflux generated during rises in bladder pressure and investigate how it is affected by the stent and ureter properties.

Analytic solution during an infusion test of the linear unsteady poroelastic equations in a spherically symmetric model of the brain

Wirth, B., Sobey, I. — 2009-02-26

This work determines the spatial and temporal distribution of cerebrospinal fluid (CSF) pressure and brain displacement during an infusion test in a spherically symmetric model of the brain. The response of CSF pressure and parenchymal displacement to blood pressure pulsations is determined in the solution. We use a spherically symmetric, three-component poroelastic model of the brain, differentiating between the solid elastic matrix, the CSF and the arterial blood compartments. The governing equations are linearized with quasi-constant poroelastic parameters. The solution does reproduce the average intracranial pressure increase during the test as well as the rise in CSF pressure pulsation amplitude due to transmission of blood pressure oscillations. In addition, the CSF flux into and out of the parenchyma is shown over time.

A family of models of angiogenesis and anti-angiogenesis anti-cancer therapy

D'Onofrio, A., Gandolfi, A. — 2009-02-26

In this paper we propose a class of models that describe the mutual interaction between tumour growth and the development of tumour vasculature and that generalize existing models. The study is mainly focused on the effect of a therapy that induces tumour vessel loss (anti-angiogenic therapy), with the aim of finding conditions that asymptotically guarantee the eradication of the disease under constant infusion or periodic administration of the drug. Furthermore, if tumour and/or vessel dynamics exhibit time delays, we derive conditions for the existence of Hopf bifurcations. The destabilizing effect of delays on achieving the tumour eradication is also investigated. Finally, global conditions for stability and eradication in the presence of delays are given for some particular cases.

Multistrain virus dynamics with mutations: a global analysis

De Leenheer, P., Pilyugin, S. S. — 2008-11-27

We consider within-host virus models with n ≥ 2 strains and allow mutation between the strains. If there is no mutation, a Lyapunov function establishes global stability of the steady state corresponding to the fittest strain. For small perturbations, this steady state persists, perhaps with small concentrations of some or all other strains, depending on the connectivity of the graph describing all possible mutations. Moreover, using a perturbation result due to Smith & Waltman (1999), we show that this steady state also preserves global stability.

Understanding post-operative temperature drop in cardiac surgery: a mathematical model

2008-11-27

A mathematical model is presented to understand heat transfer processes during the cooling and re-warming of patients during cardiac surgery. Our compartmental model is able to account for many of the qualitative features observed in the cooling of various regions of the body including the central core containing the majority of organs, the rectal region containing the intestines and the outer peripheral region of skin and muscle. In particular, we focus on the issue of afterdrop: a drop in core temperature following patient re-warming, which can lead to serious post-operative complications. Model results for a typical cooling and re-warming procedure during surgery are in qualitative agreement with experimental data in producing the afterdrop effect and the observed dynamical variation in temperature between the core, rectal and peripheral regions. The influence of heat transfer processes and the volume of each compartmental region on the afterdrop effect is discussed. We find that excess fat on the peripheral and rectal regions leads to an increase in the afterdrop effect. Our model predicts that, by allowing constant re-warming after the core temperature has been raised, the afterdrop effect will be reduced.

Fatal SIR diseases and rational exemption to vaccination

d'Onofrio, A., Manfredi, P., Salinelli, E. — 2008-11-27

A challenge to disease control in modern societies is the spread of rational exemption to vaccination as a consequence of the rational comparison between the steadily declining risk of infection and the risk of side effects from the vaccine. Here, we consider rational exemption in an susceptible-infectious-removed (SIR) model with information-dependent vaccination where individuals use information on the disease's mortality as their information set. Using suitable assumptions on the dynamics of the population, we show the dynamic implications of the interaction between rational exemption, current and delayed information and the risk of death by the disease. In particular, we illustrate the onset of the long cycles caused by rational exemption when vaccination decisions are based on delayed informations.

Evaluating the effectiveness of antiviral treatment in models for influenza pandemic

Lunelli, A., Pugliese, A. — 2008-11-27

We study the effectiveness of antiviral treatment in simple susceptible–exposed–infectious–removed models that are at the base of models used for influenza pandemic. The strategy is assessed in terms of the value of the reproductive ratio R₀. We consider a general framework and analyse six different specific cases. The same antiviral strategy is simulated in all models, but they slightly differ in the compartmental structure. These differences correspond to different underlying assumptions concerning the timing of the intervention and the selection of individuals who receive treatment. It is shown that these details can have a strong influence on the predicted effectiveness of the strategy: for instance, with R₀ = 1.8 in absence of treatment, different models predict that with treatment R₀ can become as low as 0.4 or as high as 1.3; still, in all models 70% of infected individuals are treated and the infectiousness of treated individuals is reduced by 80%. A particular assumption that can be included when modelling influenza is time-varying infectivity. We consider a specific model to verify if the predicted effectiveness of antiviral treatment is influenced by the inclusion of this assumption. We compare the results obtained with constant and variable infectivity, in relation also to the time of intervention. It is likely that existing differences in the predictions of the effect of control measures depend on such modelling details. This finding stresses the need for carefully defining the structure of models in order to obtain results useful for policymakers in pandemic planning.