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Mathematical Biosciences & Engineering

October 2018 , Volume 15 , Issue 5

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Mathematical modelling of cardiac pulse wave reflections due to arterial irregularities
Alexandre Cornet
2018, 15(5): 1055-1076 doi: 10.3934/mbe.2018047 +[Abstract](5684) +[HTML](388) +[PDF](2666.83KB)

This research aims to model cardiac pulse wave reflections due to the presence of arterial irregularities such as bifurcations, stiff arteries, stenoses or aneurysms. When an arterial pressure wave encounters an irregularity, a backward reflected wave travels upstream in the artery and a forward wave is transmitted downstream. The same process occurs at each subsequent irregularity, leading to the generation of multiple waves. An iterative algorithm is developed and applied to pathological scenarios to predict the pressure waveform of the reflected wave due to the presence of successive arterial irregularities. For an isolated stenosis, analysing the reflected pressure waveform gives information on its severity. The presence of a bifurcation after a stenosis tends do diminish the amplitude of the reflected wave, as bifurcations' reflection coefficients are relatively small compared to the ones of stenoses or aneurysms. In the case of two stenoses in series, local extrema are observed in the reflected pressure waveform which appears to be a characteristic of stenoses in series along an individual artery. Finally, we model a progressive change in stiffness in the vessel's wall and observe that the less the gradient stiffness is important, the weaker is the reflected wave.

Stochastic dynamics and survival analysis of a cell population model with random perturbations
Cristina Anton and Alan Yong
2018, 15(5): 1077-1098 doi: 10.3934/mbe.2018048 +[Abstract](5519) +[HTML](308) +[PDF](1480.8KB)

We consider a model based on the logistic equation and linear kinetics to study the effect of toxicants with various initial concentrations on a cell population. To account for parameter uncertainties, in our model the coefficients of the linear and the quadratic terms of the logistic equation are affected by noise. We show that the stochastic model has a unique positive solution and we find conditions for extinction and persistence of the cell population. In case of persistence we find the stationary distribution. The analytical results are confirmed by Monte Carlo simulations.

An age-structured vector-borne disease model with horizontal transmission in the host
Xia Wang and Yuming Chen
2018, 15(5): 1099-1116 doi: 10.3934/mbe.2018049 +[Abstract](6029) +[HTML](446) +[PDF](529.13KB)

We concern with a vector-borne disease model with horizontal transmission and infection age in the host population. With the approach of Lyapunov functionals, we establish a threshold dynamics, which is completely determined by the basic reproduction number. Roughly speaking, if the basic reproduction number is less than one then the infection-free equilibrium is globally asymptotically stable while if the basic reproduction number is larger than one then the infected equilibrium attracts all solutions with initial infection. These theoretical results are illustrated with numerical simulations.

Modelling chemistry and biology after implantation of a drug-eluting stent. Part Ⅱ: Cell proliferation
Adam Peddle, William Lee and Tuoi Vo
2018, 15(5): 1117-1135 doi: 10.3934/mbe.2018050 +[Abstract](5413) +[HTML](288) +[PDF](477.98KB)

The aim of a drug eluting stent is to prevent restenosis of arteries following percutaneous balloon angioplasty. A long term goal of research in this area is to use modelling to optimise the design of these stents to maximise their efficiency. A key obstacle to implementing this is the lack of a mathematical model of the biology of restenosis. Here we investigate whether mathematical models of cancer biology can be adapted to model the biology of restenosis and the effect of drug elution. We show that relatively simple, rate kinetic models give a good description of available data of restenosis in animal experiments, and its modification by drug elution.

Optimal control problems with time delays: Two case studies in biomedicine
Laurenz Göllmann and Helmut Maurer
2018, 15(5): 1137-1154 doi: 10.3934/mbe.2018051 +[Abstract](5191) +[HTML](474) +[PDF](708.6KB)

There exists an extensive literature on delay differential models in biology and biomedicine, but only a few papers study such models in the framework of optimal control theory. In this paper, we consider optimal control problems with multiple time delays in state and control variables and present two applications in biomedicine. After discussing the necessary optimality conditions for delayed optimal control problems with control-state constraints, we propose discretization methods by which the delayed optimal control problem is transformed into a large-scale nonlinear programming problem. The first case study is concerned with the delay differential model in [21] describing the tumour-immune response to a chemo-immuno-therapy. Assuming \begin{document}$ L^1$\end{document}-type objectives, which are linear in control, we obtain optimal controls of bang-bang type. In the second case study, we introduce a control variable in the delay differential model of Hepatitis B virus infection developed in [7]. For \begin{document}$ L^1$\end{document}-type objectives we obtain extremal controls of bang-bang type.

A stochastic model for water-vegetation systems and the effect of decreasing precipitation on semi-arid environments
Shannon Dixon, Nancy Huntly, Priscilla E. Greenwood and Luis F. Gordillo
2018, 15(5): 1155-1164 doi: 10.3934/mbe.2018052 +[Abstract](5220) +[HTML](312) +[PDF](404.09KB)

Current climate change trends are affecting the magnitude and recurrence of extreme weather events. In particular, several semi-arid regions around the planet are confronting more intense and prolonged lack of precipitation, slowly transforming part of these regions into deserts in some cases. Although it is documented that a decreasing tendency in precipitation might induce earlier disappearance of vegetation, quantifying the relationship between decrease of precipitation and vegetation endurance remains a challenging task due to the inherent complexities involved in distinct scenarios. In this paper we present a model for precipitation-vegetation dynamics in semi-arid landscapes that can be used to explore numerically the impact of decreasing precipitation trends on appearance of desertification events. The model, a stochastic differential equation approximation derived from a Markov jump process, is used to generate extensive simulations that suggest a relationship between precipitation reduction and the desertification process, which might take several years in some instances.

Quantifying the impact of early-stage contact tracing on controlling Ebola diffusion
Narges Montazeri Shahtori, Tanvir Ferdousi, Caterina Scoglio and Faryad Darabi Sahneh
2018, 15(5): 1165-1180 doi: 10.3934/mbe.2018053 +[Abstract](5344) +[HTML](370) +[PDF](447.68KB)

Recent experience of the Ebola outbreak in 2014 highlighted the importance of immediate response measure to impede transmission in the early stage. To this aim, efficient and effective allocation of limited resources is crucial. Among the standard interventions is the practice of following up with the recent physical contacts of the infected individuals -- known as contact tracing. In an effort to understand the effects of contact tracing protocols objectively, we explicitly develop a model of Ebola transmission incorporating contact tracing. Our modeling framework is individual-based, patient-centric, stochastic and parameterizable to suit early-stage Ebola transmission. Notably, we propose an activity driven network approach to contact tracing, and estimate the basic reproductive ratio of the epidemic growth in different scenarios. Exhaustive simulation experiments suggest that early contact tracing paired with rapid hospitalization can effectively impede the epidemic growth. Resource allocation needs to be carefully planned to enable early detection of the contacts and rapid hospitalization of the infected people.

Dynamics of delayed mosquitoes populations models with two different strategies of releasing sterile mosquitoes
Liming Cai, Shangbing Ai and Guihong Fan
2018, 15(5): 1181-1202 doi: 10.3934/mbe.2018054 +[Abstract](6263) +[HTML](433) +[PDF](826.66KB)

To prevent the transmissions of mosquito-borne diseases (e.g., malaria, dengue fever), recent works have considered the problem of using the sterile insect technique to reduce or eradicate the wild mosquito population. It is important to consider how reproductive advantage of the wild mosquito population offsets the success of population replacement. In this work, we explore the interactive dynamics of the wild and sterile mosquitoes by incorporating the delay in terms of the growth stage of the wild mosquitoes. We analyze (both analytically and numerically) the role of time delay in two different ways of releasing sterile mosquitoes. Our results demonstrate that in the case of constant release rate, the delay does not affect the dynamics of the system and every solution of the system approaches to an equilibrium point; while in the case of the release rate proportional to the wild mosquito populations, the delay has a large effect on the dynamics of the system, namely, for some parameter ranges, when the delay is small, every solution of the system approaches to an equilibrium point; but as the delay increases, the solutions of the system exhibit oscillatory behavior via Hopf bifurcations. Numerical examples and bifurcation diagrams are also given to demonstrate rich dynamical features of the model in the latter release case.

A dynamic model of CT scans for quantifying doubling time of ground glass opacities using histogram analysis
József Z. Farkas, Gary T. Smith and Glenn F. Webb
2018, 15(5): 1203-1224 doi: 10.3934/mbe.2018055 +[Abstract](6240) +[HTML](1400) +[PDF](5861.18KB)

We quantify a recent five-category CT histogram based classification of ground glass opacities using a dynamic mathematical model for the spatial-temporal evolution of malignant nodules. Our mathematical model takes the form of a spatially structured partial differential equation with a logistic crowding term. We present the results of extensive simulations and validate our model using patient data obtained from clinical CT images from patients with benign and malignant lesions.

Analysis of a mathematical model for brain lactate kinetics
Carole Guillevin, Rémy Guillevin, Alain Miranville and Angélique Perrillat-Mercerot
2018, 15(5): 1225-1242 doi: 10.3934/mbe.2018056 +[Abstract](4996) +[HTML](373) +[PDF](773.75KB)

The aim of this article is to study the well-posedness and properties of a fast-slow system which is related with brain lactate kinetics. In particular, we prove the existence and uniqueness of nonnegative solutions and obtain linear stability results. We also give numerical simulations with different values of the small parameter $\varepsilon$ and compare them with experimental data.

The four-dimensional Kirschner-Panetta type cancer model: How to obtain tumor eradication?
Alexander P. Krishchenko and Konstantin E. Starkov
2018, 15(5): 1243-1254 doi: 10.3934/mbe.2018057 +[Abstract](5498) +[HTML](289) +[PDF](365.9KB)

In this paper we examine ultimate dynamics of the four-dimensional model describing interactions between tumor cells, effector immune cells, interleukin -2 and transforming growth factor-beta. This model was elaborated by Arciero et al. and is obtained from the Kirschner-Panetta type model by introducing two various treatments. We provide ultimate upper bounds for all variables of this model and two lower bounds and, besides, study when dynamics of this model possesses a global attracting set. The nonexistence conditions of compact invariant sets are derived. We obtain bounds for treatment parameters \begin{document}$s_{1, 2}$\end{document} under which all trajectories in the positive orthant tend to the tumor-free equilibrium point. Conditions imposed on \begin{document}$s_{1, 2}$\end{document} under which the tumor population persists are presented as well. Finally, we compare tumor eradication/ persistence bounds and discuss our results.

The mean and noise of stochastic gene transcription with cell division
Qi Wang, Lifang Huang, Kunwen Wen and Jianshe Yu
2018, 15(5): 1255-1270 doi: 10.3934/mbe.2018058 +[Abstract](5497) +[HTML](473) +[PDF](886.45KB)

Life growth and development are driven by continuous cell divisions. Cell division is a stochastic and complex process. In this paper, we study the impact of cell division on the mean and noise of mRNA numbers by using a two-state stochastic model of transcription. Our results show that the steady-state mRNA noise with symmetric cell division is less than that with binomial inheritance with probability 0.5, but the steady-state mean transcript level with symmetric division is always equal to that with binomial inheritance with probability 0.5. Cell division except random additive inheritance always decreases mean transcript level and increases transcription noise. Inversely, random additive inheritance always increases mean transcript level and decreases transcription noise. We also show that the steady-state mean transcript level (the steady-state mRNA noise) with symmetric cell division or binomial inheritance increases (decreases) with the average cell cycle duration. But the steady-state mean transcript level (the steady-state mRNA noise) with random additive inheritance decreases (increases) with the average cell cycle duration. Our results are confirmed by Gillespie stochastic simulation using plausible parameters.

2018 Impact Factor: 1.313




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