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Mathematical Biosciences & Engineering

2017 , Volume 14 , Issue 1

Micro and macro systems in life sciences

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Avner Friedman, Mirosław Lachowicz, Urszula Ledzewicz, Monika Joanna Piotrowska and Zuzanna Szymanska
2017, 14(1): i-i doi: 10.3934/mbe.201701i +[Abstract](97) +[HTML](23) +[PDF](72.3KB)
Angiogenesis model with Erlang distributed delays
Emad Attia, Marek Bodnar and Urszula Foryś
2017, 14(1): 1-15 doi: 10.3934/mbe.2017001 +[Abstract](77) +[HTML](0) +[PDF](715.5KB)

We consider the model of angiogenesis process proposed by Bodnar and Foryś (2009) with time delays included into the vessels formation and tumour growth processes. Originally, discrete delays were considered, while in the present paper we focus on distributed delays and discuss specific results for the Erlang distributions. Analytical results concerning stability of positive steady states are illustrated by numerical results in which we also compare these results with those for discrete delays.

A singular limit for an age structured mutation problem
Jacek Banasiak and Aleksandra Falkiewicz
2017, 14(1): 17-30 doi: 10.3934/mbe.2017002 +[Abstract](71) +[HTML](0) +[PDF](398.0KB)

The spread of a particular trait in a cell population often is modelled by an appropriate system of ordinary differential equations describing how the sizes of subpopulations of the cells with the same genome change in time. On the other hand, it is recognized that cells have their own vital dynamics and mutations, leading to changes in their genome, mostly occurring during the cell division at the end of its life cycle. In this context, the process is described by a system of McKendrick type equations which resembles a network transport problem. In this paper we show that, under an appropriate scaling of the latter, these two descriptions are asymptotically equivalent.

On the initial value problem for a class of discrete velocity models
Davide Bellandi
2017, 14(1): 31-43 doi: 10.3934/mbe.2017003 +[Abstract](93) +[HTML](0) +[PDF](419.1KB)

In this paper we investigate the initial value problem for a class of hyperbolic systems relating the mathematical modeling of a class of complex phenomena, with emphasis on vehicular traffic flow. Existence and uniqueness for large times of solutions, a basic requisite both for models building and for their numerical implementation, are obtained under weak hypotheses on the terms modeling the interaction among agents. The results are then compared with the existing literature on the subject.

On the mathematical modelling of tumor-induced angiogenesis
Luis L. Bonilla, Vincenzo Capasso, Mariano Alvaro, Manuel Carretero and Filippo Terragni
2017, 14(1): 45-66 doi: 10.3934/mbe.2017004 +[Abstract](134) +[HTML](0) +[PDF](1323.2KB)

An angiogenic system is taken as an example of extremely complex ones in the field of Life Sciences, from both analytical and computational points of view, due to the strong coupling between the kinetic parameters of the relevant branching -growth -anastomosis stochastic processes of the capillary network, at the microscale, and the family of interacting underlying biochemical fields, at the macroscale. To reduce this complexity, for a conceptual stochastic model we have explored how to take advantage of the system intrinsic multiscale structure: one might describe the stochastic dynamics of the cells at the vessel tip at their natural microscale, whereas the dynamics of the underlying fields is given by a deterministic mean field approximation obtained by an averaging at a suitable mesoscale. But the outcomes of relevant numerical simulations show that the proposed model, in presence of anastomosis, is not self-averaging, so that the "propagation of chaos" assumption cannot be applied to obtain a deterministic mean field approximation. On the other hand we have shown that ensemble averages over many realizations of the stochastic system may better correspond to a deterministic reaction-diffusion system.

Network-based analysis of a small Ebola outbreak
G. Burch Mark, A. Jacobsen Karly, H. Tien Joseph and A. Rempa ła Grzegorz
2017, 14(1): 67-77 doi: 10.3934/mbe.2017005 +[Abstract](88) +[HTML](0) +[PDF](742.2KB)

We present a method for estimating epidemic parameters in network-based stochastic epidemic models when the total number of infections is assumed to be small. We illustrate the method by reanalyzing the data from the 2014 Democratic Republic of the Congo (DRC) Ebola outbreak described in Maganga et al. (2014).

Emergence of spatial patterns in a mathematical model for the co-culture dynamics of epithelial-like and mesenchymal-like cells
Marcello Delitala and Tommaso Lorenzi
2017, 14(1): 79-93 doi: 10.3934/mbe.2017006 +[Abstract](151) +[HTML](0) +[PDF](919.5KB)

Accumulating evidence indicates that the interaction between epithelial and mesenchymal cells plays a pivotal role in cancer development and metastasis formation. Here we propose an integro-differential model for the co-culture dynamics of epithelial-like and mesenchymal-like cells. Our model takes into account the effects of chemotaxis, adhesive interactions between epithelial-like cells, proliferation and competition for nutrients. We present a sample of numerical results which display the emergence of spots, stripes and honeycomb patterns, depending on parameters and initial data. These simulations also suggest that epithelial-like and mesenchymal-like cells can segregate when there is little competition for nutrients. Furthermore, our computational results provide a possible explanation for how the concerted action between epithelial-cell adhesion and mesenchymal-cell spreading can precipitate the formation of ring-like structures, which resemble the fibrous capsules frequently observed in hepatic tumours.

Controlling imported malaria cases in the United States of America
Bassidy Dembele and Abdul-Aziz Yakubu
2017, 14(1): 95-109 doi: 10.3934/mbe.2017007 +[Abstract](103) +[HTML](0) +[PDF](346.1KB)

We extend the mathematical malaria epidemic model framework of Dembele et al. and use it to “capture” the 2013 Centers for Disease Control and Prevention (CDC) reported data on the 2011 number of imported malaria cases in the USA. Furthermore, we use our “fitted” malaria models for the top 20 countries of malaria acquisition by USA residents to study the impact of protecting USA residents from malaria infection when they travel to malaria endemic areas, the impact of protecting residents of malaria endemic regions from mosquito bites and the impact of killing mosquitoes in those endemic areas on the CDC number of imported malaria cases in USA. To significantly reduce the number of imported malaria cases in USA, for each top 20 country of malaria acquisition by USA travelers, we compute the optimal proportion of USA international travelers that must be protected against malaria infection and the optimal proportion of mosquitoes that must be killed.

On application of optimal control to SEIR normalized models: Pros and cons
Rosario de Pinho Maria do and Nogueira Filipa Nunes
2017, 14(1): 111-126 doi: 10.3934/mbe.2017008 +[Abstract](69) +[HTML](0) +[PDF](709.8KB)

In this work we normalize a SEIR model that incorporates exponential natural birth and death, as well as disease-caused death. We use optimal control to control by vaccination the spread of a generic infectious disease described by a normalized model with \begin{document} $L^1$ \end{document} cost. We discuss the pros and cons of SEIR normalized models when compared with classical models when optimal control with \begin{document} $L^1$ \end{document} costs are considered. Our discussion highlights the role of the cost. Additionally, we partially validate our numerical solutions for our optimal control problem with normalized models using the Maximum Principle.

Many particle approximation of the Aw-Rascle-Zhang second order model for vehicular traffic
Marco Di Francesco, Simone Fagioli and Massimiliano D. Rosini
2017, 14(1): 127-141 doi: 10.3934/mbe.2017009 +[Abstract](93) +[HTML](0) +[PDF](365.1KB)

We consider the follow-the-leader approximation of the Aw-Rascle-Zhang (ARZ) model for traffic flow in a multi population formulation. We prove rigorous convergence to weak solutions of the ARZ system in the many particle limit in presence of vacuum. The result is based on uniform \begin{document} ${\mathbf{BV}}$ \end{document} estimates on the discrete particle velocity. We complement our result with numerical simulations of the particle method compared with some exact solutions to the Riemann problem of the ARZ system.

Mathematical modeling of liver fibrosis
Avner Friedman and Wenrui Hao
2017, 14(1): 143-164 doi: 10.3934/mbe.2017010 +[Abstract](254) +[HTML](0) +[PDF](3496.8KB)

Fibrosis is the formation of excessive fibrous connective tissue in an organ or tissue, which occurs in reparative process or in response to inflammation. Fibrotic diseases are characterized by abnormal excessive deposition of fibrous proteins, such as collagen, and the disease is most commonly progressive, leading to organ disfunction and failure. Although fibrotic diseases evolve in a similar way in all organs, differences may occur as a result of structure and function of the specific organ. In liver fibrosis, the gold standard for diagnosis and monitoring the progression of the disease is biopsy, which is invasive and cannot be repeated frequently. For this reason there is currently a great interest in identifying non-invasive biomarkers for liver fibrosis. In this paper, we develop for the first time a mathematical model of liver fibrosis by a system of partial differential equations. We use the model to explore the efficacy of potential and currently used drugs aimed at blocking the progression of liver fibrosis. We also use the model to develop a diagnostic tool based on a combination of two biomarkers.

Estimation of initial functions for systems with delays from discrete measurements
Krzysztof Fujarewicz
2017, 14(1): 165-178 doi: 10.3934/mbe.2017011 +[Abstract](52) +[HTML](0) +[PDF](586.3KB)

The work presents a gradient-based approach to estimation of initial functions of time delay elements appearing in models of dynamical systems. It is shown how to generate the gradient of the estimation objective function in the initial function space using adjoint sensitivity analysis. It is assumed that the system is continuous-time and described by ordinary differential equations with delays but the estimation is done based on discrete-time measurements of the signals appearing in the system. Results of gradient-based estimation of initial functions for exemplary models are presented and discussed.

Numerical simulations of a 3D fluid-structure interaction model for blood flow in an atherosclerotic artery
Oualid Kafi, Nader El Khatib, Jorge Tiago and Adélia Sequeira
2017, 14(1): 179-193 doi: 10.3934/mbe.2017012 +[Abstract](266) +[HTML](2) +[PDF](1905.5KB)

The inflammatory process of atherosclerosis leads to the formation of an atheromatous plaque in the intima of the blood vessel. The plaque rupture may result from the interaction between the blood and the plaque. In each cardiac cycle, blood interacts with the vessel, considered as a compliant nonlinear hyperelastic. A three dimensional idealized fluid-structure interaction (FSI) model is constructed to perform the blood-plaque and blood-vessel wall interaction studies. An absorbing boundary condition (BC) is imposed directly on the outflow in order to cope with the spurious reflexions due to the truncation of the computational domain. The difference between the Newtonian and non-Newtonian effects is highlighted. It is shown that the von Mises and wall shear stresses are significantly affected according to the rigidity of the wall. The numerical results have shown that the risk of plaque rupture is higher in the case of a moving wall, while in the case of a fixed wall the risk of progression of the atheromatous plaque is higher.

Local controllability and optimal control for a model of combined anticancer therapy with control delays
Jerzy Klamka, Helmut Maurer and Andrzej Swierniak
2017, 14(1): 195-216 doi: 10.3934/mbe.2017013 +[Abstract](103) +[HTML](0) +[PDF](567.2KB)

We study some control properties of a class of two-compartmental models of response to anticancer treatment which combines anti-angiogenic and cytotoxic drugs and take into account multiple control delays. We formulate sufficient local controllability conditions for semilinear systems resulting from these models. The control delays are related to PK/PD effects and some clinical recommendations, e.g., normalization of the vascular network. The optimized protocols of the combined therapy for the model, considered as solutions to an optimal control problem with delays in control, are found using necessary conditions of optimality and numerical computations. Our numerical approach uses dicretization and nonlinear programming methods as well as the direct optimization of switching times. The structural sensitivity of the considered control properties and optimal solutions is also discussed.

On drug resistance and metronomic chemotherapy: A mathematical modeling and optimal control approach
Urszula Ledzewicz, Shuo Wang, Heinz Schättler, Nicolas André, Marie Amélie Heng Heng and Eddy Pasquier
2017, 14(1): 217-235 doi: 10.3934/mbe.2017014 +[Abstract](139) +[HTML](2) +[PDF](676.3KB)

Effects that tumor heterogeneity and drug resistance have on the structure of chemotherapy protocols are discussed from a mathematical modeling and optimal control point of view. In the case when two compartments consisting of sensitive and resistant cells are considered, optimal protocols consist of full dose chemotherapy as long as the relative proportion of sensitive cells is high. When resistant cells become more dominant, optimal controls switch to lower dose regimens defined by so-called singular controls. The role that singular controls play in the structure of optimal therapy protocols for cell populations with a large number of traits is explored in mathematical models.

Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion
Henryk Leszczyński and Monika Wrzosek
2017, 14(1): 237-248 doi: 10.3934/mbe.2017015 +[Abstract](76) +[HTML](0) +[PDF](376.2KB)

We consider nonlinear stochastic wave equations driven by one-dimensional white noise with respect to time. The existence of solutions is proved by means of Picard iterations. Next we apply Newton's method. Moreover, a second-order convergence in a probabilistic sense is demonstrated.

Spatio-temporal models of synthetic genetic oscillators
Cicely K. Macnamara and Mark A. J. Chaplain
2017, 14(1): 249-262 doi: 10.3934/mbe.2017016 +[Abstract](173) +[HTML](2) +[PDF](2902.1KB)

Signal transduction pathways play a major role in many important aspects of cellular function e.g. cell division, apoptosis. One important class of signal transduction pathways is gene regulatory networks (GRNs). In many GRNs, proteins bind to gene sites in the nucleus thereby altering the transcription rate. Such proteins are known as transcription factors. If the binding reduces the transcription rate there is a negative feedback leading to oscillatory behaviour in mRNA and protein levels, both spatially (e.g. by observing fluorescently labelled molecules in single cells) and temporally (e.g. by observing protein/mRNA levels over time). Recent computational modelling has demonstrated that spatial movement of the molecules is a vital component of GRNs and may cause the oscillations. These numerical findings have subsequently been proved rigorously i.e. the diffusion coefficient of the protein/mRNA acts as a bifurcation parameter and gives rise to a Hopf bifurcation. In this paper we first present a model of the canonical GRN (the Hes1 protein) and show the effect of varying the spatial location of gene and protein production sites on the oscillations. We then extend the approach to examine spatio-temporal models of synthetic gene regulatory networks e.g. n-gene repressilators and activator-repressor systems.

Sufficient optimality conditions for a class of epidemic problems with control on the boundary
Miniak-Górecka Alicja and Nowakowski Andrzej
2017, 14(1): 263-275 doi: 10.3934/mbe.2017017 +[Abstract](101) +[HTML](0) +[PDF](576.9KB)

In earlier paper of V. Capasso et al it is considered a simply model of controlling an epidemic, which is described by three functionals and systems of two PDE equations having the feedback operator on the boundary. Necessary optimality conditions and two gradient-type algorithms are derived. This paper constructs dual dynamic programming method to derive sufficient optimality conditions for optimal solution as well \begin{document}$\varepsilon $\end{document}-optimality conditions in terms of dual dynamic inequalities. Approximate optimality and numerical calculations are presented too.

A criterion of collective behavior of bacteria
Roman Czapla and Vladimir V. Mityushev
2017, 14(1): 277-287 doi: 10.3934/mbe.2017018 +[Abstract](99) +[HTML](0) +[PDF](1273.2KB)

It was established in the previous works that hydrodynamic interactions between the swimmers can lead to collective motion. Its implicit evidences were confirmed by reduction in the effective viscosity. We propose a new quantitative criterion to detect such a collective behavior. Our criterion is based on a new computationally effective RVE (representative volume element) theory based on the basic statistic moments ($e$-sums or generalized Eisenstein-Rayleigh sums). The criterion can be applied to various two-phase dispersed media (biological systems, composites etc). The locations of bacteria are modeled by short segments having a small width randomly embedded in medium without overlapping. We compute the $e$-sums of the simulated disordered sets and of the observed experimental locations of Bacillus subtilis. The obtained results show a difference between these two sets that demonstrates the collective motion of bacteria.

On a mathematical model of bone marrow metastatic niche
Ana Isabel Muñoz and J. Ignacio Tello
2017, 14(1): 289-304 doi: 10.3934/mbe.2017019 +[Abstract](113) +[HTML](0) +[PDF](645.0KB)

We propose a mathematical model to describe tumor cells movement towards a metastasis location into the bone marrow considering the influence of chemotaxis inhibition due to the action of a drug. The model considers the evolution of the signaling molecules CXCL-12 secreted by osteoblasts (bone cells responsible of the mineralization of the bone) and PTHrP (secreted by tumor cells) which activates osteoblast growth. The model consists of a coupled system of second order PDEs describing the evolution of CXCL-12 and PTHrP, an ODE of logistic type to model the Osteoblasts density and an extra equation for each cancer cell. We also simulate the system to illustrate the qualitative behavior of the solutions. The numerical method of resolution is also presented in detail.

The role of TNF-$\alpha$ inhibitor in glioma virotherapy: A mathematical model
Ratajczyk Elzbieta, Ledzewicz Urszula, Leszczynski Maciej and Friedman Avner
2017, 14(1): 305-319 doi: 10.3934/mbe.2017020 +[Abstract](87) +[HTML](0) +[PDF](1012.0KB)

Virotherapy, using herpes simplex virus, represents a promising therapy of glioma. But the innate immune response, which includes TNF-α produced by macrophages, reduces the effectiveness of the treatment. Hence treatment with TNF-α inhibitor may increase the effectiveness of the virotherapy. In the present paper we develop a mathematical model that includes continuous infusion of the virus in combination with TNF-α inhibitor. We study the efficacy of the treatment under different combinations of the two drugs for different scenarios of the burst size of newly formed virus emerging from dying infected cancer cells. The model may serve as a first step toward developing an optimal strategy for the treatment of glioma by the combination of TNF-α inhibitor and oncolytic virus injection.

Optimal control of a tuberculosis model with state and control delays
Cristiana J. Silva, Helmut Maurer and Delfim F. M. Torres
2017, 14(1): 321-337 doi: 10.3934/mbe.2017021 +[Abstract](101) +[HTML](0) +[PDF](939.9KB)

We introduce delays in a tuberculosis (TB) model, representing the time delay on the diagnosis and commencement of treatment of individuals with active TB infection. The stability of the disease free and endemic equilibriums is investigated for any time delay. Corresponding optimal control problems, with time delays in both state and control variables, are formulated and studied. Although it is well-known that there is a delay between two to eight weeks between TB infection and reaction of body's immune system to tuberculin, delays for the active infected to be detected and treated, and delays on the treatment of persistent latent individuals due to clinical and patient reasons, which clearly justifies the introduction of time delays on state and control measures, our work seems to be the first to consider such time-delays for TB and apply time-delay optimal control to carry out the optimality analysis.

A male-female mathematical model of human papillomavirus (HPV) in African American population
Najat Ziyadi
2017, 14(1): 339-358 doi: 10.3934/mbe.2017022 +[Abstract](99) +[HTML](0) +[PDF](504.1KB)

We introduce mathematical human papillomavirus (HPV) epidemic models (with and without vaccination) for African American females (AAF) and African American males (AAM) with "fitted" logistic demographics and use these models to study the HPV disease dynamics. The US Census Bureau data of AAF and AAM of 16 years and older from 2000 to 2014 is used to "fit" the logistic demographic models. We compute the basic reproduction number, \begin{document} $\mathcal{R}_0$ \end{document}, and use it to show that \begin{document} $\mathcal{R}_0$ \end{document} is less than 1 in the African American (AA) population with or without implementation of HPV vaccination program. Furthermore, we obtain that adopting a HPV vaccination policy in the AAF and AAM populations lower \begin{document} $\mathcal{R}_0$ \end{document} and the number of HPV infections. Sensitivity analysis is used to illustrate the impact of each model parameter on the basic reproduction number.

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