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Mathematical Biosciences & Engineering

August 2018 , Volume 15 , Issue 4

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Closed-loop control of tumor growth by means of anti-angiogenic administration
Filippo Cacace, Valerio Cusimano, Alfredo Germani, Pasquale Palumbo and Federico Papa
2018, 15(4): 827-839 doi: 10.3934/mbe.2018037 +[Abstract](730) +[HTML](247) +[PDF](604.08KB)
Abstract:

A tumor growth model accounting for angiogenic stimulation and inhibition is here considered, and a closed-loop control law is presented with the aim of tumor volume reduction by means of anti-angiogenic administration. To this end the output-feedback linearization theory is exploited, with the feedback designed on the basis of a state observer for nonlinear systems. Measurements are supposed to be acquired at discrete sampling times, and a novel theoretical development in the area of time-delay systems is applied in order to derive a continuous-time observer in spite of the presence of sampled measurements. The overall control scheme allows to set independently the control and the observer parameters thanks to the structural properties of the tumor growth model. Simulations are carried out in order to mimic a real experimental framework on mice. These results seem extremely promising: they provide very good performances according to the measurements sampling interval suggested by the experimental literature, and show a noticeable level of robustness against the observer initial estimate, as well as against the uncertainties affecting the model parameters.

Continuous and pulsed epidemiological models for onchocerciasis with implications for eradication strategy
Glenn Ledder, Donna Sylvester, Rachelle R. Bouchat and Johann A. Thiel
2018, 15(4): 841-862 doi: 10.3934/mbe.2018038 +[Abstract](751) +[HTML](207) +[PDF](529.34KB)
Abstract:

Onchocerciasis is an endemic disease in parts of sub-Saharan Africa. Complex mathematical models are being used to assess the likely efficacy of efforts to eradicate the disease; however, their predictions have not always been borne out in practice. In this paper, we represent the immunological aspects of the disease with a single empirical parameter in order to reduce the model complexity. Asymptotic approximation allows us to reduce the vector-borne epidemiological model to a model of an infectious disease with nonlinear incidence. We then consider two versions, one with continuous treatment and a more realistic one where treatment occurs only at intervals. Thorough mathematical analysis of these models yields equilibrium solutions for the continuous case, periodic solutions for the pulsed case, and conditions for the existence of endemic disease equilibria in both cases, thereby leading to simple model criteria for eradication. The analytical results and numerical experiments show that the continuous treatment version is an excellent approximation for the pulsed version and that the current onchocerciasis eradication strategy is inadequate for regions where the incidence is highest and unacceptably slow even when the long-term behavior is the disease-free state.

A model of regulatory dynamics with threshold-type state-dependent delay
Qingwen Hu
2018, 15(4): 863-882 doi: 10.3934/mbe.2018039 +[Abstract](1008) +[HTML](205) +[PDF](803.81KB)
Abstract:

We model intracellular regulatory dynamics with threshold-type state-dependent delay and investigate the effect of the state-dependent diffusion time. A general model which is an extension of the classical differential equation models with constant or zero time delays is developed to study the stability of steady state, the occurrence and stability of periodic oscillations in regulatory dynamics. Using the method of multiple time scales, we compute the normal form of the general model and show that the state-dependent diffusion time may lead to both supercritical and subcritical Hopf bifurcations. Numerical simulations of the prototype model of Hes1 regulatory dynamics are given to illustrate the general results.

Influence of Allee effect in prey populations on the dynamics of two-prey-one-predator model
Moitri Sen, Malay Banerjee and Yasuhiro Takeuchi
2018, 15(4): 883-904 doi: 10.3934/mbe.2018040 +[Abstract](1254) +[HTML](325) +[PDF](759.18KB)
Abstract:

One of the important ecological challenges is to capture the complex dynamics and understand the underlying regulating ecological factors. Allee effect is one of the important factors in ecology and taking it into account can cause significant changes to the system dynamics. In this work we consider a two prey-one predator model where the growth of both the prey population is subjected to Allee effect, and the predator is generalist as it survives on both the prey populations. We analyze the role of Allee effect on the dynamics of the system, knowing the dynamics of the model without Allee effect. Interestingly we have observed through a comprehensive bifurcation study that incorporation of Allee effect enriches the local as well as the global dynamics of the system. Specially after a certain threshold value of the Allee effect, it has a very significant effect on the chaotic dynamics of the system. In course of the bifurcation analysis we have explored all possible bifurcations such as the existence of transcritical bifurcation, saddle-node bifurcation, Hopf-bifurcation, Bogdanov-Takens bifurcation and Bautin bifurcation and period-doubling route to chaos respectively.

EEG in neonates: Forward modeling and sensitivity analysis with respect to variations of the conductivity
Hamed Azizollahi, Marion Darbas, Mohamadou M. Diallo, Abdellatif El Badia and Stephanie Lohrengel
2018, 15(4): 905-932 doi: 10.3934/mbe.2018041 +[Abstract](954) +[HTML](316) +[PDF](1642.48KB)
Abstract:

The paper is devoted to the analysis of electroencephalography (EEG) in neonates. The goal is to investigate the impact of fontanels on EEG measurements, i.e. on the values of the electric potential on the scalp. In order to answer this clinical issue, a complete mathematical study (modeling, existence and uniqueness result, realistic simulations) is carried out. A model for the forward problem in EEG source localization is proposed. The model is able to take into account the presence and ossification process of fontanels which are characterized by a variable conductivity. From a mathematical point of view, the model consists in solving an elliptic problem with a singular source term in an inhomogeneous medium. A subtraction approach is used to deal with the singularity in the source term, and existence and uniqueness results are proved for the continuous problem. Discretization is performed with 3D Finite Elements of type P1 and error estimates are proved in the energy norm ($H^1$-norm). Numerical simulations for a three-layer spherical model as well as for a realistic neonatal head model including or not the fontanels have been obtained and corroborate the theoretical results. A mathematical tool related to the concept of Gâteau derivatives is introduced which is able to measure the sensitivity of the electric potential with respect to small variations in the fontanel conductivity. This study attests that the presence of fontanels in neonates does have an impact on EEG measurements.

The role of structural viscoelasticity in deformable porous media with incompressible constituents: Applications in biomechanics
Maurizio Verri, Giovanna Guidoboni, Lorena Bociu and Riccardo Sacco
2018, 15(4): 933-959 doi: 10.3934/mbe.2018042 +[Abstract](847) +[HTML](214) +[PDF](5319.01KB)
Abstract:

The main goal of this work is to clarify and quantify, by means of mathematical analysis, the role of structural viscoelasticity in the biomechanical response of deformable porous media with incompressible constituents to sudden changes in external applied loads. Models of deformable porous media with incompressible constituents are often utilized to describe the behavior of biological tissues, such as cartilages, bones and engineered tissue scaffolds, where viscoelastic properties may change with age, disease or by design. Here, for the first time, we show that the fluid velocity within the medium could increase tremendously, even up to infinity, should the external applied load experience sudden changes in time and the structural viscoelasticity be too small. In particular, we consider a one-dimensional poro-visco-elastic model for which we derive explicit solutions in the cases where the external applied load is characterized by a step pulse or a trapezoidal pulse in time. By means of dimensional analysis, we identify some dimensionless parameters that can aid the design of structural properties and/or experimental conditions as to ensure that the fluid velocity within the medium remains bounded below a certain given threshold, thereby preventing potential tissue damage. The application to confined compression tests for biological tissues is discussed in detail. Interestingly, the loss of viscoelastic tissue properties has been associated with various disease conditions, such as atherosclerosis, Alzheimer’s disease and glaucoma. Thus, the findings of this work may be relevant to many applications in biology and medicine.

Quantifying the survival uncertainty of Wolbachia-infected mosquitoes in a spatial model
Martin Strugarek, Nicolas Vauchelet and Jorge P. Zubelli
2018, 15(4): 961-991 doi: 10.3934/mbe.2018043 +[Abstract](672) +[HTML](216) +[PDF](977.35KB)
Abstract:

Artificial releases of Wolbachia-infected Aedes mosquitoes have been under study in the past yearsfor fighting vector-borne diseases such as dengue, chikungunya and zika.Several strains of this bacterium cause cytoplasmic incompatibility (CI) and can also affect their host's fecundity or lifespan, while highly reducing vector competence for the main arboviruses.

We consider and answer the following questions: 1) what should be the initial condition (i.e. size of the initial mosquito population) to have invasion with one mosquito release source? We note that it is hard to have an invasion in such case. 2) How many release points does one need to have sufficiently high probability of invasion? 3) What happens if one accounts for uncertainty in the release protocol (e.g. unequal spacing among release points)?

We build a framework based on existing reaction-diffusion models for the uncertainty quantification in this context,obtain both theoretical and numerical lower bounds for the probability of release successand give new quantitative results on the one dimensional case.

Optimal design for dynamical modeling of pest populations
H. T. Banks, R. A. Everett, Neha Murad, R. D. White, J. E. Banks, Bodil N. Cass and Jay A. Rosenheim
2018, 15(4): 993-1010 doi: 10.3934/mbe.2018044 +[Abstract](698) +[HTML](442) +[PDF](1031.22KB)
Abstract:

We apply SE-optimal design methodology to investigate optimal data collection procedures as a first step in investigating information content in ecoinformatics data sets. To illustrate ideas we use a simple phenomenological citrus red mite population model for pest dynamics. First the optimal sampling distributions for a varying number of data points are determined. We then analyze these optimal distributions by comparing the standard errors of parameter estimates corresponding to each distribution. This allows us to investigate how many data are required to have confidence in model parameter estimates in order to employ dynamical modeling to infer population dynamics. Our results suggest that a field researcher should collect at least 12 data points at the optimal times. Data collected according to this procedure along with dynamical modeling will allow us to estimate population dynamics from presence/absence-based data sets through the development of a scaling relationship. These Likert-type data sets are commonly collected by agricultural pest management consultants and are increasingly being used in ecoinformatics studies. By applying mathematical modeling with the relationship scale from the new data, we can then explore important integrated pest management questions using past and future presence/absence data sets.

Probability of Escherichia coli contamination spread in ground beef production
Petko M. Kitanov and Allan R. Willms
2018, 15(4): 1011-1032 doi: 10.3934/mbe.2018045 +[Abstract](867) +[HTML](205) +[PDF](505.7KB)
Abstract:

Human illness due to contamination of food by pathogenic strains of Escherichia coli is a serious public health concern and can cause significant economic losses in the food industry. Recent outbreaks of such illness sourced from ground beef production motivates the work in this paper. Most ground beef is produced in large facilities where many carcasses are butchered and various pieces of them are ground together in sequential batches. Assuming that the source of contamination is a single carcass and that downstream from the production facility ground beef from a particular batch has been identified as contaminated by E. coli, the probability that previous and subsequent batches are also contaminated is modelled. This model may help the beef industry to identify the likelihood of contamination in other batches and potentially save money by not needing to cook or recall unaffected batches of ground beef.

Role of white-tailed deer in geographic spread of the black-legged tick Ixodes scapularis : Analysis of a spatially nonlocal model
Stephen A. Gourley, Xiulan Lai, Junping Shi, Wendi Wang, Yanyu Xiao and Xingfu Zou
2018, 15(4): 1033-1054 doi: 10.3934/mbe.2018046 +[Abstract](780) +[HTML](373) +[PDF](675.6KB)
Abstract:

Lyme disease is transmitted via blacklegged ticks, the spatial spread of which is believed to be primarily via transport on white-tailed deer. In this paper, we develop a mathematical model to describe the spatial spread of blacklegged ticks due to deer dispersal. The model turns out to be a system of differential equations with a spatially non-local term accounting for the phenomenon that a questing female adult tick that attaches to a deer at one location may later drop to the ground, fully fed, at another location having been transported by the deer. We first justify the well-posedness of the model and analyze the stability of its steady states. We then explore the existence of traveling wave fronts connecting the extinction equilibrium with the positive equilibrium for the system. We derive an algebraic equation that determines a critical value $c^*$ which is at least a lower bound for the wave speed in the sense that, if $c < c^*$, there is no traveling wave front of speed $c$ connecting the extinction steady state to the positive steady state. Numerical simulations of the wave equations suggest that \begin{document}$c^*$\end{document} is the minimum wave speed. We also carry out some numerical simulations for the original spatial model system and the results seem to confirm that the actual spread rate of the tick population coincides with \begin{document}$c^*$\end{document}. We also explore the dependence of \begin{document}$c^*$\end{document} on the dispersion rate of the white tailed deer, by which one may evaluate the role of the deer's dispersion in the geographical spread of the ticks.

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