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Mathematical Biosciences & Engineering

2016 , Volume 13 , Issue 5

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Immune response in virus model structured by cell infection-age
Cameron Browne
2016, 13(5): 887-909 doi: 10.3934/mbe.2016022 +[Abstract](4438) +[PDF](1217.3KB)
This paper concerns modeling the coupled within-host population dynamics of virus and CTL (Cytotoxic T Lymphocyte) immune response. There is substantial evidence that the CTL immune response plays a crucial role in controlling HIV in infected patients. Recent experimental studies have demonstrated that certain CTL variants can recognize HIV infected cells early in the infected cell lifecycle before viral production, while other CTLs only detect viral proteins (epitopes) presented on the surface of infected cells after viral production. The kinetics of epitope presentation and immune recognition can impact the efficacy of the immune response. We extend previous virus models to include cell infection-age structure in the infected cell compartment and immune response killing/activation rates of a PDE-ODE system. We characterize solutions to our system utilizing semigroup theory, determine equilibria and reproduction numbers, and prove stability and persistence results. Numerical simulations show that ``early immune recognition'' precipitates both enhanced viral control and sustained oscillations via a Hopf bifurcation. In addition to inducing oscillatory dynamics, considering immune process rates to be functions of cell infection-age can also lead to coexistence of multiple distinct immune effector populations.
Using drinking data and pharmacokinetic modeling to calibrate transport model and blind deconvolution based data analysis software for transdermal alcohol biosensors
Zheng Dai, I.G. Rosen, Chuming Wang, Nancy Barnett and Susan E. Luczak
2016, 13(5): 911-934 doi: 10.3934/mbe.2016023 +[Abstract](2923) +[PDF](511.6KB)
Alcohol researchers/clinicians have two ways to collect subject /patient field data, standard-drink self-report and the breath analyzer, neither of which is passive or accurate because active subject participation is required. Transdermal alcohol sensors have been developed to measure transdermal alcohol concentration (TAC), but they are used primarily as abstinence monitors because converting TAC into more meaningful blood/breath alcohol concentration (BAC/BrAC) is difficult. In this paper, BAC/BrAC is estimated from TAC by first calibrating forward distributed parameter-based convolution models for ethanol transport from the blood through the skin using patient-collected drinking data for a single drinking episode and a nonlinear pharmacokinetic metabolic absorption/elimination model to estimate BAC. TAC and estimated BAC are then used to fit the forward convolution filter. Nonlinear least squares with adjoint-based gradient computation are used to fit both models. Calibration results are compared with those obtained using BAC/BrAC from alcohol challenges and from standard, linear, metabolic absorption, and zero order kinetics-based elimination models, by considering peak BAC, time of peak, and area under the BAC curve. Our models (with population parameters) could be included in a smart phone app that makes it convenient for the subject/patient to enter drinking data for a single episode in the field.
Dynamics of a diffusive age-structured HBV model with saturating incidence
Xichao Duan, Sanling Yuan and Kaifa Wang
2016, 13(5): 935-968 doi: 10.3934/mbe.2016024 +[Abstract](4031) +[PDF](472.2KB)
In this paper, we propose and investigate an age-structured hepatitis B virus (HBV) model with saturating incidence and spatial diffusion where the viral contamination process is described by the age-since-infection. We first analyze the well-posedness of the initial-boundary values problem of the model in the bounded domain $\Omega\subset\mathbb{R}^n$ and obtain an explicit formula for the basic reproductive number $R_0$ of the model. Then we investigate the global behavior of the model in terms of $R_0$: if $R_0\leq1$, then the uninfected steady state is globally asymptotically stable, whereas if $R_0>1$, then the infected steady state is globally asymptotically stable. In addition, when $R_0>1$, by constructing a suitable Lyapunov-like functional decreasing along the travelling waves to show their convergence towards two steady states as $t$ tends to $\pm\infty$, we prove the existence of traveling wave solutions. Numerical simulations are provided to illustrate the theoretical results.
Modeling the spread of bed bug infestation and optimal resource allocation for disinfestation
Ali Gharouni and Lin Wang
2016, 13(5): 969-980 doi: 10.3934/mbe.2016025 +[Abstract](3050) +[PDF](405.6KB)
A patch-structured multigroup-like $SIS$ epidemiological model is proposed to study the spread of the common bed bug infestation. It is shown that the model exhibits global threshold dynamics with the basic reproduction number as the threshold parameter. Costs associated with the disinfestation process are incorporated into setting up the optimization problems. Procedures are proposed and simulated for finding optimal resource allocation strategies to achieve the infestation free state. Our analysis and simulations provide useful insights on how to efficiently distribute the available exterminators among the infested patches for optimal disinfestation management.
Type-dependent stochastic Ising model describing the dynamics of a non-symmetric feedback module
Manuel González-Navarrete
2016, 13(5): 981-998 doi: 10.3934/mbe.2016026 +[Abstract](2699) +[PDF](1050.9KB)
We study an alternative approach to model the dynamical behaviors of biological feedback loop, that is, a type-dependent spin system, this class of stochastic models was introduced by Fernández et. al [13], and are useful since take account to inherent variability of gene expression. We analyze a non-symmetric feedback module being an extension for the repressilator, the first synthetic biological oscillator, invented by Elowitz and Leibler [7]. We consider a mean-field dynamics for a type-dependent Ising model, and then study the empirical-magnetization vector representing concentration of molecules. We apply a convergence result from stochastic jump processes to deterministic trajectories and present a bifurcation analysis for the associated dynamical system. We show that non-symmetric module under study can exhibit very rich behaviours, including the empirical oscillations described by repressilator.
Epidemic characteristics of two classic models and the dependence on the initial conditions
Jianquan Li, Yiqun Li and Yali Yang
2016, 13(5): 999-1010 doi: 10.3934/mbe.2016027 +[Abstract](2578) +[PDF](399.4KB)
The epidemic characteristics, including the epidemic final size, peak, and turning point, of two classical SIR models with disease-induced death are investigated when a small initial value of the infective population is released. The models have mass-action (i.e. bilinear), or density dependent (i.e. standard) incidence, respectively. For the two models, the conditions that determining whether the related epidemic characteristics of an epidemic outbreak appear are explicitly determine by rigorous mathematical analysis. The dependence of the epidemic final size on the initial values of the infective class is demonstrated. The peak, turning point (if it exists) and the corresponding time are found. The obtained results suggest that their basic reproduction numbers are one factor determining the epidemic characteristics, but not the only one. The characteristics of the two models depend on the initial values and proportions of various compartments as well. At last, the similarities and differences of the epidemic characteristics between the two models are discussed.
Modeling the role of healthcare access inequalities in epidemic outcomes
Oscar Patterson-Lomba, Muntaser Safan, Sherry Towers and Jay Taylor
2016, 13(5): 1011-1041 doi: 10.3934/mbe.2016028 +[Abstract](3199) +[PDF](6186.3KB)
Urban areas, with large and dense populations, offer conditions that favor the emergence and spread of certain infectious diseases. One common feature of urban populations is the existence of large socioeconomic inequalities which are often mirrored by disparities in access to healthcare. Recent empirical evidence suggests that higher levels of socioeconomic inequalities are associated with worsened public health outcomes, including higher rates of sexually transmitted diseases (STD's) and lower life expectancy. However, the reasons for these associations are still speculative. Here we formulate a mathematical model to study the effect of healthcare disparities on the spread of an infectious disease that does not confer lasting immunity, such as is true of certain STD's. Using a simple epidemic model of a population divided into two groups that differ in their recovery rates due to different levels of access to healthcare, we find that both the basic reproductive number ($\mathcal{R}_{0}$) of the disease and its endemic prevalence are increasing functions of the disparity between the two groups, in agreement with empirical evidence. Unexpectedly, this can be true even when the fraction of the population with better access to healthcare is increased if this is offset by reduced access within the disadvantaged group. Extending our model to more than two groups with different levels of access to healthcare, we find that increasing the variance of recovery rates among groups, while keeping the mean recovery rate constant, also increases $\mathcal{R}_{0}$ and disease prevalence. In addition, we show that these conclusions are sensitive to how we quantify the inequalities in our model, underscoring the importance of basing analyses on appropriate measures of inequalities. These insights shed light on the possible impact that increasing levels of inequalities in healthcare access can have on epidemic outcomes, while offering plausible explanations for the observed empirical patterns.
Development of a computational model of glucose toxicity in the progression of diabetes mellitus
Danilo T. Pérez-Rivera, Verónica L. Torres-Torres, Abraham E. Torres-Colón and Mayteé Cruz-Aponte
2016, 13(5): 1043-1058 doi: 10.3934/mbe.2016029 +[Abstract](3507) +[PDF](2312.9KB)
Diabetes mellitus is a disease characterized by a range of metabolic complications involving an individual's blood glucose levels, and its main regulator, insulin. These complications can vary largely from person to person depending on their current biophysical state. Biomedical research day-by-day makes strides to impact the lives of patients of a variety of diseases, including diabetes. One large stride that is being made is the generation of techniques to assist physicians to ``personalize medicine''. From available physiological data, biological understanding of the system, and dimensional analysis, a differential equation-based mathematical model was built in a sequential matter, to be able to elucidate clearly how each parameter correlates to the patient's current physiological state. We developed a simple mathematical model that accurately simulates the dynamics between glucose, insulin, and pancreatic $\beta$-cells throughout disease progression with constraints to maintain biological relevance. The current framework is clearly capable of tracking the patient's current progress through the disease, dependent on factors such as latent insulin resistance or an attrite $\beta$-cell population. Further interests would be to develop tools that allow the direct and feasible testing of how effective a given plan of treatment would be at returning the patient to a desirable biophysical state.
Dynamical properties and tumor clearance conditions for a nine-dimensional model of bladder cancer immunotherapy
K. E. Starkov and Svetlana Bunimovich-Mendrazitsky
2016, 13(5): 1059-1075 doi: 10.3934/mbe.2016030 +[Abstract](3084) +[PDF](400.4KB)
Understanding the global interaction dynamics between tumor and the immune system plays a key role in the advancement of cancer therapy. Bunimovich-Mendrazitsky et al. (2015) developed a mathematical model for the study of the immune system response to combined therapy for bladder cancer with Bacillus Calmette-Guérin (BCG) and interleukin-2 (IL-2) . We utilized a mathematical approach for bladder cancer treatment model for derivation of ultimate upper and lower bounds and proving dissipativity property in the sense of Levinson. Furthermore, tumor clearance conditions for BCG treatment of bladder cancer are presented. Our method is based on localization of compact invariant sets and may be exploited for a prediction of the cells populations dynamics involved into the model.
A two-sex matrix population model to represent harem structure
Anthony Tongen, María Zubillaga and Jorge E. Rabinovich
2016, 13(5): 1077-1092 doi: 10.3934/mbe.2016031 +[Abstract](3279) +[PDF](572.7KB)
Population dynamic models often include males in the calculation of population change, but even in those cases males have rarely been introduced to represent polygyny (harem social structure), where it is particularly important to include males in the reproductive performance of the population. In this article we develop an adaptable matrix population modeling framework for species that have a harem-like social structure under an assumption that the transitions from newborn to juvenile and juvenile to adult both take one time step. We are able to calculate not only the growth rates and stable stage distributions, but also the mathematical expressions for harem size for this model. We then provide applications of this model to two mammal species with slightly different harem behavior.
Heterogeneous population dynamics and scaling laws near epidemic outbreaks
Andreas Widder and Christian Kuehn
2016, 13(5): 1093-1118 doi: 10.3934/mbe.2016032 +[Abstract](4011) +[PDF](691.1KB)
In this paper, we focus on the influence of heterogeneity and stochasticity of the population on the dynamical structure of a basic susceptible-infected-susceptible (SIS) model. First we prove that, upon a suitable mathematical reformulation of the basic reproduction number, the homogeneous system and the heterogeneous system exhibit a completely analogous global behaviour. Then we consider noise terms to incorporate the fluctuation effects and the random import of the disease into the population and analyse the influence of heterogeneity on warning signs for critical transitions (or tipping points). This theory shows that one may be able to anticipate whether a bifurcation point is close before it happens. We use numerical simulations of a stochastic fast-slow heterogeneous population SIS model and show various aspects of heterogeneity have crucial influences on the scaling laws that are used as early-warning signs for the homogeneous system. Thus, although the basic structural qualitative dynamical properties are the same for both systems, the quantitative features for epidemic prediction are expected to change and care has to be taken to interpret potential warning signs for disease outbreaks correctly.

2018 Impact Factor: 1.313




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