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Mathematical Biosciences & Engineering

2013 , Volume 10 , Issue 4

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The ratio of hidden HIV infection in Cuba
Miguel Atencia , Esther García-Garaluz and  Gonzalo Joya
2013, 10(4): 959-977 doi: 10.3934/mbe.2013.10.959 +[Abstract](46) +[PDF](426.2KB)
In this work we propose the definition of the ratio of hidden infection of HIV/AIDS epidemics, as the division of the unknown infected population by the known one. The merit of the definition lies in allowing for an indirect estimation of the whole of the infected population. A dynamical model for the ratio is derived from a previous HIV/AIDS model, which was proposed for the Cuban case, where active search for infected individuals is carried out through a contact tracing program. The stability analysis proves that the model for the ratio possesses a single positive equilibrium, which turns out to be globally asymptotically stable. The sensitivity analysis provides an insight into the relative performance of various methods for detection of infected individuals. An exponential regression has been performed to fit the known infected population, owing to actual epidemiological data of HIV/AIDS epidemics in Cuba. The goodness of the obtained fit provides additional support to the proposed model.
Stability and Hopf bifurcation in a diffusive predator-prey system incorporating a prey refuge
Xiaoyuan Chang and  Junjie Wei
2013, 10(4): 979-996 doi: 10.3934/mbe.2013.10.979 +[Abstract](50) +[PDF](1783.0KB)
A diffusive predator-prey model with Holling type II functional response and the no-flux boundary condition incorporating a constant prey refuge is considered. Globally asymptotically stability of the positive equilibrium is obtained. Regarding the constant number of prey refuge $m$ as a bifurcation parameter, by analyzing the distribution of the eigenvalues, the existence of Hopf bifurcation is given. Employing the center manifold theory and normal form method, an algorithm for determining the properties of the Hopf bifurcation is derived. Some numerical simulations for illustrating the analysis results are carried out.
Modeling of the migration of endothelial cells on bioactive micropatterned polymers
Thierry Colin , Marie-Christine Durrieu , Julie Joie , Yifeng Lei , Youcef Mammeri , Clair Poignard and  Olivier Saut
2013, 10(4): 997-1015 doi: 10.3934/mbe.2013.10.997 +[Abstract](31) +[PDF](1457.3KB)
In this paper, a macroscopic model describing endothelial cell migration on bioactive micropatterned polymers is presented. It is based on a system of partial differential equations of Patlak-Keller-Segel type that describes the evolution of the cell densities. The model is studied mathematically and numerically. We prove existence and uniqueness results of the solution to the differential system. We also show that fundamental physical properties such as mass conservation, positivity and boundedness of the solution are satisfied. The numerical study allows us to show that the modeling results are in good agreement with the experiments.
Darwinian dynamics of a juvenile-adult model
J. M. Cushing and  Simon Maccracken Stump
2013, 10(4): 1017-1044 doi: 10.3934/mbe.2013.10.1017 +[Abstract](28) +[PDF](2304.3KB)
The bifurcation that occurs from the extinction equilibrium in a basic discrete time, nonlinear juvenile-adult model for semelparous populations, as the inherent net reproductive number $R_{0}$ increases through $1$, exhibits a dynamic dichotomy with two alternatives: an equilibrium with overlapping generations and a synchronous 2-cycle with non-overlapping generations. Which of the two alternatives is stable depends on the intensity of competition between juveniles and adults and on the direction of bifurcation. We study this dynamic dichotomy in an evolutionary setting by assuming adult fertility and juvenile survival are functions of a phenotypic trait $u$ subject to Darwinian evolution. Extinction equilibria for the Darwinian model exist only at traits $u^{\ast }$ that are critical points of $R_{0}\left( u\right) $. We establish the simultaneous bifurcation of positive equilibria and synchronous 2-cycles as the value of $R_{0}\left( u^{\ast }\right) $ increases through $1$ and describe how the stability of these dynamics depend on the direction of bifurcation, the intensity of between-class competition, and the extremal properties of $R_{0}\left( u\right) $ at $u^{\ast }$. These results can be equivalently stated in terms of the inherent population growth rate $r\left( u\right) $.
Model for hepatitis C virus transmissions
Elamin H. Elbasha
2013, 10(4): 1045-1065 doi: 10.3934/mbe.2013.10.1045 +[Abstract](52) +[PDF](478.7KB)
Hepatitis C virus (HCV) is a leading cause of chronic liver disease. This paper presents a deterministic model for HCV infection transmission and uses the model to assess the potential impact of antiviral therapy. The model is based on the susceptible-infective-removed-susceptible (SIRS) compartmental structure with chronic primary infection and possibility of reinfection. Important epidemiologic thresholds such as the basic and control reproduction numbers and a measure of treatment impact are derived. We find that if the control reproduction number is greater than unity, there is a locally unstable infection-free equilibrium and a unique, globally asymptotically stable endemic equilibrium. If the control reproduction number is less than unity, the infection-free equilibrium is globally asymptotically stable, and HCV will be eliminated. Numerical simulations suggest that, besides the parameters that determine the basic reproduction number, reinfection plays an important role in HCV transmissions and magnitude of the public health impact of antiviral therapy. Further, treatment regimens with better efficacy holds great promise for lowering the public health burden of HCV disease.
Parametrization of the attainable set for a nonlinear control model of a biochemical process
Ellina Grigorieva , Evgenii Khailov and  Andrei Korobeinikov
2013, 10(4): 1067-1094 doi: 10.3934/mbe.2013.10.1067 +[Abstract](27) +[PDF](8471.7KB)
In this paper, we study a three-dimensional nonlinear model of a controllable reaction $ [X] + [Y] + [Z] \rightarrow [Z] $, where the reaction rate is given by a unspecified nonlinear function. A model of this type describes a variety of real-life processes in chemical kinetics and biology; in this paper our particular interests is in its application to waste water biotreatment. For this control model, we analytically study the corresponding attainable set and parameterize it by the moments of switching of piecewise constant control functions. This allows us to visualize the attainable sets using a numerical procedure.
    These analytical results generalize the earlier findings, which were obtained for a trilinear reaction rate (which corresponds to the law of mass action) and reported in [18,19], to the case of a general rate of reaction. These results allow to reduce the problem of constructing the optimal control to a straightforward constrained finite dimensional optimization problem.
Regulation of Th1/Th2 cells in asthma development: A mathematical model
Yangjin Kim , Seongwon Lee , You-Sun Kim , Sean Lawler , Yong Song Gho , Yoon-Keun Kim and  Hyung Ju Hwang
2013, 10(4): 1095-1133 doi: 10.3934/mbe.2013.10.1095 +[Abstract](51) +[PDF](1733.0KB)
Airway exposure levels of lipopolysaccharide (LPS) determine type I versus type II helper T cell induced experimental asthma. While high LPS levels induce Th1-dominant responses, low LPS levels derive Th2 cell induced asthma. The present paper develops a mathematical model of asthma development which focuses on the relative balance of Th1 and Th2 cell induced asthma. In the present work we represent the complex network of interactions between cells and molecules by a mathematical model. The model describes the behaviors of cells (Th0, Th1, Th2 and macrophages) and regulatory molecules (IFN-$\gamma$, IL-4, IL-12, TNF-α) in response to high, intermediate, and low levels of LPS. The simulations show how variations in the levels of injected LPS affect the development of Th1 or Th2 cell responses through differential cytokine induction. The model also predicts the coexistence of these two types of response under certain biochemical and biomechanical conditions in the microenvironment.
Saturated treatments and measles resurgence episodes in South Africa: A possible linkage
Deborah Lacitignola
2013, 10(4): 1135-1157 doi: 10.3934/mbe.2013.10.1135 +[Abstract](27) +[PDF](1424.8KB)
We consider the case of measles in South Africa to show that an high vaccination coverage may be not enough - alone - to ensure measles eradication. The occurrence of certain epidemic episodes may in fact be encouraged by delays in the treatments or by not adequately fast clinical case management, which may be related to the backward bifurcation phenomenon as well as to an intriguing spiking dynamics which appears in the system for specific ranges of parameter values.
Modelling seasonal HFMD with the recessive infection in Shandong, China
Yangjun Ma , Maoxing Liu , Qiang Hou and  Jinqing Zhao
2013, 10(4): 1159-1171 doi: 10.3934/mbe.2013.10.1159 +[Abstract](48) +[PDF](449.1KB)
Hand, foot and mouth disease (HFMD) is one of the major public-health problems in China. Based on the HFMD data of the Department of Health of Shandong Province, we propose a dynamic model with periodic transmission rates to investigate the seasonal HFMD. After evaluating the basic reproduction number, we analyze the dynamical behaviors of the model and simulate the HFMD data of Shandong Province. By carrying out the sensitivity analysis of some key parameters, we conclude that the recessive subpopulation plays an important role in the spread of HFMD, and only quarantining the infected is not an effective measure in controlling the disease.
The impact of an imperfect vaccine and pap cytology screening on the transmission of human papillomavirus and occurrence of associated cervical dysplasia and cancer
Tufail Malik , Jody Reimer , Abba Gumel , Elamin H. Elbasha and  Salaheddin Mahmud
2013, 10(4): 1173-1205 doi: 10.3934/mbe.2013.10.1173 +[Abstract](29) +[PDF](839.8KB)
A mathematical model for the natural history of human papillomavirus (HPV) is designed and used to assess the impact of a hypothetical anti-HPV vaccine and Pap cytology screening on the transmission dynamics of HPV in a population. Rigorous qualitative analysis of the model reveals that it undergoes the phenomenon of backward bifurcation. It is shown that the backward bifurcation is caused by the imperfect nature of the HPV vaccine or the HPV-induced and cancer-induced mortality in females. For the case when the disease-induced and cancer-induced mortality is negligible, it is shown that the disease-free equilibrium (i.e., equilibrium in the absence of HPV and associated dysplasia) is globally-asymptotically stable if the associated reproduction number is less than unity. The model has a unique endemic equilibrium when the reproduction threshold exceeds unity. The unique endemic equilibrium is globally-asymptotically stable for a special case, where the associated HPV-induced and cancer-induced mortality is negligible. Numerical simulations of the model, using a reasonable set of parameter values, support the recent recommendations by some medical agencies and organizations in the USA to offer Pap screening on a 3-year basis (rather than annually).
Graph-theoretic conditions for zero-eigenvalue Turing instability in general chemical reaction networks
Maya Mincheva and  Gheorghe Craciun
2013, 10(4): 1207-1226 doi: 10.3934/mbe.2013.10.1207 +[Abstract](35) +[PDF](429.5KB)
We describe a necessary condition for zero-eigenvalue Turing instability, i.e., Turing instability arising from a real eigenvalue changing sign from negative to positive, for general chemical reaction networks modeled with mass-action kinetics. The reaction mechanisms are represented by the species-reaction graph (SR graph), which is a bipartite graph with different nodes representing species and reactions. If the SR graph satisfies certain conditions, similar to the conditions for ruling out multiple equilibria in spatially homogeneous differential equations systems, then the corresponding mass-action reaction-diffusion system cannot exhibit zero-eigenvalue Turing instability for any parameter values. On the other hand, if the graph-theoretic condition for ruling out zero-eigenvalue Turing instability is not satisfied, then the corresponding model may display zero-eigenvalue Turing instability for some parameter values. The technique is illustrated with a model of a bifunctional enzyme.
Mitigation of epidemics in contact networks through optimal contact adaptation
Mina Youssef and  Caterina Scoglio
2013, 10(4): 1227-1251 doi: 10.3934/mbe.2013.10.1227 +[Abstract](21) +[PDF](531.9KB)
This paper presents an optimal control problem formulation to minimize the total number of infection cases during the spread of susceptible-infected-recovered SIR epidemics in contact networks. In the new approach, contact weighted are reduced among nodes and a global minimum contact level is preserved in the network. In addition, the infection cost and the cost associated with the contact reduction are linearly combined in a single objective function. Hence, the optimal control formulation addresses the tradeoff between minimization of total infection cases and minimization of contact weights reduction. Using Pontryagin theorem, the obtained solution is a unique candidate representing the dynamical weighted contact network. To find the near-optimal solution in a decentralized way, we propose two heuristics based on Bang-Bang control function and on a piecewise nonlinear control function, respectively. We perform extensive simulations to evaluate the two heuristics on different networks. Our results show that the piecewise nonlinear control function outperforms the well-known Bang-Bang control function in minimizing both the total number of infection cases and the reduction of contact weights. Finally, our results show awareness of the infection level at which the mitigation strategies are effectively applied to the contact weights.
Heart rate variability as determinism with jump stochastic parameters
Jiongxuan Zheng , Joseph D. Skufca and  Erik M. Bollt
2013, 10(4): 1253-1264 doi: 10.3934/mbe.2013.10.1253 +[Abstract](30) +[PDF](833.5KB)
We use measured heart rate information (RR intervals) to develop a one-dimensional nonlinear map that describes short term deterministic behavior in the data. Our study suggests that there is a stochastic parameter with persistence which causes the heart rate and rhythm system to wander about a bifurcation point. We propose a modified circle map with a jump process noise term as a model which can qualitatively capture such this behavior of low dimensional transient determinism with occasional (stochastically defined) jumps from one deterministic system to another within a one parameter family of deterministic systems.

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