# American Institute of Mathematical Sciences

ISSN:
1551-0018

eISSN:
1547-1063

All Issues

## Mathematical Biosciences & Engineering

April 2017 , Volume 14 , Issue 2

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2017, 14(2): 359-376 doi: 10.3934/mbe.2017023 +[Abstract](3589) +[HTML](54) +[PDF](628.6KB)
Abstract:

Population transmission models have been helpful in studying the spread of HIV. They assess changes made at the population level for different intervention strategies.To further understand how individual changes affect the population as a whole, game-theoretical models are used to quantify the decision-making process.Investigating multiplayer nonlinear games that model HIV transmission represents a unique approach in epidemiological research. We present here 2-player and multiplayer noncooperative games where players are defined by HIV status and age and may engage in casual (sexual) encounters. The games are modelled as generalized Nash games with shared constraints, which is completely novel in the context of our applied problem. Each player's HIV status is known to potential partners, and players have personal preferences ranked via utility values of unprotected and protected sex outcomes. We model a player's strategy as their probability of being engaged in a casual unprotected sex encounter (\begin{document}$USE$\end{document}), which may lead to HIV transmission; however, we do not incorporate a transmission model here. We study the sensitivity of Nash strategies with respect to varying preference rankings, and the impact of a prophylactic vaccine introduced in players of youngest age groups. We also study the effect of these changes on the overall increase in infection level, as well as the effects that a potential prophylactic treatment may have on age-stratified groups of players. We conclude that the biggest impacts on increasing the infection levels in the overall population are given by the variation in the utilities assigned to individuals for unprotected sex with others of opposite \begin{document}$HIV$\end{document} status, while the introduction of a prophylactic vaccine in youngest age group (15-20 yr olds) slows down the increase in \begin{document}$HIV$\end{document} infection.

2017, 14(2): 377-405 doi: 10.3934/mbe.2017024 +[Abstract](5681) +[HTML](63) +[PDF](962.8KB)
Abstract:

This paper presents a mathematical model for malaria-schistosom-iasis co-infection in order to investigate their synergistic relationship in the presence of treatment. We first analyse the single infection steady states, then investigate the existence and stability of equilibria and then calculate the basic reproduction numbers. Both the single-infection models and the co-infection model exhibit backward bifurcations. We carrying out a sensitivity analysis of the co-infection model and show that schistosomiasis infection may not be associated with an increased risk of malaria. Conversely, malaria infection may be associated with an increased risk of schistosomiasis. Furthermore, we found that effective treatment and prevention of schistosomiasis infection would also assist in the effective control and eradication of malaria. Finally, we apply Pontryagin's Maximum Principle to the model in order to determine optimal strategies for control of both diseases.

2017, 14(2): 407-420 doi: 10.3934/mbe.2017025 +[Abstract](3445) +[HTML](54) +[PDF](410.7KB)
Abstract:

This paper studies the global existence and uniqueness of classicalsolutions for a generalized quasilinear parabolic equation withappropriate initial and mixed boundary conditions. Under somepracticable regularity criteria on diffusion item and nonlinearity, weestablish the local existence and uniqueness of classical solutionsbased on a contraction mapping. This local solution can be continuedfor all positive time by employing the methods of energy estimates, \begin{document}$L^{p}$\end{document}-theory, and Schauder estimate of linear parabolic equations. Astraightforward application of global existence result of classical solutions to a density-dependent diffusion model of in vitroglioblastoma growth is also presented.

2017, 14(2): 421-435 doi: 10.3934/mbe.2017026 +[Abstract](3069) +[HTML](56) +[PDF](346.6KB)
Abstract:

In this paper, we study the global stability of a multistrain SIS model with superinfection. We present an iterative procedure to calculate a sequence of reproduction numbers, and we prove that it completely determines the global dynamics of the system. We show that for any number of strains with different infectivities, the stable coexistence of any subset of the strains is possible, and we completely characterize all scenarios. As an example, we apply our method to a three-strain model.

2017, 14(2): 437-453 doi: 10.3934/mbe.2017027 +[Abstract](4937) +[HTML](60) +[PDF](1292.6KB)
Abstract:

If a given behavior of a multi-agent system restricts the phase variable to an invariant manifold, then we define a phase transition as a change of physical characteristics such as speed, coordination, and structure. We define such a phase transition as splitting an underlying manifold into two sub-manifolds with distinct dimensionalities around the singularity where the phase transition physically exists. Here, we propose a method of detecting phase transitions and splitting the manifold into phase transitions free sub-manifolds. Therein, we firstly utilize a relationship between curvature and singular value ratio of points sampled in a curve, and then extend the assertion into higher-dimensions using the shape operator. Secondly, we attest that the same phase transition can also be approximated by singular value ratios computed locally over the data in a neighborhood on the manifold. We validate the Phase Transition Detection (PTD) method using one particle simulation and three real world examples.

2017, 14(2): 455-465 doi: 10.3934/mbe.2017028 +[Abstract](3115) +[HTML](66) +[PDF](1298.0KB)
Abstract:

Altruism is typically associated with traits or behaviors that benefit the population as a whole, but are costly to the individual. We propose that, when the environment is rapidly changing, senescence (age-related deterioration) can be altruistic. According to numerical simulations of an agent-based model, while long-lived individuals can outcompete their short lived peers, populations composed of long-lived individuals are more likely to go extinct during periods of rapid environmental change. Moreover, as in many situations where other cooperative behavior arises, senescence can be stabilized in a structured population.

2017, 14(2): 467-490 doi: 10.3934/mbe.2017029 +[Abstract](3080) +[HTML](55) +[PDF](670.6KB)
Abstract:

One-dimensional logistic population models with quasi-constant-yield harvest rates are studied under the assumptions that a population inhabits a patch of dimensionless width and no members of the population can survive outside of the patch. The essential problem is to determine the size of the patch and the ranges of the harvesting rate functions under which the population survives or becomes extinct. This is the first paper which discusses such models with the Dirichlet boundary conditions and can tell the exact quantity of harvest rates of the species without having the population die out. The methodology is to establish new results on the existence of positive solutions of semi-positone Hammerstein integral equations using the fixed point index theory for compact maps defined on cones, and apply the new results to tackle the essential problem. It is expected that the established analytical results have broad applications in management of sustainable ecological systems.

2017, 14(2): 491-509 doi: 10.3934/mbe.2017030 +[Abstract](7161) +[HTML](66) +[PDF](678.9KB)
Abstract:

Drug-eluting stents have been used widely to prevent restenosis of arteries following percutaneous balloon angioplasty. Mathematical modelling plays an important role in optimising the design of these stents to maximise their efficiency. When designing a drug-eluting stent system, we expect to have a sufficient amount of drug being released into the artery wall for a sufficient period to prevent restenosis. In this paper, a simple model is considered to provide an elementary description of drug release into artery tissue from an implanted stent. From the model, we identified a parameter regime to optimise the system when preparing the polymer coating. The model provides some useful order of magnitude estimates for the key quantities of interest. From the model, we can identify the time scales over which the drug traverses the artery wall and empties from the polymer coating, as well as obtain approximate formulae for the total amount of drug in the artery tissue and the fraction of drug that has released from the polymer. The model was evaluated by comparing to in-vivo experimental data and good agreement was found.

2017, 14(2): 511-527 doi: 10.3934/mbe.2017031 +[Abstract](8458) +[HTML](3013) +[PDF](928.1KB)
Abstract:

Every performance, in an officially sanctioned meet, by a registered USA swimmer is recorded into an online database with times dating back to 1980. For the first time, statistical analysis and machine learning methods are systematically applied to 4,022,631 swim records. In this study, we investigate performance features for all strokes as a function of age and gender. The variances in performance of males and females for different ages and strokes were studied, and the correlations of performances for different ages were estimated using the Pearson correlation. Regression analysis show the performance trends for both males and females at different ages and suggest critical ages for peak training. Moreover, we assess twelve popular machine learning methods to predict or classify swimmer performance. Each method exhibited different strengths or weaknesses in different cases, indicating no one method could predict well for all strokes. To address this problem, we propose a new method by combining multiple inference methods to derive Wisdom of Crowd Classifier (WoCC). Our simulation experiments demonstrate that the WoCC is a consistent method with better overall prediction accuracy. Our study reveals several new age-dependent trends in swimming and provides an accurate method for classifying and predicting swimming times.

2017, 14(2): 529-557 doi: 10.3934/mbe.2017032 +[Abstract](4638) +[HTML](84) +[PDF](7548.0KB)
Abstract:

To study the impacts of toxin produced by phytoplankton and refuges provided for phytoplankton on phytoplankton-zooplankton interactions in lakes, we establish a simple phytoplankton-zooplankton system with Holling type Ⅱ response function. The existence and stability of positive equilibria are discussed. Bifurcation analyses are given by using normal form theory which reveals reasonably the mechanisms and nonlinear dynamics of the effects of toxin and refuges, including Hopf bifurcation, Bogdanov-Takens bifurcation of co-dimension 2 and 3. Numerical simulations are carried out to intuitively support our analytical results and help to explain the observed biological behaviors. Our findings finally show that both phytoplankton refuge and toxin have a significant impact on the occurring and terminating of algal blooms in freshwater lakes.

2017, 14(2): 559-579 doi: 10.3934/mbe.2017033 +[Abstract](4731) +[HTML](50) +[PDF](530.9KB)
Abstract:

We study the global stability issue of the reaction-convection-diffusion cholera epidemic PDE model and show that the basic reproduction number serves as a threshold parameter that predicts whether cholera will persist or become globally extinct. Specifically, when the basic reproduction number is beneath one, we show that the disease-free-equilibrium is globally attractive. On the other hand, when the basic reproduction number exceeds one, if the infectious hosts or the concentration of bacteria in the contaminated water are not initially identically zero, we prove the uniform persistence result and that there exists at least one positive steady state.

2018 Impact Factor: 1.313