ISSN:

1551-0018

eISSN:

1547-1063

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## Mathematical Biosciences & Engineering

2014 , Volume 11 , Issue 5

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2014, 11(5): 1027-1043
doi: 10.3934/mbe.2014.11.1027

*+*[Abstract](2003)*+*[PDF](474.2KB)**Abstract:**

A theoretical analysis of some of the relevant factors influencing the calcium time course and the strength and timing of release probabilities of vesicles evoked by an action potential in a calyx-type active zone is presented in this paper. In particular, our study focus on the comparison of cooperative vs non-cooperative calcium binding by the release site and the effect of the number of Ca$^{2+}$ binding sites on the calcium sensitivity for release. Regarding the comparison of cooperative and non-cooperative kinetic schemes, our simulations show that quite different results are obtained when considering one or another: a reduction in the release probability of more than a $50\,\%$ is obtained when considering the cooperative kinetic scheme. Also, a delay in the average time for release appears when using this model for the calcium sensor.

Our study also shows that a non-cooperative kinetic binding scheme gives rise to a well defined average calcium level for release assuming that the same kinetic constants are considered for all the sites. Our results also suggest that the central value of the calcium sensitivity for release depends on the number of binding sites $N$ and the dissociation constant $K_{D}$ with a scaling law depending on $N K_{D}$.

2014, 11(5): 1045-1063
doi: 10.3934/mbe.2014.11.1045

*+*[Abstract](2781)*+*[PDF](1944.2KB)**Abstract:**

For emerging diseases like pandemic influenza, several factors could impact the outcome of vaccination programs, including a delay in vaccine availability, imperfect vaccine-induced protection, and inadequate number of vaccines to sufficiently lower the susceptibility of the population by raising the level of herd immunity. We sought to investigate the effect of these factors in determining optimal vaccination strategies during an emerging influenza infection for which the population is entirely susceptible. We developed a population dynamical model of disease transmission and vaccination, and analyzed the control problem associated with an adaptive time-dependent vaccination strategy, in which the rate of vaccine distribution is optimally determined with time for minimizing the total number of infections (i.e., the epidemic final size). We simulated the model and compared the outcomes with a constant vaccination strategy in which the rate of vaccine distribution is time-independent. When vaccines are available at the onset of epidemic, our findings show that for a sufficiently high vaccine efficacy, the adaptive and constant vaccination strategies lead to comparable outcomes in terms of the epidemic final size. However, the adaptive vaccination requires a vaccine coverage higher than (or equivalent to) the constant vaccination regardless of the rate of vaccine distribution, suggesting that the latter is a more cost-effective strategy. When the vaccine efficacy is below a certain threshold, the adaptive vaccination could substantially outperform the constant vaccination, and the impact of adaptive strategy becomes more pronounced as the rate of vaccine distribution increases. We observed similar results when vaccines become available with a delay during the epidemic; however, the adaptive strategy may require a significantly higher vaccine coverage to outperform the constant vaccination strategy. The findings indicate that the vaccine efficacy is a key parameter that affects optimal control of vaccination dynamics during an epidemic, raising an important question on the trade-off between effectiveness and cost-effectiveness of vaccination policies in the context of limited vaccine quantities.

2014, 11(5): 1065-1090
doi: 10.3934/mbe.2014.11.1065

*+*[Abstract](2022)*+*[PDF](1159.2KB)**Abstract:**

Circular migrations are the periodic movement of individuals between multiple locations, observed in parts of sub-Saharan Africa. Relationships between circular migrations and HIV are complex, entailing interactions between migration frequency, partnership structure, and exposure to acute HIV infection. Mathematical modeling is a useful tool for understanding these interactions.

Two modeling classes have dominated the HIV epidemiology and policy literature for the last decade: one a form of compartmental models, the other network models. We construct models from each class, using ordinary differential equations and exponential random graph models, respectively.

Our analysis suggests that projected HIV prevalence is highly sensitive to the choice of modeling framework. Assuming initial equal HIV prevalence across locations, compartmental models show no association between migration frequency and HIV prevalence or incidence, while network models show that migrations at frequencies shorter than the acute HIV period predict greater HIV incidence and prevalence compared to longer migration periods. These differences are statistically significant when network models are extended to incorporate a requirement for migrant men's multiple partnerships to occur in different locations. In settings with circular migrations, commonly-used forms of compartmental models appear to miss key components of HIV epidemiology stemming from interactions of relational and viral dynamics.

2014, 11(5): 1091-1113
doi: 10.3934/mbe.2014.11.1091

*+*[Abstract](2401)*+*[PDF](537.2KB)**Abstract:**

In this paper, we consider the evolutionary competition between budding and lytic viral release strategies, using a delay differential equation model with distributed delay. When antibody is not established, the dynamics of competition depends on the respective basic reproductive ratios of the two viruses. If the basic reproductive ratio of budding virus is greater than that of lytic virus and one, budding virus can survive. When antibody is established for both strains but the neutralization capacities are the same for both strains, consequence of the competition also depends only on the basic reproductive ratios of the budding and lytic viruses. Using two concrete forms of the viral production functions, we are also able to conclude that budding virus will outcompete if the rates of viral production, death rates of infected cells and neutralizing capacities of the antibodies are the same for budding and lytic viruses. In this case, budding strategy would have an evolutionary advantage. However, if the antibody neutralization capacity for the budding virus is larger than that for the lytic virus, the lytic virus can outcompete the budding virus provided that its reproductive ratio is very high. An explicit threshold is derived.

Transmission dynamics and control for a brucellosis model in Hinggan League of Inner Mongolia, China

2014, 11(5): 1115-1137
doi: 10.3934/mbe.2014.11.1115

*+*[Abstract](2439)*+*[PDF](374.8KB)**Abstract:**

Brucellosis is one of the major infectious and contagious bacterial diseases in Hinggan League of Inner Mongolia, China. The number of newly infected human brucellosis data in this area has increased dramatically in the last 10 years. In this study, in order to explore effective control and prevention measures we propose a deterministic model to investigate the transmission dynamics of brucellosis in Hinggan League. The model describes the spread of brucellosis among sheep and from sheep to humans. The model simulations agree with newly infected human brucellosis data from 2001 to 2011, and the trend of newly infected human brucellosis cases is given. We estimate that the control reproduction number $\mathcal{R}_{c}$ is about $1.9789$ for the brucellosis transmission in Hinggan League and compare the effect of existing mixed cross infection between basic ewes and other sheep or not for newly infected human brucellosis cases. Our study demonstrates that combination of prohibiting mixed feeding between basic ewes and other sheep, vaccination, detection and elimination are useful strategies in controlling human brucellosis in Hinggan League.

2014, 11(5): 1139-1166
doi: 10.3934/mbe.2014.11.1139

*+*[Abstract](2020)*+*[PDF](718.7KB)**Abstract:**

We investigate the question of optimal substrate removal in a biofilm reactor with concurrent suspended growth, both with respect to the amount of substrate removed and with respect to treatment process duration. The water to be treated is fed externally from a buffer vessel to the treatment reactor. In the two-objective optimal control problem, the flow rate between the vessels is selected as the control variable. The treatment reactor is modelled by a system of three ordinary differential equations in which a two-point boundary value problem is embedded. The solution of the associated singular optimal control problem in the class of measurable functions is impractical to determine and infeasible to implement in real reactors. Instead, we solve the simpler problem to optimize reactor performance in the class of off-on functions, a choice that is motivated by the underlying biological process. These control functions start with an initial no-flow period and then switch to a constant flow rate until the buffer vessel is empty. We approximate the Pareto Front numerically and study the behaviour of the system and its dependence on reactor and initial data. Overall, the modest potential of control strategies to improve reactor performance is found to be primarily due to an initial transient period in which the bacteria have to adapt to the environmental conditions in the reactor, i.e. depends heavily on the initial state of the dynamic system. In applications, the initial state, however, is often unknown and therefore the efficiency of reactor optimization, compared to the uncontrolled system with constant flow rate, is limited.

2014, 11(5): 1167-1174
doi: 10.3934/mbe.2014.11.1167

*+*[Abstract](1909)*+*[PDF](308.9KB)**Abstract:**

The known nonlinear mathematical model of the Glassy-winged Sharpshooter is considered. It is assumed that this model is influenced by stochastic perturbations of the white noise type and these perturbations are directly proportional to the deviation of the system state from the positive equilibrium point. A necessary and sufficient condition for asymptotic mean square stability of the equilibrium point of the linear part of the considered stochastic differential equation is obtained. This condition is at the same time a sufficient one for stability in probability of the equilibrium point of the initial nonlinear equation. Numerical calculations and figures illustrate the obtained results.

2014, 11(5): 1175-1180
doi: 10.3934/mbe.2014.11.1175

*+*[Abstract](1823)*+*[PDF](280.7KB)**Abstract:**

A recent paper by L. Wang, X. Wang

*J. Theoret. Biol.*300:100--109 (2012) formulated and studied a delay differential equation model for disease dynamics in a region where a portion of the population leaves to work in a different region for an extended fixed period. Upon return, a fraction of the migrant workers have become infected with the disease. The global dynamics were not fully resolved in that paper, but are resolved here. We show that for all parameter values and all delays, the unique equilibrium is globally asymptotically stable, implying that the disease will eventually reach a constant positive level in the population.

2014, 11(5): 1181-1198
doi: 10.3934/mbe.2014.11.1181

*+*[Abstract](2568)*+*[PDF](871.0KB)**Abstract:**

In this paper, we propose a mathematical model for HIV-1 infection with intracellular delay. The model examines a viral-therapy for controlling infections through recombining HIV-1 virus with a genetically modified virus. For this model, the basic reproduction number $\mathcal{R}_0$ are identified and its threshold properties are discussed. When $\mathcal{R}_0 < 1$, the infection-free equilibrium $E_0$ is globally asymptotically stable. When $\mathcal{R}_0 > 1$, $E_0$ becomes unstable and there occurs the single-infection equilibrium $E_s$, and $E_0$ and $E_s$ exchange their stability at the transcritical point $\mathcal{R}_0 =1$. If $1< \mathcal{R}_0 < R_1$, where $R_1$ is a positive constant explicitly depending on the model parameters, $E_s$ is globally asymptotically stable, while when $\mathcal{R}_0 > R_1$, $E_s$ loses its stability to the double-infection equilibrium $E_d$. There exist a constant $R_2$ such that $E_d$ is asymptotically stable if $R_1<\mathcal R_0 < R_2$, and $E_s$ and $E_d$ exchange their stability at the transcritical point $\mathcal{R}_0 =R_1$. We use one numerical example to determine the largest range of $\mathcal R_0$ for the local stability of $E_d$ and existence of Hopf bifurcation. Some simulations are performed to support the theoretical results. These results show that the delay plays an important role in determining the dynamic behaviour of the system. In the normal range of values, the delay may change the dynamic behaviour quantitatively, such as greatly reducing the amplitudes of oscillations, or even qualitatively changes the dynamical behaviour such as revoking oscillating solutions to equilibrium solutions. This suggests that the delay is a very important fact which should not be missed in HIV-1 modelling.

2014, 11(5): 1199-1214
doi: 10.3934/mbe.2014.11.1199

*+*[Abstract](2405)*+*[PDF](2649.8KB)**Abstract:**

Vulnerable plaques are a subset of atherosclerotic plaques that are prone to rupture when high stresses occur in the cap. The roles of residual stress, plaque morphology, and cap stiffness on the cap stress are not completely understood. Here, arteries are modeled within the framework of nonlinear elasticity as incompressible cylindrical structures that are residually stressed through differential growth. These structures are assumed to have a nonlinear, anisotropic, hyperelastic response to stresses in the media and adventitia layers and an isotropic response in the intima and necrotic layers. The effect of differential growth on the peak stress is explored in a simple, concentric geometry and it is shown that axial differential growth decreases the peak stress in the inner layer. Furthermore, morphological risk factors are explored. The peak stress in residually stressed cylinders is not greatly affected by changing the thickness of the intima. The thickness of the necrotic layer is shown to be the most important morphological feature that affects the peak stress in a residually stressed vessel.

2014, 11(5): 1215-1227
doi: 10.3934/mbe.2014.11.1215

*+*[Abstract](2048)*+*[PDF](224.3KB)**Abstract:**

In this paper, we consider the spatial dynamics for a non-cooperative diffusion system arising from epidermal wound healing. We shall establish the spreading speed and existence of traveling waves and characterize the spreading speed as the slowest speed of a family of non-constant traveling wave solutions. We also construct some new types of entire solutions which are different from the traveling wave solutions and spatial variable independent solutions. The traveling wave solutions provide the healing speed and describe how wound healing process spreads from one side of the wound. The entire solution exhibits the interaction of several waves originated from different locations of the wound. To the best of knowledge of the authors, it is the first time that it is shown that there is an entire solution in the model for epidermal wound healing.

2014, 11(5): 1229-1245
doi: 10.3934/mbe.2014.11.1229

*+*[Abstract](3709)*+*[PDF](811.7KB)**Abstract:**

A compartmental deterministic model is proposed to evaluate the effectiveness of transmission-blocking vaccines of malaria, which targets at the parasite stage in the mosquito. The model is rigorously analyzed and numerical simulations are performed. The results and implications are discussed.

2018 Impact Factor: 1.313

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