# American Institute of Mathematical Sciences

ISSN:
1551-0018

eISSN:
1547-1063

All Issues

## Mathematical Biosciences & Engineering

June 2018 , Volume 15 , Issue 3

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2018, 15(3): 569-594 doi: 10.3934/mbe.2018026 +[Abstract](5230) +[HTML](772) +[PDF](468.66KB)
Abstract:

There is a growing interest to understand impacts of latent infection age and infection age on viral infection dynamics by using ordinary and partial differential equations. On one hand, activation of latently infected cells needs specificity antigen, and latently infected CD4+ T cells are often heterogeneous, which depends on how frequently they encountered antigens, how much time they need to be preferentially activated and quickly removed from the reservoir. On the other hand, infection age plays an important role in modeling the death rate and virus production rate of infected cells. By rigorous analysis for the model, this paper is devoted to the global dynamics of an HIV infection model subject to latency age and infection age from theoretical point of view, where the model formulation, basic reproduction number computation, and rigorous mathematical analysis, such as relative compactness and persistence of the solution semiflow, and existence of a global attractor are involved. By constructing Lyapunov functions, the global dynamics of a threshold type is established. The method developed here is applicable to broader contexts of investigating viral infection subject to age structure.

2018, 15(3): 595-627 doi: 10.3934/mbe.2018027 +[Abstract](4960) +[HTML](559) +[PDF](2524.87KB)
Abstract:

A diffusive intraguild predation model with delay and Beddington-DeAngelis functional response is considered. Dynamics including stability and Hopf bifurcation near the spatially homogeneous steady states are investigated in detail. Further, it is numerically demonstrated that delay can trigger the emergence of irregular spatial patterns including chaos. The impacts of diffusion and functional response on the model's dynamics are also numerically explored.

2018, 15(3): 629-652 doi: 10.3934/mbe.2018028 +[Abstract](5127) +[HTML](565) +[PDF](777.72KB)
Abstract:

We analyze a model of agent based vaccination campaign against influenza with imperfect vaccine efficacy and durability of protection. We prove the existence of a Nash equilibrium by Kakutani's fixed point theorem in the context of non-persistent immunity. Subsequently, we propose and test a novel numerical method to find the equilibrium. Various issues of the model are then discussed, such as the dependence of the optimal policy with respect to the imperfections of the vaccine, as well as the best vaccination timing. The numerical results show that, under specific circumstances, some counter-intuitive behaviors are optimal, such as, for example, an increase of the fraction of vaccinated individuals when the efficacy of the vaccine is decreasing up to a threshold. The possibility of finding optimal strategies at the individual level can help public health decision makers in designing efficient vaccination campaigns and policies.

2018, 15(3): 653-666 doi: 10.3934/mbe.2018029 +[Abstract](5402) +[HTML](665) +[PDF](574.43KB)
Abstract:

We propose an ultra-discretization for an SIR epidemic model with time delay. It is proven that the ultra-discrete model has a threshold property concerning global attractivity of equilibria as shown in differential and difference equation models. We also study an interesting convergence pattern of the solution, which is illustrated in a two-dimensional lattice.

2018, 15(3): 667-691 doi: 10.3934/mbe.2018030 +[Abstract](5743) +[HTML](626) +[PDF](904.11KB)
Abstract:

In this paper, we derive efficient drug treatment strategies for hepatitis B virus (HBV) infection by formulating a feedback control problem. We introduce and analyze a dynamic mathematical model that describes the HBV infection during antiviral therapy. We determine the reproduction number and then conduct a qualitative analysis of the model using the number. A control problem is considered to minimize the viral load with consideration for the treatment costs. In order to reflect the status of patients at both the initial time and the follow-up visits, we consider the feedback control problem based on the ensemble Kalman filter (EnKF) and differential evolution (DE). EnKF is employed to estimate full information of the state from incomplete observation data. We derive a piecewise constant drug schedule by applying DE algorithm. Numerical simulations are performed using various weights in the objective functional to suggest optimal treatment strategies in different situations.

2018, 15(3): 693-715 doi: 10.3934/mbe.2018031 +[Abstract](5646) +[HTML](628) +[PDF](600.09KB)
Abstract:

A diffusive predator-prey system with a delay and surplus killing effect subject to Neumann boundary conditions is considered. When the delay is zero, the prior estimate of positive solutions and global stability of the constant positive steady state are obtained in details. When the delay is not zero, the stability of the positive equilibrium and existence of Hopf bifurcation are established by analyzing the distribution of eigenvalues. Furthermore, an algorithm for determining the direction of Hopf bifurcation and stability of bifurcating periodic solutions is derived by using the theory of normal form and center manifold. Finally, some numerical simulations are presented to illustrate the analytical results obtained.

2018, 15(3): 717-738 doi: 10.3934/mbe.2018032 +[Abstract](6372) +[HTML](555) +[PDF](732.77KB)
Abstract:

Psoriasis is an autoimmune disorder, characterized by hyper-proli-feration of Keratinocytes for the abnormal activation of T Cells, Dendritic Cells (DCs) and cytokine signaling. Interaction of DCs and T Cells enable T Cell to differentiate into Type 1 (Th1), Type 2 (Th2) helper T Cell depending on cytokine release. Hyper-proliferation of Keratinocytes may occur due to over expression of pro-inflammatory cytokines secreted by Th1-Cells viz. Interferon gamma (\begin{document} $\mbox{IFN}-{γ}$ \end{document}), Transforming growth factor beta (\begin{document} $\mbox{TGF}-β$ \end{document}) and Tumor necrosis factor alpha (\begin{document} $\mbox{TNF}-α$ \end{document}) etc. Deregulation of epidermal happens due to signaling of anti-inflammatory cytokines like Interleukin 10 (\begin{document} $\mbox{IL}-{10}$ \end{document}), Interleukin 4 (\begin{document} $\mbox{IL}-{4}$ \end{document}) etc., released by Th2-Cells. In this article, we have constructed a set of nonlinear differential equations involving the above cell population for better understanding the impact of cytokines on Psoriasis. System is analyzed introducing therapeutic agent (Biologic / \begin{document} $\mbox{IL}-{10}$ \end{document}) for reducing the hyper-proliferation of Keratinocytes. Effect of Biologic is used as a surrogate of control parameter to reduce the psoriatic lesions. We also studied its effect both in continuous and impulsive dosing method. Our study reveals that impulsive dosing is more applicable compare with continuous dosing to prevent Psoriasis.

2018, 15(3): 739-764 doi: 10.3934/mbe.2018033 +[Abstract](6322) +[HTML](782) +[PDF](1052.69KB)
Abstract:

A Filippov epidemic model is proposed to explore the impact of capacity and limited resources of public health system on the control of epidemic diseases. The number of infected cases is chosen as an index to represent a threshold policy, that is, the capacity dependent treatment policy is implemented when the case number exceeds a critical level, and constant treatment rate is adopted otherwise. The proposed Filippov model exhibits various local sliding bifurcations, including boundary focus or node bifurcation, boundary saddle bifurcation and boundary saddle-node bifurcation, and global sliding bifurcations, including grazing bifurcation and sliding homoclinic bifurcation to pseudo-saddle. The impact of some key parameters including the threshold level on disease control is examined by numerical analysis. Our results suggest that strengthening the basic medical conditions, i.e. increasing the minimum treatment ratio, or enlarging the input of medical resources, i.e. increasing HBPR (i.e. hospital bed-population ratio) as well as the possibility and level of maximum treatment ratio, can help to contain the case number at a relatively low level when the basic reproduction number \begin{document}$R_0>1$\end{document}. If \begin{document}$R_0<1$\end{document}, implementing these strategies can help in eradicating the disease although the disease cannot always be eradicated due to the occurring of backward bifurcation in the system.

2018, 15(3): 765-773 doi: 10.3934/mbe.2018034 +[Abstract](4235) +[HTML](326) +[PDF](315.1KB)
Abstract:

2018, 15(3): 775-805 doi: 10.3934/mbe.2018035 +[Abstract](6801) +[HTML](565) +[PDF](999.16KB)
Abstract:

We propose and analyse a reaction-diffusion-advection predator-prey model in which we assume that predators move randomly but prey avoid predation by perceiving a repulsion along predator density gradient. Based on recent experimental evidence that anti-predator behaviors alone lead to a 40% reduction on prey reproduction rate, we also incorporate the cost of anti-predator responses into the local reaction terms in the model. Sufficient and necessary conditions of spatial pattern formation are obtained for various functional responses between prey and predators. By mathematical and numerical analyses, we find that small prey sensitivity to predation risk may lead to pattern formation if the Holling type Ⅱ functional response or the Beddington-DeAngelis functional response is adopted while large cost of anti-predator behaviors homogenises the system by excluding pattern formation. However, the ratio-dependent functional response gives an opposite result where large predator-taxis may lead to pattern formation but small cost of anti-predator behaviors inhibits the emergence of spatial heterogeneous solutions.

2018, 15(3): 807-825 doi: 10.3934/mbe.2018036 +[Abstract](4244) +[HTML](765) +[PDF](932.07KB)
Abstract:

Our aim is to propose a new robust and manageable technique, called multi-base harmonic balance method, to detect and characterize the periodic solutions of a nonlinear dynamical system. Our case test is the Hodgkin-Huxley model, one of the most realistic neuronal models in literature. This system, depending on the value of the external stimuli current, exhibits periodic solutions, both stable and unstable.

2018 Impact Factor: 1.313