Abstract: Antiretroviral treatment (ART) and oral pre-exposure prophylaxis (PrEP) have recently been used efficiently in management of HIV infection. Pre-exposure prophylaxis consists in the use of an antiretroviral medication to prevent the acquisition of HIV infection by uninfected individuals. We propose a new model for the transmission of HIV/AIDS including ART and PrEP. Our model can be used to test the effects of ART and of the uptake of PrEP in a given population, as we demonstrate through simulations. The model can also be used to estimate future projections of HIV prevalence. We prove global stability of the disease-free equilibrium. We also prove global stability of the endemic equilibrium for the most general case of the model, i.e., which allows for PrEP individuals to default. We include insightful simulations based on recently published South-African data. PubDate: 2018-01-11

Abstract: In this paper, a new rectifying action is combined into different proportional-α-order-derivative-type iterative learning control algorithms for a class of fractional order linear time-invariant systems. Unlike the existing fractional order iterative learning control techniques, the proposed algorithms allow the initial state value of a fractional order iterative learning control system at each iteration to shift randomly. By introducing the Lebesgue-p norm and using the method of fractional integration by parts and the generalized Young inequality of convolution integral, the tracking performances with respect to the initial state shift under the proposed algorithms are analyzed. These analyses show that the tracking errors are incurred by such a shift and improved by tuning the rectifying gain. Numerical simulations are performed to demonstrate the effectiveness of the proposed algorithms. PubDate: 2018-01-11

Abstract: Let \(I\subset\mathbb{R}\) be an open interval with \(0\in I\) , and let \(g\in C^{1}(I, (0,+\infty))\) . Let \(N\in\mathbb{N}\) be an integer with \(N\geq4\) , \([2, N-1]_{\mathbb{Z}}:=\{2, 3,\ldots,N-1\}\) . We are concerned with the existence of solutions for the discrete Neumann problem $$\textstyle\begin{cases} \nabla(k^{n-1}\frac{\triangle v_{k}}{\sqrt{1-(\triangle v_{k})^{2}}} )=nk^{n-1}[-\frac{ g'(\psi^{-1}(v_{k}))}{\sqrt{1-(\triangle v_{k})^{2}}}+g(\psi^{-1}(v_{k}))H(\psi^{-1}(v_{k}),k)],\quad k\in[2, N-1]_{\mathbb{Z}},\\ \Delta v_{1}=0=\Delta v_{N-1} \end{cases} $$ which is a discrete analogue of the Neumann problem about the rotationally symmetric spacelike graphs with a prescribed mean curvature function in some Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes, where \(\psi(s):=\int_{0}^{s}\frac{dt}{g(t)}\) , \(\psi ^{-1}\) is the inverse function of ψ, and \(H:\mathbb{R}\times[2, N-1]_{\mathbb{Z}}\to\mathbb{R}\) is continuous with respect to the first variable. The proofs of the main results are based upon the Brouwer degree theory. PubDate: 2018-01-10

Abstract: In this paper, we investigate the existence results of a fourth-order differential equation with multi-strip integral boundary conditions. Our analysis relies on the shooting method and the Sturm comparison theorem. Finally, an example is discussed for the illustration of the main work. PubDate: 2018-01-10

Abstract: In this paper, we investigate the existence and uniqueness of solutions for a fractional boundary value problem involving the p-Laplacian operator. Our analysis relies on some properties of the Green function and the Guo-Krasnoselskii fixed point theorem and the Banach contraction mapping principle. Two examples are given to illustrate our theoretical results. PubDate: 2018-01-10

Abstract: The synchronization problem for a class of fractional-order complex dynamical networks with and without time-varying delay is investigated in this paper. By utilizing generalized Barbalat’s lemma, Razumikhin-type stability theory and matrix inequality technique, some sufficient criteria ensuring synchronization under pinning control and pinning adaptive feedback control are derived. Finally, three numerical simulations are presented to demonstrate the effectiveness of the obtained results. PubDate: 2018-01-10

Abstract: In the present paper, we employ a wavelets optimization method is employed for the elucidations of fractional partial differential equations of pricing European option accompanied by a Lévy model. We apply the Legendre wavelets optimization method (LWOM) to optimize the governing problem. The novelty of the proposed method is the inclusion of differential evolution algorithm (DE) in the Legendre wavelets method for the optimized approximations of the unknown terms of the Legendre wavelets. Sequentially, the functions and components of the pricing models are discretized by utilizing the operational matrix of fractional integration of Legendre wavelets. Illustratively, the implementation of the LWOM is exemplified on a pricing European option Lévy model and successfully depicted the stock paths. Moreover, comparison analysis of the Black-Scholes model with a class of Lévy model and LWOM with q-homotopy analysis transform method (q-HATM) is also deliberated out. Accordingly, the technique is found to be appropriate for financial models that can be expressed as partial differential equations of integer and fractional orders, subjected to initial or boundary conditions. PubDate: 2018-01-10

Abstract: In this paper, we aim to investigate q-analogues of the natural transform on various elementary functions of special type. We obtain results associated with classes of q-convolution products, Heaviside functions, q-exponential functions, q-hyperbolic functions and q-trigonometric functions as well. Further, we give definitions and derive results involving some q-differential operators. PubDate: 2018-01-10

Abstract: In this paper, we put forward a fractional-order survival red blood cells model and study the dynamics through the Hopf bifurcation. When the delay transcends the threshold, a series of Hopf bifurcations occur at the positive equilibrium. Then, a fractional-order Proportional and Derivative ( \(\mathit{PD}^{\alpha} \) ) controller is applied to the proposed model for the Hopf bifurcation control. It is discovered that by setting proper parameters, the \(\mathit{PD}^{\alpha} \) controller can delay or advance the onset of Hopf bifurcations. Therefore the Hopf bifurcation of the fractional-order survival red blood cells model becomes controllable to achieve desirable behaviors. Finally, numerical examples are presented to demonstrate the theoretical analysis. PubDate: 2018-01-10

Abstract: In this paper, we investigate the existence of extremal solutions for fractional differential systems involving the p-Laplacian operator and Riemann-Liouville integral boundary conditions. We derive our results based on the monotone iterative technique, combined with the method of upper and lower solutions. An example is added to illustrate the main result. PubDate: 2018-01-08

Abstract: We study the burden of the HIV viremia and of treatment efficacy in the severity of the patterns of the HIV/HCV coinfection. For this, we derive a simple non-integer-order (fractional-order) model for the coinfection dynamics. Fractional-order models have been proved in the literature to provide good fits to real data from patients suffering from several diseases, such as HIV, dengue fever, and others. We have computed the basic reproduction number and the stability of the disease-free equilibrium of the model. The numerical results suggest that the HIV viral load impacts impressively the severity of the HCV infection. The treatment efficacy is also found to influence the natural progression of HCV on the HIV/HCV coinfection. The latter is repeated for all values of the order of the fractional derivative. Moreover, the fractional derivative may pave the way to better understanding the individuals’ patients’ adjustments to treatment and to viremia. PubDate: 2018-01-05

Abstract: In this article, we discuss a new coupled system of fractional differential equations with integral boundary conditions $$\textstyle\begin{cases} D^{\alpha}u(t)+f(t,v(t))=a,\quad 0< t< 1,\\ D^{\beta}v(t)+g(t,u(t))=b,\quad 0< t< 1,\\ u(0)=0,\qquad u(1)=\int_{0}^{1} \phi(t)u(t)\,dt,\\ v(0)=0,\qquad v(1)=\int_{0}^{1} \psi(t)v(t)\,dt, \end{cases} $$ where \(1< \alpha,\beta\le2, f,g \in C([0,1]\times(-\infty,+\infty ),(-\infty,+\infty)), \phi,\psi\in L^{1}[0,1]\) , \(a,b\) are constants and D denotes the usual Riemann-Liouville fractional derivative. Based upon a fixed point theorem of increasing φ- \((h,e)\) -concave operators, we establish the existence and uniqueness of solutions for the new coupled system dependent on two constants. And then the obtained result is well demonstrated with the aid of an interesting example. PubDate: 2018-01-04

Abstract: In this paper, we study the existence of periodic solutions for Rayleigh equation with a singularity of repulsive type $$x''(t)+f\bigl(x'(t)\bigr)+\varphi (t)x(t)-\frac{1}{x^{\alpha }(t)}=p(t), $$ where \(\alpha \geqslant 1\) is a constant, and φ and p are T-periodic functions. The proof of the main result relies on a known continuation theorem of coincidence degree theory. The interesting point is that the sign of the function \(\varphi (t)\) is allowed to change for \(t\in [0,T]\) . PubDate: 2017-12-22

Abstract: In this paper, we build a multispecies predator-prey model with mutual interference and time delays. By means of the comparison theorem, Ascoli theorem and Lebesgue dominated convergence theorem, we establish the sufficient conditions of permanence and investigate the existence of a unique almost periodic solution. By constructing a suitable Lyapunov function, we obtain that the positive almost periodic solution is globally attractive. Finally, we give numerical simulations to indicate the complex dynamical behaviors of this system. PubDate: 2017-12-21

Abstract: We are interested in the persistence in mean and extinction for a stochastic competitive Gilpin-Ayala system with regime switching. Based on the stochastic LaSalle theorem and the space-decomposition method, we derive generalized sufficient criteria on persistence in mean and extinction. By constructing a novel Lyapunov function we establish sufficient criteria on partial persistence in mean and partial extinction for the system. Finally, we provide two examples to demonstrate the feasibility and validity of our proposed methods. PubDate: 2017-12-21

Abstract: In this paper, we mainly discuss the uniqueness problem when an entire function shares 0 CM and nonzero complex constant a IM with its difference operator. We also consider the general case where they share two distinct complex constants \(a^{*}\) CM and a IM under some additional condition and give some further discussions. PubDate: 2017-12-20

Abstract: A novel stochastic turbidostat model is investigated in this paper. The stochasticity in the model comes from the maximal growth rate influenced by white noise. Firstly, the existence and uniqueness of the positive solution for the system are demonstrated. Secondly, we analyze the persistence in mean and stochastic persistence of the system, respectively. Sufficient conditions about the extinction of the microorganism are obtained. Finally, numerical simulation results are given to support the theoretical conclusions. PubDate: 2017-12-20

Abstract: In this paper, we investigate some interesting identities on the Bernoulli, Euler, Hermite and generalized Gegenbauer polynomials arising from the orthogonality of generalized Gegenbauer polynomials in the generalized inner product $$\bigl\langle {{p_{1}}(x),{p_{2}}(x)} \bigr\rangle = \int_{ - \frac{{\sqrt{\alpha q}}}{p}}^{\frac{{\sqrt{ \alpha q} }}{p}} {{\bigl(\alpha q - p^{2}{x^{2}}\bigr)}^{\lambda - \frac{1}{2}}} {p_{1}}(x){p_{2}}(x)\,dx. $$ PubDate: 2017-12-20

Abstract: Over-harvesting of forestry resource, primarily trees, through illegal logging has been exceptionally regular for decades. In the context of this worldwide issue, a mathematical model considering the joined impact of legal and illegal logging of trees from forestry biomass using delay-driven ordinary differential equations is proposed. For the set of equations, we have taken immature, mature forestry biomass, and industrial densities as three state variables. Additionally, the effect of time-lag for the conversion of immature forestry biomass to mature forestry biomass is considered. System boundedness, feasible equilibrium analysis and the stability of all the feasible equilibria is examined using the differential equation theory. From the detailed analysis of the system, it is observed that with the delay in time, the system bifurcates as it reaches the critical threshold. While without the delay, the system is asymptotically stable. Biologic and bio-economic results of the system are also interpreted for the optimal equilibrium solutions. The optimal path is obtained by constructing the Hamiltonian, which is further solved using Pontryagin”s principle associated with the control problem. Further, numerical simulation is also provided in support of analytical results. Moreover, the normalized forward sensitivity index is used to analyze the parameter sensitivity. PubDate: 2017-12-19

Abstract: In this paper, the Hopf bifurcation control for a Lotka-Volterra predator-prey model with two delays is studied by using a hybrid control strategy. By analyzing the associated characteristic equation, its local stability and the existence of Hopf bifurcation with respect to both delays are established. In addition, the onset of an inherent bifurcation is delayed. Based on the normal form theory and the center manifold theorem, explicit formulas are derived to determine the direction of Hopf bifurcation and stability of the bifurcating periodic solution. Numerical simulation results confirm that the hybrid controller is efficient in controlling Hopf bifurcation. PubDate: 2017-12-19