An unconditionally stable nonstandard finite difference method applied to a mathematical model of HIV infection

We formulate and analyze an unconditionally stable nonstandard finite difference method for a mathematical model of HIV transmission dynamics. The dynamics of this model are studied using the qualitative theory of dynamical systems. These qualitative features of the continuous model are preserved by the numerical method that we propose in this paper. This method also preserves the positivity of the solution, which is one of the essential requirements when modeling epidemic diseases. Robust numerical results confirming theoretical investigations are provided. Comparisons are also made with the other conventional approaches that are routinely used for such problems.IS

Implicit-explicit predictor-corrector methods combined with improved spectral methods for pricing European style vanilla and exotic options

In this paper we present a robust numerical method to solve several types of European style option pricing problems. The governing equations are described by variants of Black-Scholes partial differential equations (BS-PDEs) of the reaction-diffusion-advection type. To discretise these BS-PDEs numerically, we use the spectral methods in the asset (spatial) direction and couple them with a third-order implicit-explicit predictor-corrector (IMEX-PC) method for the discretisation in the time direction. The use of this high-order time integration scheme sustains the better accuracy of the spectral methods for which they are well-known. Our spectral method consists of a pseudospectral formulation of the BS-PDEs by means of an improved Lagrange formula. On the other hand, in the IMEX-PC methods, we integrate the diffusion terms implicitly whereas the reaction and advection terms are integrated explicitly. Using this combined approach, we first solve the equations for standard European options and then extend this approach to digital options, butterfly spread options, and European calls in the Heston model. Numerical experiments illustrate that our approach is highly accurate and very efficient for pricing financial options such as those described above

A robust spectral method for pricing of American put options on zero-coupon bonds

American put options on a zero-coupon bond problem is reformulated as a linear complementarity problem of the option value and approximated by a nonlinear partial differential equation. The equation is solved by an exponential time differencing method combined with a barycentric Legendre interpolation and the Krylov projection algorithm. Numerical examples shows the stability and good accuracy of the method. A bond is a financial instrument which allows an investor to loan money to an entity (a corporate or governmental) that borrows the funds for a period of time at a fixed interest rate (the coupon) and agrees to pay a fixed amount (the principal) to the investor at maturity. A zero-coupon bond is a bond that makes no periodic interest payments

A Comprehensive Survey of Intrusion Detection Systems

Alongside with digital signatures and Cryptographic protocols, Intrusion Detection Systems (IDS) are judged to be the final contour of protection to protect a system. But the major difficulty with today’s mainly admired IDSs (Intrusion Detection System) is the invention of massive quantity of false positive (FP) alerts alongside with the true positive (TP) alerts, which is an awkward assignment for the operator to examine to arrange the proper responses. So, there is an immense requirement to discover this area of study and to discover a reasonable solution. A main disadvantage of Intrusion Detection Systems (IDSs), despite of their detection method, is the vast number of alerts they produce on a daily basis that can effortlessly exhaust security supervisors. This constraint has guide researchers in the IDS society to not only extend better detection algorithms and signature tuning methods, but to also focus on determining a variety of relations between individual alerts, formally known as alert correlation. There are a variety of approaches of intrusion detection, such as Pattern Matching, Machine Learning, Data Mining, and Measure Based Methods. This paper aims towards the proper survey of IDS so that researchers can make use of it and find the new techniques towards intrusions. Keywords: Intrusion Detection System, False positive alert, KDD Cup99, Anomaly detection, misuse detection, Machine Learning

A fitted operator method for tumor cells dynamics in their micro-environment

In this paper, we consider a quasi non-linear reaction-diffusion model designed to mimic tumor cells’ proliferation and migration under the influence of their micro-environment in vitro. Since the model can be used to generate hypotheses regarding the development of drugs which confine tumor growth, then considering the composition of the model, we modify the model by incorporating realistic effects which we believe can shed more light into the original model. We do this by extending the quasi non-linear reaction-diffusion model to a system of discrete delay quasi non-linear reaction-diffusion model. Thus, we determine the steady states, provide the conditions for global stability of the steady states by using the method of upper and lower solutions and analyze the extended model for the existence of Hopf bifurcation and present the conditions for Hopf bifurcation to occur. Since it is not possible to solve the models analytically, we derive, analyze, implement a fitted operator method and present our results for the extended model. Our numerical method is analyzed for convergence and we find that is of second order accuracy. We present our numerical results for both of the models for comparison purposes

A fitted operator method for a model arising in vascular tumor dynamics

In this paper, we consider a model for the population kinetics of human tumor cells in vitro, differentiated by phases of the cell division cycle and length of time within each phase. Since it is not easy to isolate the effects of cancer treatment on the cell cycle of human cancer lines, during the process of radiotherapy or chemotherapy, therefore, we include the spatial effects of cells in each phase and analyse the extended model. The extended model is not easy to solve analytically, because perturbation by cancer therapy causes the flow cytometric profile to change in relation to one another. Hence, making it difficult for the resulting model to be solved analytically. Thus, in [16] it is reported that the non-standard schemes are reliable and propagate sharp fronts accurately, even when the advection, reaction processes are highly dominant and the initial data are not smooth. As a result, we construct a fitted operator finite difference method (FOFDM) coupled with non-standard finite difference method (NSFDM) to solve the extended model. The FOFDM and NSFDM are analyzed for convergence and are seen that they are unconditionally stable and have the accuracy of O(Dt +(Dx)2), where Dt and Dx denote time and space step-sizes, respectively. Some numerical results confirming theoretical observations are presented

Numerical solution for a problem arising in angiogenic signalling

Since the process of angiogenesis is controlled by chemical signals, which stimulate both repair of damaged blood vessels and formation of new blood vessels, then other chemical signals known as angiogenesis inhibitors interfere with blood vessels formation. This implies that the stimulating and inhibiting e ects of these chemical signals are balanced as blood vessels form only when and where they are needed. Based on this information, an optimal control problem is formulated and the arising model is a system of coupled non-linear equations with adjoint and transversality conditions. Since many of the numerical methods often fail to capture these type of models, therefore, in this paper, we carry out steady state analysis of these models before implementing the numerical computations. In this paper we analyze and present the numerical estimates as a way of providing more insight into the postvascular dormant state where stimulator and inhibitor come into balance in an optimal manner

Performance of Richardson extrapolation on some numerical methods for a singularly perturbed turning point problem whose solution has boundary layers

Investigation of the numerical solution of singularly perturbed turning point problems dates back to late 1970s. However, due to the presence of layers, not many high order schemes could be developed to solve such problems. On the other hand, one could think of applying the convergence acceleration technique to improve the performance of existing numerical methods. However, that itself posed some challenges. To this end, we design and analyze a novel fitted operator finite difference method (FOFDM) to solve this type of problems. Then we develop a fitted mesh finite difference method (FMFDM). Our detailed convergence analysis shows that this FMFDM is robust with respect to the singular perturbation parameter. Then we investigate the effect of Richardson extrapolation on both of these methods. We observe that, the accuracy is improved in both cases whereas the rate of convergence depends on the particular scheme being used

A fitted numerical method for a model arising in HIV-related cancer-immune system dynamics

The effect of diseases such as cancer and HIV among our societies is evident. Thus, from the mathematical point of view many models has been developed with the aim to contribute towards understanding the dynamics of diseases. Therefore, in this paper we believe by extending a system of delay differential equations (DDEs) model of HIV related cancer-immune system to a system of delay partial differential equations (DPDEs) model of HIV related cancer-immune dynamics, we can contribute toward understanding the dynamics more clearly. Thus, we analyse the extended models and use the qualitative features of the extended model to derive, analyse and implement a fitted operator finite difference method (FOFDM) and present our results. This FOFDM is analyzed for convergence and it is seen that it has has second-order accuracy. We present some numerical results for some cases of the the model to illustrate the reliability of our numerical method

Security of User Data in Local Connectivity Using Multicast Key Agreement

In this paper, we be trained team key contract approach a couple of parties need to create a usual secret key to be used to alternate understanding securely. The staff key contract with an arbitrary connectivity graph, where each and every consumer is simplest mindful of his neighbor and has no information about the existence of different customers. Additional, he has no knowledge concerning the community topology. We put into effect the existing approach with extra time efficient method and provide a multicast key generation server which is predicted in future scope with the aid of present authors. We replace the Diffie Hellman key trade protocol through a brand new multicast key exchange protocol that may work with one to 1 and one to many functionality. We additionally tend to put into effect a robust symmetric encryption for improving file safety within the process