On the asymptotical and practical stability of stochastic control systems

The asymptotical and practical stability in probability of stochastic control systems by means of feedback laws is provided. The main results of this work enable us to derive the sufficient conditions for the existence of control Lyapunov function that play a leading role in the existence of stabilizing feedback laws. Particularly, the sufficient conditions for practical stability in probability are established and numerical examples are also given to illustrate the usefulness of our results

Disease transmission MSEIR model with individuals traveling between patches i and i+1

In this article we want to formulate a disease transmission model, MSEIR model, for a population with individuals travelling between patches i and i + 1 and we derive an explicit formula for the basic repro- ductive number, R0, employing the spectral radius of the next generation operator. Also, in this article we show that a system of ordinary differen- tial equations for this model has a unique disease-free equilibrium and it is locally asymptotically stable if R0 1

Exponential input-to-state stability of composite stochastic systems

Sufficient conditions for the exponential input-to-state stability in probability in rth mean and for the almost sure exponential input-to-state stability in probability of a composite stochastic system are established. Illustrating example is provided to validate our results

Differential susceptible and staged progression model for HIV

In this paper we first review the mathematical formulation of the original differential infectivity DI and staged-progression SP models, then we formulate a HIV new model with differential susceptibility and staged-progression DSSP to account for variations in viral loads and in the rate of disease progression in infected individuals. Then we derive an explicit formula for the reproductive number of infection for this model, then we provide numerical example for it

Quadrupole mass filter with fuzzy initial conditions

The employment of the fuzzy method to solve differential equations has been well studied. In this article, Mathieu differential equations of the quadrupole mass filter (QMF) have been solved by using the fuzzy method. This method has not been yet investigated in the QMF with fuzzy initial conditions. We survey the physical properties of the confined ion. The results of numerical simulations are presented and discussed

The use of generation stochastic models to study an epidemic disease

Stochastic models have an important role in modeling and analyzing epidemic diseases for small size population. In this article, we study the generation of stochastic models for epidemic disease susceptible-infective-susceptible model. Here, we use the separation variable method to solve partial differential equation and the new developed modified probability generating function (PGF) of a random process to include a random catastrophe to solve the ordinary differential equations generated from partial differential equation. The results show that the probability function is too sensitive to μ, β and γ parameters

Optimum radius size between cylindrical ion trap and quadrupole ion trap

Quadrupole ion trap mass analyzer with a simplified geometry, namely, the cylindrical ion trap (CIT), has been shown to be well-suited using in miniature mass spectrometry and even in mass spectrometer arrays. Computation of stability regions is of particular importance in designing and assembling an ion trap. However, solving CIT equations are rather more difficult and complex than QIT equations, so, analytical and matrix methods have been widely used to calculate the stability regions. In this article we present the results of numerical simulations of the physical properties and the fractional mass resolutions of the confined ions in the first stability region was analyzed by the fifth order Runge-Kutta method (RKM5) at the optimum radius size for both ion traps. Because of similarity the both results, having determining the optimum radius, we can make much easier to design CIT. Also, the simulated results has been performed a high precision in the resolution of trapped ions at the optimum radius size

A new modification of the homotopy perturbation method

In this paper, a new modification of the homotopy perturbation method (HPM) is presented and applied to linear ordinary differential equations and nonlinear differential equations. A comparative study between the new modified homotopy perturbation method (MHPM) and the classical homotopy perturbation method (HPM) is conducted. Several illustrative examples are given to demonstrate the effectiveness and reliability of MHPM. The numerical results obtained from the MHPM and HPM are compared with the fourth-order Runge-Kutta method (RKM)

Behavior stability in two SIR-style models for HIV

In this article we want to compare the SIR model with the modified SIR model for HIV describing behavioral. In the modified SIR model for HIV we assume that high-infective and higher-infective individuals in infective class I to inter I1 and I2 classes. At last, we can say that the modified SIR model have bigger or equal stability region in comparison with the SIR model

Study of a quadrupole ion trap with damping force by the two-point one block method

RATIONALE: The capabilities and performances of a quadrupole ion trap under damping force based on collisional cooling is of particular importance in high-resolution mass spectrometry and should be analyzed by Mathieu's differential solutions. These solutions describe the stability and instability of the ion's trajectories confined in quadrupole devices. One of the methods for solving Mathieu's differential equation is a two-point one block method. In this case, Mathieu's stability diagram, trapping parameters az and qz and the secular frequency of the ion motion wz, can be derived in a precise manner. The two-point one block method (TPOBM) of Adams Moulton type is presented to study these parameters with and without the effect of damping force and compared to the 5th-order Runge–Kutta method (RKM5). The simulated results show that the TPOBM is more accurate and 10 times faster than the RKM5. The physical properties of the confined ions in the r and z axes are illustrated and the fractional mass resolutions m/Δm of the confined ions in the first stability region were analyzed by the RKM5 and the TPOBM. METHODS: The Lagrange interpolation polynomial was applied in the derivation of the proposed method. The proposed method will be utilized to obtain a series solution directly without reducing it to first order equations. RESULTS: The problem was tested with the ion trajectories in real time with and without the effect of damping force using constant step size. Numerical results from the two-point one block method have been compared with the fifth order Runge–Kutta method. CONCLUSIONS: The proposed two-point one block method has a potential application to solve complicated linear and nonlinear equations of the charged particle confinement in a quadrupole field especially in fine tuning accelerators, and, generally speaking, in physics of high energy