Tarbiat Modares University
Department of Maths
Tarbiat Modares University
Iran University of Science and Technology
Two Itô stochastic differential equation (SDE) systems are constructed for a Susceptible-Infected-Susceptible epidemic model with temporary vaccination. A constant number of new members enter the population and total size of the population is variable. Some conditions for disease extinction in the stochastic models are established and compared with conditions in deterministic one. It is shown that the two stochastic models are equivalent in the sense that their solutions come from same distribution. In addition, the SDE models are simulated and the equivalence of the two stochastic models is confirmed by numerical examples. The probability distribution for extinction is also obtained numerically, provided there exists a probability for disease persistence whereas the expected duration of epidemic is acquired when extinction occurs with probability 1.
A discrete-time SIS epidemic model with vaccination is introduced and formulated by a system of difference equations. Some necessary and sufficient conditions for asymptotic stability of the equilibria are obtained. Furthermore, a sufficient condition is also presented. Next, bifurcations of the model including transcritical bifurcation, period-doubling bifurcation, and the Neimark-Sacker bifurcation are considered. In addition, these issues will be studied for the corresponding model with constant population size. Dynamics of the model are also studied and compared in detail with those found theoretically by using bifurcation diagrams, analysis of eigenvalues of the Jacobian matrix, Lyapunov exponents and solutions of the models in some examples.
As far as we are aware, no research has been published about two-dimensional integral equations in the complex plane by using Haar bases or any other kinds of wavelets. We introduce a numerical method to solve two-dimensional Fredholm integral equations, using Haar wavelet bases. To attain this purpose, first, an operator and then an orthogonal projection should be defined. Regarding the characteristics of Haar wavelet, we solve an integral equation without using common mathematical methods. We prove the convergence and an upper bound that mentioned in the method by employing the Banach fixed point theorem. Moreover, the rate of convergence our method is O(n(2q)n). We present several examples of different kinds of functions and solve them by this method in this study.
Introduction: Infectious diseases threaten the public health; hence understanding their propagation mechanisms may help to control them. Mathematical models are tools that can help the scientists to understand the pathogens’ propagations and can provide strategies for their control in future. Methods: Using mathematical theorems and MATLAB software, a continuous-time model known as susceptible-infected-susceptible (SIS) for transmission of infection in a population was described and the effects of a vaccination program based on this framework was investigated. Results: It was shown that the model had two equilibria: the infection-free equilibrium and the infected equilibrium. A specific threshold in terms of model parameters was obtained and then the existence of the equilibria and asymptotic stability of the system were stated with respect to this threshold. The theoretical results were also verified numerically by providing several simulations. Conclusion: The results indicated the stability of this model which emphasized that parameters such as restricting the immigration, reducing harmful contacts between the susceptible and the infected individuals, increasing awareness level of people, and most-importantly vaccination will reduce the basic reproduction number and help to control the disease. Moreover, a relation to calculate the minimum doses for vaccinating of the new-comers and the susceptible individuals, was obtained.
In this paper, a VEISV network worm attack model is investigated. It is established that the worm-free equilibrium is locally as well as globally asymptotically stable when R0 < 1. When R0 > 1, the local and global asymptotic stability of the worm-epidemic equilibrium are derived employing the second additive compound matrix approach and the direct Lyapunov method, respectively.
We study an SIS epidemic model with a constant recruitment. The disease-related death is included in the model and total population size is variable. A vaccination program also affects both new members and susceptible individuals. Two equilibria of the model; the disease-free equilibrium (DFE) and the endemic equilibrium (EE), and the basic reproduction number R 0 , are obtained. It is shown that DFE is locally and also globally asymptotically stable if R 0 < 1 . Furthermore, it is proven that EE is locally asymptotically stable when R 0 > 1 . In addition, in this case some conditions for global asymptotic stability of EE are found by using Lyapunov’s direct method. Finally, some numerical simulations are presented to verify obtained theoretical results.
An SIS type epidemic model with variable population size is considered. The model includes a temporary vaccination program to prevent individuals from infection and to eradicate the disease. If R0 < 1, the disease-free equilibrium is locally and globally asymptotically stable i.e. the disease will be wiped out from population. When R0 > 1, the endemic equilibrium is locally asymptotically stable employing a result in stability of the second additive compound matrix. In addition, by using a geometric approach it is shown that this equilibrium is also globally asymptotically stable. So in this case, the disease will persist in population permanently. Also, a brie y discussion is made on the minimum amount of vaccination which is necessary to eradicate the disease. Finally, some numerical examples are given to conrm the obtained results.
This paper investigates an SIS epidemic model with variable population size including a vaccination program. Dynamics of the endemic equilibrium of the model are obtained, and it will be shown that this equilibrium exists and is locally asymptotically stable when R0 > 1 . In this case, the disease uniformly persists, and moreover, using a geometric approach we conclude that the model is globally asymptotically stable under some conditions. Also, a numerical discussion is given to verify the theoretical results.
در این مقاله یک مدل ریاضی اپیدمی فازی برای انتشار ویروسها در یک شبکه کامپیوتری بررسی میشود. در این کار مقایسهای بین عدد مولد کلاسیک و نوع فازی آن انجام شده است. همچنین استراتژی کنترل اپیدمی ویروسها در یک شبکه کامپیوتری تجزیه و تحلیل شده است. توصیف عددی روش با شبیه سازی و حل مدل نیز ارائه شده است
The spread of infectious diseases in populations and how to control and eliminate them from the population is an important and necessary subject. Mthematical models are introduced to study what happens when an infection enters in a popution, and under which conditions the disease will be wiped out from population or persists in population.The literature about mathematical epidemic models that have been constructed and analysed for various type of diseases is very reach. Among these models the susceptibleinfected-susceptible (SIS) epidemic models are one of the well known type of epidemic models. For the purpose of considering the effect of vaccination as a efficient strategy to control and aliminate infections, it is possible to add a compartment as the vaccination individuals to the SIS model and obtain the SIS epidemic model with vaccination namely, SIVS epidemic model. These models may be deterministic or stochastic, with constant or variable population size, and with standard or bilinear incidence rate. In this paper, we consider a discrete-time SIS epidemic model with vaccination. We shall obtain some basic properties of this model such as: the equilibria and the basic reproduction number R0. Then stability of the equilibria is given with respect to R0 and moreover, the bifurcations of model are studied. Also, the results are challenged in some numerical examples.
A continuous-time model for transmission of infection in a population is investigated. It is shown that the model has two equilibria: the infection-free equilibrium and the infected equilibrium. An specific threshold in terms of model parameters is obtained and then existence of the equilibria and asymptotic stability of the system are stated by means of this threshold. Finally, the results are also considered numerically.