Stationary distribution and density function analysis of SVIS epidemic model with saturated incidence and vaccination under stochastic environments.

In this article, we study the dynamical properties of susceptible-vaccinated-infected-susceptible (SVIS) epidemic system with saturated incidence rate and vaccination strategies. By constructing the suitable Lyapunov function, we examine the existence and uniqueness of the stochastic system. With the help of Khas'minskii theory, we set up a critical value [Formula: see text] with respect to the basic reproduction number [Formula: see text] of the deterministic system. A unique ergodic stationary distribution is investigated under the condition of [Formula: see text]. In the epidemiological study, the ergodic stationary distribution represents that the disease will persist for long-term behavior. We focus for developing the general three-dimensional Fokker-Planck equation using appropriate solving theories. Around the quasi-endemic equilibrium, the probability density function of the stochastic system is analyzed which is the main theme of our study. Under [Formula: see text], both the existence of ergodic stationary distribution and density function can elicit all the dynamical behavior of the disease persistence. The condition of disease extinction of the system is derived. For supporting theoretical study, we discuss the numerical results and the sensitivities of the biological parameters. Results and conclusions are highlighted.

An epidemic model through information-induced vaccination and treatment under fuzzy impreciseness.

In this work, we propose a nonlinear susceptible (S), vaccinated (V), infective (I), recovered (R), information level (U) (SVIRUS) model for the dynamical behavior of the contagious disease in human beings. We mainly consider the spread of information during the course of epidemic in the population. Different rate equations describe the dynamics of the information. We have developed the proposed model in crisp and fuzzy environments. In the fuzzy model, to describe the uncertainty prevailed in the dynamics, all the parameters are taken as triangular fuzzy numbers. Using graded mean integration value (GMIV) method, the fuzzy model is transformed into defuzzified model to represent the solutions avoiding the difficulties. The positivity and the boundedness of the crisp model are discussed elaborately and also the equilibrium analysis is accomplished. The stability analysis for both the infection free and the infected equilibrium are established for the crisp model. Application of optimal control of the crisp system is explored. Using Pontryagin's Maximum Principle, the optimal control is explained. The effect of vaccination is analyzed which leads the model to be complex in nature. The effect of saturation constant for information is described for the crisp model and also the effects of weight constants on control policy are discussed. Finally, it is concluded that the treatment is more fruitful and information related vaccination is more effective during the course of epidemic.