Threshold Behavior in a Stochastic SIS Epidemic Model with Standard Incidence

A stochastic SIS-type epidemic model with general nonlinear incidence and disease-induced mortality is investigated. It is proved that the dynamical behaviors of the model are determined by a certain threshold value . That is, when and together with an additional condition, the disease is extinct with probability one, and when , the disease is permanent in the mean in probability, and when there is not disease-related death, the disease oscillates stochastically about a positive number. Furthermore, when , the model admits positive recurrence and a unique stationary distribution. Particularly, the effects of the intensities of stochastic perturbation for the dynamical behaviors of the model are discussed in detail, and the dynamical behaviors for the stochastic SIS epidemic model with standard incidence are established. Finally, the numerical simulations are presented to illustrate the proposed open problems.

Classification of asymptotic behavior in a stochastic SIS epidemic model with vaccination

Stochastic epidemic metapopulation models on networks: SIS dynamics and control strategies

A stochastic SIS epidemic model with saturating contact rate

Qualitative study of a stochastic SIS epidemic model with vertical transmission

In this paper, a stochastic SIS epidemic model with birth pulses and pulse treatments is constructed and investigated. By applying Floquet theory and qualitative theory of stochastic differential equations, the existence and stability of trivial solution and infection-free periodic solution are discussed. A special endemic solution and the boundedness of infective individuals are investigated. Numerical results for phase portraits, which are illustrated with two examples, are in good agreement with the theoretical analysis.

A stochastic SEIR epidemic model with standard incidence and vertical transmission was developed in this work. The primary goal of this study was to determine whether stochastic environmental disturbances affect dynamic features of the epidemic model. The existence, uniqueness, and boundedness of global positive solutions are stated. A threshold was determined for the extinction of the infectious disease. After that, the existence and uniqueness of an ergodic stationary distribution were verified by determining the correct Lyapunov function. Ultimately, theoretical outcomes of numerical simulations are shown.

The dynamical properties of a stochastic susceptible-infected epidemic model with Logistic growth are investigated in this paper. We show that the stochastic model admits a nonnegative solution by using the Lyapunov function method. We then obtain that the infected individuals are persistent under some simple conditions. As a consequence, a simple sufficient condition that guarantees the extinction of the infected individuals is presented with a couple of illustrative examples.

Two new stochastic epidemic models, a continuous-time Markov chain model and a stochastic differential equation model, are formulated. These are based on a deterministic model that includes vaccination and is applicable to pertussis. For some parameter values, the deterministic model exhibits a backward bifurcation if the vaccine is imperfect. Thus a region of bistability exists in a subset of parameter space. The dynamics of the stochastic epidemic models are investigated in this region of bistability, and compared with those of the deterministic model. In this region the probability distribution associated with the infective population exhibits bimodality with one mode at the disease-free equilibrium and the other at the larger endemic equilibrium. For population sizes N=/>1000, the deterministic and stochastic models agree, but for small population sizes the stochastic models indicate that the backward bifurcation may have little effect on the disease dynamics.

A stochastic SIRS epidemic model with vaccination is discussed. A new stochastic threshold $ R_0^s $ is determined. When the noise is very low ($ R_0^s < 1 $), the disease becomes extinct, and if $ R_0^s > 1 $, the disease persists. Furthermore, we show that the solution of the stochastic model oscillates around the endemic equilibrium point and the intensity of the fluctuation is proportional to the intensity of the white noise. Computer simulations are used to support our findings.

Dynamic population epidemic models

This article represents a numerical analysis for a stochastic dengue epidemic model with incubation period of virus. We discuss the comparison of solutions between the stochastic dengue model and a deterministic dengue model. In this paper, we have shown that the stochastic dengue epidemic model is more realistic as compared to the deterministic dengue epidemic model. The effect of threshold number R_{1} holds in the stochastic dengue epidemic model. If R_{1} <1, then situation helps us to control the disease while R_{1} >1 shows the persistence of disease in population. Unfortunately, the numerical methods like Euler–Maruyama, stochastic Euler, and stochastic Runge–Kutta do not work for large time step sizes. The proposed framework of stochastic nonstandard finite difference scheme (SNSFD) is independent of step size and preserves all the dynamical properties like positivity, boundedness, and dynamical consistency.

Comparison of deterministic and stochastic SIS and SIR models in discrete time

Stochastic dynamics of an SIS epidemiological model with media coverage

This study employing a non-standard computational method for a stochastic anthrax epidemic model can enhance accuracy, evaluate control measures, and identify critical factors. The mathematical modeling of an anthrax disease includes the four-compartment of the population as susceptible animals (s), infected animals (i), carcasses animals (c), and grams spores of animals in the environment (a). The continuous model analysis (equilibria, reproduction number, and local stability of equilibria) is studied rigorously. The stochastic model is based on transition probabilities and parametric perturbation techniques. The fundamental properties of the model with standard computational methods such as Euler Maruyama, stochastic Euler, and stochastic Runge Kutta are studied. Unfortunately, these methods are time-dependent and even valid for short-period analysis of the disease. In comparison, the non-standard computational method, like the non-standard finite difference method nonstandard finite difference in the sense of stochastic, is designed for the given model. The non-standard computational method and its dynamical properties (positivity, boundedness, and dynamical consistency) are studied thoroughly. In the end, numerical results of the non-standard computational method with the existing standard computational methods are provided. These benefits contribute to a more comprehensive understanding of anthrax epidemiology and support effective decision-making in public health interventions.