Ergodic stationary distribution of a stochastic rumor propagation model with general incidence function

In daily lives, when emergencies occur, rumors will spread widely on the Internet. However, it is quite difficult for the netizens to distinguish the truth of the information. The main reasons are the uncertainty of netizens' behavior and attitude, which make the transmission rates of these information among social network groups being not fixed. In this paper, we propose a stochastic rumor propagation model with general incidence function. The model can be described by a stochastic differential equation. Applying the Khasminskii method via a suitable construction of Lyapunov function, we first prove the existence of a unique solution for the stochastic model with probability one. Then we show the existence of a unique ergodic stationary distribution of the rumor model, which exhibits the ergodicity. We also provide some numerical simulations to support our theoretical results. The numerical results give us some possible methods to control rumor propagation that (1)increasing noise intensity can effectively reduce rumor propagation when $\widehat{\mathcal{R}}_{0}>1$. That is, after rumors spread widely on social network platforms, government intervention and authoritative media coverage will interfere with netizens' opinions, thus reducing the degree of rumor propagation; (2) Speed up the rumor refutation, intensify efforts to refute rumors, and improve the scientific quality of netizen(i.e. increase the value of $\beta$ and decrease the value of $\alpha$ and $\gamma$ ) can effectively curb rumor propagation.

Revisiting the number of self-incompatibility alleles in finite populations: from old models to new results

Under gametophytic self-incompatibility (GSI), plants are heterozygous at the self-incompatibility locus (S-locus) and can only be fertilized by pollen with a different allele at that locus. The last century has seen a heated debate about the correct way of modeling the allele diversity in a GSI population that was never formally resolved. Starting from an individual-based model, we derive the deterministic dynamics as proposed by Fisher (1958), and compute the stationary S-allele frequency distribution. We find that the stationary distribution proposed by Wright (1964) is close to our theoretical prediction, in line with earlier numerical confirmation. Additionally, we approximate the invasion probability of a new S-allele, which scales inversely with the number of resident S-alleles. Lastly, we use the stationary allele frequency distribution to estimate the population size of a plant population from an empirically obtained allele frequency spectrum, which complements the existing estimator of the number of S-alleles. Our expression of the stationary distribution resolves the long-standing debate about the correct approximation of the number of S-alleles and paves the way to new statistical developments for the estimation of the plant population size based on S-allele frequencies.

Exact tail asymptotics for a three-dimensional Brownian-driven tandem queue with intermediate inputs

In this paper, we consider a three-dimensional Brownian-driven tandem queue with intermediate inputs, which corresponds to a three-dimensional semimartingale reflecting Brownian motion whose reflection matrix is triangular. For this three-node tandem queue, no closed form formula is known, not only for its stationary distribution but also for the corresponding transform. We are interested in exact tail asymptotics for stationary distributions. By generalizing the kernel method, and using extreme value theory and copula, we obtain exact tail asymptotics for the marginal stationary distribution of the buffer content in the third buffer and for the joint stationary distribution.

Convergence of a particle approximation for the quasi-stationary distribution of a diffusion process: uniform estimates in a compact soft case. p, li { white-space: pre-wrap; }

We establish the convergences (with respect to the simulation time $t$; the number of particles $N$; the timestep $\gamma$) of a Moran/Fleming-Viot type particle scheme toward the quasi-stationary distribution of a diffusion on the $d$-dimensional torus, killed at a smooth rate. In these conditions, quantitative bounds are obtained that, for each parameter ($t\rightarrow \infty$, $N\rightarrow \infty$ or $\gamma\rightarrow 0$) are independent from the two others. p, li { white-space: pre-wrap; }

Dynamic Analysis of a Stochastic Rumor Propagation Model with Regime Switching

We study the rumor propagation model with regime switching considering both colored and white noises. Firstly, by constructing suitable Lyapunov functions, the sufficient conditions for ergodic stationary distribution and extinction are obtained. Then we obtain the threshold Rs which guarantees the extinction and the existence of the stationary distribution of the rumor. Finally, numerical simulations are performed to verify our model. The results indicated that there is a unique ergodic stationary distribution when Rs>1. The rumor becomes extinct exponentially with probability one when Rs<1.