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The study of dynamics of epidemics, especially, double epidemics, has got paramount importance in the present days, since various health risks, an individual is exposed to due to the increasing global warming are unavoidable and/or due to limited medical resources available to handle suddenly developed disease. In view of this, we consider a study of an epidemic model with a double hypothesis, namely, SIR and SIRS mechanisms. First, we prove the local asymptotic stability of equilibrium points followed by the global stability of the disease for equilibrium established through the Lyapunov function. Next, we show the existence, extinction and persistence results of the stochastic system. Finally, some numerical examples are presented to support the theoretical results established. • Epidemic model with double hypothesis combined with SIR and SIRS transmission mechanisms. • Global existence of a positive solution. • Global stability of the disease-free equilibrium. • Extinction of the diseases. [ABSTRACT FROM AUTHOR]

In this paper, the dynamics of a stochastic SIR epidemic model with saturated incidence is investigated. Firstly, we prove that the system has a unique global positive solution with any positive initial value. Then we verify that random effect may lead the disease to extinction under a simple condition. Thirdly, we establish a sufficient condition for persistence in the mean of the disease. Moreover, we show that there is a stationary distribution to the stochastic system under certain parametric restrictions. Finally, some numerical simulations are carried out to confirm the analytical results. [ABSTRACT FROM AUTHOR]

In this paper, the deterministic and stochastic SIRS epidemic models with nonlinear incidence are introduced and investigated. For deterministic system, the basic reproductive number R 0 is obtained. Furthermore, if R 0 ≤ 1 , then the disease-free equilibrium is globally asymptotically stable and if R 0 > 1 , then there is a unique endemic equilibrium which is globally asymptotically stable. For stochastic system, to begin with, we verify that there is a unique global positive solution starting from the positive initial value. Then when R 0 > 1 , we prove that stochastic perturbations may lead the disease to extinction in scenarios where the deterministic system is persistent. When R 0 ≤ 1 , a result on fluctuation of the solution around the disease-free equilibrium of deterministic model is obtained under appropriate conditions. At last, if the intensity of the white noise is sufficiently small and R 0 > 1 , then there is a unique stationary distribution to stochastic system. [ABSTRACT FROM AUTHOR]

Abstract: A stochastic ratio-dependent predator–prey model is investigated in this paper. By the comparison theorem of stochastic equations and Itô’s formula, we obtain the global existence of a positive unique solution of the ratio-dependent model. Besides, a condition for species to be extinct is given and a persistent condition is established. We also conclude that both the prey population and the ratio-dependent function are stable in time average. In the end, numerical simulations are carried out to confirm our findings. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic SIQR epidemic model with quarantine and standard incidence is investigated. We first consider the existence and uniqueness of the global positive solution to the stochastic system. Further, we obtain conclusions that the large noise can make the disease die out exponentially, and the stationary distribution will exist under certain conditions. • A stochastic SIQR epidemic model with quarantine and standard incidence is studied. • We consider the existence and uniqueness of the global positive solution to the system. • Large noise can make the disease die out exponentially. • The stationary distribution exists under certain conditions. [ABSTRACT FROM AUTHOR]