two correlated Brownian motions to introduce correla- tion of noises in **SIS** **epidemic** **model**. Considering two correlated Brownian motions, one with linear diffusion coefficient and the other with Hölder continuous diffu- sion coefficient, is clearly different from other work on stochastic **SIS** **model**. Though Hölder continuous diffusion coefficient and correlations of white noises are often involved in stochastic financial and biological models [15], there is no related work based on deter- ministic **SIS** **model**. As a result, this paper aims to fill this gap.

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In this paper, a stochastic **SIS** **epidemic** **model** with nonlinear incidence rate and double **epidemic** hypothesis is proposed and analysed. We explain the eﬀects of stochastic disturbance on disease transmission. To this end, ﬁrstly, we investigated the dynamic properties of the system neglecting stochastic disturbance and obtained the threshold and the conditions for the extinction and the permanence of two kinds of **epidemic** diseases by considering the stability of the equilibria of the deterministic system. Secondly, we paid prime attention on the threshold dynamics of the

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Abstract—This paper deals with stability properties of the discrete numerical scheme for the **SIS** **epidemic** **model** with maturation delay. We provide the suffi- cient conditions of the numerical step-size for the nu- merical solutions to be asymptotically stable. These will be useful for choosing a suitable numerical step- size when we simulate problems with the provided numerical scheme.

There are many ways in which stochasticity can be introduced into an **epidemic** **model**. Dalal et al. [42] introduced environ- mental stochasticity into the disease transmission term in a **model** for AIDS and condom use with two distinct states. In a second paper Dalal et al. [43] introduce stochasticity into a deterministic **model** of internal HIV viral dynamics via the same technique of parameter perturbation into the death rate of healthy cells, infected cells and virus particles. Gray et al. [7] also study the **SIS** **epidemic** **model** with environmental stochasticity introduced into the disease transmission parameter. Another way to intro- duce stochasticity into deterministic models is telegraph noise where the parameters switch from one set to another according to a Markov switching process [44]. However in this paper we focus on demographic stochasticity which is a different way of approximating the differential equations which describe the spread of the disease.

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In this paper, considering the impact of stochastic environment noise on infection rate, a stochastic **SIS** **epidemic** **model** with nonlinear incidence rate is proposed and analyzed. Firstly, for the corresponding deterministic system, the threshold which determines the extinction or permanence of the disease is obtained by analyzing the stability of the equilibria. Then, for the stochastic system, the global dynamics is investigated by using the theory of stochastic diﬀerential equations; especially the threshold dynamics is explored when the stochastic environment noise is small. The results show that the condition for the **epidemic** disease to go to extinction in the stochastic system is weaker than that of the deterministic system, which implies that stochastic noise has a signiﬁcant impact on the spread of infectious diseases and the larger stochastic noise is conducive to controlling the **epidemic** diseases. To illustrate this phenomenon, we give some computer simulations with diﬀerent intensities of the stochastic noise.

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It is well known that the feedback mechanism or the individual’s intuitive response to the **epidemic** can have a vital eﬀect on the disease’s spreading. In this paper, we investigate the bifurcation behavior and the optimal feedback mechanism for an **SIS** **epidemic** **model** on heterogeneous networks. Firstly, we present the bifurcation analysis when the basic reproduction number is equal to unity. The direction of bifurcation is also determined. Secondly, diﬀerent from the constant coeﬃcient in the existing literature, we incorporate a time-varying feedback mechanism coeﬃcient. This is more reasonable since the initiative response of people is constantly changing during diﬀerent process of disease prevalence. We analyze the optimal feedback mechanism for the **SIS** **epidemic** network **model** by applying the optimal control theory. The existence and uniqueness of the optimal control strategy are obtained. Finally, a numerical example is presented to verify the eﬃciency of the obtained results. How the topology of the network aﬀects the optimal feedback mechanism is also discussed.

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deterministic and stochastic **SIS** **epidemic** models. Li, Ma and Zhu [22] analyse backward bifurcation in an **SIS** **epidemic** **model** with vaccination and Van den Driessche and Watmough [37] study backward bifurcation in an **SIS** **epidemic** **model** with hysteresis. More recently, Andersson and Lindenstrand [3] analyse an open population stochastic **SIS** **epidemic** **model** where both infectious and susceptible individuals reproduce and die. Gray et al. [16] establish the stochastic **SIS** **model** by parameter perturbation. There are many other examples of **SIS** **epidemic** models in the literature. Also, other two similar models for diseases with permanent immunity and diseases with a latent period before becoming infectious, the SIR (Susceptible-Infectious-Recovered) and the SEIR (Susceptible-Exposed-Infectious-Recovered) **model** respectively are studied by Yang et al. [39] and stochastic perturbations are introduced in these two models. Liu and Stechlinski [24] analyse the stochastic SIR **model** with contact rate being modelled by a switching parameter. Bhattacharyya and Mukhopadhyay [5] study the SI (Susceptible-Infected) **model** for prey disease with prey harvesting and predator switching. Artelejo, Economou and Lopez-Herrero [4] propose some efficient methods to obtain the distribution of the number of recovered individuals and discuss its relationship with the final **epidemic** size in the **SIS** and SIR stochastic **epidemic** models.

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Some mathematical models, for instance, see [1–5], have been employed to describe and understand **epidemic** transmission dynamics since the work of Kermack and McKendrick [6] was proposed. The classical compartment models were proposed and investigated on the ground of some restrictive assumptions including a constant total population size and a constant recruitment rate for the susceptible individuals. This assumption is relatively reasonable for a short-lasting disease. While in reality, the population sizes of human be- ings and other creatures are generally variable, instead of keeping constant for a long run. As an example of this phenomenon, Ngonghala et al. pointed out that malaria in develop- ing countries took place with growth of local population size. When it concerns the vari- able population size, some recent literature works, such as Ngonghala et al. [7], Busenberg and Driessche [8], Wang et al. [9], Zhao et al. [10], Zhu and Hu [11], Li et al. [12], had con- sidered the eﬀect of population size on the **epidemic** dynamics. We would like to mention the work by Wang et al. [9], in which they constructed an **SIS** **epidemic** **model** under the assumption that the susceptible individuals followed the Logistic growth:

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In the well-known **SIS** **epidemic** **model**, the population is always separated into two com- partments, susceptible and infective individuals. In most **SIS** **epidemic** models (see An- derson and May [1]), the incidence takes the mass-action form with bilinear interactions. However, in a practical application, to describe the transmission process more realistically, it is necessary to introduce the nonlinear contact rates [2].

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We study the global stability of endemic equilibrium of an **SIS** **epidemic** **model** with feedback mechanism on networks. The **model** was proposed by J. Zhang and J. Sun (Physica A 394:24–32, 2014), who obtained the local asymptotic stability of endemic equilibrium. Our main purpose is to show that if the feedback parameter is suﬃciently large or if the basic reproductive number belongs to the interval (1, 2], then the endemic equilibrium is globally asymptotically stable. We also present numerical simulations to illustrate the theoretical results.

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We formulate a stochastic **SIS** **epidemic** **model** with vaccination by introducing a Lévy noise and regime switching into the **epidemic** **model**. First, we prove that the stochastic **model** admits a unique global positive solution. Moreover, we study the asymptotic behavior of the stochastic regime switching **SIS** **model** with vaccination driven by Lévy noise.

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In this paper we extend the classical **SIS** **epidemic** **model** from a deterministic framework to a stochastic one, and formulate it as a stochastic differential equation (SDE) for the number of infectious individuals I (t). We then prove that this SDE has a unique global positive solution I (t) and establish conditions for extinction and persistence of I (t). We discuss perturbation by stochastic noise. In the case of persistence we show the existence of a stationary distribution and derive expressions for its mean and variance. The results are illustrated by computer simulations, including two examples based on real life diseases.

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Abstract. In this paper, we study the dynamics of a stochastic **SIS** **epidemic** **model** with vertical transmission and specific functional response. The environment variability in this work is characterized by Gaussian white noise and L´evy jump noise. We establish the existence and uniqueness of a global positive solution starting from any positive initial value. We also investigate extinction and persistence in mean of the disease. Numerical examples are presented to illustrate the theoretical results.

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previous work done on the **SIS** **epidemic** **model**, for example Hethcote [17] studied the **SIS** **epidemic** **model** involving different factors such as disease mortality and migration. The **SIS** **epidemic** **model** is the simplest possible **epidemic** **model** and has been widely studied. An **epidemic** of an infectious disease can be modelled by using either the deterministic **model** or the stochastic **model**. The deterministic **model** is often formulated as a system of differential equations where its solution is uniquely dependent on the initial value. On the other hand a stochastic **model** is a stochastic process with a collection of random variables where its solution is a probability distribution for each of the random variables. There has been much work done on deterministic models already, however there are some limitations in using these in analysing infectious diseases. A deterministic **model** is more appropriate when we are dealing with a large population. However, if we consider an **epidemic** outbreak in a small community such as school, a stochastic **model** would be more appropriate as the element of variability would become significant [6, 7, 9]. In addition, the real world is not deterministic, and there are many factors that can influence the behaviour of a disease and thus it is not always possible to predict with certainty what would happen. Consequently, a stochastic **model** is introduced to compensate for this problem. There are also many properties that are unique to the stochastic **epidemic** **model** which could enhance our understanding towards the behaviour of a particular disease. For example, the probability that an endemic will not occur, the final size distribution of an **epidemic** and the expected duration of an **epidemic** [2]. Clearly, we can see that introducing stochasticity into an **epidemic** **model** will provide some additional information that will improve the realism of our results compared to the deterministic approach.

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A discrete **SIS** **epidemic** **model** with stage structure and standard incident rate which is governed by Beverton-Holt type is studied. The suﬃcient conditions on the permanence and extinction of disease are established. The existence of the endemic equilibrium is obtained. Further, by using the method of linearization, the local asymptotical stability of the endemic equilibrium is also studied. Lastly, the examples and numerical simulations carried out to illustrate the feasibility of the main results and revealed the far richer dynamical behaviors of the discrete **epidemic** **model** compared with the corresponding continuous **epidemic** models.

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Lajmanovich and Yorke [] proposed a deterministic **model** for the spread of gonor- rhea. Since the spread of gonorrhea in a population is highly nonuniform, they developed a deterministic **SIS** (susceptible-infective-susceptible) **model** with n groups. Because of there being no immunes and negligibly few incubating the disease, they assume the pop- ulation of every subpopulation is constant in size, i.e. S k + I k = N k , where S k , I k denote the

In this paper, we have considered the features of a **SIS** **epidemic** system with the eﬀect of environmental white noise. Firstly, we show that the solution of system (.) is glob- ally positive. An important parameter is the stochastic basic reproduction number R s , which is less than the corresponding deterministic version R . We also see that R s → R

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where S(t) represents the number of individuals susceptible to the disease at time t, and I(t) represents the number of infected individuals. N is a constant input of new members into the population per unit time; β is the transmission coeﬃcient between compartments S and I; μ means the natural death rate; δ is the recovery rate from infectious individuals to the susceptible; B(t) is a standard Brownian motion on the complete probability space (, F, (F t ) t≥0 , P) with the intensity σ 2 > 0. The authors proved that this **model** has a unique

Infectious diseases have tremendous inﬂuence on human life and will bring huge panic and disaster to mankind once out of control. Every year millions of human beings suﬀer from or die of various infectious diseases. In order to predict the spreading of infectious diseases, many **epidemic** models have been proposed and analyzed in recent years (see [–]). Some new conditions should be considered into **SIS** **model** to extend the results. Li et al. (see []) studied an **SIS** **model** with bilinear incidence rate βSI and treatment. The **model** takes into account the medical conditions. The recovery of the infected rate is divided into natural and unnatural recovery rates. Because of the medical conditions, when the number of infected persons reaches a certain amount I , the unnatural recovery

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diﬀerence between the solution and the endemic equilibrium of the deterministic **model** in time average, they derived that the disease would persist. However, the authors did not consider the case when the perturbation would be large. Besides, Gray et al. in [] dis- cussed the following stochastic **SIS** **model**:

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