SIS epidemic model

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A stochastic differential equation SIS epidemic model with two correlated Brownian motions

A stochastic differential equation SIS epidemic model with two correlated Brownian motions

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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Dynamical analysis of a stochastic SIS epidemic model with nonlinear incidence rate and double epidemic hypothesis

Dynamical analysis of a stochastic SIS epidemic model with nonlinear incidence rate and double epidemic hypothesis

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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Stability of a Numerical Discretisation Scheme for the SIS Epidemic Model with a Delay

Stability of a Numerical Discretisation Scheme for the SIS Epidemic Model with a Delay

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.

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SDE SIS epidemic model with demographic stochasticity and varying population size

SDE SIS epidemic model with demographic stochasticity and varying population size

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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Dynamics of a stochastic SIS epidemic model with nonlinear incidence rates

Dynamics of a stochastic SIS epidemic model with nonlinear incidence rates

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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The bifurcation analysis and optimal feedback mechanism for an SIS epidemic model on networks

The bifurcation analysis and optimal feedback mechanism for an SIS epidemic model on networks

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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The SIS epidemic model with Markovian switching

The SIS epidemic model with Markovian switching

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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The persistence and extinction of a stochastic SIS epidemic model with Logistic growth

The persistence and extinction of a stochastic SIS epidemic model with Logistic growth

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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Complex dynamics in an SIS epidemic model with nonlinear incidence

Complex dynamics in an SIS epidemic model with nonlinear incidence

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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Global stability of an SIS epidemic model with feedback mechanism on networks

Global stability of an SIS epidemic model with feedback mechanism on networks

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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Stochastic regime switching SIS epidemic model with vaccination driven by Lévy noise

Stochastic regime switching SIS epidemic model with vaccination driven by Lévy noise

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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A stochastic differential equation SIS epidemic model

A stochastic differential equation SIS epidemic model

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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Extinction and persistence of a stochastic SIS epidemic model with vertical transmission, specific functional response and Levy jumps

Extinction and persistence of a stochastic SIS epidemic model with vertical transmission, specific functional response and Levy jumps

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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Demographic stochasticity in the SDE SIS epidemic model

Demographic stochasticity in the SDE SIS epidemic model

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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Dynamical behaviors of a discrete SIS epidemic model with standard incidence and stage structure

Dynamical behaviors of a discrete SIS epidemic model with standard incidence and stage structure

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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Asymptotic behavior of a multigroup SIS epidemic model with stochastic perturbation

Asymptotic behavior of a multigroup SIS epidemic model with stochastic perturbation

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

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The threshold of stochastic SIS epidemic model with saturated incidence rate

The threshold of stochastic SIS epidemic model with saturated incidence rate

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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Random periodic solution for a stochastic SIS epidemic model with constant population size

Random periodic solution for a stochastic SIS epidemic model with constant population size

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

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Analysis of an SIS epidemic model with treatment

Analysis of an SIS epidemic model with treatment

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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The asymptotic behavior of a stochastic SIS epidemic model with vaccination

The asymptotic behavior of a stochastic SIS epidemic model with vaccination

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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