533 research outputs found

    Extinction and recurrence of multi-group SEIR epidemic

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    In this paper, we consider a class of multi-group SEIR epidemic models with stochastic perturbations. By the method of stochastic Lyapunov functions, we study their asymptotic behavior in terms of the intensity of the stochastic perturbations and the reproductive number R0R0. When the perturbations are sufficiently large, the exposed and infective components decay exponentially to zero whilst the susceptible components converge weakly to a class of explicit stationary distributions regardless of the magnitude of R0R0. An interesting result is that, if the perturbations are sufficiently small and R0ā‰¤1R0ā‰¤1, then the exposed, infective and susceptible components have similar behaviors, respectively, as in the case of large perturbations. When the perturbations are small and R0>1R0>1, we construct a new class of stochastic Lyapunov functions to show the ergodic property and the positive recurrence, and our results reveal some cycling phenomena of recurrent diseases. Computer simulations are carried out to illustrate our analytical results

    A stochastic SIRI epidemic model with relapse and media coverage

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    This work is devoted to investigate the existence and uniqueness of a global positive solution for a stochastic epidemic model with relapse and media coverage. We also study the dynamical properties of the solution around both disease-free and endemic equilibria points of the deterministic model. Furthermore, we show the existence of a stationary distribution. Numerical simulations are presented to confirm the theoretical results.Fondo Europeo de Desarrollo RegionalMinisterio de EconomĆ­a y CompetitividadConsejerĆ­a de InnovaciĆ³n, Ciencia y Empresa (Junta de AndalucĆ­a)Faculty of Sciences (Ibn Tofail University

    Advanced Nonlinear Dynamics of Population Biology and Epidemiology

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    abstract: Modern biology and epidemiology have become more and more driven by the need of mathematical models and theory to elucidate general phenomena arising from the complexity of interactions on the numerous spatial, temporal, and hierarchical scales at which biological systems operate and diseases spread. Epidemic modeling and study of disease spread such as gonorrhea, HIV/AIDS, BSE, foot and mouth disease, measles, and rubella have had an impact on public health policy around the world which includes the United Kingdom, The Netherlands, Canada, and the United States. A wide variety of modeling approaches are involved in building up suitable models. Ordinary differential equation models, partial differential equation models, delay differential equation models, stochastic differential equation models, difference equation models, and nonautonomous models are examples of modeling approaches that are useful and capable of providing applicable strategies for the coexistence and conservation of endangered species, to prevent the overexploitation of natural resources, to control diseaseā€™s outbreak, and to make optimal dosing polices for the drug administration, and so forth.View the article as published at https://www.hindawi.com/journals/aaa/2014/214514

    A stochastic SIRI epidemic model with LĆ©vy noise

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    Some diseases such as herpes, bovine and human tuberculosis exhibit relapse in which the recovered individuals do not acquit permanent immunity but return to infectious class. Such diseases are modeled by SIRI models. In this paper, we establish the existence of a unique global positive solution for a stochastic epidemic model with relapse and jumps. We also investigate the dynamic properties of the solution around both disease-free and endemic equilibria points of the deterministic model. Furthermore, we present some numerical results to support the theoretical work

    Threshold Dynamics in Stochastic SIRS Epidemic Models with Nonlinear Incidence and Vaccination

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    In this paper, the dynamical behaviors for a stochastic SIRS epidemic model with nonlinear incidence and vaccination are investigated. In the models, the disease transmission coefficient and the removal rates are all affected by noise. Some new basic properties of the models are found. Applying these properties, we establish a series of new threshold conditions on the stochastically exponential extinction, stochastic persistence, and permanence in the mean of the disease with probability one for the models. Furthermore, we obtain a sufficient condition on the existence of unique stationary distribution for the model. Finally, a series of numerical examples are introduced to illustrate our main theoretical results and some conjectures are further proposed

    A stochastic differential equation model for the spread of HIV amongst people who inject drugs

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    In this paper, we introduce stochasticity into the deterministic differential equation model for the spread of HIV amongst people who inject drugs (PWIDs) studied by Greenhalgh and Hay [10]. This was based on the original model constructed by Kaplan [17] which analyses the behaviour of HIV/AIDS amongst a population of PWIDs. We derive a stochastic differential equation (SDE) for the fraction of PWIDs who are infected with HIV at time t. The stochasticity is introduced using the well-known standard technique of parameter perturbation. We first prove that the resulting SDE for the fraction of infected PWIDs has a unique solution in (0,1) provided that some infected PWIDs are initially present, and next construct the conditions required for extinction and persistence. Furthermore, we also show that there exists a stationary distribution for the persistence case. Simulations using realistic parameter values are then constructed to illustrate and support our theoretical results. Our results provide new insight into the spread of HIV amongst PWIDs. The results show that the introduction of stochastic noise into a model for the spread of HIV amongst PWIDs can cause the disease to die out in scenarios where deterministic models predict disease persistence. Hence in situations where stochastic noise is important predictions of control measures such as needle cleaning or reduction of needle sharing rates needed to eliminate disease may be overly conservative

    Dynamics of a Multigroup SIR Epidemic Model with Nonlinear Incidence and Stochastic Perturbation

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    We introduce stochasticity into a multigroup SIR model with nonlinear incidence. We prove that when the intensity of white noise is small, the solution of stochastic system converges weakly to a singular measure (i.e., a distribution) if ā„›0ā‰¤1 and there exists an invariant distribution which is ergodic if ā„›0>1. This is the same situation as the corresponding deterministic case. When the intensity of white noise is large, white noise controls this system. This means that the disease will extinct exponentially regardless of the magnitude of ā„›0
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