Dynamical Behaviors of a Stochastic SIQR Epidemic Model with Quarantine-Adjusted Incidence

Discrete Dynamics in Nature and Society, Mar 2018

We study the dynamics of a stochastic SIQR epidemic disease with quarantine-adjusted incidence in this article. In order to find the sufficient conditions for the ergodicity and extermination of the model, we construct suitable stochastic Lyapunov functions and find the results of the stochastic SIQR epidemic model. From the results, we find that when the white noise is relatively large, the infectious diseases will become extinct; this also shows that the intervention of white noise will play an important part in controlling the spread of the disease.

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Dynamical Behaviors of a Stochastic SIQR Epidemic Model with Quarantine-Adjusted Incidence

World Journal Dynamical Behaviors of a Stochastic SIQR Epidemic Model with Quarantine-Adjusted Incidence Zhongwei Cao Shengjuan Zhou 0 Josef Diblik 0 Department of Applied Mathematics, Jilin University of Finance and Economics , Changchun 130117 , China 1 College of Science, China University of Petroleum (East China) , Qindao 266580 , China Correspondence should be addressed to Shengjuan Zhou; We study the dynamics of a stochastic SIQR epidemic disease with quarantine-adjusted incidence in this article. In order to find the sufficient conditions for the ergodicity and extermination of the model, we construct suitable stochastic Lyapunov functions and find the results of the stochastic SIQR epidemic model. From the results, we find that when the white noise is relatively large, the infectious diseases will become extinct; this also shows that the intervention of white noise will play an important part in controlling the spread of the disease. 1. Introduction Recently, owing to the negative impact of infectious diseases on population growth, understanding the dynamic behavior of these diseases and predicting what will happened have become an important research topic (see e.g., [1?8]). Therefore, the establishment of mathematical models has become an important method to study the properties of infectious diseases. For more contagious diseases such as smallpox, measles, plague, mumps, and Ebola, the most direct and effective methods of interference are to isolate those who have already been infected, in order to decrease transmissions to susceptible individuals. From then on, one of the famous disease models, SIQR (see [9]), has been established, which can be described as follows: ? ()= ? ? ? ()= [ ? ()= + + + + ? ( + d + 2) , ( 1 ) In this model, one assumed that the infection is given a permanent immunization after recovery. is the susceptible individual. When these people are infected with the disease, some enter compartment, which will be infected; other people can quickly and completely recover and access compartment. In addition, when the susceptible individual enters compartment, it may be quarantined directly to enter compartment. Before they recover, they all will go into compartment. Here, the total population of the model varies, because vulnerable parts of the population can be received through birth or immigrants and people will die of natural and disease deaths. Besides, in this model, the incidence given by /( + + ) is the quarantine-adjusted incidence. The total contacts of a susceptible person using this form of incidence are /( ? ) = ( ? )/( ? ) = [9], so that, during quarantine process, the total number of contacts per day remains at . From the model, the parameters can be summarized in the following list: ? is the influx of people into the susceptible person?s compartment. d is the natural death rate of compartments , , , and . is transmission coefficient from compartment to compartment . is the recovery rate of infectious individuals. is the isolation rate from to . is the recovery rate of isolated individuals. 1 is the disease-caused death rate of infectious individuals. 2 is the disease-caused death rate of isolated individuals. Assume that all parameters are nonnegative parameters. In particular, ? and d are positive constants. In model ( 1 ), the quarantine reproduction number is 0 = /( + + d + 1)[9], which determines whether the disease occurs. If 0 ? 1, system ( 1 ) has a unique diseasefree equilibrium 0 = ( 0, 0, 0, 0) = (?d/, 0, 0, 0)and 0 is globally asymptotically stable in invariant set D, where D = {( , , , ) ? R4+ | + + + ? ?/ d}. T his reveals that the disease will die out and all people are susceptible to it. If 0 > 1 and 1 = 2 = 0, then 0 is locally asymptotically stable in the region D and system ( 1 ) has only a positive endemic equilibrium ? = ( ?, ?, ?, ?)which we can find in [9], where ? In addition, for some parameter values the Hopf bifurcation may occur. In real life, disease systems are often affected by white noise (see [10?17]). So it is important to include the effect of stochastic perturbation in estimation of parameters. In many cases, stochastic systems can better describe the spread of infectious diseases (see [16?18]). For instance, the stochastic model can account for the stochastic infectious contact during latent and infectious period [19]. Compared with the deterministic system, this is more practical (see [10, 20?25]). Reference [10] cleared that the stochastic systems are more adapted to the problem of extinction of the disease. Paper [21] showed that the unique equilibrium in a deterministic model may disappear in corresponding stochastic system due to stochastic fluctuations [3]. There are so many methods to introduce stochastic perturbations in this system. From biological perspective, Lemma 1. The Markov process () has a unique ergodic stationary distribution (?) if there exists a bounded domain D ? E with regular boundary ? a (...truncated)


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Zhongwei Cao, Shengjuan Zhou. Dynamical Behaviors of a Stochastic SIQR Epidemic Model with Quarantine-Adjusted Incidence, Discrete Dynamics in Nature and Society, 2018, 2018, DOI: 10.1155/2018/3693428