Threshold dynamics of a stochastic SIQR epidemic model with imperfect quarantine☆
Introduction
The outbreak of the epidemic has had a profound impact on people’s lives. For example, at the end of 2019, the COVID-19 broke out in Wuhan, China, and then spread rapidly around the world. As of April 22, 2022, there have been more than 505 million confirmed cases of the COVID-19, including 6.2 million deaths globally [1]. To this day, the COVID-19 is still severely impacting people’s lives all over the world. Throughout history, there have also been many large epidemics human life safety poses a serious threat, such as the influenza pandemic that broke out in 1918 killed about 50 million people worldwide [2], the plague kills 25 million people in Europe between 1347 and 1351 [3], and so on. Due to the inability to rapidly develop and produce an effective vaccine and the rapid mutation of the virus, quarantine is considered an effective way to control the spread of infectious diseases [4], [5], [6], [7]. For instance, people must take practical measures such as social distancing and self-isolation to prevent the further deterioration of COVID-19 and SARS. However, the effective quarantine policies also have a range of negative impacts on society. For example, the labor force in the market has decreased and the global economy has been hit dramatically [8]. Therefore, the search for quarantine strategies to prevent further development of epidemics and to minimize the serious consequences of epidemics is an important and interesting topic.
A common practice is to establish epidemic models with quarantine to assess the potential impact of quarantine on the spread of infectious diseases. A number of studies have shown that quarantine can lead to models with rich dynamical behavior, such as hopf bifurcation, homoclinic bifurcation [4], [5], [6], [7]. Besides in the real world, incidental contact between infected patients and susceptible individuals during quarantine is inevitable. Such as, in the early stages of the COVID-19 outbreak, more than 3000 healthcare workers were identified with COVID-19 in Hubei Province, China. In the 1918 influenza pandemic, a large number of health care workers were infected and caused the deaths of approximately 50 million people worldwide [2]. This type of quarantine is called imperfect quarantine. Specifically, Erdem et al. [7] studied the following model with the imperfect quarantine: where , , and stand for the number of susceptible, infected, quarantine and recovered individuals, respectively. Population size . is the effective contact rate between susceptible and infected, denotes the effective contact rate between susceptible and imperfectly quarantined, is the proportion of effectively quarantined individuals, is the birth and death rate, is the recovery rate for quarantined individuals, is the quarantine rate, and is the recovery rate for infected individuals. According to the results in [7], the disease dies out if , otherwise the disease persists if .
There is ample evidence that random changes in the environment have an enormous influence on the spread of infectious diseases. As shown in [9], the spread of epidemics is inherently random. Lowen and Steel [10] pointed out that the survival and spread of the virus are closely related to the temperature and humidity of its living environment. In reality, the effective contact rate is more responsive to random changes in the environment than other parameters of human populations [11], [12], [13], [14], [15]. In this article, we therefore investigate the effect of stochastic fluctuations in the environment on the dynamics of the epidemic transmission by introducing Gaussian white noise into the effective contact rate (i.e. ), then model (1.1) becomes where is the standard one-dimensional independent Wiener process defined over the complete probability space , is the intensity of environmental white noise.
The paper is organized as follows. The main results of this paper and their proofs are given in Section 2. In Section 3, we take full advantage of the available of weekly infected cases for northern China from the first week of 2014 to the 26th week of 2014, to estimate the parameters of model (1.1) using the least square method. We briefly discuss and summarize the main results in Section 4.
Section snippets
Main results
Throughout the paper, unless otherwise specified, we define as a complete probability space with a filtration satisfying the usual conditions. Let
We start with the existence and uniqueness of global positive solutions for stochastic model (1.2).
Theorem 2.1 There is a global solution for model (1.2) such that for any given initial condition .
Numerical simulations
The theoretical research is used in this section to assess influenza’s characteristics in northern China in 2014. The information is gathered from the Chinese National Influenza Center website [21] based on weekly infected cases for northern China from the first week of 2014 to the 26th week of 2014. We employ the least square method to fit model (1.1) and estimate the parameters. The tight agreement between the simulation time series and the data is shown in Fig. 3.1, and it is obvious that
Discussion and conclusion
In this paper, we examine the effects of imperfect quarantine and random noise on the dynamics of an SIQR model. We adopt the data of weekly infected cases for northern China from the first week of 2014 to the 26th week of 2014, to fit model (1.1) and estimate the parameters. Our research suggests that: (i) Random noise can change the outcome of deterministic model (1.1) predictions. When , the amplitude of is positively correlated with the noise intensity . As the noise intensity
References (21)
- et al.
Recurrent outbreaks of childhood diseases revisited: the impact of isolation
Math. Biosci.
(1995) - et al.
Effects of quarantine in six endemic models for infectious diseases
Math. Biosci.
(2002) - et al.
Homoclinic bifurcation in an SIQR model for childhood diseases
J. Differential Equations
(2000) - et al.
A stochastic SIRS epidemic model with infectious force under intervention strategies
J. Differential Equations
(2015) - et al.
Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching
J. Differential Equations
(2017) - et al.
Dynamics of a stochastic multigroup SIQR epidemic model with standard incidence rates
J. Franklin Inst. B
(2019) - et al.
Threshold behavior in a stochastic SIQR epidemic model with standard incidence and regime switching
Appl. Math. Comput.
(2018) - et al.
The impact of hospital resources and environmental perturbations to the dynamics of SIRS model
J. Franklin Inst. B
(2021) World health organization
(2022)- et al.
The effect of public health measures on the 1918 influenza pandemic in US cities
Proc. Natl. Acad. Sci.
(2007)
Cited by (5)
Dynamics and application of a generalized SIQR epidemic model with vaccination and treatment
2023, Applied Mathematical ModellingDynamical analysis and optimal control of a stochastic SIAR model for computer viruses
2023, European Physical Journal Plus