Threshold dynamics of a stochastic SIQR epidemic model with imperfect quarantine

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Abstract

In this paper, we study a stochastic SIQR epidemic model with imperfect quarantine. We perform a detailed mathematical analysis of the dynamics of the stochastic model. The basic reproduction number R0s turns out to be a sharp threshold, that is, if R0s<1, then the disease-free equilibrium of the stochastic model is globally stable almost surely, whereas if R0s>1, the Markov process is positive recurrence which indicates that the disease will prevail. We also adopt the data of weekly infected cases for northern China from the first week of 2014 to the 26th week of 2014, to fit the model and estimate the parameters.

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: dSdt=bNλSINρQλˆS(1ρ)QNρQbS,dIdt=λSINρQ+λˆS(1ρ)QNρQ(η+δ+b)I,dQdt=ηI(θ+b)Q,dRdt=δI+θQbR,where S, I, Q and R stand for the number of susceptible, infected, quarantine and recovered individuals, respectively. Population size N=S+I+Q+R. λ is the effective contact rate between susceptible and infected, λˆ denotes the effective contact rate between susceptible and imperfectly quarantined, ρ is the proportion of effectively quarantined individuals, b is the birth and death rate, δ is the recovery rate for quarantined individuals, η is the quarantine rate, and θ is the recovery rate for infected individuals. According to the results in [7], the disease dies out if R0λδ+b+η+λˆη(1ρ)(δ+b+η)(θ+b)<1, otherwise the disease persists if R0>1.

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 λ (i.e. λdtλdt+σdB(t)), then model (1.1) becomes dS=bNλSINρQλˆS(1ρ)QNρQbSdtσSINρQdB(t),dI=λSINρQ+λˆS(1ρ)QNρQ(η+δ+b)Idt+σSINρQdB(t),dQ=ηI(θ+b)Qdt,dR=δI+θQbRdt,where B(t) is the standard one-dimensional independent Wiener process defined over the complete probability space (Ω,,{t}t0,P), σ 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 (Ω,,{t}t0,P) as a complete probability space with a filtration {t}t0 satisfying the usual conditions. Let R+n={xRn|xi>0,i=1,2,,n},x=(x1,x2,,xn).

We start with the existence and uniqueness of global positive solutions for stochastic model (1.2).

Theorem 2.1

There is a global solution (S(t),I(t),Q(t),R(t)) for model (1.2) such that PS0,I0,Q0,R0{(S(t),I(t),Q(t),R(t))R+4}=1for allt0,for any given initial condition (S0,I0,Q0,R0)R+4.

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 R0s>1, the amplitude of I(t) is positively correlated with the noise intensity σ. As the noise intensity

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Research is supported by the National Natural Science Foundation of China (No.12071407;12171193).

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