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)