Study on the Dynamics of an SIR Epidemic Model with Saturated Growth Rate ()
1. Introduction
Since Kennack and McKendrick proposed the SIR model in 1927 (see [1]), the epidemic model has been well developed. SIR epidemic models play an important role in revealing the laws of infectious disease spread and providing a theoretical basis for prevention and control of the diseases [1] [2] [3] [4].
In 2013, Gong and Yang studied the following SIR epidemic model with saturated growth rate in [5],
(1)
where
represent the numbers of susceptible, infected and recovered individuals at time t respectively. r is the intrinsic natural growth rate, K is the environmental carrying capacity,
is the infection rate of the infectious disease, c is the recovery rate, d is the natural mortality rate,
is the removal rate,
is the psychological effect coefficient, that is, when the susceptible people know that the infected person is infected, he will take corresponding measures to affect the incidence. All parameters in the system are positive.
Since the first two equations of System (1) are independent of the third equation, it is sufficient to consider the first two equations of (1). So, we study the following simplified model
(2)
In System (2), the basic reproductive number
.
When
, the disease-free equilibrium is asymptotically stable, which means that the epidemic will disappear; when
, the disease-free equilibrium is unstable, and there is an endemic equilibrium that is globally asymptotically stable, which means that the epidemic will prevail and persist in the population.
In real life, environmental noise is ubiquitous. It is very important to study the impact of environmental noise on the spread of infectious diseases in the prevention and control of infectious diseases. In [6], Wang pointed out that every parameter in the epidemic model may be randomly perturbed by the environment, which behaves as a random fluctuation. For example, the contact rate and the disease mortality rate in the epidemic model are randomly disturbed by external factors such as age, gender, constitution, mood, climate and season of the individual. Compared with the deterministic model, that with environmental noise can provide additional realism because deterministic model does not take into account these random factors and can only roughly reflect the real situation of infectious diseases to some extent. The research on random model can also be referred to the article [7] - [13].
Inspired by the ideas in the work of [5] [8] [9] [10], we consider the System (2) with a random interference in this paper. We assume that the parameter
is affected by white noise, so, the random driving force of Brownian motion is introduced into the System (2) as a random factor, that is,
is replaced with
where
represents Brownian motion and
represents the intensity of Brownian motion. After added a random term, the system (2) is described by
(3)
The arrangement of this paper is as follows. In Section 2, we study the existence of a unique positive solution of System (3) for any positive initial value and then prove that the positive solution stays in
with probability 1. In Section 3, we establish the sufficient conditions for the extinction of the infectious diseases. In Section 4, we give the conditions for the persistence of the infectious diseases in the mean value. Finally, we give some biological explanations and prevention and control measures for the epidemic.
2. The Existence and Uniqueness of Positive Solution of System
We first give the following notations and definitions.
is a complete probability space,
is a
algebraic current on
satisfying the usual conditions.
Definition 1 (Locally Lipschitz condition).Function
is said to be locally Lipschitz continuous if there is positive constants L and r such that for
, there is
, where
.
Definition 2 (Blow-up time). If the solution of the equation exists in region
, but does not exist in region
for an arbitrarily small constant
, then
is called the blow-up time of the solution of the equation.
Definition 3 (Stopping time). If a function
satisfies condition
,
, then
is called a stopping time.
Remark 1. It is necessary to allow
to get
. For example, If
, whereB is any given Borel set, and
can be regarded as the first arrival time of
into B or the first exit time of
, then
.
Remark 2 Obviously,
(constant time) is a stopping time, which is a generalization of time.
Definition 4 (Ito Formula). Let
,
,
where
,
, then
is a Ito-process, and
Since
and
in System (3) represent the size of the susceptible and infected populations at time t respectively, they must be nonnegative. We first give the result for System (3) having a global positive solution.
Theorem 1 For any initial value
, System (3) has a unique positive solution for
and the solution stays in
with probability 1, i.e.,
is almost sure for
.
Proof Since the coefficients of System (3) are locally Lipschitz continuous, for any initial value
, System (3) has a unique local solution
,
, where
is the blow-up time (see [10]). In order to prove that the above local solution is global, it is only necessary to prove
a.s. Therefore, the stopping time
is defined as
. From the definition of stopping time, we can see that, if
a.s. can be proved, then when
,
a.s. and
a.s. Assume
, then there exists a constant
such that
.
Defining the C2 function
satisfying
and using the Ito formula (see [10]), for
and
, we have:
Here
, therefore, we obtain
Let
, then we can get
(4)
Integrating both sides of inequality (4) from 0 to t, we have
(5)
Assume that
, because
, and from the definition of the stopping time, we know that
.
Letting
in inequality (5), we get:
This is a contradiction with the assumption
. Therefore,
, which also proves that
and
will not blow up in finite time and with probability 1. We obtain the forward invariant set of System (3)
In the following, it is enough to consider the solution of System (3) in
.
3. Sufficient Conditions for Extinction of Infectious Diseases
Before giving the extinction theorem, we give a lemma which can be found in [10].
Lemma 1 Let
(
) be a locally continuous martingale with initial value
and
be a quadratic variation of
. Let
and
be two sequences of positive terms. Then, for almost all
, there exists a positive integer
such that for any
, we have
(6)
Theorem 2 Let
be a solution of System (3) with initial value
, we have
(7)
If
, then
tends to 0 exponentially with probability 1 and
, a.s.
Proof Applying I to formula to the System (3), we have
(8)
Integrating both sides of (8) from 0 to t
(9)
where
is a locally continuous martingale with quadratic variation:
(10)
Taking
,
and
in Lemma 1, for almost all
, there exists a positive integer
such that for any
, we have
(11)
From (9) and (11):
(12)
where
(13)
From (12) and (13), we get
(14)
Therefore, for
, by dividing t on both sides of inequality (14), we obtain
(15)
Let
, thus
, we have
(16)
Let
, then we have
(17)
If
holds, we get
a.s. From system (3),
(18)
then
(19)
where C is any arbitrary constant. By (19) there is
Applying L’Hopital’s rule and
, we have
almost everywhere.
4. Persistence of Infectious Diseases in the Meansense
Definition 5. If
a.s., then System (3) is persistent in the
mean sense (see [8]).
Theorem 3. If
(20)
then for any initial value
the solution of System (3) has the following properties
a.s. (21)
Proof By using the Newton-Leibniz formula, from the System (3) we obtain
(22)
Dividing both sides of inequality (22) by t, we get
(23)
So, we have
(24)
where
.
Applying the Itoformula, there is
(25)
Integrating both sides of inequality (25) from 0 to t, we get
(26)
Substituting (24) into (26) yields
Therefore
(27)
where
,
is locally continuous martingale with initial value
, and
a.s. From the
law of large numbers for martingales (see [10]), we obtain
a.s. By Theorem 1, we have
and
a.s., then
from (27), we have
Therefore, the infectious diseases are persistent in the sense of mean value.
Remark 1 Theorem 3 shows that under some conditions, the infected population is persistent on average, so it can be verified that the persistence of the susceptible population is weak, in fact,
a.s.
5. Biology Interpretation and Control Measures
From Theorem 1 and Theorem 3, we know that the System (3) is persistent only under the condition of the basic reproduction number
being greater than 1. In order to control the spread of infectious diseases, we must adopt some strategies to make it small enough during the spread of infectious diseases. Specific control measures are as follows.
1) To strengthen the control of the source of infection, the relevant public health departments should take all measures to control the source of infection. For example, during the outbreak of infectious diseases, designated hospitals should be determined for centralized treatment to strictly prevent the spread of the disease and reduce the infection rate, so as to achieve the purpose of controlling the spread of infectious diseases;
2) To establish a direct network reporting system for infectious diseases, timely detection, timely reporting, timely treatment of infectious diseases, reduce the impact of psychological effect coefficient
on the control of the spread of infectious diseases;
3) To control the size of the intensity of Brownian motion
to ensure
.
Acknowledgements
This work has been supported by the NNSF of China (Grant No. 11961021), the NSF of Guangdong province (Grant No. 2022A1515010964), the Innovation and Developing School Project (Grant No. 2019KZDXM032) and the Key Project of Science and Technology Innovation of Guangdong College Students (Grant No. pdjh 2021b0309, pdjh 2022b0320).