Calculation Method and Application of Basic Regeneration Number for
a Class of Stochastic Systems
JIAXIN SHI1, DONGWEI HUANG2
School of Mathematical Science,
Tiangong University
Tianjin, 300387,
CHINA
Abstract: - Considering the influence of random noise on SIR, SEIR and SEIAR infectious disease models, we
establish SIR, SEIR and SEIAR models with random disturbance, and deduce the calculation formula of the
basic regeneration number of the random infectious disease model in the sense of mean value by using Itô
formula. The effectiveness of the basic regeneration number calculation method is verified by numerical
simulation of the system evolution process.
Key-Words: - Random infectious disease model, Noise, Itô formula, The basic regeneration number.
Received: July 13, 2021. Revised: July 15, 2022. Accepted: August 9, 2022. Published: October 11, 2022.
1 Introduction
Infectious diseases are one of the threats to human
health and have a non-negligible impact on our
lives. Historically, infectious diseases such as
dengue[1], Severe Acute Respiratory Syndrome
(SARS)[2], pneumonia[3] threaten human life
safety. Therefore, it is of great significance for the
study of infectious diseases. By exploring the
transmission rules of infectious diseases and
predicting their development trend, it can provide a
theoretical basis for disease control. In recent years,
many mathematical scholars have studied the
dynamics behavior of epidemic models for
infections by establishing mathematical models.
Kermack and McKendrick established the SIR
epidemic model for infections by dynamics methods
in 1927 [4]; literature [5] established the SEI model
to study the impact of media reports on the
transmission and control of infectious diseases in
specific regions. The SIR model with stochastic
perturbations is discussed [6]. The global dynamics
of an SIRS epidemic model for infections with non
permanent acquired immunity was investigated [7].
Literature [8] will describe Tuberculosis
transmission using the Susceptible-Exposed-
Infected-Recovered (SEIR) model. The SEIR model
for transmission of Tuberculosis was analyzed and
performed simulations using data on the number of
TB cases in South Sulawesi.
In the epidemic model, the basic regeneration
number
0
R
is one of the important parameters to
determine the prevalence of infectious diseases. The
calculation of the basic regeneration number is
instructive for the prevention and control of
infectious diseases. Literature [9] introduces the
calculation method of the basic regeneration number
in the deterministic model. This paper mainly
introduces the basic regeneration number of several
stochastic epidemic models. When
1
0R
, the
disease disappears and
1
0R
spreads.
2 The Basic Reproduction Number of
the Stochastic Model
2.1 The SIR model
The SIR model with vaccination:
IRpbtR
IcSItI
RSISbptS
)()(
)()(
)()( 1
⑴
In the SIR model, the population is divided into
three compartments: susceptible (S), infectious (I)
and recovered with immunity (R), where
represents the rate of infection,
b
is considered as
the proportion of new individuals entering the
population, vaccination proportion coefficient is
p
,
is considered as the emigration rate, the immune
loss rate is
,
c
is considered as the emigration rate
due to illness, the recovery rate is
.
N
is the total
population size such that
)()()( tRtItSN
for
all
t
. Assuming the propagation coefficient
is
assumed to be disturbed by stochastic noise. We
define
),(tB
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where
)(tB
is the standard Brownian motion and
is the fluctuation intensity of the white noise. We
can build the following stochastic SIR model:
dtIRpbdR
tSIdBdtIcSIdI
tSIdBdtRSISbpdS
])([
)(])([
)(])[(
1
⑵
The state space of the model (2) is
0,0,0:,,
3 RISRISRX
.
Define the
2
C
function
))(),(ln())(),((: tItStItSVV
,
and using the Itô formula can be given as:
)().))(((ln tIdBdtIRSISbpSSd
221 501
⑶
)()..))(((ln tSdBdtSIcSIIId
221 50
⑷
The Eq.(3) and the Eq.(4) are transformed to
Stratonovich stochastic differential equation and
take the mean. Thus, the study of model (2) can be
turned into a study of the following systems:
dtIRpbdR
dtISIcSIdI
dtSIRSISbpdS
])([
].)([
].)[(
22
22
50
501
⑸
The disease-free equilibrium point of the model (1)
and the model (5) can be calculated:
),0,
)(
))1((
(),,( 1110
pbpb
RISESIR
.
The basic reproduction number of the model (1)
can be calculated:
))((
))1((
0
c
pb
RSIR
.
We note
T
RSIx ),,(
,the system (5) may be
represented as
)()( xxx 11 VF
,where
0
0
1
SI
x
)(F
;
IRpb
SIRSISbp
ISIc
x
)(
.)(
.)(
)( 22
22
1501
50
V
.
The Jacobian matrix of
)(),(xx 11 VF
in
SIR
E0
note
11 VF ,
, we have:
000
000
00
1
1
S
F
;
0
0050
1
2
1
2
1S
Sc
V
.
.
The next generation matrix can be calculated:
00
0
2
1
22
222
0
1
11
)()(
))((
c
pb
R
VF
SIR
,
thus, the basic reproduction number of the stochastic
SIR model (2) is:
.
)()(
))((
)(
c
pb
RVFR SIRSIR
S22
222
0
1
11 2
1
2.2 The SEIR Model
The model discussed in the previous section ignores
the disease latency. Given the latency of many
infectious diseases, many scholars introduce a latent
compartment E to indicate the latent status. After
susceptible individuals are infected, they enter the
latent status and into the infection compartment pass
the
q/1
day incubation period. The transmission
process of these infectious diseases can be
expressed in the following model:
IRpbtR
IcqEtI
EqSItE
RSISbptS
)()(
)()(
)()(
)()( 1
⑹
Similar to the reasoning in 2.1, we present the
following system of stochastic differential equations
for SEIR models with stochastic perturbations:
dtIRpbdR
dtIcqEdI
tSIdBdtEqSIdE
tSIdBdtRSISbpdS
)(
)(
)()(
)()(1
⑺
Define the
2
C
function
))(),(ln())(),((: tEtStEtSVV
,
and using the Itô formula can be expressed as:
)(].))(([ln tIdBdtIRSISbpSSd
221 501
⑻
)(].))(([ln tSIdBEdtISEIqSIEEd
122221 50
⑼
The Eq.(8) and the Eq.(9) are transformed to
Stratonovich stochastic differential equation and
take the mean.Therefore we only need to discuss
systems such as:
dtIRpbdR
dtIcqEdI
dtISEEqSIdE
dtSIRSISbpdS
])([
])([
].)([
].)[(
2221
22
50
501
⑽
Suppose
1
0
c
E
I
I
lim
⑾
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where
1
c
is constant, so the disease-free equilibrium
point of the model (6) and the model (10) can be
calculated:
).,,,
)(
))((
(),,,(
pbpb
RIESESEIR 00
1
22220
The basic reproduction number of the model (6)
is
))()((
))((
cq
pbq
RSEIR 1
0
.
We note
T
RSIEx ),,,(
,the system (10) may
be represented as:
)()( xxx 22 VF
,
where
0
0
0
2
SI
x
)(F
;
IRpb
SIRSISbp
qEIc
ISEEq
x
)(
.)(
)(
.)(
)( 22
2221
2501
50
V
.
The Jacobian matrix of
)(),(xx 22 VF
in
SEIR
E0
note
22 VF ,
, we have:
0000
0000
0000
000 2
2
S
F
;
00
0
00
0050
2
22
2
2
1
22
2
2
1
2S
cq
ScScq
V
.
.
The basic reproduction number of the stochastic
SEIR model (7) is the spectral radius of the next
generation matrix
1
22
VF
, then
222
1
22
1
NML
pbq
VFRSEIR
S
))((
)(
,
where
)(
)))(((
22
1
2
2
1pqcb
L
,
))()((qcM
2
,
)(
)))(((
2
1222
1
2
2
pccb
N
.
2.3 The SEIAR Model
Some scholars believe that asymptomatic infected
individuals can also spread the virus. Therefore, if
we add a compartment A of asymptomatic infected
individuals, the model take the form:
AIRpbtR
AqEdtA
IcdqEtI
EqSASItE
RSASISbptS
A
A
A
A
)()(
)()()(
)()(
)()(
)()(
1
1
⑿
New infections in compartment E arise by
contacts between susceptible and infected
individuals in compartments S and A at a rate
SA
A
. Individuals progress from compartment E to I at a
rate
d
. Asymptomatic individuals are recovered at a
rate
.
A
Assuming the propagation coefficient
and
A
are assumed to be disturbed by stochastic
noise. We define:
)(),(tBtB AA 2211
.
The following stochastic SEIAR model can be
established:
dtAIRpbdR
dtAqEddA
dtIcdqEdI
tSAdBtSIdBdtEqSASIdE
tSAdBtSIdBdtRSASISbpdS
A
A
A
A
)(
)()(
)(
)()()(
)()()(
1
1
2211
2211
⒀Define the
2
C
function
))(),(ln())(),((: tEtStEtSVV
,
and using the Itô formula can be expressed as:
)()()](.
))(([ln
tAdBtIdBdtASISS
RSASISbpSSd A
2211
222
2
222
1
2
1
50
1
⒁
)()(
)](.))(([ln
tSAdBEtSIdBE
dtASISEEqSASIEEd A
22
1
11
1
222
2
222
1
21 50
⒂
The Eq.(14) and Eq.(15) are transformed to
Stratonovich stochastic differential equation and
take the mean,we can study the following systems:
dtAIRpbdR
dtAqEddA
dtIcdqEdI
dtASISEEqSASIdE
dtSASIRSASISbpdS
A
A
A
A
])([
])()[(
])([
)](.)([
)](.)[(
1
50
501
222
2
222
1
1
22
2
22
1
⒃
Suppose
2
0
c
E
A
A
lim
⒄
where
2
c
is constant.
According to Eq.(11) and Eq.(17), the system
(12) and (16) has a unique disease-free equilibrium
point, with
).,,,,
)(
))((
(),,,,(
pbpb
RAIESESEIAR 000
1
333330
The basic reproduction number of the model (12) is
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.
))()()((
)))()(()()()((
A
AAA
SEIAR
cq
cdcpbq
R1
0
We note
T
RSAIEx ),,,,(
thus, the system
(16) may be represented as:
)()( xxx 33 VF
,
where
0
0
0
0
3
SASI A
F
;
AIRpb
SASIRSASISbp
qEdA
dqEIc
ASISEEq
A
A
A
)(
)(.)(
)()(
)(
)(.)(
22
2
22
1
222
2
222
1
1
3
501
1
50
V
.
The Jacobian matrix of
)(),(xx 33 VF
in
SEIAR
E0
note
33 VF ,
, we have:
00000
00000
00000
00000
000 33
3
SS
F
A
;
.
)(
)(.
00
0
0001
000
0050
33
2
2
2
32
2
1
2
31
2
2
2
2
2
1
2
1
2
3
3
A
A
A
SS
qd
cdq
ScScccSq
V
The basic reproduction number of the stochastic
SEIAR model (13) is:
333
1
33
112
NML
dcdpbq
VFR
AA
SEIAR
S
))())(()())(((
)(
where
))(()))(((
ccdqpcbL A1
2
1
2
1
2
321
,
))()(()))(((A
cdqpccbM
2
2
2
2
2
2
3121
,
)())()((qcN A
22
32
.
3 Numerical Simulation
Taking the model (2) as an example, we analyzed
the effect of the vaccination rate
p
and the noise
intensity
on
SIR
S
R
. We take
,2b
,.040
,.0250
,.010c
,.0010
90.
. Fig.1 shows
the change of the basic regeneration number
SIR
S
R
with vaccination rate
p
at
020.
, and Fig.2
shows the change of
SIR
S
R
with noise intensity
at
..50p
It indicates that
SIR
S
R
decreases
monotonically with increasing
p
and
, which is
also in line with the actual situation. In Fig. 1c we
show the results of
SIR
S
R
as a function of
p
and
.
As seen from Fig.3, the
SIR
S
R
decrease
monotonically with the increase of noise intensity
for different values of vaccination rate
p
, which
shows that the increase of noise intensity
can
effectively control the spread of the disease.
Fig. 1: The change of
SIR
S
R
with vaccination rate
p
at
020.
.
Fig. 2: The change of
SIR
S
R
with noise intensity
at
50.p
.
Fig. 3: Three-dimensional plot of
SIR
S
R
as a function
of
p
and
.
To verify the effect of the basic regeneration
number on the infectious diseases, now we will
perform some numerical simulations. The numerical
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simulations are given by the Milstein's scheme. For
the model (13), take
,.70p
,3b
,.040
,.0250
,.0150
A
,.010c
,.0010
,.90
,.950
A
,.210q
,.70d
,.010
21 cc
..020
21
By calculating
145790 .
SEIAR
S
R
,
We can see that the disease gradually goes extinct
(Fig. 4a). When we take
10.p
, therefore, the
basic reproduction number of the random system
(13)
128611 .
SEIAR
S
R
, the number of infections
continues to grow and will eventually lead to an
outbreak (Fig. 4b). Fig.4(a, b) shows the mean value
of the numerical simulation results when the
vaccination rate are equal to 0.7 and 0.1 for 100
runs, respectively.
(a):
145790 .
SEIAR
S
R
when
70.p
.
(b):
128611 .
SEIAR
S
R
when
10.p
.
Fig. 4: Evolution of infected individuals after 100
number of simulations and taking the mean value,
with
,.70p
,3b
,.040
,.0250
,.0150
A
,.010c
,.0010
..90
,.950
A
,.210q
,.70d
,.010
21 cc
..020
21
4 Conclusion
We study the SIR, SEIR, and SEIAR infectious
disease models disturbed by random noise, and give
the calculation of the basic regeneration number of
the three stochastic models, when
1
S
R
the
maximum proportion of people who can infect a
patient in the average disease period is less than 1,
the disease will not spread. If,the disease will
spread. Numerical simulation results show that:
(1) By increasing the vaccination rate
p
, the basic
regeneration number will be reduced. This means
that increasing the vaccination rate can effectively
inhibit the spread of the disease;
(2) Increased noise intensity
can also reduce the
number of infected people.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
-Dongwei Huang carried out the simulation.
-Jiaxin Shi has organized and executed the
experiments of Section 2.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
National Natural Science Foundation of China (116
72207); Tianjin Natural Science Foundation (18JCY
BJ C18900).
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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