Abstract: The ability of people to move freely between cities is thought to be a major factor in accelerating the
spread of infectious diseases. To investigate this issue, we propose a SEVIHR stochastic epidemic model, which
emphasizes the effects of transport related infections and media coverage. At the same time, the time delay
caused by the information time difference is considered. Firstly, we study the existence and uniqueness of the
global positive solution of the model by means of Lyapunov function and stopping time, and obtain sufficient
conditions for the extinction and persistence of the disease. Secondly, in order to control the spread of the disease
in time and effectively, appropriate control strategies are formulated according to the stochastic optimal theory.
Finally, the extinction and persistence of the disease were simulated by MATLAB.
Key-Words: stochastic epidemic model, media coverage, transport related infection, time delay, extinction and
persistence, optimal control
Received: May 19, 2024. Revised: October 4, 2024. Accepted: November 3, 2024. Published: December 3, 2024.
1 Introduction
In recent years, there have been frequent outbreaks
of infectious diseases around the world, such as
COVID-19, HIV/AIDS, dengue fever,Htc>1@,>2@,>3]
The outbreak of infectious diseases has threatened
the life and health of people all over the world and
caused huge economic losses, which has aroused the
wide attention of scholars all over the world, and
carried out a series of work such as modeling research
and practical investigation. Through lots of studies
and analyses, the researchers have revealed the
mechanism of disease transmission, the effectiveness
of interventions, and the impact on population
health. The infectious disease model and its related
contents studied by mathematicians and infectious
disease scientists have deepened our understanding
of infectious disease. Among them, the study
of some complex infectious disease models has
simulated the extinction and persistence of diseases,
providing valuable advice for disease prevention and
control, alleviating the impact of infectious diseases,
and ensuring public health security. Therefore,
mathematical modeling is one of the effective means
to study infectious disease control strategy. It can
both help people understand infectious diseases and
help people actively prevent them.
When a disease breaks out, the movement
of people between cities by different modes of
transportation plays a crucial role in the spread
of the disease. In order to study the impact of
transportation on the control of infectious diseases,
many researchers have established different models
of transport related infectious diseases for study[4@,>@,
>6],n reference [7], the SEIVR infectious disease
model related to transportation was considered. On
this basis, the inpatient individuals were added and
the SEVIHR infectious disease model related to
transportation was established:
dSi(t)
dt= [Λ + rSi(t)(1 −αSi(t))
−β1Si(t)Ii(t)
Ni−µ1δSj(t)Ii(t)
Nj
−(w+d1)Si(t) + γRi(t)
+υVi(t)−δ(Si(t)−Sj(t)),
dVi(t)
dt=wSi(t)−δ(Vi(t)−Vj(t))
−(υ+d1)Vi(t)−β2Vi(t)Ii(t)
Ni
−µ2δVj(t)Ii(t)
Nj,
dEi(t)
dt=β1Si(t)Ii(t)
Ni+µ1δSj(t)Ii(t)
Nj
+β2Vi(t)Ii(t)
Ni+µ2δVj(t)Ii(t)
Nj
−(θ+d1)Ei(t)−δ(Ei(t)
−Ej(t)),
dIi(t)
dt=θEi(t)−(d1+φ+fh)Ii(t)
−δ(Ii(t)−Ij(t)),
dHi(t)
dt=fhIi(t)+(−1)i+1g1I2(t)−(fr
+d1+d2)Hi(t),
dRi(t)
dt=φIi(t) + frHi(t)−(γ
+d1)Ri(t)−δ(Ri(t)−Rj(t)),
(1)
where i= 1,2and j=i−(−1)i(i= 1,2)
(i= 1 represents city A, i= 2 represents city
B). Λis the birth rate(birth rates in both cities are
expressed as average birth rates), ris the average
growth rate of the two cities, αis the inverse of
A Valid Transport Related SVEIHR Stochastic Epidemic Model with
Coverage and Time Delays
RUJIE YANG, HONG QIU
College of Science,
Civil Aviation University of China,
Tianjin 300300,
CHINA
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DOI: 10.37394/23206.2024.23.84
Rujie Yang, Hong Qiu
E-ISSN: 2224-2880
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Volume 23, 2024
carrying capacity, β1(β2) is the infection rates of Ii
to Si(Vi), wis the average vaccination rate of the
two cities, υis the rate of loss of vaccine immunity.
d1and d2are natural mortality and disease mortality,
θis the proportion of Eiwho become infectious
individuals Ii,µ1(µ2) is the infection rate of the
infected individuals in the local area to the susceptible
individuals (vaccinated individuals) in the nonnative
area. fhis the hospitalization rate, fris the recovery
rate. Due to the differences in regional medical water,
we assume that the medical level of place A is higher
than that of place B, and patients suitable for treatment
in place A will be transferred to place A, and the
transfer rate is set at g1. And the recovered population
will return to the susceptible population with a
transfer rate of γdue to loss of antibodies. Because
there is normal communication between cities, we
assume that δis the transmission rate between two
cities. φrepresents the recovery rate of the infectious
individuals without hospitalization. And divide the
population into Simeans susceptible individuals, Vi
means vaccinated individuals, Eimeans exposed
individuals, Iimeans infectious individuals, Hi
means inpatient individuals, Rimeans recovered
individuals, Nimeans total urban population.
Nowadays, with the rapid development of the
Internet, people’s life is surrounded by all kinds of
information, so people can get information about
diseases and corresponding preventive measures
through media reports[8@,>9].7he following
function was proposed in literature [10] as the
dynamic change of infection rate under the influence
of media coverage: β(I) = µe−mI , where µ
indicates the usual contact rate, mis the are the
coefficient of the influence infection rate of media
coverage, which represents how media reports affect
transmission. Since media coverage and vigilance
are not inherently deterministic factors that cause
transmission, it is reasonable to assume that m >
0. When m is m > 0and relatively small,
β(I)approaches the parameter µ. However, the
increase of mmeans that the media has carried out
more comprehensive reports to the public, increasing
the public’s awareness of the infectious disease and
taking preventive measures to further reduce the
infection rate. At the same time, it is worth noting
that from the time the danger of infectious disease
transmission is recognized and publicized, people
usually do not respond immediately, which leads to
a certain time delay, thus affecting the spread of
the disease to a certain extent[11@,>12].,n reference
[13], the extinction and persistence of asymptomatic
infection models with media coverage were studied.
In reference [14], the global asymptotic stability of
disease free homeostasis was established in a spatially
heterogeneous environment with media coverage.
Due to the delay in updating the number of cases,
the accuracy of media reports will be affected to
a certain extent, and then the infection rate will
fluctuate accordingly, so the spread of infectious
diseases will be limited to a certain extent. At the
same timeZhite noise[15@,>16@,>7]Ln nature will also
affect the spread of diseases, under the influence of
the above factors, we study the following random
model:
dS1(t) = [Λ + rS1(1 −αS1)−δ(S1−
S2)−β1S1I1
N1e−m1I1(t−τ1)
−µ1δS2I1
N2e−m2I1(t−τ1)+υV1
+γR1−(w+d1)S1]dt
+σ1S1(t)dB1(t),
dV1(t) = [wS1−(υ+d1)V1−δ(V1
−V2)−β2V1I1
N1e−m3I1(t−τ1)
−µ2δV2I1
N2e−m4I1(t−τ1)]dt
+σ2V1(t)dB2(t),
dE1(t) = [β1S1I1
N1e−m1I1(t−τ1)
+µ1δS2I1
N2e−m2I1(t−τ1)−(θ
+d1)E1+β2V1I1
N1e−m3I1(t−τ1)
+µ2δV2I1
N2e−m4I1(t−τ1)−δE1
+δE2)]dt+σ3E1(t)dB3(t),
dI1(t) = [θE1−(d1+φ+fh)I1
−δ(I1−I2)]dt+σ4I1(t)dB4(t),
dH1(t) = [fhI1+g1I2−(fr+d1
+d2)H1]dt+σ5H1(t)dB5(t),
dR1(t) = [φI1+frH1−(γ
+d1)R1−δ(R1−R2)]dt
+σ6R1(t)dB6(t),
dS2(t) = [Λ + rS2(1 −αS2)−δ(S2
−S1)−β1S2I2
N2e−m1I2(t−τ2)
−µ1δS1I2
N1e−m2I2(t−τ2)+υV2
+γR2−(w+d1)S2]dt
+σ1S2(t)dB1(t),
dV2(t) = [wS2−(υ+d1)V2−δ(V2
−V1)−β2V2I2
N2e−m3I2(t−τ2)
−µ2δV1I2
N1e−m4I2(t−τ2)]dt
+σ2V2(t)dB2(t),
dE2(t) = [β1S2I2
N2e−m1I2(t−τ2)
+µ1δS1I2
N1e−m2I2(t−τ2)−(θ
+d1)E2+β2V2I2
N2e−m3I2(t−τ2)
+µ2δV1I2
N1e−m4I2(t−τ2)+δE1
−δE2]dt+σ3E2(t)dB3(t),
dI2(t) = [θE2−(d1+φ+fh)I2
−δ(I2−I1)]dt+σ4I2(t)dB4(t),
dH2(t) = [fhI2−g1I2−(fr+d1
+d2)H2]dt+σ5H2(t)dB5(t),
dR2(t= [φI2+frH2−(γ
+d1)R2−δ(R2−R1)]dt
+σ6R2(t)dB6(t),
(2)
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while Bi(t) (i= 1,· · · ,6) are autonomous and
independent standard Brownian motions, σi(i=
1,· · · ,6) are the frequencies or intensities of the
standard Gaussian white noises. mi(i= 1,2,3,4)
are the coefficient of the influence infection rate of
media coverage. The delay of I1(I2) people count is
τ1(τ2).
In the second part of this paper, we prove the
existence and uniqueness of the global positive
solution of model (2) and obtain the sufficient
conditions for extinction. In the third part, we discuss
disease persistence and get the sufficient conditions
for disease persistence. In the fourth part, we use the
optimal control theory to establish appropriate control
strategies, so as to control the spread of the disease
better and faster under the limited resources. In the
fifth part, we use Milstein’s high order method to
simulate the results of this paper.
2 The(xistence and(xtinction of
'isease
2.1 The(xistence and8niqueness of3ositive
6olution
Theorem 2.1. For any initial condition (S1(0), V1(0),
E1(0), I1(0), H1(0), R1(0), S2(0), V2(0), E2(0), I2(0),
H2(0), R2(0)) ∈R12
+. The stochastic model (2)
admits a unique solution (S1(t), V1(t), E1(t), I1(t),
H1(t), R1(t), S2(t), V2(t),E2(t), I2(t), H2(t), R2(t))
for t≥ −[τ1∨τ2]and the solution will remain in
R12
+with probability one.
Proof. The proof of this theorem is similar to
reference [18]. Therefore, we omit this proof.
2.2 Extinction
For the convenience of expression in this paper, we
define Z(t) = (Z1(t), Z2(t)), where Zi(t) = (Si(t),
Vi(t), Ei(t), Ii(t), Hi(t), Ri(t)) (i= 1,2). And
define the following symbols ⟨Q(t)⟩=1
tt
0Q(s)ds,
where Q(t)is any integral function defined on [0,∞].
Lemma 2.1. Let Z(t)be the solution of model
(2) give any initial value Z(0). Then for i∈1,2, we
have
lim
t→∞
Si(t)
t= 0,lim
t→∞
Vi(t)
t= 0,lim
t→∞
Ei(t)
t= 0,
lim
t→∞
Ii(t)
t= 0,lim
t→∞
Hi(t)
t= 0,lim
t→∞
Ri(t)
t= 0.
lemma 2.2. Let Z(t)be the solution of model (2) give
any initial value Z(0) ∈R12
+. Then for i∈1,2, we
have
lim
t→∞ t
0Si(r)dB1(r)
t= 0,lim
t→∞ t
0Vi(r)dB2(r)
t= 0,
lim
t→∞ t
0Ei(r)dB3(r)
t= 0,lim
t→∞ t
0Ii(r)dB4(r)
t= 0,
lim
t→∞ t
0Hi(r)dB5(r)
t= 0,lim
t→∞ t
0Ri(r)dB6(r)
t= 0.
Theorem 2.2. Let Z(t)be the solution of model (2)
for any initial value Z(0) ∈R12
+. Thus, in the case of
Rjp
0<1, the following property holds:
lim
t→∞ sup ln(I1(t) + I2(t))
t≤P(Rjp
0−1) <0a.s.
and have limt→∞ I1(t) = 0,limt→∞ I2(t) = 0 a.s,
where Rjp
0=β1+δµ1+β2+δµ2
d1+φ+fh−(σ3∧σ4)2
2(d1+φ+fh),
P=d1+φ+fh
Proof. Let G1(t) = E1(t) + E2(t) + I1(t) + I2(t).
According to Itˆo′s formula and model (2), we can get
dG1(t) = LG1dt+σ3(E1+E2)dB3(t) + σ4(I1+
I2)dB4(t), where
LG1=β1S1I1
N1
e−m1I1(t−τ1)+µ1δS2I1
N2
e−m2I1(t−τ1)
+β2V1I1
N1
e−m3I1(t−τ1)+µ2δV2I1
N2
e−m4I1(t−τ1)
+β1S2I2
N2
e−m1I2(t−τ2)+µ1δS1I2
N1
e−m2I2(t−τ2)
+β2V2I2
N2
e−m3I2(t−τ2)+µ2δV1I2
N1
e−m4I2(t−τ2)
−(θ+d1)E2−(θ+d1)E1+θ(E1
+E2)−(d1+φ+fh)(I1+I2)
≤β1I1+µ1δI1+β2I1+µ2δI1+β1I2
+µ1δI2+β2I2+µ2δI2−d1(E1+E2)
−(d1+φ+fh)(I1+I2)
= (β1+δµ1+β2+δµ2)(I1+I2)−d1(E1
+E2)−(d1+φ+fh)(I1+I2).
Define differentiable mapping G2,G2=ln[E1(t)
+E2(t) + I1(t) + I2(t)]. On the basis of Itˆo′s formula
and model (2), we have
dG2=LG2dt+σ3
E1+E2
I1+I2+E1+E2
dB3(t)
+σ4
I1+I2
I1+I2+E1+E2
dB4(t),
where
LG2≤1
I1+I2+E1+E2
[β1I1+µ1δI1+β2I1
+µ2δI1+β1I2+µ1δI2+β2I2+µ2δI2
−d1(E1+E2)−(d1+φ+fh)(I1+I2)]
−1
2(σ3∧σ4)2
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=1
I1+I2+E1+E2
((β1+δµ1+β2
+δµ2)(I1+I2)−d1(E1+E2)−(d1+φ
+fh)(I1+I2)) −1
2(σ3∧σ4)2
≤(β1+δµ1+β2+δµ2)−(d1+φ+fh)
−1
2(σ3∧σ4)2.
Therfore, we have
dG2≤[(β1+δµ1+β2+δµ2)−1
2(σ3∧σ4)2
−(d1+φ+fh)]dt+σ3(E1+E2)
I1+I2+E1+E2
dB3(t)
+σ4
I1+I2
I1+I2+E1+E2
dB4(t)
≤[(β1+δµ1+β2+δµ2)−1
2(σ3∧σ4)2
−(d1+φ+fh)]dt +σ3dB3(t) + σ4dB4(t).
Integrate the above formula from 0to tand multiply
by 1
t, we can get the following inequality
ln(I1(t) + I2(t) + E1(t) + E2(t))
t
≤ln(I1(0) + I2(0) + E1(0) + E2(0))
t+ (β1
+δµ1+β2+δµ2)−(d1+φ+fh)−1
2(σ3∧σ4)2
+1
tt
0
σ3dB3(s) + 1
tt
0
σ4dB4(s),
using the result of Lemma 2.1 and Lemma 2.2, and
according to Rjp
0<1we can get the following
conclusion
lim
t→∞
ln(I1(t) + I2(t) + E1(t) + E2(t))
t
≤(β1+δµ1+β2+δµ2)−1
2(σ3∧σ4)2
−(d1+φ+fh)
≤P(Rjp
0−1)
≤0.
where P=d1+φ+fh. Then, we deduce that
lim
t→∞ sup ln(I1(t))
t
≤lim
t→∞
ln(I1(t) + I2(t) + E1(t) + E2(t))
t<0,
lim
t→∞ sup ln(I2(t))
t
≤lim
t→∞
ln(I1(t) + I2(t) + E1(t) + E2(t))
t<0.
The aforementioned results lead to the conclusion that
lim
t→∞ I1(t) = lim
t→∞ I2(t) = 0 a.s.
3 Persistence
Theorem 3.1 For any initial value Z(0) ∈R12
+,
Z(t)is the solution of model (2). Therefore, if the
condition Rp
0>1is hold, the disease in the two cities
persists in the mean. Moreover, the following hold:
(i) limt→∞ inf⟨I1(t) + I2(t)⟩ ≥ κ(Rp
0−1) = I >
0a.s.
(ii) S≤limt→∞⟨S1+S2⟩ ≤ ¯
S,
(iii) V≤limt→∞⟨V1+V2⟩ ≤ ¯
V,
(iv) E≤limt→∞⟨E1+E2⟩ ≤ ¯
E,
(v) H≤limt→∞⟨H1+H2⟩ ≤ ¯
H,
(vi) R≤limt→∞⟨R1+R2⟩ ≤ ¯
R,
whereRp
0=d1+θ+σ2
3
2
δ+θ,S=θ+d1
w¯
E−β1+δµ1
wI,
¯
S=2Λ
w+d1−r+2υN∗
w+d1−r+γ
w+d1−rlimt→∞ inf⟨R1+
R2⟩,V=w
υ+d1
¯
S−β2+δµ2
υ+d1I,¯
V=w
υ+d1S,
E=e−(m1∨m2∨m3∨m4)N∗
N∗
β1+δµ1+β2+δµ2
θ+d1I,
¯
H=fh+φ
d1+d2I,¯
E=β1+δµ1+β2+δµ2
θ+d1I,
H=fh
d1+d2I−γ+d1
d1+d2
¯
R,¯
R=fh+φ
γ+d1I.
R=w+d1−r
γS−υ
γ¯
V−2Λ
γ,κ=δ+θ
β1+β2+µ1δ+µ2δ.
Proof. According to Theorem 2.2, we can get
G3(t) = E1(t) + E2(t), and using Itˆo′s formula to
get dG3(t) = LG3dt+σ3(E1+E2)dB3(t), where
LG3=β1S1I1
N1
e−m1I1(t−τ1)+µ1δS2I1
N2
e−m2I1(t−τ1)
+β2V1I1
N1
e−m3I1(t−τ1)+µ2δV2I1
N2
e−m4I1(t−τ1)
−(θ+d1)E1+β1S2I2
N2
e−m1I2(t−τ2)
+µ1δS1I2
N1
e−m2I2(t−τ2)+β2V2I2
N2
e−m3I2(t−τ2)
+µ2δV1I2
N1
e−m4I2(t−τ2)−(θ+d1)E2
≤ −(θ+d1)(E1+E2)+(β1+β2+µ1δ
+µ2δ)I1+ (β1+β2+µ1δ+µ2δ)I2
= (β1+β2+µ1δ+µ2δ)(I1+I2)
−(θ+d1)(E1+E2).
We define G4=ln[E1(t) + E2(t)], we can obtain
dG4≤[1
E1+E2
((β1+β2+µ1δ+µ2δ)(I1+I2)
−(θ+d1)(E1+E2)) −σ2
3
2]dt+σ3dB3(t)
≤[(δ+θ)+(β1+β2+µ1δ+µ2δ)(I1+I2)
−(θ+d1+σ2
3
2)]dt+σ3dB3(t),
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integrate the above formula from 0 to t, we have
ln(E1(t) + E2(t)) −ln(E1(0) + E2(0))
t
≤(δ+θ)+(β1+β2+µ1δ+µ2δ)< I1+I2>
−(θ+d1+σ2
3
2) + 1
tt
0
σ3dB3(t),
on the basis of [19], we can get
⟨I1+I2⟩≥− (δ+θ)
β1+β2+µ1δ+µ2δ
+ln(E1(0) + E2(0)) −ln(E1(t) + E2(t))
t(β1+β2+µ1δ+µ2δ)
+(θ+d1+σ2
3
2)
β1+β2+µ1δ+µ2δ
+1
β1+β2+µ1δ+µ2δ·1
tt
0
σ3dB3(t).
Therefore
lim
t→∞⟨I1+I2⟩
≥ − (δ+θ)
β1+β2+µ1δ+µ2δ+(θ+d1+σ2
3
2)
β1+β2+µ1δ+µ2δ
=(δ+θ)
β1+β2+µ1δ+µ2δ(d1+θ+σ2
3
2
δ+θ−1)
=κ(Rp
0−1),
where κ=δ+θ
β1+β2+µ1δ+µ2δ.
Since Rp
0>1, we can get the following results:
lim
t→∞ inf⟨I1(t) + I2(t)⟩ ≥ κ(Rp
0−1) = I > 0a.s.
Define differentiable mapping W1,W1=S1(t) +
S2(t). On the basis of Itˆo′s formula and model (2), we
have dW1(t) = LW1dt+σ1(S1+S2)dB1(t), where
LW1= Λ + rS1(1 −αS1)−β1S1I1
N1
e−m1I1(t−τ1)
−µ1δS2I1
N2
e−m2I1(t−τ1)−(w+d1)S1+γR1
+υV1−δ(S1−S2)+Λ+rS2(1 −αS2)
−β1S2I2
N2
e−m1I2(t−τ2)−µ1δS1I2
N1
e−m2I2(t−τ2)
−(w+d1)S2+γR2+υV2−δ(S2−S1)
≤2Λ −(w+d1−r)(S1+S2) + γ(R1+R2)
+υ(V1+V2).
Integrate over the above equation from 0to tand
multiply by 1
t
(S1(t) + S2(t)) −(S1(0) + S2(0))
t
≤2Λ −(w+d1−r)⟨S1+S2⟩+γ⟨R1+R2⟩
+υ(V1+V2) + σ1
tt
0
(S1+S2)dB1(s),
then, using the result of Lemma 2.1 and Lemma 2.2
we can obtain
lim
t→∞⟨S1+S2⟩ ≤ 2Λ
w+d1−r+2υN∗
w+d1−r
+γ
w+d1−rlim
t→∞ inf⟨R1+R2⟩
.
=¯
S.
where N∗is the maximum environmental capacity.
Define differentiable mapping W2,W2=E1(t)+
E2(t) + V1(t) + V2(t). On the basis of Itˆo′s formula
and model (2), we have dW2(t) = LW2dt+σ2(V1+
V2)dB2(t) + σ3(E1+E2)dB3(t), where
LW2(t) = w(S1+S2)−(υ+d1)(V1+V2)−(θ
+d1)(E1+E2) + β1S1I1
N1
e−m1I1(t−τ1)
+µ1δS2I1
N2
e−m2I1(t−τ1)+β1S2I2
N2
e−m1I2(t−τ2)
+µ1δS1I2
N1
e−m2I2(t−τ2)
≤w(S1+S2)−(θ+d1)(E1+E2)+(β1
+µ1δ)(I1+I2).
Integrate from 0to t
2
i=1[(Ei(t) + Vi(t)) −(Ei(0) + Vi(0))]
t
≤w⟨S1+S2⟩ − (θ+d1)⟨E1+E2⟩+ (β1
+µ1δ)⟨I1+I2⟩+σ2
tt
0
(V1+V2)dB2(s)
+σ3
tt
0
(E1+E2)dB3(s).
So, using the result of Lemma 2.1 and Lemma 2.2, we
can get
lim
t→∞⟨S1+S2⟩ ≥ θ+d1
wlim
t→∞ sup⟨E1+E2⟩
−β1+δµ1
wlim
t→∞ inf⟨I1+I2⟩
=θ+d1
w¯
E−β1+δµ1
wI.
=S.
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Therefore, we can get
S≤lim
t→∞⟨S1+S2⟩ ≤ ¯
S,
where S=θ+d1
w¯
E−β1+δµ1
wI,¯
S=2Λ
w+d1−r+
2υN∗
w+d1−r+γ
w+d1−rlimt→∞ inf⟨R1+R2⟩.
Define differentiable mapping W3,W3=V1(t) +
V2(t). On the basis of Itˆo′s formula and model (2), we
have dW3(t) = LW3dt+σ2(V1+V2)dB2(t), where
LW3(t) = w(S1+S2)−(υ+d1)(V1+V2)
−β2V1I1
N1
e−m3I1(t−τ1)−µ2δV2I1
N2
e−m4I1(t−τ1)
−β2V2I2
N2
e−m3I2(t−τ2)−µ2δV1I2
N1
e−m4I2(t−τ2)
≤w(S1+S2)−(υ+d1)(V1+V2),
integrate from 0to t and using the result of Lemma
2.1 and Lemma 2.2, we can get
lim
t→∞⟨V1+V2⟩ ≤ w
υ+d1
lim
t→∞ inf⟨S1+S2⟩
=w
υ+d1
S.
=¯
V .
On the other hand
LW3(t)≥w(S1+S2)−(υ+d1)(V1+V2)
−β2V1I1
N1
−µ2δV2I1
N2
−β2V2I2
N2
−µ2δV1I2
N1
≥w(S1+S2)−(υ+d1)(V1+V2)
−(β2+δµ2)(I1+I2)
The same can be obtained
lim
t→∞⟨V1+V2⟩
≥w
υ+d1
lim
t→∞ sup⟨S1+S2⟩
−β2+δµ2
υ+d1
lim
t→∞ inf⟨I1+I2⟩
=w
υ+d1
¯
S−β2+δµ2
υ+d1
I.
=V .
So, we can obtain
V≤lim
t→∞⟨V1+V2⟩ ≤ ¯
V ,
where V=w
υ+d1
¯
S−β2+δµ2
υ+d1I,¯
V=w
υ+d1S.
Define differentiable mapping W4,W4=E1(t)+
E2(t). On the basis of Itˆo′s formula and model (2), we
have dW4(t) = LW4dt+σ3(E1+E2)dB3(t), where
LW4(t)
=β1S1I1
N1
e−m1I1(t−τ1)+µ1δS2I1
N2
e−m2I1(t−τ1)
+β2V1I1
N1
e−m3I1(t−τ1)+µ2δV2I1
N2
e−m4I1(t−τ1)
+β1S2I2
N2
e−m1I2(t−τ2)+µ1δS1I2
N1
e−m2I2(t−τ2)
+β2V2I2
N2
e−m3I2(t−τ2)+µ2δV1I2
N1
e−m4I2(t−τ2)
−(θ+d1)(E1+E2)
≤(β1+δµ1+β2+δµ2)(I1+I2)
−(θ+d1)(E1+E2),
then
dW4(t)≤[(β1+δµ1+β2+δµ2)(I1+I2)−(θ
+d1)(E1+E2)]dt+σ3(E1+E2)dB3(t),
integrate from 0to t and using the result of Lemma
2.1 and Lemma 2.2, we can get
lim
t→∞⟨E1+E2⟩
≤β1+δµ1+β2+δµ2
θ+d1
lim
t→∞ inf⟨I1+I2⟩
=β1+δµ1+β2+δµ2
θ+d1
I
.
=¯
E.
On the other hand
LW4(t)≥β1S1I1
N1
e−m1N∗+µ1δS2I1
N2
e−m2N∗
+β2V1I1
N1
e−m3N∗+µ2δV2I1
N2
e−m4N∗
+β1S2I2
N2
e−m1N∗+µ1δS1I2
N1
e−m2N∗
+β2V2I2
N2
e−m3N∗+µ2δV1I2
N1
e−m4N∗
−(θ+d1)(E1+E2)
≥e−(m1∨m2∨m3∨m4)N∗
N∗(β1+δµ1+β2
+δµ2)(I1+I2)−(θ+d1)(E1+E2).
Since people’s immunity is one of the factors to
resist infectious diseases, and there are always people
with low immunity in the population, people with
low immunity are usually classified as susceptible
groups, so we assume that Si>1. Also, let’s assume
that Vi>1. After the integral, can be obtained
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lim
t→∞⟨E1+E2⟩
≥e−(m1∨m2∨m3∨m4)N∗
N∗(β1+δµ1+β2
θ+d1
+δµ2
θ+d1
)lim
t→∞ inf⟨I1+I2⟩
=e−(m1∨m2∨m3∨m4)N∗
N∗
β1+δµ1+β2+δµ2
θ+d1
I
.
=E.
So, we can get
E≤lim
t→∞⟨E1+E2⟩ ≤ ¯
E,
where E=e−(m1∨m2∨m3∨m4)N∗
N∗
β1+δµ1+β2+δµ2
θ+d1I,
¯
E=β1+δµ1+β2+δµ2
θ+d1I.
Define differentiable mapping W5,W5=
2
i=1(Hi(t) + Ri(t)). On the basis of Itˆo′s formula
and model (2), we have dW5(t) = LW5dt+σ5(H1+
H2)dB5(t) + σ6(R1+R2)d65(t), where
LW5= (fh+φ)(I1+I2)−(d1+d2)(H1+H2)
−(γ+d1)(R1+R2)
≤(fh+φ)(I1+I2)−(d1+d2)(H1+H2).
After integrating, using the result of lemma 2.1 and
lemma 2.2, we can get
lim
t→∞⟨H1+H2⟩ ≤ fh+φ
d1+d2
lim
t→∞ inf⟨I1+I2⟩
=fh+φ
d1+d2
I
.
=¯
H,
the same can be obtained
lim
t→∞⟨R1+R2⟩ ≤ fh+φ
γ+d1
I.
=¯
R.
Take the limit to get
lim
t→∞⟨H1+H2⟩ ≥ fh
d1+d2
lim
t→∞ inf⟨I1+I2⟩
−γ+d1
d1+d2
lim
t→∞ inf⟨R1+R2⟩
=fh
d1+d2
I−γ+d1
d1+d2
¯
R
.
=H.
According to the W1, we have
LW1≤2Λ + r(S1+S2)−(w+d1)(S1+S2)
+υ(V1+V2) + γ(R1+R2),
then, dW5(t)can be obtained after integration
γ⟨R1+R2⟩ ≥ (w+d1−r)⟨S1+S2⟩ − υ⟨V1
+V2⟩ − 2Λ −σ1
tt
0
dB1(t)
+(S1(t) + S2(t)) −(S1(0) + S2(0))
t,
using the result of Lemma 2.1 and Lemma 2.2, we can
get
lim
t→∞⟨R1+R2⟩ ≥ w+d1−r
γlim
t→∞ inf⟨S1+S2⟩
−υ
γlim
t→∞ sup⟨V1+V2⟩ − 2Λ
γ
=w+d1−r
γS−υ
γ¯
V−2Λ
γ
.
=R.
Then, we can get
H≤lim
t→∞⟨H1+H2⟩ ≤ ¯
H,
R≤lim
t→∞⟨R1+R2⟩ ≤ ¯
R,
where H=fh
d1+d2I−γ+d1
d1+d2
¯
R,¯
H=fh+φ
d1+d2I,R=
w+d1−r
γS−υ
γ¯
V−2Λ
γ,¯
R=fh+φ
γ+d1I.
The proof completes here.
4 Optimal&ontrol
For infectious diseases, understanding how to
control and eliminate/eradicate infectious diseases is
one of the main goals of mathematical epidemiology
and public health[20]. To this end, we studied the
impact of vaccination and aggressive treatment on
reducing disease transmission. At the same time,
we actively call for a reduction in the movement of
people everywhere. To this effect, we introduce into
the model (2) a set of time dependent control variables
u(t) = (u1(t), u2(t), u3(t), u4(t)), where
(a) u1(t)represents the infection rate of infectious
individuals (Ii) to susceptible individuals (Si).
(b) u2(t)represents the infection rate of local infected
individuals to nonnative susceptible individuals,
(c) u3(t)represents the recovery rate, the cure of
disease,
(d) u4(t)represents the vaccination of susceptible
individuals.
A stochastic control system with control variables
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u(t) = (u1(t), u2(t), u3(t), u4(t)) is as follows:
dSi(t) = [Λ + rSi(1 −αSi) + γRi+υVi
−u1(t)SiIi
Nie−m1Ii(t−τi)−(u4(t)
+d1)Si−u2(t)δSjIi
Nje−m2Ii(t−τi)
−δ(Si−Sj)]dt+σ1SidB1(t),
dVi(t) = [u4(t)Si−β2ViIi
Nie−m3Ii(t−τi)
−µ2δVjIi
Nje−m4Ii(t−τi)−(υ
+d1)Vi−δ(Vi−Vj)]dt
+σ2VidB2(t),
dEi(t) = [u1(t)SiIi
Nie−m1Ii(t−τi)−(θ
+d1)Ei+u2(t)δSjIi
Nje−m2Ii(t−τi)
+β2ViIi
Nie−m3Ii(t−τi)−δ(Ei
−Ej+µ2δVjIi
Nje−m4Ii(t−τi)]dt
+σ3EidB3(t),
dIi(t) = [θEi−(d1+φ+fh)Ii−δ(Ii
−Ij)]dt+σ4IidB4(t),
dHi(t) = [fhIi+ (−1)i+1g1I2−(u3(t)
+d1+d2)Hi]dt+σ5HidB5(t),
dRi(t) = [φIi+u3(t)Hi−(γ+d1)Ri
−δ(Ri−Rj)]dt+σ6RidB6(t),
(3)
where i= 1,2and j=i−(−1)i(i= 1,2).
Our control problem is to minimize the number
of symptomatic individuals as well as minimizing the
cost of treatment via minimization of the following
cost functional, thus our objective function is as
follows
J(u1, u2, u3, u4) = Etf
0
[
2
i=1
(A1Ii+A2Hi)(4)
+1
2(B1u1(t)2+B2u2(t)2+B3u3(t)2+B4u4(t)2)]dt,
where tfis the final time, Ai, Bj(i= 1,2; j=
1,2,3,4) are balancing cost factors. In order to
optimize our control strategy, we need to save as
much money as possible while controlling the spread
of the disease. Thus, we seek to find an optimal
control, u∗= (u∗
1, u∗
2, u∗
3, u∗
4), such that the optimal
control function
J(u∗) = inf
UJ(u),(5)
where U={(u1, u2, u3, u4)∈(L4(0, tf))2|ai≤
ui(t)≤bi, t ∈[0, tf]} is the control set, and
ai, bi(i= 1,2,3,4) are fixed positive.
4.1 Existence of an2ptimal&ontrol
Using the results obtained in reference [21], it
is possible to prove the existence of the solution of
model (3) in the finite time interval of the given
control in the admissible control set U. And there are
the following results.
Theorem 4.1 Given any control (u1, u2, u3, u4)∈U,
there exists a bounded solution to model (3).
Since the state variables and the controls are
uniformly bounded, existence of an optimal
control follows boundedness of solutions and
their derivatives of the model (3) for a finite time
interval. The boundedness and convexity of the
target functional provide sufficient compactness for
the existence of the optimal control [22@>23].7hus,
with the objective functional Jin Eq. (4) subject
to the control set U, there exists an optimal control
u∗∈Usuch that J(u∗) = minu∈UJ(u).
4.2 Specific'escription of2ptimal&ontrol
Pontryagin’s Maximum Principle is a necessary
condition for optimal control. This principle converts
(3) and (4) into a problem of minimizing pointwise a
Hamiltonian H, with respect to u= (u1, u2, u3, u4).
Firstly, we obtain the Hamiltonian from the target
functional (4) and the stochastic system (3), and thus
obtain the optimality condition. So, by processing the
Hamiltonian function, we can obtain
H(t)
=
2
i=1
(A1Ii+A2Hi) + 1
2(B1u1(t)2+B2u2(t)2
+B3u3(t)2+B4u4(t)2) +
2
i=1
{λSi[Λ + rSi(1
−αSi)−u1(t)SiIi
Ni
e−m1Ii(t−τi)−(u4(t)
+d1)Si
u2(t)δSjIi
Nj
e−m2Ii(t−τi)+γRi+υVi
−δ(Si−Sj)] + λVi[u4(t)Si−β2ViIi
Ni
e−m3Ii(t−τi)
−µ2δVjIi
Nj
e−m4Ii(t−τi)−(υ+d1)Vi−δ(Vi−Vj)]
+λEi[u1(t)SiIi
Ni
e−m1Ii(t−τi)+u2(t)δSjIi
Nj
e−m2Ii(t−τi)
+β2ViIi
Ni
e−m3Ii(t−τi)+µ2δVjIi
Nj
e−m4Ii(t−τi)−(θ
+d1)Ei−δ(Ei−Ej)] + λIi[θEi−(d1+φ+fh)Ii
−δ(Ii−Ij)] + λHi[fhIi+ (−1)i+1g1I2−(u3(t)
+d1+d2)Hi] + λRi[φIi+u3(t)Hi−(γ+d1)Ri
−δ(Ri−Rj)]}+σ1(S1q1+S2q2) + σ2(V1q3
+V2q4) + σ3(E1q5+E2q6) + σ4(I1q7+I2q8)
+σ5(H1q9+H2q10) + σ6(R1q11 +R2q12),
where the λSi, λVi, λEi, λIi, λHi,λRi(i= 1,2)
are the associated adjoints for the state variables
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Si, Vi, Ei, Ii,Hi, Ri. The sys of equations is found
by taking the appropriate partial derivatives of the
Hamiltonian with respect to the associated state
variable.
Given the optimal control u∗= (u∗
1, u∗
2, u∗
3, u∗
4)
and the corresponding state solutions S∗
i,V∗
i,E∗
i,
I∗
i,H∗
i,R∗
i,(i= 1,2) of system (3) that
minimizes J(u1, u2,u3, u4)over U, there exists
λSi, λVi, λEi, λIi, λHi, λRi(i= 1,2) satisfying the
adjoint system
dλS1
dt= (λS1−λE1)e−m1I1(t−τ1)u1(t)I1
N1
+σ1q1+ (λS1
−λV1)u4(t)+(λS2−λE2)e−m2I2(t−τ2)u2(t)δI2
N1
+δ(λS1−λS2) + λS1(d1−r(1 −2αS1)),
dλS2
dt= (λS2−λE2)e−m1I2(t−τ2)u1(t)I2
N2
+σ1q2+ (λS2
−λV2)u4(t)+(λS1−λE1)e−m2I1(t−τ1)u2(t)δI1
N2
+δ(λS2−λS1) + λS2(d1−r(1 −2αS2)),
dλV1
dt= (λV1−λE1)e−m3I1(t−τ1)β2I1
N1
+ (λV2
−λE2)e−m4I2(t−τ2)µ2δI2
N1
+ (λV1−λS1)υ
+d1λV1+δ(λV1−λV2) + σ2q3,
dλV2
dt= (λV2−λE2)e−m3I2(t−τ12) β2I2
N2
+ (λV1
−λE21)e−m4I1(t−τ1)µ2δI1
N2
+d1λV2
+ (λV2−λS2)υ+δ(λV2−λV1) + σ2q4,
dλE1
dt= (λE1−λE2)δ+ (λE1−λI1)θ+λE1d1
+σ3q5,
dλE2
dt= (λE2−λE1)δ+ (λE2−λI2)θ+λE2d1
+σ3q6,
dλI1
dt=−A1+σ4q7+ (λI1−λI2)δ+λI1d1
+ (λS1−λE1)(u1(t)e−m1I1(t−τ1)S1
N1
+δu2(t)e−m2I1(t−τ1)S2
N2
)+(λI1−λR1)φ
+ (λI1−λH1)fh+ (λV1
−λE1)(β2e−m3I1(t−τ1)V1
N1
+δµ2e−m4I1(t−τ1)V2
N2
)−χ[0,T −τ1]{[λE1(t+τ1)
−λS1(t+τ1)](u1(t)m1e−m1I1(t−τ1)S1I1
N1
+e−m2I1(t−τ1)u2(t)m2δS2I1
N2
)
+ [λE1(t+τ1)
−λV1(t+τ1)](e−m3I1(t−τ1)m3β2V1I1
N1
+e−m4I1(t−τ1)m4δµ2V2I1
N2
)}
dλI2
dt=−A1+σ4q8+ (λI2−λI1)δ+λI1d1
+ (λS2−λE2)(u1(t)e−m1I2(t−τ2)S2
N2
+δu2(t)e−m2I2(t−τ2)S1
N1
)+(λI2−λR2)φ
+ (λI2−λH2)fh+ (λH2−λH1)g1+ (λV2
−λE2)(β2e−m3I2(t−τ2)V2
N2
+δµ2e−m4I2(t−τ2)V1
N1
)−χ[0,T −τ2]{[λE2(t+τ2)
−λS2(t+τ2)](u1(t)m1e−m1I2(t−τ2)S2I2
N2
+u2(t)δm2e−m2I2(t−τ2)S1I2
N1
)
+ [λE2(t+τ2)
−λV2(t+τ2)](e−m3I2(t−τ2)m3β2V2I2
N2
+e−m4I2(t−τ2)m4δµ2V1I2
N1
)},
dλH1
dt=−A2+ (λH1−λR1)u3(t) + λH1(d1+d2)
+σ5q9,
dλH2
dt=−A2+ (λH2−λR2)u3(t) + λH2(d1+d2)
+σ5q10,
dλR1
dt= (λR1−λS1)γ+λR1d1+ (λR1−λR2)δ
+σ6q11,
dλR2
dt= (λR2−λS2)γ+λR2d1+ (λR2−λR1)δ
+σ6q12,
with transversality conditions λSi(tf) = 0,λVi(tf) =
0,λEi(tf) = 0,λIi(tf) = 0,λHi(tf) = 0,λRi(tf) =
0,where χ[0,T −τi]are indicator functions on [0, T −
τi] (i= 1,2)satisfying
χ[0,T −τi)=1t∈[0, T −τi)
0t/∈[0, T −τi).(6)
Considering the optimality conditions, the
Hamiltonian function is differentiated with respect to
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the control variables resulting in
∂H
∂u1
=B1u1(t)−(λS1−λE1)e−m1I1(t−τ1)S1I1
N1
−(λS2−λE2)e−m1I2(t−τ2)S2I2
N2
= 0,
∂H
∂u2
=B2u2(t)−(λS1−λE1)e−m2I1(t−τ1)δS2I1
N2
−(λS2−λE2)e−m2I2(t−τ2)δS1I2
N1
= 0,
∂H
∂u3
=B3u3(t)−(λH1−λR1)H1−(λH2−λR2)H2
= 0,
∂H
∂u4
=B4u4(t)−(λS1−λV1)S1−(λS2−λV2)S2
= 0,
on the interior of the control set U. So we can get
an unique solution ¯u(t) = (¯u1(t),¯u2(t),¯u3(t),¯u4(t))
in terms of state variables and adjoint variable, as
following
¯u1(t) = 1
B1
[(λS1−λE1)e−m1I1(t−τ1)S1I1
N1
+ (λS2−λE2)e−m1I2(t−τ2)S2I2
N2
],
¯u2(t) = 1
B2
[(λS1−λE1)e−m2I1(t−τ1)δS2I1
N2
+ (λS2−λE2)e−m2I2(t−τ2)δS1I2
N1
],
¯u3(t) = 1
B3
[(λH1−λR1)H1+ (λH2−λR2)H2],
¯u4(t) = 1
B4
[(λS1−λV1)S1−(λS2−λV2)S2],
Finally, we get the solution shown below
u∗
1=min{b, max[a, ¯u1(t)]},
u∗
2=min{b, max[a, ¯u2(t)]},
u∗
3=min{b, max[a, ¯u3(t)]},
u∗
4=min{b, max[a, ¯u4(t)]}.
5 Numerical6imulations
In this section, we use Milsteins Higher Order
Method for discretization and the RK4 techniques
for iteration to perform numerical simulations. The
following three examples are numerically simulated
for extinction, persistence, and optimal control of
diseases.
Example 1. According to Theorem 2.2, we obtain
a sufficient condition for disease extinction, which
is Rjp
0<1.As shown in Fig)LJ2, it can
EHintuitively seen that when Rjp
0<1is satisfied, the
number of patients increases to a certain number and
then gradually decreases until it approaches 0, which
represents the gradual extinction of the disease.
0 5 10 15 20 25 30
0
5
10
15
20
25
30
I1
Fig.ure 1: I1(t)-infected individuals in
the A place
0 10 20 30 40 50 60 70 80 90
1
2
3
4
5
6
7
8
I2
Fig.ure 2: I2(t)-infected individuals in
the B place
Example 2. According to Theorem 3.1, we obtain
a sufficient condition for the persistence of disease,
which is Rp
0>1. As shown in Fig)LJ4, it can
be intuitively seen that under the premise of meeting
Rp
0>1, although the number of patients in the image
is decreasing, it does not approach 0, which indicates
that the disease will always exist for a long time.
Example 3. As can be seen from Fig.5 & Fig.6,
implementing appropriate control strategies after a
disease outbreak can better control the spread of the
disease than not implementing measures.
6 Conclusion
By incorporating urban transport correlation,
media coverage, and time delays into the SVEIHR
epidemic model, we can gain a more realistic
understanding of disease transmission. In this paper,
we use suitable Lyapunov functions to study the
existence of global positive solutions and establish the
conditions for disease extinction or persistence. At
the same time, we use the stochastic optimal theory to
establish the optimal control strategy. In addition, we
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.84
Rujie Yang, Hong Qiu
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0 20 40 60 80 100
0
50
100
150
200
250
300
350
400
I1
Fig.ure 3: I1(t)-infected individuals in
the A place
0 20 40 60 80 100
0
10
20
30
40
50
60
I2
Fig.ure 4: I2(t)-infected individuals in
the B place
construct an efficient numerical format based on the
Milstein method to support our results. Our analysis
also highlights the impact of time delay on model
dynamics, which provides an aspect of disease control
that needs attention. According to our research, the
movement of people between two cities can increase
the spread of disease. Therefore, in the outbreak of
infectious diseases, we should pay attention to the
control of the flow of people, timely and rational use
of the media, and do our best to quickly control the
spread of the disease and protect people’s physical
and mental health.
References:
[1] XY. Bai, SJ. Ma, Stochastic dynamical behavior
of COVID-19 model based on secondary vaccina-
tion, Mathematical biosciences and engineering :
MBE, 20(2) (2023) 2980-2997.
[2] HT. Song, RF Wang, et al. Global stability
and optimal control for a COVID-19 model
with vaccination and isolation delays, Results in
physics, 42 (2022) 106011.
[3] G. Liu, NR. Mugo, C. Bayer, et al. Impact of
catch-up human papillomavirus vaccination
on cervical cancer incidence in Kenya: A
0 5 10 15 20 25 30
0
0.5
1
1.5
2
2.5
3
without control
with control
Fig.ure 5: I1(t)-infected individuals in
the A place
0 5 10 15 20 25 30
0
0.5
1
1.5
2
2.5
3
without control
with control
Fig.ure 6: I2(t)-infected individuals in
the B place
mathematical modeling evaluation of HPV
vaccination strategies in the context of moderate
HIV prevalence, Eclinicalmedicine, 45 (2022)
101306.
[4] Y. Chen, M. Yan, ZY. Xiang, An epidemic model
with transport-related infection incorporating
awareness and screening, Journal of applied
mathematics and computing, 68(5) (2022)
3107-3146.
[5] R. Liu, GR. Liu, Analysis of a stochastic SIS
epidemic model with transport-related infection,
Journal of applied analysis and computation,
11(3) (2021) 1296-1321.
[6] A. Alkhazzan, JA. Wang, YF. Nie, et al. A
stochastic SIRS modeling of transport-related
infection with three types of noises, Alexandria
engineering journal, 76 (2023) 557-572.
[7] A. Yousef, A fractional-order dynamical model of
the SEIVRS epidemic with temporary immunity
and transport-related infection, Alexandria
engineering journal, 78 (2023) 592-613.
[8] JJ. Xian, ZH. Zhang, ZY. Li, et al. Coupled
Information-Epidemic Spreading Dynamics with
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.84
Rujie Yang, Hong Qiu
E-ISSN: 2224-2880
825
Volume 23, 2024
Selective Mass Media, Entropy, 25(6) (2023)
927.
[9] JM. Tchuenche, N. Dube, CP. Bhunu, RJ. Smith,
CT. Bauch, The impact of media coverage on the
transmission dynamics of human influenza, BMC
Public Health, 11 (2011) S5.
[10] JG. Cui, YH. Sun, HP. Zhu, The impact of media
on the control of infectious diseases, Journal of
dynamics and differential equations, 20(1) (2008)
31-53.
[11] T. Kuniya, Recurrent epidemic waves in a
delayed epidemic model with quarantine, Journal
of biological dynamics, 16(1) (2022) 619-639.
[12] GO. Agaba, YN. Kyrychko, KB. Blyuss,
Dynamics of vaccination in a time-delayed
epidemic model with awareness, Mathematical
bioscience, 294 (2017) 92-99.
[13] XY. Shi, XW. Gao, XY. Zhou, et al. Analysis of
an SQEIAR epidemic model with media coverage
and asymptomatic infection, AIMS Mathematics,
6(11) (2021) 12298-12320.
[14] PF. Liu, YT. Luo, ZD. Teng, Role of media
coverage in a SVEIR-I epidemic model with
nonlinear incidence and spatial heterogeneous
environment, Mathematical biosciences and
engineering, 20 (9) (2023) 15641-15671.
[15] XY. Wang, Switching vaccination schemes for
epidemic models with distributed time delay
and two types of noise, Advances in difference
equations, 2021(1) (2021) 150.
[16] T. Khan, A. Khan, G. Zaman, A stochastic SIS
epidemic infectious diseases model with double
stochastic perturbations, International journal of
biomathematics, 17(4) (2024) 2350040.
[17] H. Chen, XW. Tan, J. Wang, et al. Stochastic
Dynamics of a Virus Variant Epidemic Model
with Double Inoculations, Mathematics, 11(7)
(2023) 1712.
[18] A. Khan, R. Ikram, et al. Stochastic COVID-19
SEIQ epidemic model with time-delay, Results in
physics, 30 (2021) 104775.
[19] CY. Ji, DQ. Jiang, Threshold behaviour of
a stochastic SIR model. Applied mathematical
modelling, 38 (2014) 5067-5679.
[20] JL. Ma, Z. Ma, Epidemic threshold conditions
for seasonally forced SEIR models, Mathematical
biosciences and engineering, 3(1) (2006)
161-172.
[21] DL. Lukes, Differential Equations: Classical
to Controlled, Mathematics in Science and
Engineering, Academic Press, New York, 1982.
[22] WH. Fleming, RW. Rishel, Deterministic and
Stochastic Optimal Control, Springer Verlag,
New York, 1975.
[23] MR. Kelly, JH. Tien, MC. Eisenberg, S.
Lenhart, The impact of spatial arrangements
on epidemic disease dynamics and intervention
strategies, Journal of biological dynamics, 10(1)
(2016) 222-249.
Contribution of individual authors to
the creation of a scientific article
(ghostwriting policy)
Hong Qiu, instructed and checked the reasonableness
and correctness of the article. Rujie Yang is
responsible for the derivation of calculations,
simulation design and the writing of articles.
Sources of funding for research
presented in a scientific article or
scientific article itself
The Scientific Research Project of Tianjin Municipal
Educational Commission (Grant No. 2022KJ069).
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
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WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.84
Rujie Yang, Hong Qiu
E-ISSN: 2224-2880
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Volume 23, 2024