An-optimal-control-problem-for-dengue-tran_2023_Communications-in-Nonlinear-.pdf

Communications in Nonlinear Science and Numerical Simulation 116 (2023) 106856
Contents lists available at ScienceDirect
Communications in Nonlinear Science and
Numerical Simulation
journal homepage: www.elsevier.com/locate/cnsns
Research paper
An optimal control problem for dengue transmission model
with Wolbachia and vaccination
Jian Zhang a
, Lili Liu a,∗
, Yazhi Li b
, Yan Wang c
a
Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis on Disease Control and Prevention, Complex Systems
Research Center, Shanxi University, Taiyuan 030006, China
b
Key Laboratory of Complex Systems and Intelligent Optimization of Qiannan, School of Mathematics and Statistics, Qiannan
Normal University for Nationalities, Guizhou Duyun 558000, China
c
Key Laboratory of Eco-environments in Three Gorges Reservoir Region (Ministry of Education), School of Mathematics and
Statistics, Southwest University, Chongqing 400715, China
a r t i c l e i n f o
Article history:
Received 26 December 2021
Received in revised form 29 July 2022
Accepted 29 August 2022
Available online 2 September 2022
Keywords:
Dengue
Wolbachia
Vaccination
Optimal control
a b s t r a c t
The release of Wolbachia-infected mosquitoes into the wild mosquitoes population is
an excellent biological control strategy which can be effective against mosquito-borne
infections. In this work, we propose a dengue transmission model that incorporates
releasing Wolbachia into the wild mosquito population and vaccinating human popu-
lation. We analyze the basic reproduction number, the existence and stability of the
equilibria, and carry out the sensitivity analysis to quantify the effects of parameters
on the basic reproduction number. The analysis shows that the basic reproduction
number determines the stability of two equilibria, and two transmission probabilities are
positively correlated with the basic reproduction number, especially, the transmission
probability of human being infected byWolbachia-infected mosquitoes is more sensitive
than that of wild mosquitoes. Based on an integrated control strategy for the prevention
of dengue fever (vaccination, using mosquito nets, improved treatment of dengue
and spraying insecticides), we derive the necessary conditions for the optimal control
problem using Pontryagin’s Maximum Principle and carry out numerical simulations by
MATLAB. Finally, the cost-effectiveness analysis of several control strategies is examined
to determine the most feasible strategy. It shows that the most cost-effective integrated
strategy is vaccination, using mosquito nets and spraying insecticides.
© 2022 Elsevier B.V. All rights reserved.
1. Introduction
Dengue is an acute vector-borne disease caused by the dengue virus, the primary vector is the Aedes aegypti
mosquito [1]. Dengue is endemic in at least 100 countries in Asia, Pacific, Americas, Africa and Caribbean, and more
than 40% of the global population live in areas at the risk of dengue transmission [2]. According to the World Health
Organization (WHO) [3], it is estimated that more than 100–400 million people worldwide are infected with dengue fever
each year. With the increase of urbanization and climate change, Aedes aegypti, as the main mosquito vector of dengue,
continues to spread widely and invades new habitats [4]. This causes successive dengue and focuses on the prevention
for humans and mosquito population control.
∗ Corresponding author.
E-mail address: liulili03@sxu.edu.cn (L. Liu).
https://doi.org/10.1016/j.cnsns.2022.106856
1007-5704/© 2022 Elsevier B.V. All rights reserved.

J. Zhang, L. Liu, Y. Li et al. Communications in Nonlinear Science and Numerical Simulation 116 (2023) 106856
Vaccination is the best strategy for humans to prevent and control dengue transmission. Dengue vaccine has been
under development since the 1940s [5]. The development of an effective vaccine remains a challenge because there are
four main strains (DEN1-DEN4). In December 2015, the first dengue vaccine was produced by Sanofi Pasteur. Subsequently,
several dengue endemic regions such as Philippines, Brazil and Paraguay applied this vaccine [6]. Although all the vaccines
for dengue have been proven to be ineffective, they still have attracted a large number of scholars to work on them. Garba
et al. [7] studied a class of dengue model with an imperfect vaccine and showed that an imperfect dengue vaccine would
have an important epidemiological impact on the effective control of dengue. Shim [8] showed that an increase in vaccine
coverage reduces the cost-effectiveness of dengue vaccination. Chao et al. [9] developed a stochastic model of dengue
transmission and calibrated it with local data on dengue serotype-specific infections and hospitalizations. The simulation
results suggested that children should be vaccinated at a high priority, and adults also need to be vaccinated if they want
to prevent dengue infection. Maier et al. [10] studied the optimal age for dengue vaccination based on data for Brazil and
noted that the optimal age for vaccination is 9–45 years old.
In terms of mosquito prevention and control, one of common methods is to reduce the number of mosquitoes by
cleaning the mosquito habitats and spraying insecticides with certain chemicals to kill immature and adult mosquitoes.
However, since mosquitoes are active in the summer and fall, these methods can only control the mosquito population
for a short time. In recent years, biological measures to control mosquito populations have emerged, such as releasing
Wolbachia-carrying mosquitoes, which has attracted more attention. Wolbachia is an endosymbiotic bacteria that lives
within the cells of its invertebrate hosts [11]. When male Wolbachia-infected mosquitoes mate with wild females, their
eggs do not hatch [12]. This phenomenon is called cytoplasmic incompatibility (CI). In addition, if female mosquitoes
carry Wolbachia, their offspring also carry Wolbachia. Therefore, dengue transmission can be mitigated by continuous
release male mosquitoes infected with Wolbachia [13]. Nowadays, the transmission dynamics of Wolbachia has become
an important research topic. Zheng et al. [14] showed that a combination of radiation-based insect sterilization techniques
and CI effect caused by Wolbachia could eliminate Aedes albopictus, which is the world’s most invasive mosquito species. Li
et al. [15] developed a mathematical ODE model to analyze the transmission of dengue between humans and mosquitoes.
It is pointed out that the persistence of Wolbachia depends on its suitability for mosquitoes, and Wolbachia significantly
reduces the transmission of dengue. Hughes et al. [16] pointed out that if ℜ0 is not particularly large, Wolbachia is
theoretically able to control dengue fever regionally, however, if ℜ0 ≫ 1, this approach can only reduce the number of
infected but cannot eradicate dengue. Taghikhani et al. [17] designed a new two-sex mathematical model of the population
ecology of dengue mosquitoes and dengue fever disease. They conclude that only releasing Wolbachia-infected adult males
is more effective than releasing adult female mosquitoes infected with Wolbachia. Cardona et al. [18] showed that releasing
Wolbachia-carrying mosquitoes daily can reduce the number of humans infected with dengue.
Optimal control is a core element of modern control theory. Nowadays it is more widely applied to the research and
strategic control of various infectious diseases, such as tuberculosis [19,20] and Zika [21,22] et al. In general, people focus
on the effects of disease reduction while ignoring the cost-effectiveness of these interventions. As a result, there are
relatively few studies on cost-effectiveness analysis of control strategies [23–25]. The aim of this study is to investigate
the joint impacts and cost-effectiveness of four possible combinations of dengue control strategies: (i) vaccination, (ii)
using mosquito nets, (iii) improved treatment of dengue and (iv) spraying insecticides. To this end, we first consider
control parameters as constants to study the stability of the model and sensitivity, and then consider control parameters
as time-dependent functions to find the optimal control solution and explore the impact and cost-effectiveness analysis
of different combinations of these controls on dengue transmission.
The rest of this paper is organized as follows. In Section 2, we mainly formulate the model and set some basic
assumptions. In Section 3 and Section 4, we present stability analysis of the model and sensitivity analysis based on
the basic reproduction number, respectively. In Section 5, we work on the optimal control problem for the proposed four
strategies, while in Section 6, we numerically simulate the control problem and discuss the cost-effectiveness of four
integrated strategies. Finally, conclusion and discussion are given in Section 7.
2. Model formulation and the basic property
Mathematical modeling is useful to illustrate the dynamics of Wolbachia in mosquito populations. It generally
divides mosquito populations (Nm) into two types: wild mosquito populations (Nmu) and Wolbachia-carrying mosquito
populations (Nmi). In 2010, Farkas et al. [26] developed a model of Wolbachia transmission in mosquito populations
⎧
⎪
⎪
⎨
⎪
⎪
⎩
dNmi
dt
= qrmNmi − (µm + µ̄m)NmNmi,
dNmu
dt
= (1 − q)rmNmi +
(
1 −
pNmi
Nm
)
Nmu − (µm + µ̄m)NmNmu,
(1)
where rm is the birth rate of mosquitoes, µm and µ̄m denote mortality and adaptation rates, respectively. q ∈ [0, 1] is the
probability of maternal transmission, and p ∈ [0, 1] is the probability of CI. Walker et al. [27] implied that the maternal
transmission and CI are almost perfect. Hence, in our paper, we make the assumption that q = 1 and p = 1.
For the transmission of dengue fever in mosquito populations, we consider that mosquitoes have extrinsic incubation
period (EIP), which is due to the fact that EIP of dengue virus in the mosquito is about 4–10 days before they have the
2

Fig. 1. Schematic diagram of compartmental dengue transmission model (2) with Wolbachia and vaccination.
ability to infect others. We assume that the mosquitoes cannot recover. Therefore, the mosquito populations are modeled
as a SEI structure, where S, E and I are susceptible group, incubation group and infected group, respectively. Obviously,
Nmu = Smu + Emu + Imu and Nmi = Smi + Emi + Imi.
For human populations, we include the vaccine compartment and ignore the intrinsic incubation period (IIP). Although
the IIP for the patient infected the dengue virus is about 4–5 days, which is too short compared with the lifetime of
humans. Therefore, the human populations Nh are modeled as a SVIR structure, where Sh, Vh, Ih and Rh are susceptible
humans, vaccinated humans, infected humans and recovered humans, respectively.
Combining the above factors, we develop the following model with the flow chart shown in Fig. 1 and the parameters
are listed in Table 1.
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
dSmi
dt
= r̄mNmi − µmNmSmi −
βmSmiIh
Nh
,
dEmi
dt
=
βmSmiIh
Nh
− (µmNm + km)Emi,
dImi
dt
= kmEmi − µmNmImi,
dSmu
dt
= rm
(
1 −
Nmi
Nm
)
Nmu − µmNmSmu −
βmSmuIh
Nh
,
dEmu
dt
=
βmSmuIh
Nh
− (µmNm + km)Emu,
dImu
dt
= kmEmu − µmNmImu,
dSh
dt
= Λ − (µh + ph)Sh −
(β̄hImi + βhImu)Sh
Nh
+ σhVh,
dVh
dt
= phSh − (σh + µh)Vh,
dIh
dt
=
Nh
− (µh + γh)Ih,
dRh
dt
= γhIh − µhRh.
(2)
The reproductive capacity of mosquitoes decreases with Wolbachia infection [28–30]. Hence, we consider r̄m < rm. The
probability of a human being bitten by a mosquito regardless of whether the mosquito carries Wolbachia is essentially
the same. We make the assumption that infected humans have the same probability of infecting two types of susceptible
mosquitoes, denoted by βm. Moreover, Wolbachia is able to reduce or block the replication of dengue virus in mosquitoes,
thereby decreasing the ability of the virus to be transmitted in mosquitoes [27,31,32]. Therefore, we hypothesize that
the transmission rate from wild mosquitoes to humans (βh) is different from that of Wolbachia-infected mosquitoes to
humans (β̄h), and βh > β̄h.
The model is biologically feasible only when the solutions are non-negative. The following result is regarding positivity
and boundedness of the solution to model (2).
3

Table 1
Definitons of model variables and parameters.
Parameters Definitions Values Range Source
rm Birth rate of W-free mosquitoes (day−1
) 0.13 0.02–0.27 [15]
r̄m Birth rate of W-infected mosquitoes (day−1
) 2rm/3 – [18]
µm Natural death rate of mosquitoes (day−1
) 2.0 × 10−5
1.0 × 10−6
–1.0 × 10−3
[33]
βm Human-to-mosquito transmission probability 0.24 0.01–0.27 [15,33]
km Rate at which exposed mosquitoes 1/7 0.05–0.1 [34]
become infectious (day−1
)
Λ Recruiting rate of humans (day−1
) 100 0 − ∞ [25]
µh Natural death rate of humans (day−1
) 3.5 × 10−5
1.0 × 10−6
–1.0 × 10−3
[15,34]
ph Vaccination rate of humans (day−1
) 0.2 0–1 [35]
βh W-free mosquito-to-human 0.48 0.072–0.64 [33]
transmission probability
β̄h transmission probability of 0.026 – Assumed
W-infected mosquito-to-human
σh Waning rate of immunity (day−1
) 1/720 0 − ∞ [35]
γh Recovery rate of infectious humans (day−1
) 1/4 1/14–1/3 [36]
‘‘W-’’ stands for the abbreviation of ‘‘Wolbachia-’’.
Theorem 2.1. For non-negative initial condition, all solutions of system (2) remain non-negative and ultimately bounded for
all t ≥ 0.
Proof. The non-negativity is obvious. We only need to prove that the solutions are bounded.
Adding up the first six equations of system (2), we can see that the size of mosquito population Nm satisfies
dNm
dt
= r̄mNmi − µmNmNmi + rm
(
1 −
Nmi
Nm
)
Nmu − µmNmNmu ≤ max{r̄m, rm}Nm − µmN2
m.
Similarly, summing up the last four equations of system (2), we have dNh/dt = Λ−µhNh. Hence, lim supt→∞ Nm =
max{r̄m, rm}/µm and lim supt→∞ Nh = Λ/µh.
By Theorem 2.1, the positively invariant set for system (2) can be defined by
Ω =
{
(Smi, Emi, Imi, Smu, Emu, Imu, Sh, Vh, Ih, Rh) ∈ R10
+ : Nm ≤ max{r̄m, rm}/µm, Nh ≤ Λ/µh
}
.
3. Stability analysis
In this section, we focus on the analysis of system (2), including the basic reproduction number, the existence
and stability of the equilibria. It is noted that here we are interested in two extreme cases: wild-dominated and
Wolbachia-dominated, which correspond to whether the Wolbachia-dominated population invades successfully or not.
3.1. The existence of equilibria
3.1.1. Mosquito-free equilibrium
The mosquito-free and dengue-free equilibrium is E0 =
(
0, 0, 0, 0, 0, 0, S0
h , V0
h , 0, 0
)
, where
S0
h =
Λ(σh + µh)
µh(σh + µh + ph)
, V0
h =
Λph
µh(σh + µh + ph)
. (3)
This equilibrium is ecologically difficult to hold, since eliminating mosquitoes is nearly impossible. Therefore, we do
not focus on the property of E0.
3.1.2. Wild-dominated equilibrium
The wild-dominated and dengue-free equilibrium is E1 =
(
0, 0, 0, S0
mu, 0, 0, S0
h , V0
h , 0, 0
)
, where S0
mu = rm/µm, S0
h and
V0
h are defined in (3).
The next generation matrix approach [37] is applied to obtain the basic reproduction number (see Appendix A for
details). It is given by
ℜ01 =
√
ℜhℜm,
4

where
ℜh =
km
µmN0
m + km
·
1
µmN0
m
·
βh
N0
h
· S0
h =
kmβh(σh + µh)
rm(rm + km)(σh + µh + ph)
,
ℜm =
1
µh + γh
·
βm
N0
h
· S0
mu =
rmβmµh
Λµm(µh + γh)
.
Here, N0
h = S0
h +V0
h , N0
m = S0
mi +S0
mu. Note that, km/(µmN0
m + km) is the probability of an exposed mosquito being infectious.
1/µmN0
m is the average lifespan of infectious mosquitoes. The infection rate from infected mosquitoes to susceptible
humans is βh/N0
h . Thus, ℜh is the average number of an infectious mosquito infecting the susceptible humans during its
infectious period. Similarly, 1/(µh + γh) is the average lifespan of infectious mosquitoes. The infection rate from infected
humans to susceptible mosquitoes is βm/N0
h . Thus, ℜm is the average number of the susceptible mosquitoes infected by
an infectious human during his infectious period.
The wild-dominated and dengue-infected equilibrium is denoted by
E∗
1 = (0, 0, 0, S∗
mu, E∗
mu, I∗
mu, S∗
h1, V∗
h1, I∗
h1, R∗
h1), (4)
where
S∗
mu = τ2 − τ3I∗
mu, E∗
mu =
rm
km
I∗
mu, I∗
mu =
τ4ζ4(ℜ2
01 − 1)
ζ1(τ1τ3ζ2 + τ4ζ3)
,
S∗
h1 = ζ2 − ζ3I∗
h1, V∗
h1 =
phζ2
σh + µh
−
phζ3
σh + µh
I∗
h1, I∗
h1 =
τ4ζ4(ℜ2
01 − 1)
τ1(τ2ζ1ζ3 + τ3ζ4)
, R∗
h1 =
γh
µh
I∗
h1.
The detailed calculation of E∗
1 is shown in Appendix B. The notations τi and ζi (i = 1, 2, 3, 4) are given in (B.3). From the
biological meaning, E∗
1 stands for the invasion failure of Wolbachia-infected mosquito populations.
According to Appendices C and D, we obtain the existence and local stability of E1 and E∗
1 by the following theorems.
Theorem 3.1. There always exists a wild-dominated and dengue-free equilibrium E1, and it is locally asymptotically stable if
ℜ01 < 1.
Theorem 3.2. If ℜ01 > 1, there exists a wild-dominated and dengue-infected equilibrium E∗
1 , which is locally asymptotically
stable.
3.1.3. Wolbachia -dominated equilibrium
The Wolbachia-dominated and dengue-free equilibrium is E2 = (S0
mi, 0, 0, 0, 0, 0, S0
h , V0
h , 0, 0), where S0
mi = r̄m/µm.
Under this case, we can compute the basic reproduction number as
ℜ02 =
√
ℜ̄hℜm,
where
ℜ̄h =
km
µmN0
m + km
·
1
µmN0
m
·
β̄h
N0
h
· S0
h =
kmβ̄h(σh + µh)
r̄m(r̄m + km)(σh + µh + ph)
.
The biological meaning of ℜ02 is similar to that of ℜ01.
The Wolbachia-dominated and dengue-infected equilibrium is denoted by E2, which is
E∗
2 = (S∗
mi, E∗
mi, I∗
mi, 0, 0, 0, S∗
h2, V∗
h2, I∗
h2, R∗
h2), (5)
where
S∗
mi = τ̄2 − τ̄3I∗
mi, E∗
mi =
r̄m
km
I∗
mi, I∗
mi =
τ̄4ζ4(ℜ2
02 − 1)
¯
ζ1(τ1τ̄3ζ2 + τ̄4ζ3)
,
S∗
h2 = ζ2 − ζ3I∗
h2, V∗
h2 =
phζ2
σh + µh
−
phζ3
σh + µh
I∗
h2, I∗
h2 =
τ̄4ζ4(ℜ2
02 − 1)
τ1(τ̄2ζ̄1ζ3 + τ̄3ζ4)
, R∗
h2 =
γh
µh
I∗
h2.
Detailed calculation of E∗
2 is given in Appendix B. The notations τ1, τ̄i, ζ̄1 and ζi (i = 2, 3, 4) are given in (B.3) and (B.4).
The biological meaning of E∗
1 is that Wolbachia-dominated mosquito population invades successfully.
Following Appendices E and F, the following theorems on the existence and local stability of E2 and E∗
2 are derived.
Theorem 3.3. There always exists a Wolbachia-dominated and dengue-free equilibrium E2, and it is locally asymptotically
stable if ℜ02 < 1.
5

Table 2
Sensitivity indices of ℜ01 and ℜ02 evaluated at the baseline parameter values in Table 1.
Parameter ℜ01 ℜ02 Parameter ℜ01 ℜ02
Λ −0.5 −0.5 µm −0.5 −0.5
km 0.23821 0.18880 βm 0.5 0.5
µh 0.51212 0.51212 σh 0.48426 0.48426
ph −0.49647 −0.49647 γh −0.4999 −0.4999
rm −0.23821 – r̄m – −0.18880
βh 0.5 – β̄h – 0.5
Theorem 3.4. If ℜ02 > 1, there exists a Wolbachia-dominated and dengue-infected equilibrium E∗
2 , which is locally
asymptotically stable.
4. Sensitivity analysis
Sensitivity analysis reveals how important each parameter relates to disease transmission. Such information is crucial
not only for experimental design, but also for data assimilation and reduction of complex nonlinear models [38]. Two
methods of sensitivity analysis are available, namely, local and global sensitivity analysis. The local sensitivity analysis
examines the local response of the output by varying one input parameter at a time by holding the other parameters at
central values [39]. The global sensitivity analysis perturbs the model input parameters within a wide range, so as to
quantify the overall impact of the model inputs on the model output [40]. In this section, two types of sensitivity analysis
methods are performed to evaluate the relative importance of parameters on the basic reproduction number.
4.1. Local sensitivity analysis
The local sensitivity analysis can be quantified by the sensitivity index, which is defined as follows.
Definition 4.1 ([33]). The normalized forward sensitivity index of ℜ0, that depends differentially on a parameter k, is
defined as
Υ
ℜ0
k =
∂ℜ0
∂k
k
ℜ0
.
The values of the sensitivity indices for ℜ01 and ℜ02 are presented in Table 2 and shown in Fig. 2. The analytical
expressions of each parameter sensitivity index for ℜ01 are as follows.
Υ
ℜ01
βm
=
1
2
, Υ
ℜ01
βh
=
1
2
, Υ
ℜ01
Λ = −
1
2
, Υ ℜ01
µm
= −
1
2
, Υ
ℜ01
km
=
rm
2(rm + km)
,
Υ
ℜ01
rm
= −
rm
2(rm + km)
, Υ ℜ01
µh
=
γh(σh + 2µh)(σh + ph) + µ2
h(ph + γh)
2(σh + µh)(σh + µh + ph)(µh + γh)
,
Υ ℜ01
σh
=
σhph
2(σh + µh)(σh + µh + ph)
, Υ
ℜ01
ph
= −
ph
2(σh + µh + ph)
, Υ ℜ01
γh
= −
γh
2(µh + γh)
.
Similarly, we obtain the analytical expressions of each parameter sensitivity index for ℜ02:
Υ
ℜ02
βm
=
1
2
, Υ
ℜ02
β̄h
=
1
2
, Υ
ℜ02
Λ = −
1
2
, Υ ℜ02
µm
= −
1
2
, Υ
ℜ02
km
=
r̄m
2(r̄m + km)
,
Υ
ℜ02
r̄m
= −
r̄m
2(r̄m + km)
, Υ ℜ02
µh
=
γh(σh + 2µh)(σh + ph) + µ2
h(ph + γh)
2(σh + µh)(σh + µh + ph)(µh + γh)
,
Υ ℜ02
σh
=
σhph
2(σh + µh)(σh + µh + ph)
, Υ
ℜ02
ph
= −
ph
2(σh + µh + ph)
, Υ ℜ02
γh
= −
γh
2(µh + γh)
.
It is worth noting that some of the sensitivity indices depend on one or more parameters, while some of them can be
constant. For example, Υ
ℜ01
βh
= 0.5 implies that increasing (decreasing) βh by a given percentage (such as 20%) will
increase (decrease) ℜ01 by half of its percentage (that is, 10%).
From Table 2, we conclude that for ℜ01 and ℜ02 at this set of baseline values given in Table 1, the parameters km,
βm, µh, σh have a positive sensitivity index and Λ, µm, ph, γh have a negative sensitivity index. βh and β̄h have positive
sensitivity indices for ℜ01 and ℜ02, respectively, meanwhile, rm and r̄m have negative sensitivity indices for ℜ01 and ℜ02,
respectively. The positive sensitivity indices indicate that ℜ01 and ℜ02 increase with the increasing of the parameters.
Conversely, the negative sensitivity indices mean that ℜ01 and ℜ02 decrease as the parameters increase.
6

Fig. 2. A histogram for the sensitivity indices of ℜ01 and ℜ02.
4.2. Global sensitivity analysis
Compared with local sensitivity analysis, global sensitivity analysis expands the range of model input parameters. We
perform global sensitivity analysis by calculating the partial rank correlation coefficient (PRCC) [41–43] for each parameter
(sampled by the LHS scheme) of ℜ0i (i = 1, 2), respectively. Taking the parameter values in Table 1, we can compute the
PRCC values for ℜ01 and ℜ02 which are shown in Fig. 3.
Fig. 3 shows that the parameters km, βh(β̄h), βm, µh, σh are positively correlated with ℜ01(ℜ02), while the parameters
Λ, µm, rm(r̄m), ph and γh are negatively correlated with ℜ01(ℜ02). The most sensitive parameters to ℜ01 and ℜ02 are µh
and Λ. In particular, an increase in the value of µh by 50% increases ℜ01 and ℜ02 by 95.02% and 89.85%, respectively.
In contrast, an increase in the value of Λ by 50% decreases ℜ01 by 87.82% and ℜ02 by 88.04%. Comparing the PRCC of
the parameters of ℜ01 and ℜ02, β̄h and r̄m are more sensitive than βh and rm, which implies that releasing Wolbachia can
exactly reduce the value of ℜ01 and ℜ02.
5. Optimal control strategies
To reduce the spread of dengue, in this section, we explore four time-dependent control strategies simultaneously: (i)
vaccination u1(t), (ii) using mosquito nets u2(t), (iii) improved treatment of dengue u3(t) and (iv) spraying insecticides
u4(t). Hence, system (2) with integrated control strategy is given by the following state system of differential equations:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
dSmi
dt
= r̄mNmi − µmNmSmi − u4(t)Smi −
(1 − u2(t))βmSmiIh
Nh
,
dEmi
dt
=
(1 − u2(t))βmSmiIh
Nh
− (µmNm + km)Emi − u4(t)Emi,
dImi
dt
= kmEmi − µmNmImi − u4(t)Imi,
dSmu
dt
= rm
(
1 −
Nmi
Nm
)
Nmu − µmNmSmu −
(1 − u2(t))βmSmuIh
Nh
− u4(t)Smu,
dEmu
dt
=
(1 − u2(t))βmSmuIh
Nh
− (µmNm + km)Emu − u4(t)Emu,
dImu
dt
= kmEmu − µmNmImu − u4(t)Imu,
dSh
dt
= Λ − µhSh − (1 + u1(t))phSh −
(1 − u2(t))Sh(β̄hImi + βhImu)
Nh
+ σhVh,
dVh
dt
= (1 + u1(t))phSh − (σh + µh)Vh,
dIh
dt
=
(1 − u2(t))Sh(β̄hImi + βhImu)
Nh
− µhIh − (1 + u3(t))γhIh,
dRh
dt
= (1 + u3(t))γhIh − µhRh.
(6)
7

Fig. 3. PRCC of ℜ01 and ℜ02 for model (2).
The objective of optimal control is to find the solutions of control variables u1(t), u2(t), u3(t) and u4(t) to minimize
the number of infected humans as well as the cost of control strategies, that is, for U(t) = (u1(t), u2(t), u3(t), u4(t)) ∈ U
where
U = {U ∈ (L∞
(0, T))4
|0 ≤ ui ≤ uf , 0 ≤ uf ≤ 1, i = 1, 2, 3, 4},
define the objective functional
J(U∗
) = min
U∈U
J(U), (7)
where
J(U) =
∫ T
0
[
A1Ih(t) +
1
2
(
C1u2
1(t) + C2u2
2(t) + C3u2
3(t) + C4u2
4(t)
)
]
dt.
Here, A1 represents the weight for the number of infected humans, Ci (i = 1, 2, 3, 4) stand for weight coefficients and
express the unit costs of the controls at different levels. We assume that the cost of treatment is nonlinear.
5.1. Existence of optimal control
Theorem 5.1. There exists an optimal control U∗
= (u∗
1, u∗
2, u∗
3, u∗
4) such that J(U∗
) = minU∈U J(U).
Proof. We apply the results in [44] to prove this theorem. Note that (i) The state variables and control variables are non-
negative. (ii) The control set U is closed and convex. (iii) The optimal system is bounded, which implies the compactness of
the optimal control. (iv) The integrand of the objective functional J(U) is convex on U . (v) There exist constants a1, a2 > 0
and q > 1 such that the integrand of the objective functional J(U) satisfies
J(U) ≥ a1(|u1|2
+ |u2|2
+ |u3|2
+ |u4|2
)
q
2 + a2.
Therefore, there exists an optimal control U∗
= (u∗
1, u∗
2, u∗
3, u∗
4) such that J(U∗
) = minU∈U J(U).
5.2. Characterization of optimal control
We will apply the Pontryagin’s maximum principle [45] to derive the necessary conditions of the optimal control U∗
.
Theorem 5.2. Let (Smi, Emi, Imi, Smu, Emu, Imu, Sh, Vh, Ih, Rh) be the state solutions of Problem (6)–(7) and u∗
i (i = 1, 2, 3, 4) be
the given optimal control, then there exist adjoint functions satisfying
dλ1
dt
= −
∂H
∂Smi
= λ1
(
−r̄m + µm(Nm + Smi) + u4(t) + (1 − u2(t))βm
Ih
Nh
)
+ λ2
(
−(1 − u2(t))βm
Ih
Nh
+ µmEmi
)
+ λ3µmImi + λ4
(
rm
N2
mu
N2
m
+ µmSmu
)
+ λ5µmEmu + λ6µmImu,
8

dλ2
dt
= −
∂H
∂Emi
= λ1 (−r̄m + µmSmi) + λ2 (µm(Nm + Emi) + km + u4(t)) + λ3 (−km + µmImi) + λ4
(
rm
N2
mu
N2
m
+ µmSmu
)
+ λ5µmEmu + λ6µmImu,
dλ3
dt
= −
∂H
∂Imi
= λ1 (−r̄m + µmSmi) + λ2µmEmi + λ3 (µm(Nm + Imi) + u4(t)) + λ4
(
rm
N2
mu
N2
m
+ µmSmu
)
+ λ5µmEmu
+ λ6µmImu + λ7(1 − u2(t))β̄h
Sh
Nh
− λ9(1 − u2(t))β̄h
Sh
Nh
,
dλ4
dt
= −
∂H
∂Smu
= λ1µmSmi + λ2µmEmi + λ3µmImi + λ4
(
−rm + rm
N2
mi
N2
m
+ µm(Nm + Smu) + u4(t) + (1 − u2(t))βm
Ih
Nh
)
+ λ5
(
µmEmi − (1 − u2(t))βm
Ih
Nh
)
+ λ6µmImu,
dλ5
dt
= −
∂H
∂Emu
(
−rm + rm
N2
mi
N2
m
+ µmSmu
)
+ λ5 (µm(Nm + Emu) + km + u4(t))
+ λ6 (−km + µmImu) ,
dλ6
dt
= −
∂H
∂Imu
(
−rm + rm
N2
mi
N2
m
+ µmSmu
)
+ λ5µmEmu
+ λ6 (µm(Nm + Imu) + u4(t)) + λ7(1 − u2(t))βh
Sh
Nh
− λ9(1 − u2(t))βh
Sh
Nh
,
dλ7
dt
= −
∂H
∂Sh
= −λ1(1 − u2(t))βmSmi
Ih
N2
h
+ λ2(1 − u2(t))βmSmi
Ih
N2
h
− λ4(1 − u2(t))βmSmu
Ih
N2
h
+ λ5(1 − u2(t))βmSmu
Ih
N2
h
+ λ7
(
µh + (1 + u1(t))ph + (1 − u2(t))(β̄hImi + βhImu)
Vh + Ih + Rh
N2
h
)
− λ8(1 + u1(t))ph
− λ9(1 − u2(t))(β̄hImi + βhImu)
Vh + Ih + Rh
N2
h
,
dλ8
dt
= −
∂H
∂Vh
= −λ1(1 − u2(t))βmSmi
Ih
N2
h
Ih
N2
h
Ih
N2
h
Ih
N2
h
+ λ7
(
−(1 − u2(t))(β̄hImi + βhImu)
Sh
N2
h
− σh
)
+ λ8 (σh + µh) + λ9(1 − u2(t))(β̄hImi + βhImu)
Sh
N2
h
,
dλ9
dt
= −
∂H
∂Ih
= λ1(1 − u2(t))βmSmi
Sh + Vh + Rh
N2
h
− λ2(1 − u2(t))βmSmi
Sh + Vh + Rh
N2
h
Sh + Vh + Rh
N2
h
Sh + Vh + Rh
N2
h
Sh
N2
h
+ λ9
(
(1 − u2(t))(β̄hImi + βhImu)
Sh
N2
h
+ µh + (1 + u3(t))γh
)
− λ10(1 + u3(t))γh − A1,
dλ10
dt
= −
∂H
∂Rh
= −λ1(1 − u2(t))βmSmi
Ih
N2
h
Ih
N2
h
Ih
N2
h
Ih
N2
h
Sh
N2
h
+ λ9(1 − u2(t))(β̄hImi + βhImu)
Sh
N2
h
+ µhλ10,
and the transversality conditions are λi(T) = 0, i = 1, · · · , 10. Furthermore, the optimal control U∗
(·) = (u∗
1, u∗
2, u∗
3, u∗
4) are
given in compact form as
u∗
1 = min{1, max{0, uc
1}}, u∗
2}},
u∗
3}}, u∗
4}},
9

where
uc
1 =
(λ7 − λ8)phSh
C1
, uc
2 =
(λ2 − λ1)
βmSmiIh
Nh
+ (λ5 − λ4)
βmSmuIh
Nh
+ (λ9 − λ7)
Sh(β̄hImi+βhImu)
Nh
C2
,
uc
3 =
(λ9 − λ10)γhIh
C3
, uc
4 =
λ1Smi + λ2Emi + λ3Imi + λ4Smu + λ5Emu + λ6Imu
C4
. (8)
Proof. According to Pontryagin’s maximum principle, if U∗
(·) ∈ U is an optimal solution of problem (6)–(7), then there
exists a continuous mapping λ : [0, T] → R, where λ(t) = (λ1(t), . . . , λ10(t)) is called the adjoint vector, which is derived
from
dxi
dt
=
∂H
∂λi
,
where xi is the ith element of vector x = (Smi, Emi, Imi, Smu, Emu, Imu, Sh, Vh, Ih, Rh). That is, dλi/dt = −∂H/∂xi. Here, the
Hamiltonian function H is as follows
H = A1Ih +
10
∑
i=1
λi
dxi
dt
+
1
2
4
∑
j=1
Cju2
j .
Thus, we can derive the adjoint system. In order to find the characterization of the optimal control U∗
, we only need to
minimize the Hamiltonian function H on the interior of the control set U , that is,
∂H
∂ui
= 0, i = 1, 2, 3, 4. (9)
Thus, we can obtain the characteristic of optimal control (8).
6. Numerical results of optimal control
6.1. Four different control strategies
In this section, we perform numerical simulations of the control problem using MATLAB by the forward–backward
sweep method [46] and the fourth order Runge–Kutta scheme. Since research on control strategies using mosquito nets
and insecticides is rich, we focus on vaccination and treatment. Therefore, we focus on the following control strategies.
• Strategy A (u2, u4): combination of using mosquito nets (u2) and spraying insecticides (u4).
• Strategy B (u1, u2, u4): combination of vaccination (u1), using mosquito nets (u2) and spraying insecticides (u4).
• Strategy C (u2, u3, u4): combination of using mosquito nets (u2), improved treatment of dengue (u3) and spraying
insecticides (u4).
• Strategy D (u1, u2, u3, u4): combination of vaccination (u1), using mosquito nets (u2), improved treatment of dengue
(u3) and spraying insecticides (u4).
In the numerical simulations, we assume the control period with 20 days, and all parameters are defined and taken
from Table 1. Furthermore, we assume that A1 = 25, C1 = 20, C2 = 20, C3 = 60, C4 = 20. The cost associated with
treatment C3 includes the costs of medical examinations, medications, and hospitalization. Hence, C3 is greater than the
other controls. The initial state variables are chosen as Smi = 20000, Emi = 0, Imi = 800, Smu = 20000, Emu = 0, Imu = 800,
Sh = 8000, Vh = 0, Ih = 20, Rh = 0.
6.1.1. Strategy A
With Strategy A, a positive effect is observed in human population. Fig. 4(a) shows that both the controls for using
mosquito nets and spraying insecticides are at the upper bound at the beginning. The optimal control profile of mosquito
nets (u2) starts to decrease slowly from the seventh day and reaches the lower bound at the 11th day. The optimal control
profile of insecticides (u4) starts to decrease slowly from the ninth day and reaches the lower bound at the 17th day. In
Fig. 4(b), we observe that the number of infected humans under Strategy A significantly decreases compared with the
case without Strategy A.
6.1.2. Strategy B
Here we use Strategy B including vaccination, using mosquito nets and spraying insecticides to optimize the objective
functional J. The numerical results are shown in Fig. 5. For the optimal control profile of vaccination (u1), it is at the upper
bound on the third day of the intervention period and then drop rapidly to reach the lower bound on the tenth day. The
optimal control profile of mosquito nets (u2) is at the upper bound for only the first 5 days and then drop to the lower
bound on the tenth day as well. In contrast, the optimal control profile of treatment (u4) is at the upper bound for a longer
period of time, maintaining it for a total of 9 days and reaching the lower bound on the 17th day. Under Strategy B, and
the number of infected humans becomes smaller.
10

Fig. 4. Simulations on the effect of Strategy A.
Fig. 5. Simulations on the effect of Strategy B.
Fig. 6. Simulations on the effect of Strategy C.
6.1.3. Strategy C
In Fig. 6(a), we observe that different control profiles all drop from the upper bound to the lower bound eventually,
but at different rates. Fig. 6(b) depicts the change in the number of infected humans with control and without control.
Under this strategy, the number of infected humans no longer has a rising trend, but keeps decreasing until it drops to
zero after the sixth day.
6.1.4. Strategy D
Under this strategy, we optimize the objective functional J using all the four controls. We observe the change of the
control profile from Fig. 7(a). The optimal control profile of vaccination (u1) and using mosquito nets (u2) are at the upper
bound for a short period of time and drop quickly to the lower bound. Although the optimal control profile of treatment
(u3) is at the upper bound for a shorter period of time, it lasts throughout the cycle. Different from the optimal control
11

Fig. 7. Simulations on the effect of Strategy D.
Fig. 8. The number of infected humans with different strategies.
profile of spraying insecticides (u4), which is at the upper bound for 8 days, it slowly decreases and reaches the lower
bound on the 17th day. Fig. 7(b) shows that the number of infected humans is significantly reduced in Strategy D.
6.2. Control effect analysis
The comparison of each strategy is shown in Fig. 8. Compared with Strategy A, Strategy B adds the vaccination strategy.
The peak of infected humans is reduced. The number of infected humans calculated are 15.10 and 10.95 at the end of the
fifth day, respectively, which implies that adding vaccination is more effective. Compared with Strategy A, here, Strategy
C introduces the improved treatment of dengue. We can obtain the number of infected humans is 5.74 at the end of
the fifth day under Strategy C. Hence, we conclude that the introduction of improved treatment has great influence on
the transmission of dengue. Based on Strategy A, Strategy D incorporates both vaccination and improved treatment for
dengue. In this strategy, the number of infected humans on the fifth day is 3.92. Compared with all strategies, lower peaks
and faster stable speeds are obtained by taking Strategy D. In conclusion, Strategy D is the most effective strategy in
reducing the number of infected humans.
6.3. Cost-effectiveness analysis
The benefits of an intervention require consideration not only its effectiveness, but also its cost. Therefore, we measure
the cost-effectiveness of different measures using the incremental cost-effectiveness ratio (ICER), which is calculated as
follows [23].
ICER =
Cost of Strategy Y − Cost of Strategy X
Number of infections averted under Strategy Y − Number of infections averted under Strategy X
.
It is worth noting that the number of infections avoided according to strategy Y can be obtained from Figs. 4(b)–7(b).
Meanwhile, it can be obtained as
Number of infections averted under strategy = γh
∫ T
0
(Ĩh(t) − Īh(t)) dt,
12

where Ĩh(t) denotes the trajectory without control, while Īh(t) refers to the trajectory under the strategy. Cost of strategy
represents the total consumption of each control in the policy over the control time, that is,
Cost of strategy =
∫ T
0
(C1u1 + C2u2 + C3u3 + C4u4) dt.
Based on the simulation results of the model, we rank the results by the number of infections averted.
Strategies Total infections averted Total costs ICER
Strategy A 4455.19 16 553.57 3.72
Strategy B 6530.69 17 782.13 0.59
Strategy C 8668.09 41 398.76 11.05
Strategy D 9732.11 40 549.69 −0.80
The ICER of four strategies is calculated as follows:
ICER(A) =
16553.57 − 0
4455.19 − 0
= 3.72, ICER(B) =
17782.13 − 16553.57
6530.69 − 4455.19
= 0.59,
ICER(C) =
41398.76 − 17782.13
8668.09 − 6530.69
= 11.05, ICER(D) =
40549.69 − 41398.76
9732.11 − 8668.09
= −0.80.
The comparison between Strategies B and C shows that Strategy C has higher costs and lower benefits than Strategy
B. Therefore, Strategy C is excluded from the alternatives.
We calculate the ICER for the remaining strategies
Strategy A 4455.19 16 553.57 3.72
Strategy B 6530.69 17 782.13 0.59
Strategy D 9732.11 40 549.69 7.11
The comparison between Strategies B and D shows that Strategy B costs less to achieve better effects. Hence, Strategy D
is excluded from the alternatives.
Strategies A and B are compared as follows.
Strategy A 4455.19 16 553.57 3.72
Strategy B 6530.69 17 782.13 0.59
The comparison between Strategies A and B shows that Strategy A costs more than Strategy B. Therefore, Strategy B is
the most cost-effective among the four strategies listed.
7. Conclusion and discussion
This paper proposes a dengue transmission model to explore the joint effects of releasing Wolbachia-infected
mosquitoes and vaccination. Firstly, we performed a rigorous analysis on the model, i.e., the dynamical behavior of the
invasion success and failure of Wolbachia-infected mosquitoes (See Section 3). It is shown that if mosquitoes infected
with Wolbachia win in the population competition, then the Wild-dominated and dengue-infected equilibrium (E∗
1 ) is
locally asymptotically stable when ℜ01 > 1 (See Theorem 3.2), otherwise, the Wolbachia-dominated and dengue-infected
equilibrium (E∗
2 ) is locally asymptotically stable when ℜ02 > 1 (See Theorem 3.4). This phenomenon illustrates the
practicality of controlling dengue by releasing Wolbachia-carrying mosquitoes.
The basic reproduction number is an important threshold for assessing whether a disease can be transmitted in a
population. The sensitivity analysis for the basic reproduction number shows how important each parameter is to dengue
transmission (See Section 4). If the correlation coefficient is positive, we can make the basic reproduction number smaller
by decreasing the value of the parameter. Likewise, if the correlation coefficient is negative, the basic reproduction number
can be reduced by increasing its value. For example, the use of long sleeves, long pants and mosquito nets can reduce
the contact frequency between mosquitoes and humans, and spraying insecticides can increase the death of mosquito
populations. It is worth noting that the nearer the absolute value of the correlation coefficient is to 1, the larger the effect
of parameter values on the basic reproduction number. Hence, we design four more common and practical strategies
(vaccination, using mosquito nets, improved treatment of dengue and spraying insecticides) to investigate the effects
of control strategies on preventing the spread of dengue (See Section 5). Then we combine the strategies in different
combinations. Numerical simulations show that all control combinations can effectively reduce the number of infected
humans, which means that the control is meaningful, then control effects of four strategies are compared (See Sections 6.1
13

and 6.2). Furthermore, we apply cost-effectiveness analysis (ICER) to evaluate the proposed four strategies in order to
explore the most economically efficient combination of intervention options (See Section 6.3). We conclude that Strategy
B (combination of vaccination, using mosquito nets and spraying insecticides) is the most cost-effective from the view of
cost and results.
Our analysis provides guidance to public health authorities on how to allocate finite resources to mitigate the spread
of dengue. However, how to effectively release Wolbachia-dominated mosquitoes is still a challenging work, which will
be our further work.
CRediT authorship contribution statement
Jian Zhang: Writing, Formal analysis. Lili Liu: Resources, Formal analysis, Writing – review & editing, Supervision.
Yazhi Li: Writing – review & editing. Yan Wang: Review, Visualization.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have
appeared to influence the work reported in this paper.
Data availability
No data was used for the research described in the article.
Acknowledgments
We are grateful to the handling editor and anonymous referees for their valuable and helpful comments which led to
an improvement of our manuscript. L. Liu is supported by National Natural Science Foundation of China (Nos. 12126349,
11601293). Y. Li is supported by National Natural Science Foundation of China (No. 11901326), the Science and Technology
Top Talent Project of Department of Education of Guizhou Province, China (Qian Jiao He KY Zi[2019]063). Y. Wang is
supported by National Natural Science Foundation of China (No. 12101513), China Postdoctoral Science Foundation (No.
2021M702704), and Natural Science Foundation of Chongqing, China (cstc2021jcyj-bshX0070).
Appendix A. Calculation for the basic reproduction number
To calculate the basic reproduction number for system (2), we consider the following subsystem
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
dEmi
dt
=
βmSmiIh
Nh
− (µmNm + km)Emi,
dImi
dt
= kmEmi − µmNmImi,
dEmu
dt
=
βmSmuIh
Nh
− (µmNm + km)Emu,
dImu
dt
= kmEmu − µmNmImu,
dIh
dt
=
Nh
− (µh + γh)Ih.
(A.1)
Hence, we have
F =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
0 0 0 0
βmS0
mi
N0
h
0 0 0 0 0
0 0 0 0
βmS0
mu
N0
h
0 0 0 0 0
0
β̄hS0
h
N0
h
0
βhS0
h
N0
h
0
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
, V =
⎛
⎜
⎜
⎜
⎝
µmN0
m + km 0 0 0 0
−km µmN0
m 0 0 0
0 0 µmN0
m + km 0 0
0 0 −km µmN0
m 0
0 0 0 0 µh + γh
⎞
⎟
⎟
⎟
⎠
,
where N0
h = S0
h + V0
h and N0
m = S0
mi + S0
mu. Therefore, for E1, one has
ℜ01 = ρ(FV−1
⏐
⏐E1
) =
√
kmβhβmµh(σh + µh)
Λµm(rm + km)(σh + µh + ph)(µh + γh)
.
14

For E2,
ℜ02 = ρ(FV−1
⏐
⏐E2
) =
√
kmβ̄hβmµh(σh + µh)
.
Appendix B. Calculation for two extreme boundary equilibria
The steady state E∗
1 = (0, 0, 0, S∗
mu, E∗
mu, I∗
mu, S∗
h , V∗
h , I∗
h , R∗
h) satisfies the following equalities
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
0 = rmNmu − µmNmuSmu −
βmSmuIh
Nh
,
0 =
βmSmuIh
Nh
− (µmNmu + km)Emu,
0 = kmEmu − µmNmuImu,
0 = Λ − (µh + ph)Sh −
βhShImu
Nh
+ σhVh,
0 = phSh − (σh + µh)Vh,
0 =
βhShImu
Nh
− (µh + γh)Ih,
0 = γhIh − µhRh.
(B.1)
Summing up the first three equations and the last four equations, respectively, one has rmNmu − µmN2
mu = 0 and
Λ − µhNh = 0. Thus, we obtain N∗
mu = rm/µm and N∗
h = Λ/µh. By some simplification of (B.1), we obtain the equations
as follows
{
τ1Ih(τ2 − τ3Imu) = τ4Imu,
ζ1Imu(ζ2 − ζ3Ih) = ζ4Ih,
(B.2)
where
τ1 =
βmµh
Λ
, τ2 =
rm
µm
, τ3 =
rm + km
km
, τ4 =
rm(rm + km)
km
,
ζ1 =
βhµh
Λ
, ζ2 =
Λ(σh + µh)
µh(σh + µh + ph)
, ζ3 =
(µh + γh)(σh + µh)
µh(σh + µh + ph)
, ζ4 = µh + γh.
(B.3)
Here, (B.2) has a unique root
I∗
mu =
τ1τ2ζ1ζ2 − τ4ζ4
ζ1(τ1τ3ζ2 + τ4ζ3)
, I∗
h1 =
τ1τ2ζ1ζ2 − τ4ζ4
τ1(τ2ζ1ζ3 + τ3ζ4)
.
Recall that
ℜ2
01 =
,
then we have
τ1τ2ζ1ζ3 − τ4ζ4 =
rm(rm + km)(µh + γh)(ℜ2
01 − 1)
km
= τ4ζ4(ℜ2
01 − 1).
Hence, I∗
mu and I∗
h can be rewritten as
I∗
mu =
τ4ζ4(ℜ2
01 − 1)
ζ1(τ1τ3ζ2 + τ4ζ3)
, I∗
h1 =
τ4ζ4(ℜ2
01 − 1)
τ1(τ2ζ1ζ3 + τ3ζ4)
.
Therefore, if ℜ01 > 1, then a unique boundary equilibrium E∗
1 = (0, 0, 0, S∗
mu, E∗
mu, I∗
mu, S∗
h1, V∗
h1, I∗
h1, R∗
h1) exists with the
following form
S∗
mu = τ2 − τ3I∗
mu, E∗
mu =
rm
km
I∗
mu, S∗
h1 = ζ2 − ζ3I∗
h1, V∗
h1 =
phζ2
σh + µh
−
phζ3
σh + µh
I∗
h1, R∗
h1 =
γh
µh
I∗
h1,
which coincide with (4) presented in Section 3.1.
15

Similarly, if ℜ02 > 1, there exists the another boundary equilibrium E∗
2 = (S∗
mi, E∗
mi, I∗
mi, 0, 0, 0, S∗
h2, V∗
h2, I∗
h2, R∗
h2), where
I∗
mi =
τ̄4ζ4(ℜ2
02 − 1)
¯
ζ1(τ1τ̄3ζ2 + τ̄4ζ3)
, I∗
h2 =
τ̄4ζ4(ℜ2
02 − 1)
τ1(τ̄2
¯
ζ1ζ3 + τ̄3ζ4)
,
S∗
mi = τ̄2 − τ̄3I∗
mi, E∗
mi =
r̄m
km
I∗
mi, S∗
h2 = ζ2 − ζ3I∗
h2, V∗
h2 =
phζ2
σh + µh
−
phζ3
σh + µh
I∗
h2, R∗
h2 =
γh
µh
I∗
h2,
τ̄2 =
r̄m
µm
, τ̄3 =
r̄m + km
km
, τ̄4 =
r̄m(r̄m + km)
km
, ζ̄1 =
β̄hµh
Λ
.
(B.4)
Appendix C. The proof of Theorem 3.1
The Jacobian matrix of system (2) at E1 is
J(E1) =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
r̄m − rm r̄m r̄m 0 0 0 0 0 0 0
0 −rm − km 0 0 0 0 0 0 0 0
0 km −rm 0 0 0 0 0 0 0
−2rm −2rm −2rm −rm 0 0 0 0 −rmβmµh
Λµm
0
0 0 0 0 −rm − km 0 0 0
rmβmµh
Λµm
0
0 0 0 0 km −rm 0 0 0 0
0 0 − β̄h(σh+µh)
σh+µh+ph
0 0 − βh(σh+µh)
σh+µh+ph
−µh − ph σh 0 0
0 0 0 0 0 0 ph −σh − µh 0 0
0 0
β̄h(σh+µh)
σh+µh+ph
0 0
βh(σh+µh)
σh+µh+ph
0 0 −µh − γh 0
0 0 0 0 0 0 0 0 γh −µh
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.
The associated characteristic equation of E1 is given by
(λ + (rm − r̄m))(λ + rm + km)(λ + µh + σh + ph)(λ + rm)2
(λ + µh)2
(λ3
+ ψ1λ2
+ ψ2λ + ψ3) = 0,
where
ψ1 = 2rm + km + µh + γh,
ψ2 = rm(µh + γh) + (rm + km)(rm + µh + γh),
ψ3 = rm(rm + km)(µh + γh)
(
1 −
)
= rm(rm + km)(µh + γh)(1 − ℜ2
01).
Note that ψ3 > 0 when ℜ01 < 1. Denote L1(λ) = λ3
+ ψ1λ2
+ ψ2λ + ψ3. It is easy to get
H1 = ψ1 > 0, H2 =
⏐
⏐
⏐
⏐
ψ1 ψ3
1 ψ2
⏐
⏐
⏐
⏐ = ψ1ψ2 − ψ3 > 0, H3 =
⏐
⏐
⏐
⏐
⏐
ψ1 ψ3 0
1 ψ2 0
0 ψ1 ψ3
⏐
⏐
⏐
⏐
⏐
= ψ3(ψ1ψ2 − ψ3).
Since
ψ1ψ2 −ψ3 = (2rm +km +µh +γh) (rm(rm + km) + rm(µh + γh) + (rm + km)(µh + γh))+rm(rm +km)(µh +γh)ℜ2
01 > 0,
we have H3 > 0. By Hurwitz criteria, it is clear that all eigenvalues of L1(λ) have negative real parts. Moreover,
(λ + (rm − r̄m))(λ + rm + km)(λ + µh + σh + ph)(λ + rm)2
(λ + µh)2
= 0 has seven negative roots. Hence, E1 is locally
asymptotically stable if ℜ01 < 1.
Appendix D. The proof of Theorem 3.2
The Jacobian matrix of system (2) at E∗
1 is
J(E∗
1 ) =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
a11 a12 a13 0 0 0 0 0 0 0
a21 a22 0 0 0 0 0 0 0 0
0 a32 a33 0 0 0 0 0 0 0
a41 a42 a43 a44 a45 a46 a47 a48 a49 a410
a51 a52 a53 a54 a55 a56 a57 a58 a59 a510
a61 a62 a63 a64 a65 a66 0 0 0 0
0 0 a73 0 0 a76 a77 a78 a79 a710
0 0 0 0 0 0 a87 a88 0 0
0 0 a93 0 0 a96 a97 a98 a99 a910
0 0 0 0 0 0 0 0 a109 a1010
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
16

where
a11 = r̄m − rm −
βmµh
Λ
I∗
h , a12 = r̄m, a13 = r̄m, a21 =
βmµh
Λ
I∗
h , a22 = −rm − km,
a32 = km, a33 = −rm, a41 = −2rm +
µm(rm + km)
km
I∗
mu, a42 = −2rm +
µm(rm + km)
km
I∗
mu,
a43 = −2rm +
µm(rm + km)
km
I∗
mu, a44 = −rm +
µm(rm + km)
km
I∗
mu −
βmµh
Λ
I∗
h ,
a45 =
µm(rm + km)
km
I∗
mu, a46 =
µm(rm + km)
km
I∗
mu, a47 =
βmµ2
h
Λ2
I∗
h
(
rm
µm
−
rm + km
km
I∗
mu
)
,
a48 =
βmµ2
h
Λ2
I∗
h
(
rm
µm
−
rm + km
km
I∗
mu
)
, a49 = −βm
(
µh
Λ
−
µ2
h
Λ2
I∗
h
) (
rm
µm
−
rm + km
km
I∗
mu
)
,
a410 =
βmµ2
h
Λ2
I∗
h
(
rm
µm
−
rm + km
km
I∗
mu
)
, a51 = −
rmµm
km
I∗
mu, a52 = −
rmµm
km
I∗
mu, a53 = −
rmµm
km
I∗
mu,
a54 =
βmµh
Λ
I∗
h −
rmµm
km
I∗
mu, a55 = −rm − km −
rmµm
km
I∗
mu, a56 = −
rmµm
km
I∗
mu,
a57 = −
βmµ2
h
Λ2
I∗
h
(
rm
µm
−
rm + km
km
I∗
mu
)
, a58 = −
βmµ2
h
Λ2
I∗
h
(
rm
µm
−
rm + km
km
I∗
mu
)
,
a59 = βm
(
µh
Λ
−
µ2
h
Λ2
I∗
h
) (
rm
µm
−
rm + km
km
I∗
mu
)
, a510 = −
βmµ2
h
Λ2
I∗
h
(
rm
µm
−
rm + km
km
I∗
mu
)
,
a61 = −µmI∗
mu, a62 = −µmI∗
mu, a65 = km − µmI∗
mu,
a66 = −rm − µmI∗
mu, a73 = −
β̄h(σh + µh)
σh + µh + ph
(
1 −
µh + γh
Λ
)
, a76 = −
βh(σh + µh)
σh + µh + ph
(
1 −
µh + γh
Λ
)
,
a77 = −µh − ph −
βhµh
Λ2(σh + µh + ph)
I∗
mu
(
Λph + (σh + µh)(µh + γh)I∗
h
)
,
a78 = σh +
βhµh(σh + µh)
Λ2(σh + µh + ph)
I∗
mu
(
Λ − (µh + γh)I∗
h
)
, a79 =
βhµh(σh + µh)
Λ2(σh + µh + ph)
I∗
mu
(
h
)
,
a710 =
βhµh(σh + µh)
Λ2(σh + µh + ph)
I∗
mu
(
h
)
, a87 = ph, a88 = −σh − µh,
a93 =
β̄h(σh + µh)
σh + µh + ph
(
1 −
µh + γh
Λ
)
, a96 =
βh(σh + µh)
σh + µh + ph
(
1 −
µh + γh
Λ
)
,
a97 =
βhµh
Λ2(σh + µh + ph)
I∗
mu
(
h
)
, a98 = −
βhµh(σh + µh)
Λ2(σh + µh + ph)
I∗
mu
(
h
)
,
a99 = −µh − γh −
βhµh(σh + µh)
Λ2(σh + µh + ph)
I∗
mu
(
h
)
, a910 = −
βhµh(σh + µh)
Λ2(σh + µh + ph)
I∗
mu
(
h
)
,
a109 = γh, a1010 = −µh.
The associated characteristic equation of E∗
1 is given by
f (E∗
1 ) = g1(λ)g2(λ),
where
g1(λ) =
⏐
⏐
⏐
⏐
⏐
λ − a11 −a12 −a13
−a21 λ − a22 0
0 −a32 −λ − a33
⏐
⏐
⏐
⏐
⏐
,
g2(λ) =
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
λ − a44 −a45 −a46 −a47 −a48 −a49 −a410
−a54 λ − a55 −a56 −a57 −a58 −a59 −a510
−a64 −a65 λ − a66 0 0 0 0
0 0 −a76 λ − a77 −a78 −a79 −a710
0 0 0 −a87 λ − a88 0 0
0 0 −a96 −a97 −a98 λ − a99 −a910
0 0 0 0 0 −a109 λ − a1010
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
.
For g1(λ), it can be written as g1(λ) = λ3
+ g11λ2
+ g12λ + g13, where g11 = −(a11 + a22 + a33), g12 = a11a22 + a11a33 +
a22a33 −a12a21, g13 = a12a21a33 −a11a22a33 −a13a21a32. Clearly, g11 > 0, g13 > 0 and g11g12 −g13 > 0, then all eigenvalues
of g1(λ) have negative real parts by Hurwitz criteria.
17

By transforming the rows and columns of g2(λ), we obtain g2(λ) = (λ + rm)(λ + µh)g3(λ), where
g3(λ) =
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
λ + a54 − a55 a54 − a56 −a58 −a59 −a510
a64 − a65 λ + a64 − a66 0 0 0
0 0 λ + a87 − a88 a87 a87
0 −a96 a97 − a98 λ + a97 − a99 a97 − a910
0 0 0 −a109 λ − a1010
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
.
That is, g3(λ) = λ5
+ g31λ4
+ g32λ3
+ g33λ2
+ g34λ + g35, where
g31 = (a54 − a55) + (a64 − a66) + (a87 − a88) + (a97 − a99) − a1010,
g32 = (a54 − a55)[(a64 − a66) + (a87 − a88) + (a97 − a99) − a1010] + (a64 − a66)[(a87 − a88) + (a97 − a99)
− a1010] + (a87 − a88)[(a97 − a99) − a1010] − (a54 − a56)(a64 − a65) − a87(a97 − a98) − a1010(a97 − a99)
+ a109(a97 − a910),
g33 = (a54 − a55)[(a64 − a66)((a87 − a88) + (a97 − a99) − a1010) + (a87 − a88)((a97 − a99) − a1010)
− a87(a97 − a98) − a1010(a97 − a99) + a109(a97 − a910)] − (a54 − a56)(a64 − a65)((a87 − a88)
+ (a97 − a99) − a1010) + a59a96(a64 − a65) + (a64 − a66)[(a87 − a88)((a97 − a99) − a1010)
− a87(a97 − a98) − a1010(a97 − a99) + a109(a97 − a910)] + (a87 − a88)[−a1010(a97 − a99)
+ a109(a97 − a910)] + a87(a97 − a98)(a1010 − a109),
g34 = (a54 − a55)[(a87 − a88)(−a1010(a97 − a99) + a109(a97 − a910) + a87(a97 − a98)(a1010 − a109))]
+ [(a54 − a55)(a64 − a66) − (a54 − a56)(a64 − a65)][((a87 − a88)((a97 − a99) − a1010)
− a87(a97 − a98) − a1010(a97 − a99) + a109(a97 − a910))] + a96(a64 − a65)[a59((a87 − a88) − a1010)
− a58a87 + a510a96] + (a54 − a56)[a109(−a87(a97 − a98) + (a87 − a88)(a97 − a910))
+ a1010(a87(a97 − a98) − (a87 − a88)(a97 − a99))],
g35 = (a54 − a55)(a64 − a66)[a109(−a87(a97 − a98) + (a87 − a88)(a97 − a910)) + a1010(a87(a97 − a98)
− (a87 − a88)(a97 − a99))] + (a64 − a65)[a1010((a87 − a88)(a510a96 − (a54 − a56)(a97 − a910)))
+ a87((a54 − a56)(a97 − a98) − a58a96)] + a1010[(a87 − a88)((a54 − a56)(a97 − a99) − a59a96)
+ a87(a58a96 − (a54 − a56)(a97 − a98))].
It is easy to verify that
H1 = g31 > 0, H2 =
⏐
⏐
⏐
⏐
g31 g33
1 g32
⏐
⏐
⏐
⏐ = g31g32 − g33 > 0, H3 =
⏐
⏐
⏐
⏐
⏐
g31 g33 0
1 g32 0
0 g31 g33
⏐
⏐
⏐
⏐
⏐
= g33(g31g32 − g33) > 0,
H4 =
⏐
⏐
⏐
⏐
⏐
⏐
⏐
g31 g33 0 0
1 g32 g34 0
0 g31 g33 0
0 1 g32 g34
⏐
⏐
⏐
⏐
⏐
⏐
⏐
= g34[g31(g32g33 − g31g34) − g2
33] > 0,
H5 =
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
g31 g33 g35 0 0
1 g32 g34 0 0
0 g31 g33 g35 0
0 1 g32 g34 0
0 0 g31 g33 g35
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
= g35[(g31g32 − g33)(g33g34 − g32g35) − (g31g34 − g35)2
] > 0.
Hence, all eigenvalues of g3(λ) have negative real parts by Hurwitz criteria. To sum up, f (E∗
1 ) has seven negative roots,
which imply that E∗
1 is locally asymptotically stable if ℜ01 > 1.
18

Appendix E. The proof of Theorem 3.3
The Jacobian matrix of system (2) at E2 is
J(E2) =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
−r̄m 0 0 −r̄m −r̄m −r̄m 0 0 −r̄mβmµh
Λµm
0
0 −r̄m − km 0 0 0 0 0 0
r̄mβmµh
Λµm
0
0 km −r̄m 0 0 0 0 0 0 0
0 0 0 −r̄m 0 0 0 0 0 0
0 0 0 0 −r̄m − km 0 0 0 0 0
0 0 0 0 km −r̄m 0 0 0 0
0 0 − β̄h(σh+µh)
σh+µh+ph
0 0 − βh(σh+µh)
σh+µh+ph
−µh − ph σh 0 0
0 0 0 0 0 0 ph −σh − µh 0 0
0 0
β̄h(σh+µh)
σh+µh+ph
0 0
βh(σh+µh)
σh+µh+ph
0 0 −µh − γh 0
0 0 0 0 0 0 0 0 γh −µh
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.
The associated characteristic equation of E2 is given by
f (E2) = (λ + r̄m + km)(λ + µh + σh + ph)(λ + µh)2
(λ + r̄m)3
(λ3
+ ψ̄1λ2
+ ψ̄2λ + ψ̄3),
where
ψ̄1 = 2r̄m + km + µh + γh,
ψ̄2 = r̄m(µh + γh) + (r̄m + km)(r̄m + µh + γh),
ψ̄3 = r̄m(r̄m + km)(µh + γh)
(
1 −
kmβ̄hβmµh(σh + µh)
Λµm(r̄m + km)(σh + µh + ph)(µh + γh)
)
= r̄m(r̄m + km)(µh + γh)(1 − ℜ2
02).
Note that ψ̄3 > 0 when ℜ02 < 1. Denote L̄1(λ) = λ3
+ ψ̄1λ2
+ ψ̄2λ + ψ̄3. It is easy to get
H1 = ψ̄1 > 0, H2 =
⏐
⏐
⏐
⏐
ψ̄1 ψ̄3
1 ψ̄2
⏐
⏐
⏐
⏐ = ψ̄1ψ̄2 − ψ̄3, H3 =
⏐
⏐
⏐
⏐
⏐
⏐
ψ̄1 ψ̄3 0
1 ψ̄2 0
0 ψ̄1 ψ̄3
⏐
⏐
⏐
⏐
⏐
⏐
= ψ̄3(ψ̄1ψ̄2 − ψ̄3).
Since
ψ̄1ψ̄2 −ψ̄3 = (2r̄m +km +µh +γh) (r̄m(r̄m + km) + r̄m(µh + γh) + (r̄m + km)(µh + γh))+r̄m(r̄m +km)(µh +γh)ℜ2
02 > 0,
we have H2 > 0 and H3 > 0. By Hurwitz criteria, it is clear that all eigenvalues of L̄1(λ) have negative real parts. Moreover,
(λ + r̄m + km)(λ + µh + σh + ph)(λ + µh)2
(λ + r̄m)3
= 0 has seven negative roots. Hence, E2 is locally asymptotically stable
if ℜ02 < 1.
Appendix F. The proof of Theorem 3.4
The Jacobian matrix of system (2) at E∗
2 is
J(E∗
2 ) =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
c11 c12 c13 c14 c15 c16 c17 c18 c19 c110
c21 c22 c23 c24 c25 c26 c27 c28 c29 c210
c31 c32 c33 c34 c35 c36 0 0 0 0
0 0 0 c44 0 0 0 0 0 0
0 0 0 c54 c55 0 0 0 0 0
0 0 0 0 c65 c66 0 0 0 0
0 0 c73 0 0 c76 c77 c78 c79 c710
0 0 0 0 0 0 c87 c88 0 0
0 0 c93 0 0 c96 c97 c98 c99 c910
0 0 0 0 0 0 0 0 c109 c1010
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
where
c11 = −r̄m +
µm(r̄m + km)
km
I∗
mi −
βmµh
Λ
I∗
h , c12 =
µm(r̄m + km)
km
I∗
mi, c13 =
µm(r̄m + km)
km
I∗
mi,
c14 = −r̄m +
µm(r̄m + km)
km
I∗
mi, c15 = −r̄m +
µm(r̄m + km)
km
I∗
mi, c16 = −r̄m +
µm(r̄m + km)
km
I∗
mi,
c17 =
βmµ2
h
Λ2
I∗
h
(
r̄m
µm
−
r̄m + km
km
I∗
mi
)
, c18 =
βmµ2
h
Λ2
I∗
h
(
r̄m
µm
−
r̄m + km
km
I∗
mi
)
,
19

c19 = −βm
(
µh
Λ
−
µ2
h
Λ2
I∗
h
) (
r̄m
µm
−
r̄m + km
km
I∗
mi
)
, c110 =
βmµ2
h
Λ2
I∗
h
(
r̄m
µm
−
r̄m + km
km
I∗
mi
)
,
c21 =
βmµh
Λ
I∗
h −
r̄mµm
km
I∗
mi, c22 = −r̄m − km −
r̄mµm
km
I∗
mi, c23 = −
r̄mµm
km
I∗
mi,
c24 = −
r̄mµm
km
I∗
mi, c25 = −
r̄mµm
km
I∗
mi, c26 = −
r̄mµm
km
I∗
mi, c27 = −
βmµ2
h
Λ2
I∗
h
(
r̄m
µm
−
r̄m + km
km
I∗
mi
)
,
c28 = −
βmµ2
h
Λ2
I∗
h
(
r̄m
µm
−
r̄m + km
km
I∗
mi
)
, c29 = βm
(
µh
Λ
−
µ2
h
Λ2
I∗
h
) (
r̄m
µm
−
r̄m + km
km
I∗
mi
)
,
c210 =
βmµ2
h
Λ2
I∗
h
(
r̄m
µm
−
r̄m + km
km
I∗
mi
)
, c31 = −µmI∗
mi, c32 = km − µmI∗
mi, c33 = −r̄m − µmI∗
mi,
c34 = −µmI∗
mi, c35 = −µmI∗
mi, c36 = −µmI∗
mi, c44 = −r̄m −
βmµh
Λ
I∗
h , c54 =
βmµh
Λ
I∗
h ,
c55 = −r̄m − km, c65 = km, c66 = −r̄m, c73 = −
β̄h(σh + µh)
σh + µh + ph
(
1 −
µh + γh
Λ
)
,
c76 = −
βh(σh + µh)
σh + µh + ph
(
1 −
µh + γh
Λ
)
, c77 = −µh − ph −
β̄hµh
Λ2(σh + µh + ph)
I∗
mi
(
h
)
,
c78 = σh +
β̄hµh(σh + µh)
Λ2(σh + µh + ph)
I∗
mi
(
h
)
, c79 =
β̄hµh(σh + µh)
Λ2(σh + µh + ph)
I∗
mi
(
h
)
,
c710 =
β̄hµh(σh + µh)
Λ2(σh + µh + ph)
I∗
mi
(
h
)
, c87 = ph, c88 = −σh − µh,
c93 =
β̄h(σh + µh)
σh + µh + ph
(
1 −
µh + γh
Λ
)
,
c96 =
βh(σh + µh)
σh + µh + ph
(
1 −
µh + γh
Λ
)
, c97 =
β̄hµh
Λ2(σh + µh + ph)
I∗
mi
(
h
)
,
c98 = −
β̄hµh(σh + µh)
Λ2(σh + µh + ph)
I∗
mi
(
h
)
, c99 = −µh − γh −
β̄hµh(σh + µh)
Λ2(σh + µh + ph)
I∗
mi
(
h
)
,
c910 = −
β̄hµh(σh + µh)
Λ2(σh + µh + ph)
I∗
mi
(
h
)
, c109 = γh, c1010 = −µh.
The associated characteristic equation of E∗
2 is given by f (E∗
2 ) = (λ − c44)(λ − c55)(λ − c66)g4(λ), where
g4(λ) =
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
λ − c11 −c12 −c13 −c47 −c48 −c49 −c410
−c21 λ − c22 −c23 −c27 −c28 −c29 −c210
−c31 −c32 λ − c33 0 0 0 0
0 0 −c73 λ − c77 −c78 −c79 −c710
0 0 0 −c87 λ − c88 0 0
0 0 −c93 −c97 −c98 λ − c99 −c910
0 0 0 0 0 −c109 λ − c1010
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
.
Note that c44, c55, c66 < 0, thus, (λ − c44)(λ − c55)(λ − c66) = 0 has three negative roots.
On the other hand, we use a similar approach with g3(λ) analyze g4(λ), and get that all characteristic roots of g4(λ) = 0
have negative real parts. In conclusion, E∗
2 is locally asymptotically stable if ℜ02 > 1.
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