Dynamic analysis and optimal control of a mosquito-borne infectious disease model under the influence of biodiversity dilution effect

Abstract

In this paper, biodiversity and a saturation treatment rate are introduced into a type of mosquito-borne infectious disease model with an incubation time delay. Through the dynamical analysis of the model, conditions for the existence and stability of disease-free and endemic equilibria are determined. The basic reproduction number of the model is calculated, and the condition for the existence of backward bifurcation is outlined. The study finds that under the dual influence of biodiversity and saturation treatment, the threshold characteristic of the basic reproduction number becomes invalidated. When $R_{0}$ is less than 1, the model may exhibit four equilibrium states, with both the disease-free equilibrium and the endemic equilibrium being locally stable. In this scenario, whether the virus will become extinct depends on the initial conditions. The study also finds that when the basic reproduction number $R_{0}$ is greater than 1, the stability of the model is influenced by the time delay, with Hopf bifurcation occurring at a specific time delay. In addition, another novel contribution of this paper is the formulation of an optimal control problem that takes into account the minimization of damage caused by humans to biodiversity. Based on the Pontryagin’s maximum principle, the specific characteristics of the optimal control measures are given, and the optimal strategy is derived by comparing five groups of control strategies. The optimal control results highlight the synergistic effect of multiple control measures with biodiversity. Under optimal control, a significant complementary effect between medical inputs and the dilution effect of biodiversity is evident. The findings imply that maintaining high biodiversity levels can decrease the demand for medical resources in mosquito-borne disease control efforts.

1 Introduction

Mosquito-borne viruses (MBVs), primarily transmitted to humans and other vertebrates through mosquito bites, have given rise to a significant incidence of mosquito-borne diseases [1]. Notably, diseases such as malaria, dengue fever, and yellow fever have emerged, thereby causing substantial public health challenges [2]. According to the most recent World Malaria Report released in December 2022, there were 608,000 malaria-related fatalities in 2022, which marked a slight decrease from the 610,000 recorded in 2021. Additionally, there were an estimated 249 million cases of malaria worldwide in 2022, compared to 244 million in 2021 [3]. Dengue fever afflicts nearly 400 million individuals annually, and 3,900 million people are now at high risk of infection with the dengue virus [4]. Moreover, the infection or sustained transmission of yellow fever, chikungunya, and Zika viruses also exerts a substantial impact on public health, endangering more than 40% of the global population [5].

There is no denying that globalization and accelerated population movements are significant contributors to the frequent spread of viruses. Human activities and environmental changes create opportunities for viral transmission. Deforestation and urban expansion lead to the destruction of animal habitats and the fragmentation of ecosystems. This, in turn, increased contact between viruses and humans. The impact of biodiversity on the spread of viruses among populations becoming increasingly evident. For instance, in their study on hantavirus transmission, Luis et al. [6] observed that in a more diverse rodent community habitat, increased competition among rodent species of the same type led to a decrease in the overall rat population. Consequently, this resulted in a lower prevalence of hantavirus caused by deer mice. Similarly, Keesing et al. [7] noted in their research that reduced biodiversity facilitates the mosquito-borne transmission of Lyme and West Nile viruses. Peixoto et al. [8] employed a rodent–aliens model and introduced a class of exotic species in their study on the effects of biodiversity on Hantavirus. This showed that competitive pressures may lessen or even completely eradicate infection epidemics. Yue et al. [9], building on this framework, developed a Zika virus model affected by biodiversity through a competition mechanism. They found that viral infections were ultimately reduced due to competition limiting the size of the vector population. In the context of malaria transmission, Laporta et al. [10] highlighted the effect of biodiversity on mosquito population size, suggesting a positive effect of biodiversity on malaria transmission. This phenomenon, where virus transmission is facilitated by a reduction in biodiversity, is known as the dilution effect of biodiversity. Hence, the decline in biodiversity emerges as an undeniable factor contributing significantly to the proliferation of mosquito-borne diseases.

Conversely, in the concerted effort to control these diseases, investing in medical resources and other vital aspects becomes a critical expenditure. However, due to disparate economic conditions in various regions, the investment in healthcare remains limited, constraining the capacity to effectively manage larger-scale virus outbreaks. Many mathematical models of mosquito-borne infectious diseases operate under the assumption that the cure rate is directly proportional to the number of patients. This implies an exponential increase in healthcare resource allocation targeted at the virus as the number of patients grows, which is evidently unrealistic. Zhang et al. [11] introduced the concept of the saturation cure rate, which sets a maximum limit on the number of individuals receiving treatment within a specific timeframe. This concept is mathematically expressed as $h(I)=rI/(1+\alpha I)$, where r is the cure rate and α characterizes the effect of delaying treatment for medically constrained infected patients. Consequently, several scholars redirected their focus toward exploring the saturation cure rate [12–16]. For example, Srivastav et al. [17] proposed an SIRS model to study the transmission dynamics of malaria with saturated treatment. Li et al. [18] proposed an SIR epidemic model with logistic growth and saturation treatment to account for limited medical resources. Wang et al. [19] created a model focusing on the Zika virus under a class of saturation treatment rate. Through comprehensive model analysis, these studies unveiled intricate dynamical behaviors, offering substantial insights into disease control strategies. Optimal control theory is integral to modern control theory, focusing on identifying the best control strategy to maximize model performance while adhering to given constraints. Revelle, Lynn, and Feldmann [20] made a significant contribution to the field of infectious disease control by applying optimal control theory to the study of tuberculosis transmission. Their work laid the foundation for using mathematical modeling and optimization techniques to identify strategies that minimize control costs and the spread of infections. This approach has since been widely adopted in the field of epidemiology to develop optimal control strategies for managing various infectious diseases [21–24].

This study builds upon the work of Yue and Yusof [9] by further investigating the optimal control problem of vector-borne infectious diseases in the context of biodiversity. By examining this issue, it becomes clearer how biodiversity plays a synergistic role in many measures of mosquito-borne disease control. The contribution of this study focuses on exploring the dynamics and optimal control of a novel infectious disease model that takes into account time delays and two key factors: biodiversity and saturation cure rate. Yue and Yusof [9] emphasize the moderating role of biodiversity in disease transmission. However, this paper will focus on the specific control measures where biodiversity has the greatest impact on disease control, which can be an important guide in controlling mosquito-borne diseases.

The paper unfolds in a sequential manner. Section 2 delineates the model employed in this study. Moving forward, Sect. 3 engages in a comprehensive analysis, exploring the existence and stability of equilibrium points while scrutinizing the emergence of backward and Hopf bifurcations. Meanwhile, we discuss the consistency and persistence of the model under study. In Sect. 4, based on the actual situation, we propose the control strategy, establish an optimal control model, and provide the optimal control analysis. Finally, Sect. 5 presents a series of numerical simulations. Conclusions are given in Sect. 6.

2 Model formulation

Consider a total population denoted as $N_{H}(t)$, which is further divided into four segments: the susceptible population $S_{H}(t)$, the infected population $I_{H}(t)$, individuals infected and receiving treatment $T_{H}(t)$, and the recovered population $R_{H}(t)$. Similarly, the total mosquito population is represented as $N_{V}(t)$, categorized into susceptible mosquitoes $S_{V}(t)$ and infected mosquitoes $I_{V}(t)$. In line with the research in [9], we incorporate aliens denoted as $Z(t)$ into our model. Here, “aliens” refers to the array of all species within the expansive ecological setting that share resources with mosquitoes and engage in competitive interactions.

Given the presence of an incubation period for a disease, we assume that a susceptible individual gets bitten by an infected mosquito at time $t-\tau$, subsequently becoming infected at the time t. The probability of this transmission event occurring, where a susceptible person gets bitten by an infected mosquito and contracts the infection, is determined as $\beta _{H} S_{H}(t-\tau ) I_{V}(t-\tau )$. This probability indicates the interaction of susceptible humans with infected mosquitoes at different times, which leads to the transmission of the disease.

Thus the following model is developed:

$$ \textstyle\begin{cases} \frac{dS_{H}}{dt}=\Lambda _{H}-\beta _{H} S_{H}(t-\tau )I_{V}(t- \tau )-\mu _{H}S_{H}, \\ \frac{dI_{H}}{dt}=\beta _{H}S_{H}(t-\tau )I_{V}(t-\tau )-f(I_{H})-( \mu _{H}+\delta )I_{H}, \\ \frac{dT_{H}}{dt}=f(I_{H})-(r+\mu _{H})T_{H}, \\ \frac{dR_{H}}{dt}=r T_{H}-\mu _{H}R_{H}, \\ \frac{dS_{V}}{dt}=\Lambda _{V}-\beta _{V} S_{V} I_{H}-\mu _{V}S_{V}- \frac{S_{V}}{K}(N_{V}+qZ), \\ \frac{dI_{V}}{dt}=\beta _{V} S_{V} I_{H}-\mu _{V}I_{V}- \frac{I_{V}}{K}(N_{V}+qZ), \\ \frac{dZ}{dt}=a Z-\frac{Z}{k}(Z+\varepsilon N_{V}). \end{cases} $$

(1)

All parameters are positive, as described in Table 1, while $f(I_{H})=\frac{\alpha I_{H}}{1+I_{H}}$ denotes the number of people treated per unit of time. Considering the biological significance of the model, the initial conditions of model (1) are appropriately fulfilled:

$$\begin{aligned}& S_{H}(\theta )=\varphi _{1}(\theta ),\qquad I_{H}(\theta )=\varphi _{2}( \theta ), \qquad T_{H}( \theta )=\varphi _{3}(\theta ), \\& R_{H}(\theta )= \varphi _{4}(\theta ), \qquad S_{V}(\theta )=\varphi _{5}( \theta ), \\& I_{V}(\theta )=\varphi _{6}(\theta ),\qquad Z(\theta )= \varphi _{7}( \theta ), \\& \varphi _{i}(\theta )\ge 0,\qquad t\in [-\tau ,0],\qquad \varphi _{i}(0)>0 \qquad (i=1,\dots ,7), \end{aligned}$$

where

$$ W= \bigl(\varphi _{1}(\theta ),\varphi _{2}(\theta ), \varphi _{3}(\theta ), \varphi _{4}(\theta ),\varphi _{5}(\theta ),\varphi _{6}(\theta ), \varphi _{7}(\theta ) \bigr)\in C \bigl([-\tau ,0],R_{+}^{7} \bigr), $$

$R_{+}^{7}=\{(x_{1},x_{2},\dots ,x_{7}):x_{i}\geq 0,i=1,2,\dots ,7\}$, and C is the Banach space of continuous maps from $[-\tau ,0]$ to $R_{+}^{7}$.

Table 1 The description of the parameters used in model (1)

Lemma 1

The solution $(S_{H},I_{H},T_{H},R_{H},S_{V},I_{V},Z)$ of model (1) is nonnegative and bounded for all $t>0$.

Proof

The total population is

$$ \frac{d N_{H}}{dt}=\Lambda _{H}-\mu _{H} N_{H}- \delta I_{H}, $$

and then we get $0\leq N_{H}\leq \frac{\Lambda _{H}}{\mu _{H}}$ as $t\to \infty$.

The mosquito population satisfies

$$\frac{d N_{V}}{dt}=\Lambda _{V}-\mu _{V} N_{V}- \frac{N_{V}}{K}(N_{V}+qZ), $$

so similarly as for $N_{H}$, we have $0\leq N_{V}\leq \frac{\Lambda _{V}}{\mu _{V}}$ as $t\to \infty$.

For the aliens, we have

$$\frac{dZ}{dt}=aZ-\frac{Z}{k}(Z+\varepsilon N_{V})\leq aZ- \frac{Z^{2}}{k}=aZ\biggl(1-\frac{Z}{ka}\biggr), $$

thus

$$ \limsup_{t\to \infty} N_{H}(t)\leq \frac{\Lambda _{H}}{\mu _{H}}, \qquad \limsup_{t\to \infty}N_{V}(t)\leq \frac{\Lambda _{V}}{\mu _{V}}, \qquad \limsup_{t\to \infty}Z(t)\leq ka. $$

Letting $c(t)=N_{H}(t)+N_{V}(t)+Z(t)$, the derivative with respect to t satisfies

$$\frac{dc(t)}{dt}\leq \Lambda _{H}+\Lambda _{V}+ \frac{ka^{2}}{4} \triangleq M. $$

According to the principle of differential inequality, for all $t\ge T\ge 0$, we have

$$0\leq c(t)\leq M-\bigl(M-c(T)\bigr)e^{-(t-T)}. $$

So, $\limsup_{t\to \infty} (N_{H}+N_{V}+Z)\leq M$.

Let

$$ \begin{aligned} \Omega ={}& \biggl\{ (S_{H},I_{H},T_{H},R_{H},S_{V},I_{V},Z )\in R_{+}^{7}| 0\leq S_{H}+I_{H}+T_{H}+R_{H} \leq \frac{\Lambda _{H}}{\mu _{H}}, \\ & {}0\leq S_{V}+I_{V}\leq \frac{\Lambda _{V}}{\mu _{V}},0\leq Z\leq ka \biggr\} . \end{aligned} $$

According to Lemma 1, Ω is a positively invariant set for which model (1) has nonnegative initial conditions. □

Without considering infectious diseases, the following model illuminates the interaction between mosquitoes and aliens:

$$ \textstyle\begin{cases} \frac{dN_{V}}{dt}=\Lambda _{V}-\mu _{V} N_{V}-\frac{N_{V}}{K}(N_{V}+qZ), \\ \frac{dZ}{dt}=a Z-\frac{Z}{k}(Z+\varepsilon N_{V}). \end{cases} $$

(2)

Denote the positive equilibrium of model (2) as $\overline{E}_{i}^{*}=(N^{*}_{Vi},Z_{i}^{*})$, where $Z_{i}^{*}=ak-\varepsilon N^{*}_{Vi}$ ($i=0,1,2$). Reference [9] provides the following conclusion regarding the positive equilibrium of model (2):

• (A1) If $a>\frac{\varepsilon N^{*}_{V0}}{k}$ and $\varepsilon q\leq 1$, then the positive equilibrium $\overline{E}_{0}^{*}=(N^{*}_{V0},Z_{0}^{*})$ is globally asymptotically stable.

• (A2) The positive equilibrium $\overline{E}_{2}^{*}=(N^{*}_{V2},Z_{2}^{*})$ is globally asymptotically stable if $\frac{\varepsilon N^{*}_{V2}}{k}<\frac{\varepsilon N_{V0}}{k}<a< \frac{\varepsilon N^{*}_{V1}}{k}$, $\varepsilon q>1$, and $\Delta _{1}>0$.

• (A3) If $a>\frac{\varepsilon N^{*}_{V1}}{k}$, $\varepsilon q>1$, and $\Delta _{1}>0$, then the positive equilibrium $\overline{E}_{1}^{*}=(N^{*}_{V1},Z_{1}^{*})$ is unstable.

Above we used $\Delta _{1}=(K\mu _{V}+kaq)^{2}-4K \Lambda _{V}(\varepsilon q-1)$, and

$$ \begin{aligned} &N^{*}_{V0}=\frac{K\Lambda _{V}}{K\mu _{V}+kaq}, \qquad N^{*}_{V1,2}= \frac{(K\mu _{V}+kaq)\pm \sqrt{\Delta _{1}}}{2(\varepsilon q-1)}, \\ &N_{V0}= \frac{-K\mu _{V}+\sqrt{K^{2}\mu _{V}^{2}+4\Lambda _{V} K}}{2}. \end{aligned} $$

Since $\overline{E}_{1}^{*}$ in (A3) is unstable, the equilibrium point corresponding to $\overline{E}_{1}^{*}$ will no longer be taken into account in model (1).

3 Stability of model (1) and the basic reproduction number

3.1 Existence of equilibrium points

3.1.1 Existence of disease-free equilibrium and the basic reproduction number

Let $n_{1}=\frac{S_{V}}{N_{V}}$ and $n_{2}=\frac{I_{V}}{N_{V}}$. Then (1) is equivalent to the following model:

$$ \textstyle\begin{cases}\frac{dS_{H}}{dt}=\Lambda _{H}-\beta _{H}S_{H}(t-\tau )u_{2}(t- \tau )N_{V}-\mu _{H} S_{H}, \\ \frac{dI_{H}}{dt}=\beta _{H}S_{H}(t-\tau )n_{2}(t-\tau )N_{V}-f(I_{H})-( \mu _{H}+\delta )I_{H}, \\ \frac{dT_{H}}{dt}=f(I_{H})-(r+\mu _{H})T_{H}, \\ \frac{dR_{H}}{dt}=r T_{H}-\mu _{H}R_{H}, \\ \frac{dn_{2}}{dt}=\beta _{V} (1-n_{2})I_{H}- \frac{n_{2}}{N_{V}} \Lambda _{V}, \\ \frac{dN_{V}}{dt}=\Lambda _{V}-\mu _{V} N_{V}-\frac{N_{V}}{K}(N_{V}+qZ), \\ \frac{dZ}{dt}=a Z-\frac{Z}{k}(Z+\varepsilon N_{V}). \end{cases} $$

(3)

We consider the situation $Z\neq 0$, thus

$$\begin{aligned}& \lim_{t\to \infty}N_{V}(t)=N^{*}_{V}, \qquad N^{*}_{V}= \textstyle\begin{cases} N^{*}_{V2},&\varepsilon q>1, \Delta _{1}>0, \\ N^{*}_{V0},&\varepsilon q\leq 1, \end{cases}\displaystyle \end{aligned}$$

(4)

$$\begin{aligned}& \lim_{t\to \infty}Z(t)=Z^{*},\qquad Z^{*}= \textstyle\begin{cases} Z^{*}_{2},&N^{*}_{V}=N^{*}_{V2}, a>\frac{\varepsilon N^{*}_{V2}}{k}, \\ Z^{*}_{0},&N^{*}_{V}=N^{*}_{V0}, a>\frac{\varepsilon N^{*}_{V0}}{k}. \end{cases}\displaystyle \end{aligned}$$

(5)

It is easy to see that there exists a disease-free equilibrium

$$ E_{0}= \biggl(\frac{\Lambda _{H}}{\mu _{H}},0,0,0,0,N^{*}_{V},Z^{*} \biggr) $$

for model (3).

Using the regeneration matrix method [25], we have

$$F=\left(\begin{array}{cc}0& \frac{{\beta }_H{\mathrm{\Lambda }}_H{N}_V^{\ast }}{{\mu }_H}\\ {\beta }_V& 0\end{array}\right),V=\left(\begin{array}{cc}\alpha +\delta +{\mu }_H& 0\\ 0& \frac{{\mathrm{\Lambda }}_V}{N_V^{\ast }}\end{array}\right),$$

and the basic reproduction number of model (3) can be obtained as

$$ R_{0}=\rho \bigl(FV^{-1} \bigr)=\sqrt { \frac{\beta _{H}\beta _{V}\Lambda _{H}(N^{*}_{V})^{2}}{(\alpha +\delta +\mu _{H})\mu _{H}\Lambda _{V}}}. $$

(6)

3.1.2 Existence of endemic equilibrium

Now, we will determine some of the conditions for the existence of endemic equilibrium. To this end, let the right-hand side of model (3) be equal to zero, and then the endemic equilibrium point $E^{*}=(S_{H}^{*},I_{H}^{*},T_{H}^{*},R_{H}^{*},n_{2}^{*},N^{*}_{V},Z^{*})$ can be determined from the following system of equations:

$$ \textstyle\begin{cases} \Lambda _{H}-\beta _{H} S_{H}(t-\tau )n_{2}(t-\tau )N_{V}-\mu _{H} S_{H}=0, \\ \beta _{H} S_{H}(t-\tau )n_{2}(t-\tau )N_{V}-f(I_{H})-(\mu _{H}+ \delta )I_{H}=0, \\ f(I_{H})-(r+\mu _{H})S_{H}=0, \\ r T_{H}-\mu _{H}R_{H}=0, \\ \beta _{V} (1-n_{2})I_{H}-\frac{n_{2}}{N_{V}} \Lambda _{V}=0, \\ \Lambda _{V}-\mu _{V} N_{V}- \frac{N_{V}}{K}(N_{V}+qZ)=0, \\ a Z-\frac{Z}{k}(Z+\varepsilon N_{V})=0. \end{cases} $$

(7)

Letting $m_{1}=\mu _{H}+\delta$, $m_{2}=\mu _{H}+r$, we get

$$\begin{aligned}& T_{H}^{*}=\frac{\alpha I_{H}^{*}}{m_{2} (1+I_{H}^{*} )}, \qquad R_{H}^{*}= \frac{r\alpha I_{H}^{*}}{m_{2}\mu _{H} (1+I_{H}^{*} )}, \\& n_{2}^{*}= \frac{\beta _{V} I_{H}^{*}N_{V}^{*}}{\beta _{V} I_{H}^{*}N_{V}^{*} +\Lambda _{V}},\qquad Z^{*}=ak- \varepsilon N_{V}^{*}, \\& S_{H}^{*}= \frac{\Lambda _{H}(1+I_{H}^{*})- [m_{1}(1+I_{H}^{*})+\alpha ]I_{H}^{*}}{\mu _{H}(1+I_{H}^{*})}, \end{aligned}$$

(8)

$$\begin{aligned}& N_{V}^{*}n_{2}^{*}= \frac{(m_{1}+\alpha +m_{1}I_{H}^{*})\mu _{H}I_{H}^{*}}{\beta _{H} [\Lambda _{H}(1+I_{H}^{*})-(m_{1}(1+I_{H}^{*})+\alpha )I_{H}^{*} ]}. \end{aligned}$$

(9)

Since $n_{1}=\frac{S_{V}}{N_{V}}$, $n_{1}+n_{2}=1$, it follows that

$$ \frac{dn_{1}}{dt}=\frac{(1-n_{1})\Lambda _{V}}{N_{V}}-\beta _{V} n_{1}I_{H}, $$

(10)

thus $n_{1}^{*}= \frac{\Lambda _{V}}{\beta _{V} I_{H}^{*}N_{V}^{*}+\Lambda _{V}}$ and

$$ n_{2}^{*}N_{V}^{*}= \bigl(1-u_{1}^{*} \bigr)N_{V}^{*}= \frac{\beta _{V}(N_{V}^{*})^{2}I_{H}^{*}}{\beta _{V}N_{V}^{*}I_{H}^{*}+\Lambda _{V}}. $$

(11)

Remark 1

From Eq. (11), $n_{2}^{*}N_{V}^{*}>0$, combining Eqs. (8) and (9), we get $S_{H}>0$.

Note that $I_{H}^{*}$ satisfies the following one-dimensional cubic equation according to Eqs. (9) and (11):

$$ I_{H}^{*} \bigl(B_{1} \bigl(I_{H}^{*} \bigr)^{2}+B_{2}I_{H}^{*}+B_{3} \bigr)=0, $$

(12)

where

$$ \begin{aligned} &B_{1}=m_{1}\mu _{H} \beta _{V}N_{V}^{*}+m_{1}\beta _{H} \beta _{V} \bigl(N_{V}^{*} \bigr)^{2}>0, \\ &B_{2}=\mu _{H}\Lambda _{V}(m_{1}+ \alpha ) \biggl[ \frac{m_{1}\mu _{H}\Lambda _{V}+\beta _{V}N_{V}^{*}(m_{1}+\alpha )(\mu _{H}+\beta _{H}N_{V}^{*})}{(m_{1}+\alpha )\mu _{H}\Lambda _{V}}-R_{0}^{2} \biggr], \\ &B_{3}=(m_{1}+\alpha )\mu _{H}\Lambda _{V}-\beta _{H}\beta _{V} \Lambda _{H} \bigl(N_{V}^{*} \bigr)^{2}=\mu _{H}\Lambda _{V}(m_{1}+\alpha ) \bigl(1-R_{0}^{2} \bigr). \end{aligned} $$

So, we can express the three roots of Eq. (12) respectively as

$$ I_{H-1}^{*}=0,\qquad I_{H-2}^{*}= \frac{-B_{2}+\sqrt{B_{2}^{2}-4B_{1}B_{3}}}{2B_{1}},\qquad I_{H-3}^{*}= \frac{-B_{2}-\sqrt{B_{2}^{2}-4B_{1}B_{3}}}{2B_{1}}. $$

It is clear that $B_{3}<0\Longleftrightarrow R_{0}>1$, $B_{3}\ge 0\Longleftrightarrow R_{0} \leq 1$. If we denote

$$ \alpha ^{*}= \frac{\beta _{H}\beta _{V}\Lambda _{H}(N_{V}^{*})^{2}-m_{1}\mu _{H}\Lambda _{V}-m_{1}\beta _{V} N_{V}^{*}(\mu _{H}+\beta _{H} N_{V}^{*})}{\beta _{V} N_{V}^{*}(\mu _{H}+\beta _{H} N_{V}^{*})}, $$

then $B_{2}<0\Longleftrightarrow \alpha <\alpha ^{*}$, $B_{2}>0 \Longleftrightarrow \alpha >\alpha ^{*}$.

Set $\Delta _{2}=B_{2}^{2}-4B_{1}B_{3}$, then

$$\begin{aligned}& \Delta _{2}< 0\quad \Longleftrightarrow\quad R_{0}^{2}< R_{0}^{c} \Bigl(R_{0}< \sqrt{R_{0}^{c}} \Bigr), \\& \Delta _{2}=0\quad \Longleftrightarrow\quad R_{0}^{2}=R_{0}^{c} \Bigl(R_{0}=\sqrt{R_{0}^{c}} \Bigr), \\& \Delta _{2}>0\quad \Longleftrightarrow\quad R_{0}^{2}>R_{0}^{c} \Bigl(R_{0}>\sqrt{R_{0}^{c}} \Bigr), \end{aligned}$$

where

$$ R_{0}^{c}=1- \frac{B_{2}^{2}}{4B_{1}\mu _{H}\Lambda _{V}(m_{1}+\alpha )}. $$

Thus, the roots of Eq. (12) fall into the following five cases:

• (i) When $R_{0}>1$, $I_{H-2}^{*}I_{H-3}^{*}=\frac{B_{3}}{B_{1}}<0$, Eq. (12) has only one positive real root $I_{H}^{*}=I_{H-2}^{*}$.

• (ii) When $R_{0}=1$ and $\alpha <\alpha ^{*}$, Eq. (12) has a positive real root $I_{H}^{*}=-\frac{B_{2}}{B_{1}}$.

• (iii) When $R_{0}=\sqrt{R_{0}^{c}}$ and $\alpha <\alpha ^{*}$, Eq. (12) has one positive real root $I_{H}^{*}=-\frac{B_{2}}{2B_{1}}$.

• (iv) When $\sqrt{R_{0}^{c}}< R_{0}<1$ and $\alpha <\alpha ^{*}$, Eq. (12) has two positive real roots $I_{H-2}^{*}$ and $I_{H-3}^{*}$, where $I_{H-2}^{*}>I_{H-3}^{*}$.

• (v) When $R_{0}<\sqrt{R_{0}^{c}}$ and $\alpha <\alpha ^{*}$ or $R_{0}\leq 1$ and $\alpha >\alpha ^{*}$, Eq. (12) has no positive real roots.

Hence, the existence of endemic equilibrium of model (3) is given by the following theorem:

Theorem 1

For model (3), assuming (A1) or (A2) holds,

• (1) there exists a unique endemic equilibrium if $R_{0}>1$ or $R_{0}=1$ and $\alpha <\alpha ^{*}$ or $R_{0}=\sqrt{R_{0}^{c}}$ and $\alpha <\alpha ^{*}$;

• (2) if $\sqrt{R_{0}^{c}}< R_{0}<1$ and $\alpha <\alpha ^{*}$, there are two endemic equilibria, namely $E_{2}^{*}=(S_{H-2}^{*},I_{H-2}^{*},T_{H-2}^{*},R_{H-2}^{*},n_{2-2}^{*},N^{*}_{V},Z^{*})$ and $E_{3}^{*}=(S_{H-3}^{*},I_{H-3}^{*}, T_{H-3}^{*},R_{H-3}^{*},n_{2-3}^{*},N^{*}_{V},Z^{*})$;

• (3) there is no endemic equilibrium if $R_{0}<\sqrt{R_{0}^{c}}$ and $\alpha <\alpha ^{*}$ or $R_{0}\leq 1$ and $\alpha >\alpha ^{*}$.

3.2 Stability of the equilibrium

In this section, we study the local and global stability of the equilibrium point of model (3) and further discuss the type of bifurcation (forward or backward) that occurs near the DFE when $R_{0}=1$ and $\alpha <\alpha ^{*}$.

3.2.1 Stability of the DFE and backward bifurcation

The characteristic equation of model (3) at $E_{0}$ is $\det (\lambda E-L(e^{\lambda \theta}))=0$, i.e.,

$$\left|\begin{array}{ccccc}\lambda +{\mu }_H& 0& 0& 0& \frac{{\beta }_H{\mathrm{\Lambda }}_H{N}_V^{\ast }}{{\mu }_H}{e}^{-\lambda \tau }\\ 0& \lambda +(\delta +{\mu }_H+\alpha )& 0& 0& -\frac{{\beta }_H{\mathrm{\Lambda }}_H{N}_V^{\ast }}{{\mu }_H}{e}^{-\lambda \tau }\\ 0& -\alpha & \lambda +{\mu }_H+r& 0& 0\\ 0& 0& -r& \lambda +{\mu }_H& 0\\ 0& -{\beta }_V& 0& 0& \lambda +\frac{{\mathrm{\Lambda }}_V}{N_V^{\ast }}\end{array}\right|=0.$$

It is easy to get three eigenvalues: $\lambda _{1}=\lambda _{2}=-\mu _{H}$, $\lambda _{3}=-(\mu _{H}+r)$. The remaining eigenvalues are the roots of the following equation:

$$ p(\lambda )=\lambda ^{2}+a_{1}\lambda -b_{1}e^{-\lambda \tau}+a_{2}=0, $$

(13)

where

$$ a_{1}=(\alpha +\delta +\mu _{H})+\frac{\Lambda _{V}}{N_{V}^{*}},\qquad a_{2}=(\alpha +\delta +\mu _{H})\frac{\Lambda _{V}}{N_{V}^{*}},\qquad b_{1}= \frac{\beta _{V}\beta _{H}\Lambda _{H}N_{V}^{*}}{\mu _{H}}. $$

Case 1. If $\tau =0$, then Eq. (13) becomes

$$ p(\lambda )=\lambda ^{2}+a_{1}\lambda +(a_{2}-b_{1})=0, $$

(14)

where

$$\begin{aligned}& a_{1}=(\alpha +\delta +\mu _{H})+ \frac{\Lambda _{V}}{N_{V}^{*}}>0, \\& a_{2}-b_{1}=(\alpha +\delta +\mu _{H}) \frac{\Lambda _{V}}{N_{V}^{*}}- \frac{\beta _{V}\beta _{H}\Lambda _{H}N_{V}^{*}}{\mu _{H}}= \bigl(1-R_{0}^{2} \bigr) ( \alpha +\delta +\mu _{H})\frac{\Lambda _{V}}{N_{V}^{*}}. \end{aligned}$$

When $R_{0}<1$, we have $\lambda _{1}+\lambda _{2}=-a_{1}<0$, $\lambda _{1}\lambda _{2}=a_{2}-b_{1}>0$.

It is determined that all the characteristic roots of Eq. (14) exhibit negative real parts. Consequently, under the conditions $R_{0}<1$ and $\tau =0$, the disease-free equilibrium $E_{0}$ can be deemed locally asymptotically stable.

Case 2. If $\tau >0$, assume $\lambda =i\omega$ ($\omega >0$) represents a purely imaginary root of Eq. (13). Upon substituting $\lambda =i\omega$ into Eq. (13), we obtain

$$ -\omega ^{2}+a_{1}i\omega -b_{1}\cos \omega \tau +b_{1}i\sin \omega \tau +a_{2}=0. $$

By separating the real and imaginary parts, we get

$$ \textstyle\begin{cases}a_{1}\omega =-b_{1}\sin \omega \tau , \\ a_{2}-\omega ^{2}=b_{1}\cos \omega \tau . \end{cases} $$

(15)

Adding the squares of the two equations in Eq. (15), we derive

$$ \omega ^{4}+ \bigl(a_{1}^{2}-2a_{2} \bigr)\omega ^{2}+a_{2}^{2}-b_{1}^{2}=0. $$

(16)

In Eq. (16), substituting y with $\omega ^{2}$ leads to the transformation of Eq. (16) into

$$ y^{2}+ \bigl(a_{1}^{2}-2a_{2} \bigr)y+a_{2}^{2}-b_{1}^{2}=0, $$

(17)

where

$$\begin{aligned} & a_{1}^{2}-2a_{1}=( \alpha +\delta + \mu _{H})+ \biggl( \frac{\Lambda _{V}}{N_{V}^{*}} \biggr)^{2}>0, \\ &a_{2}^{2}-b_{1}^{2}=(\alpha +\delta + \mu _{H})^{2} \biggl( \frac{\Lambda _{V}}{N_{V}^{*}} \biggr)^{2}- \biggl( \frac{\beta _{V}\beta _{H}\Lambda _{H}N_{V}^{*}}{\mu _{H}} \biggr)^{2} \\ &\hphantom{a_{2}^{2}-b_{1}^{2}}= \bigl[1- \bigl(R_{0}^{2} \bigr)^{2} \bigr](\alpha +\delta +\mu _{H})^{2} \biggl( \frac{\Lambda _{V}}{N_{V}^{*}} \biggr)^{2}. \end{aligned}$$

If $R_{0}<1$, $a_{2}^{2}-b_{1}^{2}>0$, there are no positive real roots of Eq. (17), so we obtain the following conclusion:

Theorem 2

For model (3), suppose that (A1) or (A2) holds. Then the disease-free equilibrium $E_{0}$ is locally asymptotically stable when $R_{0}<1$ and unstable when $R_{0}>1$.

Next, we consider the global asymptotic stability of model (3) at $E_{0}$. After substituting Eqs. (4) and (5) into model (3), we have

$$ \textstyle\begin{cases}\frac{dS_{H}}{dt}=\Lambda _{H}-\beta _{H} S_{H}(t-\tau )n_{2}(t- \tau )N_{V}^{*}-\mu _{H} S_{H}, \\ \frac{dI_{H}}{dt}=\beta _{H} S_{H}(t-\tau )n_{2}(t-\tau )N_{V}^{*}-f(I_{H})-( \mu _{H}+\delta )I_{H}, \\ \frac{dT_{H}}{dt}=f(I_{H})-(r+\mu _{H})T_{H}, \\ \frac{dR_{H}}{dt}=r T_{H}-\mu _{H}R_{H}, \\ \frac{dn_{2}}{dt}=\beta _{V} (1-n_{2})I_{H}- \frac{n_{2}}{N_{V}^{*}}\Lambda _{V}. \end{cases} $$

(18)

If model (18) is globally asymptotically stable at $E_{0}^{*}= (\frac{\Lambda _{H}}{\mu _{H}},0,0,0,0 )$, then model (3) is globally asymptotically stable at $E_{0}= (\frac{\Lambda _{H}}{\mu _{H}},0,0,0,0,N^{*}_{V},Z^{*} )$. Here, we define $R_{0}^{*}=\sqrt{ \frac{\beta _{H}\beta _{V}\Lambda _{H}(N^{*}_{V})^{2}}{(\delta +\mu _{H})\mu _{H}\Lambda _{V}}}$. Obviously, if $\alpha =0$ then $R_{0}^{*}=R_{0}$, and if $\alpha >0$ then $R_{0}^{*}>R_{0}$.

Theorem 3

For $R_{0}^{*}\leq 1$, model (3) is globally asymptotically stable at $E_{0}$ if it satisfies (A1) or (A2).

Proof

We define the following Lyapunov function [26]:

$$ V(t)=g_{1} \biggl(S_{H}-S_{H}^{0}-S_{H}^{0} \ln \frac{S_{H}}{S_{H}^{0}} \biggr)+g_{2}I_{H}+g_{3}T_{H}+g_{4}R_{H}+g_{5}n_{2}. $$

Then the time derivative of along model (18) is

$$ \begin{aligned} \frac{dV(t)}{dt} ={}&g_{1} \biggl( \frac{S_{H}-S_{H}^{0}}{S_{H}} \biggr) \frac{dS_{H}}{dt}+g_{2}\frac{dI_{H}}{dt}+g_{3} \frac{dT_{H}}{dt}+g_{4} \frac{dR_{H}}{dt}+g_{5} \frac{dn_{2}}{dt} \\ ={}&g_{1} \biggl(\frac{S_{H}-S_{H}^{0}}{S_{H}} \biggr) \bigl[\Lambda _{H}- \beta _{H} S_{H}(t-\tau )n_{2}(t-\tau )N_{V}^{*}-\mu _{H}S_{H} \bigr] \\ & {}+g_{2} \bigl[\beta _{H} S_{H}(t-\tau )n_{2}(t-\tau )N_{V}^{*}-f(I_{H})-( \mu _{H}+\delta )I_{H} \bigr] \\ &{} +g_{3} \bigl[f(I_{H})-(r+\mu _{H})T_{H} \bigr]+g_{4} (rT_{H}- \mu _{H}R_{H} ) \\ & {}+g_{5} \biggl[\beta _{V}(1-n_{2})I_{H}- \frac{n_{2}}{N_{V}^{*}} \Lambda _{V} \biggr]. \end{aligned} $$

At $E_{0}^{*}= (\frac{\Lambda _{H}}{\mu _{H}},0,0,0,0 )$, we have $S_{H}^{0}=\frac{\Lambda _{H}}{\mu _{H}}$. Thus

$$ \begin{aligned} \frac{dV(t)}{dt} ={}&-\mu _{H}g_{1} \frac{(S_{H}-S_{H}^{0})^{2}}{S_{H}}-(g_{1}-g_{2})\beta _{H}S_{H}(t- \tau )n_{2}(t-\tau )N_{V}^{*} \\ &{} - \biggl[ \biggl(\mu _{H}+\delta +\frac{\alpha}{1+I_{H}} \biggr)g_{2}- \frac{\alpha}{1+I_{H}}g_{3}-\beta _{V} g_{3} \biggr]I_{H} \\ & {}- \bigl[(r+\mu _{H})g_{3}-rg_{4} \bigr]T_{H}-g_{4}\mu _{H}R_{H}-g_{5} \beta _{V}n_{2}I_{H} \\ & {}+ \biggl[\frac{g_{1}\Lambda _{H}}{\mu _{H}}\beta _{H}N_{V}^{*}- \frac{\Lambda _{V}}{N_{V}^{*}}g_{5} \biggr]n_{2}. \end{aligned} $$

We take

$$ g_{1}=g_{2}=g_{3}=1, \qquad g_{5}= \frac{\beta _{H}\Lambda _{H}(N_{V}^{*})^{2}}{\mu _{H}\Lambda _{V}}, $$

so $\frac{dV(t)}{dt}=-\mu _{H}\frac{(S_{H}-S_{H}^{0})^{2}}{S_{H}}-(m_{1}+ \alpha )[1-(R_{0}^{*})^{2}]I_{H}-\frac{\alpha I_{H}}{1+I_{H}}- \frac{\beta _{H}\Lambda _{H}(N_{V}^{*})^{2}}{\mu _{H}\Lambda _{V}}n_{2}I_{H}$, thus $\frac{dV(t)}{dt}<0$ if $R_{0}^{*}\leq 1$. Moreover, $\frac{dV(t)}{dt}=0$ if and only if $(S_{H},I_{H},T_{H},R_{H},n_{2})= (\frac{\Lambda _{H}}{\mu _{H}},0,0,0,0 )$. Hence, the largest compact invariant set in Φ is the single point set ${E_{0}^{*}}$. According to LaSalle’s invariance principle [24], model (18) is globally asymptotically stable inside Φ at $E_{0}^{*}$, where

$$ \Phi = \biggl\{ (S_{H},I_{H},T_{H},R_{H},n_{2} )\in R_{+}^{5} |0\le S_{H}+I_{H}+T_{H}+R_{H} \le \frac{\Lambda _{H}}{\mu _{H}},0< n_{2}< 1 \biggr\} . $$

Hence, model (3) is globally asymptotically stable at $E_{0}$ when $R_{0}^{*}\leq 1$. □

Remark 2

When $R_{0}^{*}<1$, $R_{0}<1$ always holds.

Theorem 1 shows that for $\alpha <\alpha ^{*}$, $R_{0}=1$ is a bifurcation value. In fact, the stability of the disease-free equilibrium changes at $R_{0}=1$. We examine model (3) and explore the characteristics of the bifurcation related to the disease-free equilibrium $E_{0}$ when $R_{0}=1$. In the case of $R_{0}=1$, we regard $\beta _{H}$ as a branching parameter, leading us to

$$ \beta _{H}^{*}= \frac{(\alpha +\delta +\mu _{H})\mu _{H}\Lambda _{V}}{\beta _{V}\Lambda _{H}(N_{V}^{*})^{2}}. $$

Thus $R_{0}<1$ if $\beta _{H}<\beta _{H}^{*}$ and $R_{0}>1$ if $\beta _{H}>\beta _{H}^{*}$.

Introducing $S_{H}=x_{1}$, $I_{H}=x_{2}$, $T_{H}=x_{3}$, $R_{H}=x_{4}$, $n_{2}=x_{5}$, model (3) reduces to

$$ \textstyle\begin{cases}\frac{dx_{1}}{dt}=\Lambda _{H}-\beta _{H} x_{1}x_{5}N_{V}^{*}- \mu _{H} x_{1}:=f_{1}, \\ \frac{dx_{2}}{dt}=\beta _{H} x_{1}x_{5}N_{V}^{*}-f(x_{2})-( \mu _{H}+ \delta )x_{2}:=f_{2}, \\ \frac{dx_{3}}{dt}=f(x_{2})-(r+\mu _{H})x_{3}:=f_{3}, \\ \frac{dx_{4}}{dt}=r x_{3}-\mu _{H}x_{4}:=f_{4}, \\ \frac{dx_{5}}{dt}=\beta _{V} (1-x_{5})x_{2}- \frac{x_{5}}{N_{V}^{*}}\Lambda _{V}:=f_{5}. \end{cases} $$

(19)

The Jacobian matrix of model (19) at the disease-free equilibrium $E_{0}$ is

$$J=\left(\begin{array}{ccccc}-{\mu }_H& 0& 0& 0& -\frac{(\alpha +\delta +{\mu }_H){\mathrm{\Lambda }}_V}{{\beta }_V{N}_V^{\ast }}\\ 0& -(\alpha +\delta +{\mu }_H)& 0& 0& \frac{(\alpha +\delta +{\mu }_H){\mathrm{\Lambda }}_V}{{\beta }_V{N}_V^{\ast }}\\ 0& \alpha & -({\mu }_H+r)& 0& 0\\ 0& 0& r& -{\mu }_H& 0\\ 0& {\beta }_V& 0& 0& -\frac{{\mathrm{\Lambda }}_V}{N_V^{\ast }}\end{array}\right).$$

The eigenequation is

$$ \lambda (\lambda +\mu _{H})^{2} \bigl[\lambda +(\mu _{H}+r) \bigr] \biggl[\lambda + \biggl(\alpha +\delta +\mu _{H}+\frac{\Lambda _{V}}{N_{V}^{*}} \biggr) \biggr]=0. $$

Therefore, the characteristics are as follows:

$$ \lambda _{1}=0,\qquad \lambda _{2,3}=-\mu _{H}, \qquad \lambda _{4}=-(\mu _{H}+r),\qquad \lambda _{5}=- \biggl(\alpha +\delta +\mu _{H}+ \frac{\Lambda _{V}}{N_{V}^{*}} \biggr). $$

Hence, $\lambda _{1}=0$ represents a zero eigenvalue of J, and all remaining eigenvalues possess negative real parts. Consequently, an application of Center Manifold Theory [27] is appropriate. The right eigenvector $w=(w_{1},w_{2},w_{3},w_{4},w_{5})^{T}$ associated with the eigenvalue 0 is specifically determined by the following equation:

$$ \textstyle\begin{cases}\mu _{H}w_{1}+ \frac{(\alpha +\delta +\mu _{H})\Lambda _{V}}{\beta _{V}N_{V}^{*}}w_{5}=0, \\ (\alpha +\delta +\mu _{H})w_{2}- \frac{(\alpha +\delta +\mu _{H})\Lambda _{V}}{\beta _{V}N_{V}^{*}}w_{5}=0, \\ -\alpha w_{2}+(\mu _{H}+r)w_{3}=0, \\ -rw_{3}+\mu _{H}w_{4}=0, \\ -\beta _{V}w_{2}+\frac{\Lambda _{V}}{N_{V}^{*}}w_{5}=0. \end{cases} $$

Consequently, through the calculation process, it is established that

$$ w= \biggl(-\frac{\alpha +\delta +\mu _{H}}{\mu _{H}}w_{2},w_{2}, \frac{\alpha}{\mu _{H}+r}w_{2},\frac{r\alpha}{(\mu _{H}+r)\mu _{H}}w_{2}, \frac{\beta _{V}N_{V}^{*}}{\Lambda _{V}}w_{2} \biggr)^{T}. $$

(20)

Similarly, we get the left eigenvector as

$$ v= \left ( 0,\frac{\beta _{V}}{\alpha +\delta +\mu _{H}}v_{5},0,0,v_{5} \right ) . $$

(21)

Given $v\cdot w=1$, it follows that $w_{2}=\frac{\Lambda _{V}}{\beta _{V}N_{V}^{*}}$ and $v_{5}= \frac{(\alpha +\delta +\mu _{H})N_{V}^{*}}{\Lambda _{V}+(\alpha +\delta +\mu _{H})N_{V}^{*}}$.

Next, we take the partial derivatives at $E_{0}$,

$$\begin{aligned}& \frac{\partial ^{2}f_{1}}{\partial x_{1}\partial x_{5}} = \frac{\partial ^{2}f_{1}}{\partial x_{5}\partial x_{1}}=-\beta _{H} N_{V}^{*},\qquad \frac{\partial ^{2}f_{2}}{\partial x_{1}\partial x_{5}}= \frac{\partial ^{2}f_{2}}{\partial x_{5}\partial x_{1}}= \beta _{H} N_{V}^{*}, \\& \frac{\partial ^{2}f_{5}}{\partial x_{2}\partial x_{5}} = \frac{\partial ^{2}f_{5}}{\partial x_{5}\partial x_{2}}=-\beta _{V},\qquad \frac{\partial ^{2}f_{2}}{\partial x_{2}^{2}}=-2\alpha , \qquad \frac{\partial ^{2}f_{3}}{\partial x_{2}^{2}}=2\alpha , \\& \frac{\partial ^{2}f_{1}}{\partial x_{5}\partial \beta _{H}} = \frac{\partial ^{2}f_{1}}{\partial \beta _{H}\partial x_{5}}=- \frac{N_{V}^{*}\Lambda _{H}}{\mu _{H}},\qquad \frac{\partial ^{2}f_{2}}{\partial x_{5}\partial \beta _{H}}= \frac{\partial ^{2}f_{2}}{\partial \beta _{H}\partial x_{5}}= \frac{N_{V}^{*}\Lambda _{H}}{\mu _{H}}, \end{aligned}$$

and all other partial derivatives of second order are equal to 0.

From this we get

$$ \begin{aligned} &a=2v_{2}w_{1}w_{5} \frac{\partial ^{2}f_{2}}{\partial x_{1}\partial x_{5}} \bigl(E_{0}^{*}, \beta _{H} \bigr)+2v_{5} w_{2}w_{5} \frac{\partial ^{2}f_{5}}{\partial x_{2}\partial x_{5}} \bigl(E_{0}^{*}, \beta _{H} \bigr)+\nu _{2}w_{2}^{2} \frac{\partial ^{2}f_{2}}{\partial x_{2}^{2}} \bigl(E_{0}^{*},\beta _{H} \bigr), \\ &b=v_{2}w_{5} \frac{\partial ^{2}f_{2}}{\partial x_{5}\partial \beta _{H}} \bigl(E_{0}^{*}, \beta _{H} \bigr). \end{aligned} $$

By substituting (20) and (21), we get

$$ \begin{aligned} &a= \frac{2\Lambda _{V}}{\Lambda _{V}+(m_{1}+\alpha )N_{V}^{*}} \biggl( \frac{\alpha \Lambda _{V}}{\beta _{V}N_{V}^{*}}-(m_{1}+ \alpha )- \frac{(m_{1}+\alpha )\beta _{H}N_{V}^{*}}{\mu _{H}} \biggr), \\ &b= \frac{\beta _{V}\Lambda _{H}(N_{V}^{*})^{2}}{\mu _{H} [\Lambda _{V}+(m_{1}+\alpha )N_{V}^{*} ]}. \end{aligned} $$

It is obvious that $b>0$. Since $\alpha <\alpha ^{*}$, it follows that $a>0$. According to Castillo-Chavez and Song lemma in [27], we get the following result:

Theorem 4

Model (3) undergoes backward bifurcation at the disease-free equilibrium $E_{0}$ when $R_{0}=1$ and $\alpha <\alpha ^{*}$. The disease-free equilibrium $E_{0}$ is globally asymptotically stable when $R_{0}<\sqrt{R_{0}^{c}}$ and $\alpha <\alpha ^{*}$. However, when $\sqrt{R_{0}^{c}}< R_{0}<1$ and $\alpha <\alpha ^{*}$, there are two endemic equilibria, $E_{2}^{*}$ and $E_{3}^{*}$, for model (3), where $E_{2}^{*}$ is stable and $E_{3}^{*}$ is unstable (see Fig. 1).

Figure 1

Backward bifurcation diagram. Setting $\beta _{H}=0.2$, $\beta _{V}=0.09$, $\Lambda _{H}=0.5$, $\Lambda _{V}=0.5$, $N_{V}^{*}=2.3$, $\alpha =0.12$, $m_{1}=\delta +\mu _{H}=0.2$, and $\mu _{H}=0.04$, we can obtain $R_{0}^{c}=0.7923$. In this diagram, the solid lines indicate stability, and the dashed lines indicate instability

From the examples depicted in Fig. 1, it is evident that $R_{0}^{c}<1$ and $R_{0}^{*}>1$. Therefore, it is clear that the condition $R_{0}^{*}<1$ stated in Theorem 3 is comparatively stringent.

According to Theorem 1, for model (3), backward bifurcation disappears at $\alpha >\alpha ^{*}$, and there is no endemic equilibrium if $R_{0}<1$.

3.2.2 Stability of the endemic equilibrium

Next, we consider the stability of the endemic equilibrium for model (3) when $R_{0}>1$. From Theorem 1, it follows that there exists an unique endemic equilibrium for model (3) when $R_{0}>1$. Let us denote this equilibrium point as $E^{*}=(S_{H}^{*},I_{H}^{*},T_{H}^{*},R_{H}^{*},n_{2}^{*},N^{*}_{V},Z^{*})$, where $I_{H}=I_{H-2}^{*}$. The characteristic equation at $E^{*}$ is given by

$$\left|\begin{array}{ccccc}\lambda +{\mu }_H+k_1{e}^{-\lambda \tau }& 0& 0& 0& k_5{e}^{-\lambda \tau }\\ -k_1{e}^{-\lambda \tau }& \lambda +k_2& 0& 0& -k_5{e}^{-\lambda \tau }\\ 0& -\frac{\alpha }{{(1+I_H^{\ast })}^2}& \lambda +k_3& 0& 0\\ 0& 0& -r& \lambda +{\mu }_H& 0\\ 0& -k_6& 0& 0& \lambda +k_4\end{array}\right|=0,$$

where

$$ \begin{aligned} &k_{1}=\beta _{H}n_{2}^{*}N_{V}^{*}, \qquad k_{2}=m_{1}+ \frac{\alpha}{(1+I_{H}^{*})^{2}}, \qquad k_{3}=\mu _{H}+r, \\ &k_{4}=\beta _{V}I_{H}^{*}+ \frac{\Lambda _{V}}{N_{V}^{*}},\qquad k_{5}= \beta _{H}S_{H}^{*}N_{V}^{*}, \qquad k_{6}=\beta _{V} \bigl(1-n_{2}^{*} \bigr). \end{aligned} $$

It is easy to get $\lambda _{1}=-\mu _{H}$, $\lambda _{2}=-(r+\mu _{H})$, while the remaining eigenvalues are the roots of the following equation:

$$ q(\lambda )=\lambda ^{3}+c_{1}\lambda ^{2}+c_{2}\lambda + \bigl(d_{1} \lambda ^{2}+d_{2}\lambda +d_{3} \bigr)e^{-\lambda \tau}+c_{3}=0, $$

(22)

where

$$ \begin{aligned} &c_{1}=k_{2}+k_{4}+ \mu _{H},\qquad c_{2}=k_{2}k_{4}+\mu _{H}(k_{2}+k_{4}),\qquad c_{3}=\mu _{H}k_{2}k_{4}, \\ &d_{1}=k_{1},\qquad d_{2}=k_{1}k_{2}+k_{1}k_{4}-k_{5}k_{6}, \qquad d_{3}=k_{1}k_{2}k_{4}-k_{5}k_{6} \mu _{H}. \end{aligned} $$

Case 1. For $\tau =0$, Eq. (22) becomes

$$ q(\lambda )=\lambda ^{3}+D_{1}\lambda ^{2}+D_{2}\lambda +D_{3}=0, $$

(23)

where

$$ \begin{aligned} &D_{1}=k_{1}+k_{2}+k_{4}+ \mu _{H}, \\ &D_{2}=k_{1}k_{4}+\mu _{H}(k_{2}+k_{4})+k_{1}k_{2}+k_{2}k_{4}-k_{5}k_{6}, \\ &D_{3}=k_{1}k_{2}k_{4}+\mu _{H}(k_{2}k_{4}-k_{5}k_{6}). \end{aligned} $$

Since $D_{i}>0$ ($i=1,2,3$), $D_{1}D_{2}-D_{3}>0$, so applying the Routh–Hurwitz criterion [26], all roots of Eq. (23) have negative real parts. Hence, we get:

Theorem 5

For model (3), assuming (A1) or (A2) holds, the endemic equilibrium $E^{*}$ is locally asymptotically stable when $R_{0}>1$ and $\tau =0$.

We observe that all solutions of Eq. (23) possess negative real parts when $\tau >0$. Moreover, for $\tau >0$, if a root of Eq. (23) emerges in the right half-plane, it must inevitably traverse the imaginary axis.

Case 2. For $\tau >0$, we consider that Eq. (22) has a purely imaginary root of the form $\lambda =i\omega$ ($\omega >0$). By substituting $\lambda =i\omega$ into Eq. (22), we can derive:

$$ \omega ^{6}+C_{1}\omega ^{4}+C_{2}\omega ^{2}+C_{3}=0, $$

(24)

where

$$ \begin{aligned} C_{1}=c_{1}^{2}-2c_{2}-d_{1}^{2}, \qquad C_{2}=c_{2}^{2}-2c_{1}c_{3}+2d_{1}d_{3}-d_{2}^{2}, \qquad C_{3}=c_{3}^{2}-d_{3}^{2}. \end{aligned} $$

Letting $z=\omega ^{2}$, Eq. (24) becomes

$$ z^{3}+C_{1}z^{2}+C_{2}z+C_{3}=0. $$

(25)

If we denote $h(z)=z^{3}+C_{1}z^{2}+C_{2}z+C_{3}$, then $h{'}(z)=3z^{2}+2C_{1}z+C_{2}=0$ has two roots $z_{1}^{*}=\frac{-C_{1}+\sqrt{\Delta _{3}}}{3}$, $z_{2}^{*}= \frac{-C_{1}-\sqrt{\Delta _{3}}}{3}$, where $\Delta _{3}=C_{1}^{2}-3C_{2}$. For the situation of the roots of Eq. (25), we give the following conclusion:

Lemma 2

• (1) If $C_{3}<0$, Eq. (25) has three positive roots or two negative roots and one positive root;

• (2) If $C_{3}\ge 0$ and $\Delta _{3}\leq 0$, Eq. (25) has no positive roots;

• (3) When $C_{3}\ge 0$ and $\Delta _{3}>0$, if $z_{1}^{*}>0$ such that $h(z_{1}^{*})\leq 0$, then Eq. (25) has two positive roots.

Theorem 6

For model (3), assuming (A1) or (A2) holds, if Eq. (25) satisfies $C_{3}\ge 0$ and $\Delta _{3}\leq 0$, the endemic equilibrium $E^{*}$ is locally asymptotically stable for all $\tau >0$ when $R_{0}>1$.

Proof

For $R_{0}>1$, if $C_{3}\ge 0$, then $h(0)=C_{3}\ge 0$, and when $\Delta _{3}=C_{1}^{2}-3C_{2}\leq 0$, it holds that

$$ h{'}(z)=3z^{2}+2C_{1}z+C_{2}>0. $$

Hence, $h(z)$ is monotonically increasing on the interval $(-\infty ,+\infty )$, meaning that for all $z>0$, $h(z)>0$. This implies that Eq. (25) has no positive roots, consequently leading to all characteristic roots of (22) having negative real parts. Therefore, $E^{*}$ is locally asymptotically stable. □

3.3 Hopf bifurcation

In this section, we consider that model (3) has a Hopf bifurcation around the endemic equilibrium $E^{*}$. Consequently, we choose the time delay τ as the bifurcation parameter. According to Lemma 2, Eq. (25) exists at least one positive root. Assuming that Eq. (25) has three positive roots $z_{1}$, $z_{2}$, $z_{3}$, Eq. (24) has three positive roots $\omega _{k}=\sqrt{z_{k}}$, $k=1,2,3$.

For each $\omega _{k}$, the corresponding time delay is

$$\begin{aligned}& \tau _{k}^{(j)}=\frac{1}{\omega _{k}} \biggl[ \arccos \frac{(c_{3}-c_{1}\omega _{k}^{2})(d_{1}\omega _{k}^{2}-d_{3})+d_{2}\omega _{k}(\omega _{k}^{3}-c_{2}\omega _{k})}{ (d_{1}\omega _{k}^{2}-d_{3})^{2}+(d_{2}\omega _{k})^{2}}+2j\pi \biggr], \\& j=0,1,2,\dots ; k=1,2,3. \end{aligned}$$

Define $\tau _{0}=\min \{\tau _{k}^{(0)},k=1,2,3 \}$, then $\omega _{0}=\omega _{k} |_{\tau =\tau _{0}}$. When $\tau =\tau _{0}$, there exist a pair of pure imaginary roots $\pm i\omega _{0}$ of Eq. (22), where

$$ \tau _{0}=\frac{1}{\omega _{0}} \biggl[\arccos \frac{(c_{3}-c_{1}\omega _{0}^{2})(d_{1}\omega _{0}^{2}-d_{3})+d_{2}\omega _{0}(\omega _{k}^{3}-c_{2}\omega _{0})}{ (d_{1}\omega _{0}^{2}-d_{3})^{2}+(d_{2}\omega _{0})^{2}} \biggr]. $$

Taking the derivatives with respect to τ on both sides of Eq. (22), we have

$$ \biggl(\frac{d\lambda}{d\tau} \biggr)^{-1} = \frac{ (3\lambda ^{2}+2c_{1}\lambda +c_{2} )e^{\lambda \tau}}{\lambda (d_{1}\lambda ^{2}+d_{2}\lambda +d_{3} )} + \frac{2d_{1}\lambda +d_{2}}{\lambda (d_{1}\lambda ^{2}+d_{2}\lambda +d_{3} )}- \frac{\tau}{\lambda}. $$

When $\tau =\tau _{0}$, $\lambda =i\omega _{0}$, we get

$$ \operatorname{sign} \biggl\{ \frac{d (\operatorname{Re} {\lambda} )}{d\tau}|_{ \tau =\tau _{0},\lambda =i\omega _{0}} \biggr\} = \operatorname{sign} \biggl\{ \operatorname{Re} \biggl(\frac{d{\lambda}}{d\tau} \biggr)^{-1} \biggr\} _{\lambda =i \omega _{0}}, $$

and

$$ \begin{aligned} \operatorname{Re} \biggl(\frac{d\lambda}{d\tau} \biggr)^{-1}_{\lambda =i\omega _{0}} &=\operatorname{Re} \biggl\{ \frac{(3\lambda ^{2}+2c_{1}\lambda +c_{2})e^{\lambda \tau}}{\lambda (d_{1}\lambda ^{2}+d_{2}\lambda +d_{3})} + \frac{2d_{1}\lambda +d_{2}}{\lambda (d_{1}\lambda ^{2}+d_{2}\lambda +d_{3})}- \frac{\tau}{\lambda} \biggr\} _{\lambda =i\omega _{0}} \\ &= \frac{3\omega _{0}^{6}+(2c_{1}^{2}-4c_{2})\omega _{0}^{4}+(c_{2}^{2}-2c_{1}c_{3})\omega _{0}^{2}}{(\omega _{0}^{3}-c_{2}\omega _{0})^{2}\omega _{0}^{2}+(c_{3}-c_{1}\omega _{0}^{2})^{2}\omega _{0}^{2}} + \frac{(2d_{1}d_{3}-d_{2}^{2})\omega _{0}^{2}-2d_{1}^{2}\omega _{0}^{4}}{d_{2}^{2}\omega _{0}^{4}+\omega _{0}^{2}(d_{3}-d_{1}\omega _{0}^{2})^{2}} \\ &= \frac{3\omega _{0}^{6}+2(c_{1}^{2}-2c_{2}-d_{1}^{2})\omega _{0}^{4}+(c_{2}^{2}-2c_{1}c_{3}+2d_{1}d_{3}-d_{2}^{2})\omega _{0}^{2}}{d_{2}^{2}\omega _{0}^{4}+\omega _{0}^{2}(d_{3}-d_{1}\omega _{0}^{2})^{2}} \\ &=\omega _{0}^{2} \frac{3\omega _{0}^{4}+2(c_{1}^{2}-2c_{2}-d_{1}^{2})\omega _{k}^{2}+(c_{2}^{2}-2c_{1}c_{3}+2d_{1}d_{3}-d_{2}^{2})}{d_{2}^{2}\omega _{0}^{4}+\omega _{0}^{2}(d_{3}-d_{1}\omega _{0}^{2})^{2}} \\ &= \frac{\omega _{0}^{2}h{'}(\omega _{0}^{2})}{d_{2}^{2}\omega _{0}^{4}+\omega _{0}^{2}(d_{3}-d_{1}\omega _{0}^{2})^{2}}. \end{aligned} $$

Thus, we can derive

$$ \begin{aligned} \operatorname{sign} \biggl\{ \operatorname{Re} \biggl( \frac{d\lambda}{d\tau} \biggr)^{-1}_{ \lambda =i\omega _{0}} \biggr\} = \operatorname{sign} \biggl\{ \frac{h{'}(\omega _{0}^{2})}{d_{2}^{2}\omega _{0}^{2}+(d_{3}-d_{1}\omega _{0}^{2})^{2}} \biggr\} =\operatorname{sign} \biggl\{ \frac{h{'}(z_{k})}{d_{2}^{2}\omega _{0}^{2}+(d_{3}-d_{1}\omega _{0}^{2})^{2}} \biggr\} . \end{aligned} $$

If $h{'}(z_{k})\neq 0$, then the transversality condition holds and a Hopf bifurcation occurs at $\tau =\tau _{0}$. According to the Hopf bifurcation theorem for generalized differential equations [28], we obtain:

Theorem 7

For model (3), if (A1) or (A2) is satisfied and $R_{0}>1$, then the following conclusion holds:

• (i) when either $C_{3}<0$ or $C_{3}\ge 0$, $\Delta _{3}>0$, and $h(z_{1}^{*})\leq 0$, the endemic equilibrium $E^{*}$ is asymptotically stable for $\tau \in [0,\tau _{0})$, and $E^{*}$ is unstable for $\tau >\tau _{0}$;

• (ii) when the conditions for (i) are satisfied and $h{'}(z_{k})\neq 0$, model (3) experiences a Hopf bifurcation at the endemic equilibrium $E^{*}$ for $\tau =\tau _{0}$.

3.4 Consistent persistence

In this section, we will discuss the consistent persistence of model (3).

Consider the metric space $X=X^{0}\cup \partial X^{0}$, where $X^{0}$ is an open set and $\partial X^{0}$ is the boundary. The $C^{0}$-semigroup $T(t)$ over X is defined as

$$ T(t):X^{0}\to X^{0},\qquad T(t):\partial X^{0}\to \partial X^{0}. $$

(26)

Denote $T_{0}(t)=T(t)|_{X^{0}}$ and $T_{\partial}(t)=T(t)|_{\partial X^{0}}$, where $A_{\partial}$ is a global attractor with respect to $T_{\partial}(t)$. The ω limit set is defined by $\omega (x)=\bigcap_{\tau \ge 0}C1\bigcup_{t\ge \tau}{\{T(t)x\}}$.

Lemma 3

([29])

• (1) There exists $t_{0}>0$ such that when $t>t_{0}$, $T(t)$ is compact;

• (2) $T(t)$ is point dissipative in X;

• (3) $\widetilde{A_{\partial}}=\bigcup \omega (x)$ ($x\in A_{\partial}$) is isolated and there exists an acyclic covering Ñ, where $\widetilde{N}={N_{1},N_{2},N_{3},\ldots ,N_{n}}$;

• (4) $W^{s}(N_{i})\cap X^{0}=\emptyset$, where $i=1,2,\ldots ,n$.

Assuming $T(t)$ satisfies (26) and the aforementioned conditions hold, $T(t)$ is uniformly continuous: for any $x\in X^{0}$, there exists $\varepsilon >0$ such that

$$ \liminf_{t\to \infty}d \bigl(T(t)x,\partial X^{0} \bigr)\ge \varepsilon . $$

Theorem 8

Model (3) is consistently persistent when $R_{0}>1$.

Proof

Define

$$\begin{aligned}& X:= \bigl\{ \bigl(\varphi _{1}(\theta ),\varphi _{2}(\theta ),\varphi _{3}( \theta ),\varphi _{4}(\theta ),\varphi _{5}(\theta ),\varphi _{6}( \theta ),\varphi _{7}(\theta ) \bigr)\in C| \\& \hphantom{X:=}{}\varphi _{i}(\theta )>0,\varphi _{j}( \theta )>0,i=2,3,4,5,j=1,6,7 \bigr\} , \\& X^{0}:= \bigl\{ \bigl(\varphi _{1}(\theta ),\varphi _{2}(\theta ),\varphi _{3}( \theta ),\varphi _{4}(\theta ), \varphi _{5}(\theta ),\varphi _{6}( \theta ),\varphi _{7}(\theta ) \bigr)\in X|\varphi _{i}(\theta )>0,i=2,3,4,5 \bigr\} , \\& \partial X^{0}:=X\setminus X^{0}, \end{aligned}$$

where $C=C([-\tau ,0],R^{7}_{+0})$. Assume that $T(t)$ ($t>0$) is a solution semiflow of model (3). Verifying that $T(t)$ satisfies condition (2) of Lemma 3 is straightforward. As the function on the right side of model (3) is continuously differentiable, it can be shown that condition (1) in Lemma 3 is valid by using Theorem 2.2 and the Theory of Smoothness of Solutions of Time Delay Differential Equations [29, 30].

It is obvious that $\widetilde{A_{\partial}}=\bigcup \omega (x) (x\in A_{\partial})=E_{0}$, i.e., ${E_{0}}$ is an isolated tightly invariant set, thus condition (3) of Lemma 2 is true. Next, we will prove that $W^{s}(N_{i})\cap X^{0}=\emptyset$. If not, then there is a positive solution

$$ \begin{aligned} \bigl(S_{H}(t),I_{H}(t),T_{H}(t),R_{H}(t),n_{2}(t),N_{V}(t),Z(t) \bigr) \end{aligned} $$

such that

$$ \begin{aligned} \lim_{t\to \infty} \bigl(S_{H}(t),I_{H}(t),T_{H}(t),R_{H}(t),n_{2}(t),N_{V}(t),Z(t) \bigr)= \bigl(S_{H}^{0},0,0,0,0,N_{V}^{0},Z^{0} \bigr), \end{aligned} $$

i.e.,

$$ \lim_{t\to \infty} \bigl\Vert \bigl(S_{H}(t),I_{H}(t),T_{H}(t),R_{H}(t),n_{2}(t),N_{V}(t),Z(t) \bigr)-E_{0} \bigr\Vert < \eta _{1}, $$

where $\eta _{1}$ is a sufficiently small positive number. Thus there exists a $t_{0}>0$ such that for $t>t_{0}$, we have

$$ \begin{aligned} &S_{H}^{0}-\eta _{1}< S_{H}(t)< S_{H}^{0}+\eta _{1},\qquad I_{H}(t)< \eta _{1},\qquad T_{H}(t)< \eta _{1}, \qquad R_{H}(t)< \eta _{1}, \\ &n_{2}(t)< \eta _{1}, \qquad N_{V}^{0}- \eta _{1}< N_{V}(t)< N_{V}^{0}+\eta _{1}, \qquad Z^{0}-\eta _{1}< Z(t)< Z^{0}+ \eta _{1}. \end{aligned} $$

Substituting this into model (3), when $t\ge t_{0}$, we obtain

$$ \textstyle\begin{cases}I'_{H}\ge \beta _{H} (S_{H}^{0}-\eta _{1} ) (N_{V}^{0}-\eta _{1} )n_{2}- \frac{\alpha}{1+\eta _{1}}I_{H}-(\mu _{H}+\delta )I_{H}, \\ T'_{H}\ge \frac{\alpha}{1+\eta _{1}}I_{H}- (r+\mu _{H} )T_{H}, \\ R'_{H}\ge rT_{H}-\mu _{H} R_{H}, \\ n'_{2}\ge \beta _{V}(1-\eta _{1})I_{H}- \frac{\Lambda _{V}}{N_{V}^{0}-\eta _{1}}n_{2}. \end{cases} $$

Further, we consider the following auxiliary system:

$$ \textstyle\begin{cases} \widetilde{I_{H}}'=\beta _{H}(S_{H}^{0}-\eta _{1})(N_{V}^{0}-\eta _{1}) \widetilde{n}_{2}-\frac{\alpha}{1+\eta _{1}}\widetilde{I}_{H}-(\mu _{H}+ \delta )\widetilde{I}_{H}, \\ \widetilde{T_{H}}'=\frac{\alpha}{1+\eta _{1}}\widetilde{I}_{H}- (r+ \mu _{H} )\widetilde{T}_{H}, \\ \widetilde{R_{H}}'=r\widetilde{T}_{H}-\mu _{H} \widetilde{R}_{H}, \\ \widetilde{n_{2}}'=\beta _{V}(1-\eta _{1})\widetilde{I}_{H}- \frac{\Lambda _{V}}{N_{V}^{0}-\eta _{1}}\widetilde{n}_{2}. \end{cases} $$

Letting

$$M({\eta }_1)=\left(\begin{array}{cccc}-\frac{\alpha }{1+{\eta }_1}-({\mu }_H+\delta )& 0& 0& {\beta }_H(S_H^0-{\eta }_1)(N_V^0-{\eta }_1)\\ \frac{\alpha }{1+{\eta }_1}& -(r+{\mu }_H)& 0& 0\\ 0& -r& -{\mu }_H& 0\\ {\beta }_V(1-{\eta }_1)& 0& 0& -\frac{{\mathrm{\Lambda }}_V}{N_V^0-{\eta }_1}\end{array}\right),$$

the auxiliary system can be rewritten as $U'=M(\eta _{1})U$, where

$$ U=(\widetilde{I}_{H},\widetilde{T}_{H}, \widetilde{R}_{H}, \widetilde {n}_{2}). $$

Let us assume that the eigenvalues of the eigenvalue equation corresponding to the matrix $M(\eta _{1})$ are denoted as $\xi _{i}$ ($i=1,2,3,4$). Specifically, $\xi _{1}=\mu _{H}$, $\xi _{2}=r+\mu _{H}$, and the remaining eigenvalues are determined by the following equation:

$$ \biggl[\xi +\frac{\alpha}{1+\eta _{1}}+ (\mu _{H}+\delta ) \biggr] \biggl( \xi +\frac{\Lambda _{V}}{N_{V}^{0}-\eta _{1}} \biggr)- \beta _{H}\beta _{V}(1- \eta _{1}) \bigl(S_{H}^{0}-\eta _{1} \bigr) \bigl(N_{V}^{0}- \eta _{1} \bigr)=0. $$

Take $R_{0}^{\eta}=\sqrt{ \frac{\beta _{H}\beta _{V}(1-\eta _{1}^{2})(S_{H}^{0}-\eta _{1})(N_{V}^{0}-\eta _{1})^{2}}{ [\alpha +(1+\eta _{1})(\mu _{H}+\delta ) ]\Lambda _{V}}}$. Based on the relationship between the eigenroots and the coefficients of the characteristic equation, it is clear that

$$ \begin{aligned} &\xi _{3}+\xi _{4}=- \biggl[ \frac{\alpha}{1+\eta _{1}}+ (\mu _{H}+\delta )+ \frac{\Lambda _{V}}{N_{V}^{0}-\eta _{1}} \biggr]< 0, \\ &\xi _{3}\xi _{4}= \bigl[1- \bigl(R_{0}^{\eta} \bigr)^{2} \bigr] \bigl[\alpha +(1+ \eta _{1}) (\mu _{H}+\delta ) \bigr]\Lambda _{V} \frac{\Lambda _{V}}{ (N_{V}^{0}-\eta _{1} ) (1+\eta _{1} )}. \end{aligned} $$

When $R_{0}>1$, there definitely exists a sufficiently small $\eta _{1}$ such that $R_{0}^{\eta}>1$. Thus, $\xi _{3}\xi _{4}<0$ if $R_{0}>1$. That is, one of the characteristic eigenvalues $\xi _{3}$ and $\xi _{4}$ is always positive.

Denote $s(M(\eta _{1}))=\max \{\operatorname{Re} \xi _{i} \vert \xi _{i} \text{ is the characteristic eigenvalue of } M(\eta _{1}),i=3,4\}$. Thus $s(M(\eta _{1}))>0$ when $R_{0}>1$. Notice that $s(M(\cdot ))$ is continuous, so for a sufficiently small $\eta _{1}>0$ there exists $s(M(\eta _{1}))>0$.

Denote by $(\Psi _{1},\Psi _{2},\Psi _{3},\Psi _{4})^{T}$ the strongly positive eigenvector corresponding to $s(M(\eta _{1}))$. Then

$$ (\widetilde{I}_{H},\widetilde{T}_{H},\widetilde{R}_{H}, \widetilde {n}_{2})=e^{s(M( \eta _{1}))t}(\Psi _{1},\Psi _{2},\Psi _{3},\Psi _{4})^{T} $$

is a solution of model (3).

Consequently, for any $I_{H}(t_{0})>0$, $T_{H}(t_{0})>0$, $R_{H}(t_{0})>0$, $n_{2}(t_{0})>0$, there exists an $\eta _{2}>0$ such that

$$ \bigl(I_{H}(t_{0}),T_{H}(t_{0}),R_{H}(t_{0}),n_{2}(t_{0}) \bigr)^{T}\ge \eta _{2} \bigl( \widetilde{I}_{H}(t_{0}), \widetilde{T}_{H}(t_{0}),\widetilde{R}_{H}(t_{0}), \widetilde{n}_{2}(t_{0}) \bigr)^{T}. $$

It further follows from the comparison principle that for any $t>t_{0}$, we have

$$ \bigl(I_{H}(t),T_{H}(t),R_{H}(t),n_{2}(t) \bigr)^{T}\ge \eta _{2}e^{s(M(\eta _{1}))t}( \Psi _{1},\Psi _{2},\Psi _{3},\Psi _{4})^{T}. $$

Since $s(M(\eta _{1}))>0$, we get that $I_{H}(t)$, $T_{H}(t)$, $R_{H}(t)$, and $n_{2}(t)$ are unbounded, which contradicts the fact that $I_{H}(t)$, $T_{H}(t)$, $R_{H}(t)$, and $n_{2}(t)$ are bounded. Therefore $W^{s}(E_{0})\cap X^{0}=\emptyset$, and from Lemma 3, model (3) is consistently persistent when $R_{0}>1$. □

4 Optimal control problem

In this section, in order to reduce the number of infected mosquitoes and control the spread of the disease, based on the previous analysis combined with the actual situation, we consider an optimal control problem of model (1).

The methods of mosquito-borne disease control mainly include preventing mosquito-borne virus infections (e.g., using mosquito nets, mosquito repellents, etc.), eliminating mosquitoes by spraying insecticides, and improving the rate of treatment for the population with drugs. We assume that in a more ideal and favorable ecological environment, human activities are appropriately allowed to disrupt biodiversity. However, we must control the extent of this disruption. Thus, we propose the following four control strategies:

• (1) Controlling mosquito–human transmission through mosquito nets and mosquito repellents, denoted by $u_{1}$.

• (2) Treatment through medicine, denoted by $u_{2}$.

• (3) Insecticide spraying to control mosquito populations, denoted by $u_{3}$.

• (4) Consideration of the impact of human activities on biodiversity, denoted by $u_{4}$.

Consequently, the optimal control model with time delay is given by the following system of equations:

$$ \textstyle\begin{cases}\frac{dS_{H}}{dt}=\Lambda _{H}-\beta _{H}(1-u_{1}) S_{H}(t-\tau )I_{V}(t- \tau )-\mu _{H}S_{H}, \\ \frac{dI_{H}}{dt}=\beta _{H}(1-u_{1})S_{H}(t- \tau )I_{V}(t-\tau )- \frac{(\alpha _{0} u_{2}+\alpha )I_{H}}{1+I_{H}}-(\mu _{H}+\delta )I_{H}, \\ \frac{dT_{H}}{dt}=\frac{(\alpha _{0} u_{2}+\alpha )I_{H}}{1+I_{H}}-(r+ \mu _{H})T_{H}, \\ \frac{dR_{H}}{dt}=r T_{H}-\mu _{H}R_{H}, \\ \frac{dS_{V}}{dt}=\Lambda _{V}-\beta _{V}(1-u_{1}) S_{V} I_{H}-( \mu _{V}+c_{0}u_{3})S_{V}- \frac{S_{V}}{K} ((S_{V}+I_{V})+qZ ), \\ \frac{dI_{V}}{dt}=\beta _{V}(1-u_{1}) S_{V} I_{H}-(\mu _{V}+c_{0}u_{3})I_{V}- \frac{I_{V}}{K} ((S_{V}+I_{V})+qZ ), \\ \frac{dZ}{dt}=a(1-u_{4})Z-\frac{Z}{k} (Z+\varepsilon (S_{V}+I_{V}) ). \end{cases} $$

(27)

Define the following objective functional:

$$ J(u_{1},u_{2},u_{3},u_{4})= \int _{0}^{T} \Biggl[\omega _{I_{H}}I_{H}- \omega _{T_{H}}T_{H}-\omega _{I_{V}}I_{V}+ \sum_{i=1}^{4} \frac{W_{i}}{2}{u_{i}}^{2} \Biggr]\,dt, $$

(28)

where $\omega _{I_{H}}$, $\omega _{T_{H}}$, $\omega _{I_{V}}$, $W_{1}$, $W_{2}$, $W_{3}$, and $W_{4}$ represent the weight coefficients of the control variables, which are designed to maintain a balance among the items of the integration function so that no dominant individual term emerges; T is the terminal moment when the control policy is implemented. The Lagrange function is

$$ \begin{aligned} L(I_{H},T_{H},I_{V},u_{1},u_{2},u_{3},u_{4})= \omega _{I_{H}}I_{H}- \omega _{T_{H}}T_{H}- \omega _{I_{V}}I_{V}+\sum_{i=1}^{4} \frac{W_{i}}{2}{u_{i}}^{2}, \end{aligned} $$

and the control parameters $u_{1}$, $u_{2}$, $u_{3}$, $u_{4}$ are taken from the control set $U_{ad}= \{(u_{1},u_{2}, u_{3},u_{4})\in L(0,T)|0\leq u_{i}(t) \leq 1,i=1,2,3,4 \}$. Note that the right-hand side of model (28) is bounded and the objective function is convex, so there exists $(u_{1}^{*},u_{2}^{*},u_{3}^{*},u_{4}^{*})$ satisfying $J(u_{1}^{*},u_{2}^{*},u_{3}^{*},u_{4}^{*})=\min \{J(u_{1},u_{2},u_{3},u_{4}):(u_{1},u_{2},u_{3},u_{4}) \in U_{ad}\}$.

Define the Hamiltonian function as

$$\begin{aligned} H =&\omega _{I_{H}}I_{H}-\omega _{T_{H}}T_{H}-\omega _{I_{V}}I_{V}+ \frac{W_{1}}{2}{u_{1}}^{2}+\frac{W_{2}}{2}{u_{2}}^{2}+ \frac{W_{3}}{2}{u_{3}}^{2}+\frac{W_{4}}{2}{u_{4}}^{2} \\ &{} +\lambda _{S_{H}} \bigl[\Lambda _{H}-\beta _{H}(1-u_{1}) S_{H}(t- \tau )I_{V}(t- \tau )-\mu _{H}S_{H} \bigr] \\ &{} +\lambda _{I_{H}} \biggl[\beta _{H}(1-u_{1})S_{H}(t- \tau )I_{V}(t- \tau )-\frac{(\alpha _{0} u_{2}+\alpha )I_{H}}{1+I_{H}}-(\mu _{H}+ \delta )I_{H} \biggr] \\ &{} +\lambda _{T_{H}} \biggl[ \frac{(\alpha _{0} u_{2}+\alpha )I_{H}}{1+I_{H}}-(r+\mu _{H})T_{H} \biggr]+\lambda _{R_{H}} [r T_{H}-\mu _{H}R_{H} ] \\ &{} +\lambda _{S_{V}} \biggl[\Lambda _{V}-\beta _{V}(1-u_{1}) S_{V} I_{H}-( \mu _{V}+c_{0}u_{3})S_{V}- \frac{S_{V}}{K} \bigl((S_{V}+I_{V})+qZ \bigr) \biggr] \\ &{} +\lambda _{I_{V}} \biggl[\beta _{V}(1-u_{1}) S_{V} I_{H}-(\mu _{V}+c_{0}u_{3})I_{V}- \frac{I_{V}}{K} \bigl((S_{V}+I_{V})+qZ \bigr) \biggr] \\ &{} +\lambda _{Z} \biggl[a(1-u_{4})Z-\frac{Z}{k} \bigl(Z+\varepsilon (S_{V}+I_{V}) \bigr) \biggr]. \end{aligned}$$

The function $\chi _{[0,T-\tau ]}(t)$ represents an indicator function and has the form

$$ \chi _{[0,T-\tau ]}(t)= \textstyle\begin{cases} 1,&x\in [0,T-\tau ], \\ 0,&x\notin [0,T-\tau ]. \end{cases} $$

According to the Pontryagin maximum principle [31], the control set $u(t)$ should satisfy the following necessary conditions:

$$ \begin{aligned} &\lambda _{S_{H}}'= \chi _{[0,T-\tau ]}(t) \bigl[\lambda _{S_{H}}(t+\tau )- \lambda _{I_{H}}(t+\tau ) \bigr]\beta _{H}(1-u_{1})I_{V}+ \lambda _{S_{H}} \mu _{H}, \\ &\lambda _{I_{H}}'=-\omega _{I_{H}}+(\mu _{H}+\delta )\lambda _{I_{H}}+ \frac{\alpha _{0} u_{2}+\alpha}{(1+I_{H})^{2}}(\lambda _{I_{H}}- \lambda _{T_{H}})+\beta _{V}(1-u_{1}) S_{V}(\lambda _{S_{V}}-\lambda _{I_{V}}), \\ &\lambda _{T_{H}}'=\omega _{T_{H}}+r(\lambda _{T_{H}}-\lambda _{R_{H}})+ \mu _{H}\lambda _{T_{H}}, \\ &\lambda _{R_{H}}'=\mu _{H}\lambda _{R_{H}}, \\ &\lambda _{S_{V}}'=\beta _{V}(1-u_{1})I_{H}( \lambda _{S_{V}}-\lambda _{I_{V}})+( \mu _{V}+c_{0}u_{3}) \lambda _{S_{V}} \\ &\hphantom{\lambda _{S_{V}}'=}{} +\frac{1}{K} \bigl[(2S_{V}+I_{V}+qZ) \lambda _{S_{V}}+I_{V}\lambda _{I_{V}} \bigr]+ \frac{\varepsilon Z}{k}\lambda _{Z}, \\ &\lambda _{I_{V}}'=\omega _{I_{V}}+\chi _{[0,T-\tau ]}(t) \bigl[\lambda _{S_{H}}(t+ \tau )-\lambda _{I_{H}}(t+\tau ) \bigr]\beta _{H}(1-u_{1})S_{H} \\ &\hphantom{\lambda _{I_{V}}'=}{} +(\mu _{V}+c_{0}u_{3}) \lambda _{I_{V}}+\frac{1}{K} \bigl[(2I_{V}+S_{V}+qZ) \lambda _{I_{V}}+S_{V}\lambda _{S_{V}} \bigr]+ \frac{\varepsilon Z}{k} \lambda _{Z}, \\ &\lambda _{Z}'=-a(1-u_{4})\lambda _{Z}+\frac{1}{k} \bigl[2Z+\varepsilon (S_{V}+I_{V}) \bigr] \lambda _{Z}+\frac{1}{K}[qI_{V}\lambda _{I_{V}}+qS_{V}\lambda _{S_{V}}]. \end{aligned} $$

(29)

Theorem 9

There is a set of $u_{1}$, $u_{2}$, $u_{3}$, $u_{4}$ such that $J(u_{1}^{*},u_{2}^{*},u_{3}^{*})=\min J(u_{1},u_{2},u_{3})$. The optimal controls are as follows:

$$ \begin{aligned} &u_{1}^{*}=\max \bigl\{ \min \{u_{1},1 \},0 \bigr\} , \\ &u_{2}^{*}=\max \bigl\{ \min \{u_{2},1 \},0 \bigr\} , \\ &u_{3}^{*}=\max \bigl\{ \min \{u_{3},1 \},0 \bigr\} , \\ &u_{4}^{*}=\max \bigl\{ \min \{u_{4},1 \},0 \bigr\} . \end{aligned} $$

(30)

where

$$ \begin{aligned} &u_{1}= \frac{[\beta _{H}(1-u_{1})S_{H}(t-\tau )I_{V}(t-\tau )(\lambda _{I_{H}}-\lambda _{S_{H}}) +\beta _{V}(1-u_{1})I_{H}S_{V}(\lambda _{I_{V}}-\lambda _{S_{V}})]}{W_{1}}, \\ &u_{2}= \frac{\alpha _{0}I_{H}(\lambda _{I_{H}}-\lambda _{T_{H}})}{(1+I_{H})W_{2}}, \qquad u_{3}= \frac{c_{0}S_{V}\lambda _{S_{V}}+c_{0}I_{V}\lambda _{I_{V}}}{W_{3}}, \qquad u_{4}=\frac{aZ\lambda _{Z}}{W_{4}}. \end{aligned} $$

Proof

According to the extremum condition of the control equation, we can derive

$$\begin{aligned}& \frac{\partial H}{\partial u_{1}} =W_{1}u_{1}+ \beta _{H}S_{H}(t- \tau )I_{V}(t-\tau ) (\lambda _{S_{H}}-\lambda _{I_{H}}) \\& \hphantom{\frac{\partial H}{\partial u_{1}} =}{} +\beta _{V}(1-u_{1})I_{H}S_{V}( \lambda _{S_{V}}-\lambda _{I_{V}})=0, \\& \frac{\partial H}{\partial u_{2}} =W_{2}u_{2}+ \frac{\alpha _{0}I_{H}(\lambda _{T_{H}}-\lambda _{I_{H}})}{(1+I_{H})}=0, \\& \frac{\partial H}{\partial u_{3}} =W_{3}u_{3}-c_{0}S_{V} \lambda _{S_{V}}-c_{0}I_{V} \lambda _{I_{V}}=0, \\& \frac{\partial H}{\partial u_{4}} =W_{4}u_{4}-aZ\lambda _{Z}=0. \end{aligned}$$

By solving the above equation, we have

$$ \begin{aligned} u_{1}&=\frac{1}{W_{1}} \bigl[\beta _{H}(1-u_{1})S_{H}(t-\tau )I_{V}(t- \tau ) (\lambda _{I_{H}}-\lambda _{S_{H}}) +\beta _{V}(1-u_{1})I_{H}S_{V}( \lambda _{I_{V}}-\lambda _{S_{V}}) \bigr], \\ u_{2}&= \frac{\alpha _{0}I_{H}(\lambda _{I_{H}}-\lambda _{T_{H}})}{(1+I_{H})W_{2}}, \qquad u_{3}= \frac{c_{0}S_{V}\lambda _{S_{V}}+c_{0}I_{V}\lambda _{I_{V}}}{W_{3}}, \qquad u_{4}=\frac{aZ\lambda _{Z}}{W_{4}}. \end{aligned} $$

Therefore, the optimal control solution can be expressed as Eq. (30). □

5 Numerical simulation and discussions

In this section, we present numerical simulations, with Table 2 providing the values of certain fixed parameters.

Table 2 Constant parameter values in numerical simulations of model (1)

5.1 The effect of τ on the dynamic behavior of model (1)

Assume $\Lambda _{H}=0.2$, $\Lambda _{V}=0.5$, $\mu _{V}=0.19$, $\alpha =0.35$, $K=40$, $k=30$, $a=0.2$, $q=1$, and $\varepsilon =0.2$. Under these conditions, $R_{0}=27.6005>1$, leading to the emergence of a unique positive equilibrium $E^{*}=(3,1.33,0.39,3986,1.03,0.33,5.73)$. Additional numerical calculations reveal the critical time delay $\tau _{0}=5.5955$. It is known that the positive equilibrium $E^{*}$ is locally asymptotically stable for $0<\tau <\tau _{0}$ and unstable when $\tau >\tau _{0}$.

Figure 2 shows the solution curves and the phase planes when $\tau <\tau _{0}$. No matter which set of initial value conditions is considered, the trajectory of the solution of model (1) converges to $(3,1.32,0.39,1.03,0.33,5.73)$, indicating that the endemic equilibrium $E^{*}$ is locally asymptotically stable if $\tau =4<\tau _{0}$.

Figure 2

Select $\Lambda _{H}=0.2$, $\Lambda _{V}=0.5$, $\mu _{V}=0.19$, $\alpha =0.35$, $K=40$, $k=30$, $a=0.2$, $q=1$, $\varepsilon =0.2$, and $\tau =4$. (a) When the initial value is $[2,1.5,0.3,1,0.2,3,1.2]$, the positive equilibrium $E^{*}$ is asymptotically stable. (b) The initial values chosen for the blue line is $[2,1.5,0.3,1,0.2,3,1.2]$, and the initial values for the red line is $[1,0.5,0.3,1.2,0.8,3,2]$

Figure 3 shows the solution curves and phase diagrams when $\tau >\tau _{0}$. We can observe that the endemic equilibrium $E^{*}$ is unstable when $\tau =7.5>\tau _{0}$. Figure 4 presents a bifurcation diagram of the model with varying time delays.

Figure 3

Let $\Lambda _{H}=0.2$, $\Lambda _{V}=0.5$, $\mu _{V}=0.19$, $\alpha =0.35$, $K=40$, $k=30$, $a=0.2$, $q=1$, $\varepsilon =0.2$, and $\tau =7.5$. (a) The positive equilibrium $E^{*}$ is unstable for the initial value $[2,1.5,0.3,1,0.2,3,1.2]$. (b) The corresponding initial values of blue line and red line are $[2,1.5,0.3,1,0.2,3,1.2]$ and $[1,0.5,0.3,1.2,0.8,3,2]$, respectively

Figure 4

Set $\Lambda _{H}=0.2$, $\Lambda _{V}=0.5$, $\mu _{V}=0.19$, $\alpha =0.35$, $K=40$, $k=30$, $a=0.2$, $q=1$, $\varepsilon =0.2$, and let the initial value be $[2,1.5,0.3,1,0.2,3,1.2]$. Hopf bifurcation diagram of model (1)

5.2 The impact of saturation treatment rate

The critical parameter representing the saturation treatment rate is α, symbolizing the maximum treatment rate. From the analysis in the preceding sections, it is evident that the most significant impact of saturation treatment is the emergence of backward bifurcations in the model. Consequently, when $R_{0}<1$ and $\alpha <\alpha ^{*}$, the model may exhibit four stable states. As shown in Fig. 5, apart from two disease-free equilibria, two local endemic equilibrium points also emerge, all of which are locally asymptotically stable. Furthermore, in such a scenario, the four equilibrium states depend on the choice of initial conditions. When $R_{0}<1$, $\alpha >\alpha ^{*}$, only two disease-free equilibria remain (see Fig. 6). This highlights the significance of $\alpha ^{*}$, which represents a threshold value for the maximum treatment rate. If medical resources are limited, the virus may continue to spread even when $R_{0}<1$.

Figure 5

Taking $\Lambda _{H}=2.5$, $\Lambda _{V}=0.5$, $\beta _{V}=0.006$, $\mu _{H}=0.09$, $\mu _{V}=0.07$, we get $\alpha ^{*}=0.9285$ and $\alpha <\alpha ^{*}$, $R_{0}<1$. (a) When the initial value is $[1,0.5,0.3,1.1,0.2,1.2,2]$, the number of infected people ($I_{H}$ converges to 6.1) and the aliens (Z converges to 3.2) eventually are not 0; (b) When the initial value is $[0.05,0.02,0.02,0.1,0.05,0.15,0.2]$, the aliens face extinction and the number of infected people $(I_{H})$ eventually tends to 19.5; (c) When the initial value is $[3,2,2,5,2,5,1]$, the number of infected persons $(I_{H})$ eventually tends to 0, but the aliens will survive; (d) When the initial value is $[0.04,0.02,0.02,0.05,0.02,0.08,0.07]$, the populations of aliens $(Z)$ and infected $(I_{H})$ ultimately converge to 0

Figure 6

Taking $\Lambda _{H}=2.5$, $\Lambda _{V}=0.5$, $\beta _{V}=0.006$, $\mu _{H}=0.09$, $\mu _{V}=0.07$, we get $\alpha ^{*}=0.9285$ and $\alpha >\alpha ^{*}$, $R_{0}<1$. (a) The initial value is $[1,0.5,0.3,1.1,0.2,1.2,2]$; (b) The initial value is $[0.04,0.02,0.02,0.05,0.02,0.08,0.07]$

5.3 Optimal control results and discussion

In this part, we use the forward–backward Runge–Kutta method [33] and from a numerical point of view examine the effects of optimal control strategies. Under the assumption that this is a desirable, well-developed ecosystem, the value of a will be taken to avoid areas where aliens could become extinct. We take $\Lambda _{H}=1$, $\Lambda _{V}=0.5$, $\mu _{V}=0.21$, $r=0.1$, $\alpha =0.5$, $K=70$, $k=30$, $q=3$, $\varepsilon =0.1$, $a=0.2$, $\tau =1$; more parameter values are presented in Table 2. The weight functions used in the simulation of the control problem are $\omega _{IH}=20$, $\omega _{TH}=30$, and $\omega _{IV}=5$. Also, for the control factors, the following weights are utilized: $W_{1}=3$, $W_{2}=4$, $W_{3}=2$, $W_{4}=1$. The initial values are also set as $S_{H0}=1$, $I_{H0}=0.3$, $T_{H0}=0.1$, $R_{H0}=0.1$, $S_{V0}=1$, $I_{V0}=0.1$, and $Z_{0}=0.1$.

Figure 7(a)–(c) demonstrates the population size changes of $I_{H}$, $I_{V}$, Z with and without optimal control. By comparison, it can be seen that under optimal control, the number of infected humans decreased rapidly; also the number of infected mosquitoes decreased significantly. The presence of aliens has obviously decreased in the current context of excellent ecological diversity and minimal allowable disruption control. Figure 7(d) illustrates the change process of the optimal control over time. As it can be seen from the chart, there is a significant complementary effect between medical inputs and the dilution effect of biodiversity. In other words, maintaining good biodiversity can reduce the demand for medical resources for mosquito-borne disease control. Next, we aim to compare several control strategies as the following descriptions:

Figure 7

Set $\Lambda _{H}=1$, $\Lambda _{V}=0.5$, $\mu _{V}=0.21$, $r=0.1$, $\alpha =0.5$, $K=70$, $k=30$, $q=3$, $\varepsilon =0.1$, $a=0.2$, and $\tau =1$. (a)–(c) Populations $I_{H}$, $I_{V}$, Z with and without optimal controls. (d) Optimal control strategy $u_{1}$, $u_{2}$, $u_{3}$, $u_{4}$

Strategy A: Imposing all measures $(u_{1}(t)\neq 0,u_{2}(t)\neq 0,u_{3}(t)\neq 0,u_{4}(t)\neq 0)$.

Strategy B: Improving treatment, using insecticide and measures that control the degree of damage to biodiversity $(u_{1}(t)=0,u_{2}(t)\neq 0,u_{3}(t)\neq 0,u_{4}(t)\neq 0)$.

Strategy C: Using mosquito nets, spraying insecticide, and making use of measures that control the degree of damage to biodiversity $(u_{1}(t)\neq 0,u_{2}(t)=0,u_{3}(t)\neq 0,u_{4}(t)\neq 0)$.

Strategy D: Using mosquito nets, improving treatment, and utilizing measures that control the degree of damage to biodiversity $(u_{1}(t)\neq 0,u_{2}(t)\neq 0,u_{3}(t)=0,u_{4}(t)\neq 0)$.

Strategy E: Using mosquito nets, improving treatment, and spraying insecticide $(u_{1}(t)\neq 0,u_{2}(t)\neq 0,u_{3}(t)\neq 0,u_{4}(t)=0)$.

In Fig. 8, it can be observed that Strategy E is the most efficient and effective, indicating that maintaining the current good ecological environment, along with employing the other three strategies (reducing mosquito bites, promoting healthcare, and spraying insecticides), can effectively control the spread of the virus. In cases where a reduction in biodiversity is unavoidable, comparing the other three measures:

• (1) The combined use of strategies performs relatively well, appropriately delaying the peak (see Strategy A).

• (2) Due to a higher cost of healthcare compared to insecticide usage, Strategy C achieves faster results when medical investment is not taken into account. However, behind Strategy C lies an increased reliance on insecticides.

• (3) Achieving control goals without using insecticides is also possible, with slightly higher peak in infection numbers but a significant increase in medical investment (see Strategy D).

• (4) Reducing mosquito bites is crucial – disregarding this measure results in the poorest control effectiveness (see Strategy B).

Figure 8

Select $\Lambda _{H}=1$, $\Lambda _{V}=0.5$, $\mu _{V}=0.21$, $r=0.1$, $\alpha =0.5$, $K=70$, $k=30$, $q=3$, $\varepsilon =0.1$, $a=0.2$, and $\tau =1$. Change of the infected people $I_{H}$ with five different control strategies

On the other hand, as shown in Fig. 9, by increasing the time delay, the nodes of the control measures shift backward, extending the duration until the number of infections reaches zero. And from Fig. 10, for a smaller delay, $I_{H}$ and $I_{V}$ achieve stability in a shorter time.

Figure 9

Let $\Lambda _{H}=1$, $\Lambda _{V}=0.5$, $\mu _{V}=0.21$, $r=0.1$, $\alpha =0.5$, $K=70$, $k=30$, $q=3$, $\varepsilon =0.1$, and $a=0.2$. The optimal control corresponding to $\tau =1$, $\tau =5$, and $\tau =9$, respectively

Figure 10

Take $\Lambda _{H}=1$, $\Lambda _{V}=0.5$, $\mu _{V}=0.21$, $r=0.1$, $\alpha =0.5$, $K=70$, $k=30$, $q=3$, $\varepsilon =0.1$, and $a=0.2$. The populations of infected persons and infected mosquitoes with optimal controls when $\tau =5$, $\tau =7$, and $\tau =9$, respectively

6 Conclusion

In our study, a mosquito-borne infectious disease model with time delay, saturated treatment function, and biodiversity is considered. We investigate the effect of the time delay on the dynamical behavior of our model: there exists a critical time delay $\tau _{0}$ such that the endemic equilibrium is locally and asymptotically stable if $\tau <\tau _{0}$ and unstable if $\tau >\tau _{0}$. We also find that there is a Hopf bifurcation as τ increases, implying that the larger the τ, the greater the risk of mosquito-borne disease occurrence. Numerical simulations and characterization analysis have revealed that the augmentation of the natural growth rate of aliens can mitigate the risk of mosquito-borne infections to some extent. Moreover, the competition between aliens and mosquitoes can regulate the overall vector population under specific circumstances.

We investigated the optimal control of mosquito-borne infectious diseases in the context of good ecological diversity, and the results reflect a significant impact of biodiversity on optimal control. Maintaining high biodiversity levels reduces the need for medical resources and decreases reliance on insecticides. If the dilution effect of biodiversity is fully utilized in the control of mosquito-borne diseases, simulations of optimal control strategies show that it is possible to effectively reduce the intensity of other measures and achieve better control outcomes.

In addition, our model describes the transmission dynamics of mosquito-borne infectious diseases, which include but are not limited to dengue fever, malaria, Japanese encephalitis, Zika virus, and so on. The growth, development, and bites of vector mosquitoes can be influenced by climatic factors like temperature and precipitation in mosquito-borne disease transmission. Seasonal variations have a significant impact on the dynamic behavior of mosquitoes and the transmission of mosquito-borne diseases, which we will explore in the next study.