Modeling and analysis of the transmission dynamics of cystic echinococcosis: Effects of increasing the number of sheep

1.   Introduction

Cystic echinococcosis (CE) is a worldwide parasitic disease produced by echinococcus granulosus. In the life cycle of echinococcosis granulosus (Figure 1), dogs are the primary definitive host, and livestock (such as sheep, goats and swine) constitute the major intermediate host. In usual, CE mainly transmits between dogs and livestock. Humans as accidental hosts of CE, infected by eggs in the environment but not participating in the spread of the disease. This disease mainly occurs in animal husbandry countries and causes about USD 2 billion dollars of economic losses to the livestock industry every year ^[1,2]. The World Health Organization recognizes it as an ignored tropical disease and lists it as a priority disease ^[3,4].

It is a crucial global public health goal to prevent and control the spread of CE ^[5]. Mathematical modeling is a powerful tool to solve such problems. A great quantity of mathematical models on CE have been developed and analyzed ^[6,7,8]. Roberts el al. ^[9,10] and Gemmell et al. ^[11,12,13] explored a mathematical model of the life cycle of CE in dogs and sheep to discuss previously published experimental and survey data. Considering that CE cannot spread without eggs in the environment, Wang et al. ^[14] constructed a new deterministic model incorporating dogs, sheep, humans and eggs in the environment, and they used it to examine the transmission dynamics of echinococcosis in Xinjiang. Recently, Rong et al. ^[15,16] improved the compartmental model presented in ^[14] and used the new model to investigate the effect of stray dogs and searched the optimal control measures. Zhao and Yang^[17] considered an echinococcosis model that contains four control measures and discussed the optimal control problem to decrease and eliminate the spread of echinococcosis between dogs and livestock with comprehensive interventions. Cui ^[18] analyzed the prevention and control effects of CE in Pengyang County, predicted the epidemic trend of CE in Pengyang County and evaluated the impact of prevention and control measures on the spread of the disease. For more details, the reader can refer to ^[19,20,21].

However, these studies disregard the effect of the number of sheep on the prevention and control of echinococcosis. In fact, the demand for sheep is increasing as the population increases. For example, the production of mutton in China increased from 4.45 million tons in 2012 to 5.14 million tons in 2021, and more than 146.2 million sheep were slaughtered in 2017, which is almost 20 million more than six years ago (http://ncpscxx.moa.gov.cn/). Moreover, sheep, as the main intermediate host, represent the key link in the spread of echinococcosis, and they are also the main income of many farmers and herders. Specifically, the importance of considering the effects of the prevalence of echinococcus in sheep was once mentioned by Yang et al. ^[22]. They studied the prevalence of CE in slaughtered sheep in Emin County, Xinjiang, China. In addition, Xiao et al. ^[23] investigated the epidemiology of sheep echinococcosis in Kashi, Xinjiang during 2014–2017. Recently, Gao et al. ^[24] investigated the epidemic situation of sheep echinococcus in China from 1983 to 2020 by using meta-analysis. These studies focused on the epidemiological survey of sheep echinococcosis, and there is no clear qualitative analysis of the effect of sheep on the spread of CE.

Thus, in this paper, we attempt to elucidate the impact of increasing sheep on the spread of CE by constructing and analyzing a mathematical model. The remainder of this paper is organized as follows. Section 2 presents the development of a CE model to highlight the logistic growth of sheep in reality. The basic reproduction number and the dynamical behaviors of the model are presented in Section 3. In the next section, the effects of sheep and other control measures are investigated through numerical simulations. A brief conclusion and discussion are presented in the last section.

2.   Model formulation

In this section, a dynamical model based on the transmission of CE among animals is formulated. Based on the idea of compartmental modeling, the definitive host dogs are divided into susceptible dogs and infected dogs, and the corresponding numbers at time $t$ are respectively denoted by ${{S}_{D}}\left(t \right)$ and ${{I}_{D}}\left(t \right)$; the intermediate host sheep is divided into susceptible, immune and infected individuals, and the corresponding numbers at time $t$ are recorded as ${{S}_{L}}\left(t \right)$, ${{V}_{L}}\left(t \right)$ and ${{I}_{L}}\left(t \right)$, respectively; and the number of parasite eggs in the environment is denoted by $E\left(t \right)$ at time t. Let $N_L(t) = S_L(t)+V_L(t)+I_L(t)$ be the total number of intermediate host sheep that satisfies

where $b_2$ represents the natural birth rate of sheep, $d_2$ denotes the natural death rate of sheep, $K$ represents the maximum environmental capacity of sheep and $\theta$ is the slaughter rate of sheep. To model $N_L(t)$ as increasing in reality, we assume that ${{b}_{2}}\ge {{d}_{2}}+\theta$. Based on this compartmental scheme (Figure 2), a dog-sheep-egg life cycle transmission dynamics model can be given as

For the definitive host dogs, parameter ${{A}_{1}}$ represents the number of dogs born per year, ${{d}_{1}}$ is the natural mortality rate of dogs, $\sigma$ is the deworming recovery rate for infectious dogs, $\beta_1$ is the dogs' transmission rate and ${{\beta }_{1}}{{S}_{D}}{{I}_{L}}$ represents the transmission of echinococcosis to dogs by ingesting cyst-containing organs of infectious sheep. For the intermediate host sheep, $\omega$ represents the vaccination rate of sheep, $\delta$ is the immune failure rate of sheep, $\beta_2$ is the transmission rate of sheep and ${{\beta }_{2}}{{S}_{L}}E$ denotes the transmission of echinococcosis to sheep by ingesting parasite eggs in the environment. Moreover, $\alpha$ represents the release rate of parasite eggs in the environment, and ${{d}_{e}}$ denotes the natural extinction rate of parasite eggs.

It should be noted that the release rate of parasite eggs in the environment, $\alpha$, is determined by the average annual amount of parasites in each dog, $q$, the number of eggs laid per adult per unit time, ${{b}_{e}}$, the average life span of a dog, ${{t}_{d}}$, and adult mortality in dogs, $d$. It follows from literature ^[6] that $\alpha = {{b}_{e}}q\left(1-{{e}^{-d{{t}_{d}}}} \right)/d$.

Based on the biological meaning of the model, all solutions of the system (2.2) are assumed to satisfy the following initial conditions:

Define

We have the following result.

Theorem 1. All of the solutions satisfying the initial conditions of the system (2.2) are positive; $\Gamma$ is the positively invariant set of system (2.2).

Proof. Let $x\left(t \right) = \left({{S}_{D}}\left(t \right), {{I}_{D}}\left(t \right), {{S}_{L}}\left(t \right), {{V}_{L}}\left(t \right), {{I}_{L}}\left(t \right), E\left(t \right) \right)$ be the solution of the system (2.2) satisfying the initial conditions given by (2.3). We first prove that $x\left(t \right)$ is the positive solution. Assume that the conclusion is not true; then, there exists a constant ${{t}_{1}} > 0$ such that

and

If ${{x}^{0}}\left({{t}_{1}} \right) = {{I}_{D}}\left({{t}_{1}} \right),$ by the second equation of system (2.2), we have

One can easily calculate that

This is inconsistent with ${{x}^{0}}\left({{t}_{1}} \right) = {{I}_{D}}\left({{t}_{1}} \right).$

If ${{x}^{0}}\left({{t}_{1}} \right) = E\left({{t}_{1}} \right),$ it can be known from the last equation of system (2.2) that

Then, we have

This is inconsistent with ${{x}^{0}}\left({{t}_{1}} \right) = E\left({{t}_{1}} \right).$

Similarly results can be obtained when ${{S}_{D}^{0}}\left({{t}_{1}} \right) = {{S}_{D}}\left({{t}_{1}} \right), {{S}_{L}^{0}}\left({{t}_{1}} \right) = {{S}_{L}}\left({{t}_{1}} \right), {{I}_{L}^{0}}\left({{t}_{1}} \right) = {{I}_{L}}\left({{t}_{1}} \right)$ and ${{V}_{L}^{0}}\left({{t}_{1}} \right) = {{V}_{L}}\left({{t}_{1}} \right)$. Therefore, all of the solutions satisfying the initial conditions given by (2.3) are positive solutions for system (2.2).

Next, let us prove that $\Gamma$ is the positively invariant set of the system (2.2). From the first two equations of system (2.2), we have

Simple calculation implies that

Similarly, we have

Therefore, $\Gamma$ is the positively invariant set of system (2.2).

3.   Mathematical analysis

In this section, we first investigate the existence of the disease-free equilibrium and endemic equilibrium and then discuss the global dynamics of system (2.2).

3.1.   Equilibrium and basic reproduction number

Set the right-hand side of system (2.2) as zero; then, we have the following algebraic equation

In the absence of disease, it follows from Eq (3.1) that system (2.2) always has a disease-free equilibrium ${{E}_{dfe}} = \left(S_{D}^{0}, 0, S_{L}^{0}, V_{L}^{0}, 0, 0 \right)$, where

According to the next-generation operator approach proposed by Diekmann et al. ^[25] and van den Driessche and Watmough ^[26], we define

Note that the disease-free equilibrium of system (2.2) is ${{E}_{dfe}}$; then,

Hence, the next generation matrix reads as

Then the basic reproduction number, which is calculated from $\rho \left(F{{V}^{-1}} \right)$, is as follows:

Here, ${{R}_{0}}$ represents the average number of new infections produced by infectious dogs during their respective infection period ^[26,27]. In the non-negative neighborhood of the disease-free equilibrium, the density of echinococcus eggs released by each infectious dog is ${\alpha }/{{{d}_{e}}}.$ In the dogs' expected infectious period ${1}/{\left({{d}_{1}}+\sigma \right)},$ the susceptible sheep ${\left[h\left({{b}_{2}}-{{d}_{2}}-\theta \right)K \right]}/{\left({{b}_{2}}-{{d}_{2}} \right)}$ are infected by contacting parasites eggs with the probability ${{\beta }_{2}}$. Meanwhile, in the sheep expected infectious period ${1}/{{{b}_{2}}}$, the total dogs (number: ${{{A}_{1}}}/{{{d}_{1}}}$) are infected by ingesting infectious cyst-containing organs of sheep with the probability ${{\beta }_{1}}$.

If there exist infected dogs or sheep, it follows from Eq (3.1) that system (2.2) has an endemic equilibrium $E_{e}^{*} = \left(S_{D}^{*}, I_{D}^{*}, S_{L}^{*}, V_{L}^{*}, I_{L}^{*}, {{E}^{*}} \right)$ when ${{R}_{0}} > 1$, where

3.2.   Global dynamical behaviors

Based on Theorem 2 presented by van den Driessche and Watmough ^[26], when $R_0 < 1$, the disease-free equilibrium $E_{dfe}$ of system (2.2) is locally asymptotically stable, and unstable when $R_0 > 1$. The global asymptotic stability of the disease-free equilibrium ${{E}_{dfe}}$ can be given by performing a direct Lyapunov approach.

Theorem 2. If ${{R}_{0}}\le 1$, the disease-free equilibrium ${{E}_{dfe}}$ of system (2.2) is globally asymptotically stable.

Proof. Let $\left({{S}_{D}}\left(t \right), {{I}_{D}}\left(t \right), {{S}_{L}}\left(t \right), {{V}_{L}}\left(t \right), {{I}_{L}}\left(t \right), E\left(t \right) \right)$ be any solution of the system (2.2) that satisfies

for $\forall t\ge {{t}_{0}}, \; \text{where} \; \exists {{t}_{0}} > N, \; N\in {{\mathbb{R}}^{+}}.$

Define

By applying its derivative along the solutions of system (2.2) and using the expression of $R_0$, we calculate that

Therefore, ${dV}/{dt}\le 0$ if $R_0\le 1$ and ${dV}/{dt}\; = 0$ if and only if ${{I}_{L}} = 0.$ It is not hard to verify that $\left(S_{D}^{0}, 0, S_{L}^{0}, V_{L}^{0}, 0, 0 \right)$ is the unique invariant set of system (2.2). According to Lasalle's invariant principle ^[28], the disease-free equilibrium $E_{dfe}$ is globally asymptotically stable.

The proof of Theorem 1 implies that system (2.1) has a unique positive equilibrium $N_{L}^{0}$ and one can easily verify that $N_{L}^{0}$ is globally asymptotically stable in $\Gamma$, that is,

Using the results from Castillo-Chavez and Thieme ^[29] and Mischaikow et al. ^[30], we can obtain the analytical results by considering the following limit system of system (2.2) in which the sheep population is assumed to be constant $N_{L}^{0}$:

Theorem 3. If ${{R}_{0}} > 1,$ the endemic equilibrium $E_{e}^{*}$ is globally asymptotically stable.

Proof. Let $g\left(x \right) = x-1-\ln x, x > 0.$ Obviously, $g\left(x \right)\ge 0$, if and only if $x = 1$, $g\left(x \right) = 0$. Note that $\left(x-1 \right)\left(1-y \right) = g\left(x \right)+g\left(y \right)-g\left(xy \right)$ for all $x, y\in {{\mathbb{R}}_{+}}.$ Let ${{V}_{\#}} = {{\#}^{*}}g\left(\frac{\#}{{{\#}^{*}}} \right),$ where $\#$ represents ${{S}_{D}}, {{I}_{D}}, {{S}_{L}}, {{V}_{L}}, {{I}_{L}}$ and $E.$

Then, using the equilibrium equation ${{A}_{1}}-{{\beta }_{1}}S_{D}^{*}I_{L}^{*}-{{d}_{1}}S_{D}^{*}+\sigma I_{D}^{*} = 0$ and differentiating ${{V}_{{{S}_{D}}}}$ along system (2.2), one has

Using the equilibrium equation ${{\beta }_{1}}S_{D}^{*}I_{L}^{*}-\left({{d}_{1}}+\sigma \right)I_{D}^{*} = 0$ and differentiating ${{V}_{{{I}_{D}}}}$ along system (2.2), one has

Similarly, we have

Then, consider the following Lyapunov candidate function:

Add the derivatives of ${{V}_{{{S}_{D}}}}, {{V}_{{{I}_{D}}}}, \ldots, {{V}_{{E}}}$ along system (2.2); then, combining with $\sigma = 0$, we have

Therefore, ${d\bar{V}}/{dt\le 0}\;$and ${d\bar{V}}/{dt = 0}\;$if and only if ${{S}_{D}} = S_{D}^{*}, {{I}_{D}} = I_{D}^{*}, {{S}_{L}} = S_{L}^{*}, {{V}_{L}} = V_{L}^{*}, {{I}_{L}} = I_{L}^{*}, E = {{E}^{*}}$. Then, the maximal invariant set of system (2.2) in the set $\left\{ \left({{S}_{D}}, {{I}_{D}}, {{S}_{L}}, {{V}_{L}}, {{I}_{L}}, E \right)\left| {d\bar{V}}/{dt}\; \right. \right.$ $\left. = 0 \right\}$ is $\left(S_{D}^{*}, I_{D}^{*}, S_{L}^{*}, V_{L}^{*}, I_{L}^{*}, {{E}^{*}} \right).$ Additionally, because the Lyapunov function $\bar{V}$ has a lower bound of 0 on ${{\mathbb{R}}^{6}}$, when $\left\| \left({{S}_{D}}, {{I}_{D}}, {{S}_{L}}, {{V}_{L}}, {{I}_{L}}, E \right) \right\|\to +\infty,$ we have that $\bar{V}\to +\infty.$ It follows from LaSalle's invariant principle ^[28] that $E_{e}^{*} = \left(S_{D}^{*}, I_{D}^{*}, S_{L}^{*}, V_{L}^{*}, I_{L}^{*}, {{E}^{*}} \right)$ is globally asymptotically stable.

4.   Numerical simulations

In this section, numerical simulation is used to further illustrate the effects of sheep on the spread of CE.

4.1.   Sensitivity analysis and uncertainty analysis of $R_0$

First, we fit the parameters of the model based on the statistical yearbook of Ningxia Hui Autonomous Region ^[31] and surveillance indicators of echinococcosis in Pengyang County, Ningxia Hui Autonomous Region, from 2011 to 2018 ^[32]. The monitoring indicators included the infection rate of sheep (the ratio of infected sheep to the total number of tested sheep) and the infection rate of dogs (the ratio of canine antigen positive feces to tested feces). Some parameters in the model were assumed based on actual conditions, and other parameters were selected from the literature. The biological significance and values of parameters in the model are shown in Table 1. The software used in this study was Mathematica.

Based on the parameter values in Table 1, the infection rates of sheep and dogs in Pengyang County from 2011 to 2018 are shown in Figure 3. Figure 3 shows that the fitting value is basically consistent with the actual value. The correlation coefficient ${{R}^{2}}$ in the statistical index is often used to measure the accuracy of fitting. The closer the value is to 1, the better the fitting effect. In this paper, the ${{R}^{2}}$ values obtained by using the parameters in Table 1 are 0.9699 and 0.9905, respectively, which proves that the fitting effect meets the requirements. DISO values (distance between indices of simulation and observation) can easily and quantitatively obtain the accuracy between models. Models with small DISO values indicate high overall performances, and vice versa ^[34]. In this study, ${{DISO}_{sheep}} = 0.0438$ and ${{DISO}_{dog}} = 0.05488$; the DISO values are close to 0, which also indicates a good fitting effect. According to the "Health Industry Standards of the People's Republic of China" issued by the National Health Commission, the prevalence of echinococcosis in livestock should be less than 1% and the infection rate of dogs should be less than 1%. According to this standard, it can be seen from Figure 3 that Pengyang County reached the national control standard in 2016. In addition, by 2030, the infection rate of sheep will be controlled to 0.0068%, and the infection rate of dogs will be controlled to 0.0058%, which is far lower than the national standard, indicating that the prevention and control measures of echinococcosis in Pengyang County are very effective.

As the theoretical results have shown, the basic reproduction number $R_0$ is a significant threshold for the dynamics of CE transmission. Therefore, we first present the sensitivity analysis and uncertainty analysis of $R_0$. We used the partial rank correlation coefficient (PRCC) method to study the sensitivity analysis of $R_0$, which illustrates the degree of influence of different parameters on ${{R}_{0}}$ (Figure 4). Figure 4(a) shows that $R_0$ has strong negative correlation with the dogs deworming recovery rate ($\sigma$) and the fraction of annual slaughtered sheep ($\theta$). Moreover, it also illustrates that $R_0$ has strong positive correlation with the dog infection rate $\beta_1$ and the sheep infection rate ${{\beta }_{2}}$. Birhan et al. ^[20] also pointed out that the most sensitive parameter of the spread of CE is the transmission rate $\beta_2$. That is, control measures must be taken for dogs and sheep, and infection routes must be cut off to reduce the risk of CE. It should be noted that $R_0$ also has strong positive correlation with the maximum environmental capacity of sheep ($K$). This implies that the maximum environmental capacity of sheep is a crucial factor affecting the spread of CE. However, this important fact has been neglected for a long time. Uncertainty analysis of $R_0$ was conducted by using the Latin hypercube method, and it was used to quantify the influence of parameter uncertainty on ${{R}_{0}}$ (Figure 4(b)). We assumed that the parameters follow a uniform distribution in their range as given in Table 1; then, the distribution histogram of ${{R}_{0}}$ can be given by using Mathematica software (sample number: 1000). The results show that the range of $R_0$ is $[0.05, 2.8]$, with a mean of 0.899 and a variance of 0.507 (Figure 4(b)). Combined with the theoretical analysis, it is possible to control the epidemic of echinococcosis and even eliminate it completely.

4.2.   Contour plot of $R_0$

The sensitivity analysis in this paper indicates that the vaccination rate of sheep ($\omega$), the dogs' deworming recovery rate ($\sigma$) and the sheep slaughter rate ($\theta$) have strong negative correlations with $R_0$. Therefore, we constructed the contour plots of $R_0$ with these three parameters to better understand their respective influences on $R_0$. The results show that, if we want to make the basic reproduction number less than 1 ($R_0\le1$), we can increase the dogs' deworming recovery rate to equal or be more than 0.185 ($\sigma\ge0.185$) (Figure 5(a)) or increase the slaughter rate of sheep to equal or be more than 0.15 ($\theta\ge0.15$) (Figure 5(b)) when the vaccination rate of sheep is equal to 0.65 ($\omega = 0.65$). It also shows that we should increase the dogs' deworming recovery rate to decrease the value of the basic reproduction number to less than 1 if we fix the slaughter rate of sheep, and vice versa (Figure 5(c)). Figure 5 also shows the effect of control patterns, with an emphasis on canine deworming and a secondary focus on sheep immunization, on preventing and controlling the spread of CE. It also shows the important role of the slaughter rate of sheep on the spread of CE, which is always ignored in the theoretical studies.

Except the transmission rates $\beta_1, \beta_2$ that have strong positive correlations with $R_0$, the maximum environmental capacity of sheep, $K$, also has strong correlation with $R_0$. In former studies, researchers usually assumed the total number of sheep to be a constant ^{[14,15,16,20,21]}. There is a lack of studies that investigate how $K$ affects the spread of CE. Combined the slaughter rate of sheep ($\theta$) and the deworming recovery rate of infectious dogs ($\sigma$) are the main factors that influence the value of $R_0$ (Figures 4 and 5). We also show the contour plots of $R_0$ at the $(\theta, \sigma)$ plane when the maximum environmental capacity of sheep, $K$, is different (Figure 6). Figure 6 shows that both the sheep slaughter rate and the dogs deworming recovery rate increase as the the maximum environmental capacity of sheep increases if we want to keep $R_0 = 1$. For example, if we fix $\theta = 0.49$, the value of $\sigma$ increases from 0.155 to 0.18 and then to 0.21 when $K$ respectively increases to 4.5 and 5 from 4. Similar results can be obtained if we fix $\sigma$. That is, the premise of expanding the number of sheep is to increase the deworming strength for dogs and the slaughter rate of sheep.

4.3.   Time plots of infected hosts

Figure 7 shows that the number of sheep tends to be stable over time, and that, when the maximum environmental capacity of sheep increases, the growth rate of the sheep population becomes faster. Time plots of infected sheep $I_l$ and infected dogs $I_d$ (Figure 8(a), (b)) indicate that, if we only expand the number of sheep without any additional control measures, reducing the sheep and dog infection rates below national standards takes longer and is more difficult to control. Due to the sheep slaughter rate having strong correlation with ${R_0}$, here, we evaluate the effect of the sheep slaughter rate on CE. As shown in Figure 8(c), (d), when the slaughter rate of sheep increases, it becomes less difficult to reduce the sheep and dog infection rates below the national standard. When $K = 9$, that is, when the number of sheep increases, the control difficulty increases. At this time, the slaughter rate of sheep increases to 1.25 times and 1.5 times the original rate ($\theta = 0.49$); then, the time to control the sheep infection rate is significantly shortened. The results show that the premise of increasing sheep size is to increase the slaughter rate of sheep at the same time, which could effectively control the transmission of CE and reduce the difficulty of controlling CE transmission.

Finally, in order to evaluate the priorities of CE prevention and control in a region with different breeding scales, we analyzed the effects of different intensities of dog deworming and sheep immunization measures on CE prevention and control. When $K = 4.5$, the original dog deworming rate ($\sigma = 0.74$) and sheep immunity rate ($\omega = 0.65$) are the baselines, the results are as shown by the red solid lines in Figure 9. When $K$ increases to 9, if we do not change the strength of dog deworming and increase the vaccination rate of sheep by 1.25 and 1.5 times, and then for sheep populations, the greater the strength of sheep immunization, the shorter the time it takes to reduce sheep infection rate to below the national standard; for dog populations, there is little change in the dog infection rate (see yellow and blue dashed lines in the Figure 9). If the immunity intensity of sheep is not changed and the deworming rate of dogs is increased to 1.25 times and 1.5 times of the original, and then for sheep populations, the greater the deworming rate of dogs, the less difficult it is to reduce the infection rate of sheep to below the national standard; for dog populations, the increase of the deworming intensity of dogs leads to a rapid decline in the infection rate of dogs; it significantly reduces the time required to control the infection rate in dogs (see pink and purple dashed lines in Figure 9). This indicates that when the number of sheep increases, improving dogs deworming rate allows for better control of the transmission of CE. If the dog deworming strength and sheep immunity strength are increased at the same time, the infection rates of sheep and dogs and the time required to control CE are reduced more quickly (see gray and cyan dashed lines in Figure 9). The results indicate that the dog deworming rate is more effective in controlling echinococcosis than the sheep vaccination rate, which theoretically verified the rationality of the control strategy of placing an emphasis on canine deworming and a secondary focus on sheep immunization to prevent and control the spread of CE.

5.   Conclusions

In this study, we attempted to reveal the influence of sheep in the prevention and control of CE. Based on the mechanisms for transmission of CE among dogs, sheep and eggs in the environment, we established a mathematical model and then analyzed its global dynamical behaviors. We computed the basic reproduction number of the model and obtained that the transmission of CE is mainly determined by $R_0$. The results showed that, when $R_0 < 1$, the disease-free equilibrium $E_{dfe}$ is globally asymptotically stable, and that, when $R_0 > 1$, the endemic equilibrium $E_{e}^{*}$ is globally asymptotically stable. That is, when $R_0 < 1$, the disease gradually dies out, and the disease continuous to persist if $R_0 > 1$. Sensitivity analysis and uncertainty analysis of $R_0$ were conducted to reveal the important influential factors of $R_0$. The results theoretically validate control patterns with an emphasis on dog deworming and a secondary focus on sheep immunization aimed at preventing and controlling CE (Figure 4). Figure 4 also indicated that we cannot ignore the effect of the slaughter rate of sheep and the maximum environmental capacity of sheep in the control of CE. Contour plots of $R_0$ further illustrated that the important roles of the slaughter rate of sheep and dog deworming on the spread of CE (Figure 5). In addition, with an increasing number of sheep, Figure 6 showed that we should increase the sheep slaughter rate and the deworming rate of dogs to control CE. That is, the risk of CE will be underestimated if we do not consider the increasing flock of sheep.

Use of AI tools declaration

The authors declare that they have not used artificial intelligence tools in the creation of this article.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 12001305), the National Natural Science Foundation of Ningxia (Grant No. 2023AAC03089), the Alliance of International Science Organizations (Grant No. ANSO-CR-KP-2021-02), the Natural Science Project of Ningxia Higher Institutions (Grant No. NGY2020005), the National Natural Science Foundation of China (Grant No. 12201274), and the Shenzhen Science and Technology Innovations Committee (JCYJ20210324101406019).

Conflict of interest

The authors declare that they have no competing interests.