Modeling and analysis of Cystic Echinococcosis epidemic model with health education

1.   Introduction

Cystic Echinococcosis (CE) is one of the most widespread parasitic diseases in China, especially in western China ^[1,2]. It is estimated that approximately 380,000 people are affected by Echinococcoses, and 50 million are at risk of infection nationwide ^[3]. The transmission of CE seriously threatens the health of farmers and herdsmen and hampers the development of local animal husbandry. Moreover, it is also considered as a disease that can drive farmers and herdsmen into poverty or return them to poverty ^[4].

CE always in remote rural areas and urban slums across the world. These places most often have no safe drinking water, poor sanitation and limited access to basic health care and it is impossible to control CE just relying on treatment ^[5]. The most important means of prevention is to change people's attitude towards CE from the way of thinking and living habits, and then control its root cause. Actually, health education can popularize knowledge, modify attitudes, and change behaviors among the populations ^[6]. Regarding history, Iceland's success in eradicating CE relied heavily on health education ^[7,8]. Recently, diverse health education products have been designed and applied in the control of CE in China, and some of them have garnered first-class awards ^[2,9]. For example, the health education film Galsang Flowers in Blossom depicts the journey of a big family of Tibetan compatriots who did not know about Echinococcosis as they learned with the help of the government. It showed that this film does play a positive role in improving the knowledge, attitude, and behavior intention of CE prevention among the high-risk groups ^[10,11].

This paper highlights the importance of mathematical modeling and operations research in the understanding of the dynamics of the transmission of CE, and, in particular, of the impact of health education. The proposed study was designed to contribute to existing knowledge by comprehensively analysing the mathematical model and providing insights that can inform effective CE control and prevention strategies. Compartmental modelling is a powerful tool for the qualitative and quantitative analysis of the dynamics of the transmission of infectious diseases. Based on the mechanism of disease transmission, these models typically divide the population into different compartments ^{[12,13,14,15,16]}. Several mathematical models have been developed to study the transmission dynamics and control of CE, such as ^{[17,18,19,20,21,22,23,24,25,26]}. Wang et al. ^[27] proposed a deterministic model centered around dogs, livestock, humans and eggs in the environment to study the dynamics of the transmission of echinococcosis in Xinjiang. He et al. ^[28] formulated a mathematical model analyzing livestock and dogs to reveal the effect of increasing the sheep number. Sun et al. ^[29] studied the dynamics of stochastic echinococcosis infection with environmental noise. They presented the sufficient condition for ergodic stochasticity. However, there are very few models that have been developed to examine the impact of health education on the spread of CE. Rong et al. ^[30,31] studied the potential role of free-roaming dogs in transmitting echinococcosis and provided effective measures to control free-roaming dogs. Zhang and Xiao ^[32] proposed an impulsive intervention mathematical model with periodic transmission to describe multi-host echinococcosis transmission dynamics and explore the effectiveness of control and prevention measures. Their findings suggest that key strategies for controlling the spread of echinococcosis in humans include culling wild dog populations and implementing environmental sanitation.

Based on the works of Wang et al. ^[27], Rong et al. ^[30] and Zhao et al. Zhao and Yang ^[24] in the field of mathematical modeling, our study formulated a compartmental model of echinococcosis with health education to find effective health education products that can control the spread of echinococcosis. We have extended the existing model to include a death compartment and incorporated the most important aspects of health education. This allows us to qualitatively assess the impact of health education on CE transmission dynamics. The research directly addresses a critical issue in zoonoses: CE control. CE remains a global burden for several countries, and understanding its dynamics through analysis is in line with the interests of prevention and control of parasitic diseases.

To the best of our knowledge, this is the first study to consider the combination of the death compartment and the main aspects of health education on CE. Our main objectives are to assess the impact of health education, and to investigate how health education influences the transmission dynamics of CE. This research represents a significant advance the understanding of the dynamics of CE and may contribute to the development of more effective strategies to control the spread of the disease. The rest of the paper is organized as follows: A CE model with health education and control measures will be formulated in the next section. The existence and stability of equilibria will be analyzed in Section 3, and numerical simulations are given in Section 4. The discussion of our theoretical and numerical results will be presented in the final section.

2.   Model formulation

CE is a multi-host parasitic disease, which is produced by Echinococcus granulosus. In Echinococcus granulosus's life cycle (Figure 1 (left)), most adult worms inhabit the small intestine of dogs; the E. granulosus eggs produced by an adult worm are released through the feces of dogs, ultimately polluting the water source, grassland, and so on; after viable E. granulosus eggs are ingested by a suitable host (e.g., under natural conditions: sheep, goat, and other livestock; under contingent condition: human), then the oncosphere emerges from the egg, penetrates the intestinal mucosa, and reaches different organs via the bloodstream, mainly, the liver and lungs. In the organ tissues, the oncosphere develops into a cystic larva. The cycle is completed when the dog eats the infected organs of dead livestock. It is obvious that E. granulosus eggs in the environment are an integral part of the life cycle of Echinococcus granulosus. Moreover, in the life cycle of Echinococcus granulosus, dogs are considered as the definitive host, and livestock and humans are considered as the intermediate hosts. It is worth noting that the human being, as an intermediate host of the accident, does not take part in the spread of the CE.

By referencing the modeling ideas of existing works ^[24,27,30], in this paper, we divide definitive host dogs into two classes (susceptible dogs $S_d$ and infected dogs $I_d$), intermediate host livestock into four classes (susceptible livestock $S_l$, vaccinated livestock $V_l$, infected livestock $I_l$, and dead livestock $D_l$), and humans into three classes (susceptible human $S_h$, exposed human $E_h$, and infected human $I_h$); we also denote the E. granulosus eggs in the environment as $x$.

In the model, the probability that one person is infected by ingesting a viable E. granulosus egg is assumed to be $\beta$ and the rate of ingestion of viable E. granulosus eggs by a person is assumed to be $r$. Then, the transmission rate for a person who ingests an E. granulosus egg should be $\beta \cdot r$, which is simply denoted as $\beta_h$. Similarly, $\beta_l$ is the transmission rate of an E. granulosus egg via ingestion by livestock.

In reality, health education primarily involves assisting individuals in the practice of good hygiene (e.g., washing hands before meals, consuming boiled water, eating thoroughly cooked meat) and proper disposal of livestock carcasses (e.g., through burial or incineration, or by not feeding them to dogs) ^[10,33]. To better incorporate these parts of health education in our model, we make the following assumptions for each host:

First, health education can help people to form good health habits such as, washing hands before meals or after playing with dogs or cats, drinking boiled water, and so on. Good health habits cut off the contact between humans and E. granulosus eggs; that is, good health habits can decrease the infection rate for humans. If we denote coefficient of influence of health education on people as $p$, then the transmission rate from E. granulosus eggs to humans is $(1-p)\beta_h$.

Second, health education helps people to develop the habit of deeply burying the livestock carcasses, especially the internal organs of dead livestock. To better describe this situation, we chose to add a dead compartment class ($D_l$) in the model, which is considered in a CE model for the first time, and it is reasonable and necessary, as it could better understand the effect of livestock carcass disposal. If we denote $b$ as the buried rate of livestock carcasses, then the dead livestock $D_l$ will reduce by $bD_l$ per unit time. And, it is also convenient to describe the habit of not feeding dead livestock organs to dogs. As health education raises the awareness of humans who do not feed livestock carcasses to dogs. This decreases the transmission rate from dead livestock to susceptible dogs. If we denote the coefficient of influence of health education on dogs as $q$, then the effective transmission rate from dead livestock ($D_l$) to susceptible dogs ($S_d$) is $(1-q)\beta_d$.

Finally, dog deworming and sheep immunization, as two primary control measures, are also considered in the model. The recovery rate $\gamma_d$ of dogs can reflect the degree of dog deworming. It was observed that the canine was prone to reinfection after deworming ^[34]; this can be modeled by returning from compartment $I_d$ to $S_d$. The vaccination rate $\mu$ and the rate of loss of immunity $\sigma$ of livestock can reflect the immune sheep. As E. granulosus eggs can remain viable for several weeks or months in a water source and grassland, we also consider environmental disinfection in the model. Environmental disinfection can reduce the density of E. granulosus eggs. We denote $c$ as the environmental disinfection rate; then, the density of eggs will reduce $cx$ per unit time.

Based on the above assumptions and the flowchart shown in Figure 1 (right), the CE model with health education can be given by the following system of differential equations:

where $\Lambda_h$ is the annual recruitment rate for human population, $\beta_h$ is the transmission rate from the environment to humans, $p$ is the coefficient of influence of health education on humans, $\delta$ is the rate of transition rate from exposed humans to infectious humans, $\gamma_h$ is the treatment rate for infectious humans, $d_h$ is the natural death rate of humans, $\Lambda_l$ is the annual recruitment rate for livestock, $\beta_l$ is the rate of transmission from hte environment to livestock, $q$ is incidence of infected livestock viscera that is not fed to dogs, $\nu$ is the vaccination rate of livestock, $\sigma$ is the invalid livestock vaccination rate, $\kappa$ is the fraction of annual slaughtered livestock, $b$ is the burial rate for infected livestock carcasses, $d_l$ is the natural death rate of livestock, $\Lambda_d$ is the annual recruitment rate for dogs, $\beta_d$ is the rate of transmission from livestock carcasses with CE to dogs, $\gamma_d$ is the deworming recovery rate for infectious dogs, $d$ is the natural death rate of dogs, $m$ is the release rate from infectious dogs, $c$ is the cleaning/disinfection rate of E. granulosus eggs in the environment, and $d_x$ is the natural death rate of E. granulosus eggs in the environment. All parameters in this paper are assumed to be non-negative.

Based on the biological meaning, we assume that all solutions of model (2.1) satisfy the following positive initial conditions:

Let $(S_h(t), E_h(t), I_h(t), S_l(t), V_l(t), I_l(t), D_l(t), S_d(t), I_d(t), x(t))$ be any solution of model (2.1) with the initial conditions given by (2.2); using a similar argument as in ^[27], one can prove that $S_h(t) > 0, E_h(t) > 0, I_h(t) > 0, S_l(t) > 0, V_l(t) > 0, I_l(t) > 0, D_l(t) > 0, S_d(t) > 0, I_d(t) > 0$, and $x(t) > 0$ for all $t > 0$.

Define

It is not difficult to show that $\Gamma$ is positively invariant with respect to model (2.1), such that any solution with positive initial values will ultimately enter $\Gamma$ as $t\to+\infty$.

3.   Dynamical behaviors of model (2.1)

Let all of the right-hand sides of model (2.1) equal to zero; one can calculate that model (2.1) always has a unique disease-free equilibrium $E_0 = (S_h^0, 0, 0, S_l^0, V_l^0, 0, 0, S_d^0, 0, 0)$, where

It follows from the concepts of the next-generation matrix provided by Diekmann et al. ^[35] and van den Driessche and Watmough ^[36] that the basic reproduction number can be given as

The basic reproduction number $(R_0$) denotes the expected number of secondary cases from an infected individual in the susceptible population ^[35]. That is, $R_0$ represents the number of susceptible people that an infected person can infect during the infectious period. Here, $R_0$ is the number of new infected human/sheep that are produced by infected dogs during the infectious period. Each infected dog releases E. granulosus eggs at the density $m/(\gamma_d+d)$. All susceptible livestock, i.e., $\epsilon\Lambda_l/(\kappa+d_l)$, are infected via contact with E. granulosus eggs at the rate $\beta_l$ during the dogs' expected infectious period $1/(c+d_x)$. Moreover, the total number of dogs $\Lambda_d/d$ that are infected at the rate $(1-q)\beta_d$ via ingestion of infectious cyst-containing organs of dead livestock during the livestock's burial period $1/b$. The cube root depicts a complete transmission cycle of echinococcosis, that is, the transmission from infectious dogs to E. granulosus eggs, to livestock, and then to infectious dogs again.

Besides the disease-free equilibrium $E_0$, there also exists a unique endemic equilibrium $E^* = (S_h^*, E_h^*, I_h^*, S_l^*, V_l^*, I_l^*, D_l^*, S_d^*, I_d^*, x^*)$ when $R_0 > 1$, where

For the stability of disease-free equilibrium $E_0$ and endemic equilibrium $E^*$, we have the following results.

Theorem 1. If $R_0 < 1$, then the disease-free equilibrium $E_0$ of model (2.1) is globally asymptotically stable in $\Gamma$.

Proof. The Jacobian matrix at $E_0$ is

where

□

The corresponding characteristic equation is

where

When $R_0 < 1$, direct calculation yields

Meanwhile, $H_1: = a_1 > 0$ and $H_4: = a_4H_3 > 0$. Therefore, by Routh-Hurwitz criteria, all roots of $\Phi(\lambda)$ have negative real parts; hence $E_0$ is locally stable.

Notice that the first three equations of model (2.1) are independent of the remaining equations; we then split it up into two subsystems:

and

To proof the global attractivity of $E_0$, we first consider subsystem (3.3). Let $g(x) = x-1-\ln x$, $x > 0$. One can easily verify that $g(x)\ge0$ with $g(x) = 0$ if and only if $x = 1$ and $(x-1)(y-1) = g(x)+g(y)-g(xy)$ for any $x, y\in \mathbb{R}_+$. Define

Differentiating $V_1$ along subsystem (3.3) and using $S_l^0$ and $V_l^0$ is the solution of subsystem (3.3); simple calculation implies that

Let $(S_l(t), V_l(t), I_l(t), D_l(t), S_d(t), I_d(t), x(t))$ be any solution of system (3.3) in $\Gamma$; then, for any $t\ge0$, we have that $S_d(t)\le\Lambda_d/d = S_d^0.$ Define

Differentiating $V_2(t)$ along subsystem (3.3), one can derive

Consider the following Lyapunov candidate function:

Then, combining (3.4) and (3.5) and using the expression of $R_0$, we have

Therefore, if $R_0\le1$, then $dV/dt\le0$ and $dV/dt = 0$ if and only if $S_l = S_l^0, V_l = V_l^0$, and $I_l = 0$. It is not difficult to verify that $(S_l^0, V_l^0, 0, 0, S_d^0, 0, 0)$ is the only invariant set of subsystem (3.3). By LaSalle's invariant principle, $(S_l^0, V_l^0, 0, 0, S_d^0, 0, 0)$ is globally asymptotically stable.

Now, we consider the subsystem (3.2). Since $x\to 0$ as $t\to+\infty$, it is not difficult to show that $S_h(t)\to\Lambda_h/d_h, E_h(t)\to0$, and $I_h(t)\to0$ as $t\to+\infty$. Hence, $(\Lambda_h/d, 0, 0)$ is attractive with respect to subsystem (3.2). Thus, according to the theory of asymptotic autonomous systems ^[37], the disease-free equilibrium $E_0$ of system (2.1) is globally asymptotically stable when $R_0\le1$.

This theorem suggests that the propagation of CE can be eliminated when the basic reproduction number $R_0$ is less than one. In other words, it is necessary to keep the value of the basic reproduction number as less than unity to ensure the extinction of echinococcosis. The lower the value, the better the prevention and control of the CE.

Theorem 2. If $R_0 > 1$, then the endemic equilibrium $E^*$ of model (2.1) is globally asymptotically stable in $\Gamma$.

Proof. First, we investigate the global asymptotic stability of the endemic equilibrium of subsystem (3.3). Let $g(x) = x-1-\ln x$ and

where $\sharp$ represents $S_l, V_l, I_l, D_l, S_d, I_d$, and $x$. Note that $(x-1)(1-y) = g(x)+g(y)-g(xy)$ for all $x, y\in\mathbb{R}_+$. Then, using the equilibrium equation $\Lambda_l-\beta_lx^*S_l^*+\sigma V_l^*-(\nu+\kappa+d_l)S_l^* = 0$ and differentiating $V_{S_l}$ along subsystem (3.3), one has

Using the equilibrium equation $\nu S_l^*-(\sigma+\kappa+d_l)V_l^* = 0$ and differentiating $V_{V_l}$ along subsystem (3.3), one can derive

Similarly, we have

Then, consider the following Lyapunov candidate function:

It follows from the equations $\beta_lx^*S_l^*-(\kappa+d_l)I_l^* = 0$ and $(1-q)\beta_dS_d^*D_l^*-(\gamma_d+d)I_d^* = 0$ that

And, $D_l^* = \frac{\kappa+d_l}{b}I_l^*$ and $x^* = \frac{m}{c+d_x}I_d^*$ imply that

Combining (3.7) (3.8), and (3.9) and using (3.11), we have

Therefore, $d\bar{V}/dt\le0$ and $d\bar{V}/dt = 0$ if and only if $S_l = S_l^*, V_l = V_l^*, I_l = I_l^*, D_l = D_l^*, S_d = S_d^*, I_d = I_d^*$, and $x = x^*$, that is, $(S_l^*, V_l^*, I_l^*, D_l^*, S_d^*, I_d^*, x^*)$ is the only invariant set of subsystem (3.3). It follows from LaSalle's invariant principle that $(S_l^*, V_l^*, I_l^*, D_l^*, S_d^*, I_d^*, x^*)$ is globally asymptotically stable.

The pervious proof implies that $x\to x^*$ as $t\to+\infty$. Therefore, the limiting system of subsystem (3.2) can be given as

Using the Lyapunov candidate function

one can easily derive that $d\tilde{V}/dt|_{(3.13)}\le0$ and $d\tilde{V}/dt|_{(3.13)} = 0$ if and only if $S_h = S_h^*, E_h = E_h^*$, and $I_h = I_h^*$. Therefore, $(S_h^*, E_h^*, I_h^*)$ is the only invariant set of (3.13), and it is globally asymptotically stable based on LaSalle's invariant principle. Hence, according to the theory of asymptotic autonomous systems ^[37], if $R_0 > 1$, the unique endemic equilibrium $E^*$ of system (2.1) is globally asymptotically stable. □

A simple interpretation and epidemiological implication of Theorem 2 is that CE will persist among humans, livestock, and dogs when the basic reproduction number is larger than one. In other words, CE will never become extinct when $R_0 > 1$.

4.   Numerical simulation

In this paper, the data were mainly taken from the Data Center of China Public Health Science (https://www.phsciencedata.cn/) and Ningxia Statistical Yearbook (NSY) (http://nxdata.com.cn/). Nevertheless, part of the data could not be acquired easily. We had to rely on published papers and our estimation, such as the number of E. granulosus eggs in the environment and the number of infected dogs in Ningxia Hui Autonomous Region (NHAR) in China. Wolfram Mathematica 10.2 was employed for the calculation, curve fitting, and sensitivity analysis. The values of the model parameters were estimated based on the following facts and assumptions:

(a) The number of dogs in NHAR was estimated to be 515000 in 2004 (NSY). The average life span of dogs was set as 12.5 years ^[27]. Thus, the natural death rate of dogs depends on the life span of dogs, with $d = 1/12.5 = 0.08$. The annual recruitment rate for dogs, $\Lambda_d$, equals the total number of dogs times the natural death rate with $\Lambda_d = 515000\times0.08 = 41200$. The deworming recovery rate for infectious dogs $\gamma_d$ was estimated as $0.91$, and the coefficient of influence $q$ on dogs has been assumed to be $0.1$.

(b) The natural death rate of sheep $d_l$ was set as $0.152$ ^[31]. Based on the number of livestock (only considering the sheep in this study, as sheep constitute the main livestock infected with CE) slaughtered and handled in the NSY, the fraction of the annual slaughtered rate for livestock $\kappa$ was calculated as 0.49, and the annual recruitment rate for livestock $\Lambda_l$ was estimated to be $7.48\times10^6$. According to the assumption of immune effect on livestock, the vaccination rate of livestock $\nu$ and the invalid vaccination rate among livestock $\sigma$ were assumed to be $0.92$ and $0.14$, respectively. In addition, the burial rate $b$ of infected livestock carcasses was set as $0.09$.

(c) The life span of E. granulosus eggs was assumed to be $35$ days ^[38]; then, $d_x = 365/35 = 10.42$. We supposed that the farmers perform a major disinfection once a year, which means that $c = 1/1 = 1$. Based on the assumptions in ^[39], the release rate for infectious dogs can be expressed as $m = \phi(h/\eta)(1-\exp(-\eta t_d)$, where $\phi$ denotes the number of E. granulosus eggs released by adult worms per year, $h$ is the average number of E. granulosus eggs in the small intestines of dogs during a one-year period, $t_d$ represents the average life span of dogs, and $\eta$ means the natural mortality rate of adult worms. From Budke et al. ^[40] and Torgerson and Heath ^[41], the E. granulosus egg release rate for one worm per unit time was $42$, the worm production rate per dog per year was set as $560$, the E. granulosus egg mortality rate in the dogs' body is $12/5$, and the average life span of dogs was set as $5$; thus, $m = 9799$.

(d) The human annual birth population in NHAR was applied as $1.57\times10^5$ (NSY). Then, we have that $\Lambda_h = 1.57\times10^5$. The average life span of people in NHAR is 76.5 years; thus, $d_h = 1/76.5$. The recovery rate $\gamma_h$ was estimated to be $35\%$, and the coefficient of influence of health education on humans was assumed to be $0.2$. The incubation period of CE can be months to years, even decades; we chose 14 years ^[27]; thus, $\delta = 1/14$. And, the parameters $\beta_h, \beta_l$, and $\beta_d$ were fitted by using the data in NHAR.

For clarity, we summarize the description, default values, and references of the parameters of model (2.1) in Table 1.

Using the parameter values in Table 1, the time plot for infected human cases in NHAR was constructed as presented in Figure 2. One can see that our model provides a relatively good match with the reported data in NHAR. That means that our studies based on this model are rational. In what follows, we will use model (2.1) and the parameter values listed in Table 1 to investigate the effect of health education on the spread of CE.

Notice the important role of the basic reproduction number $R_0$. We first use the partial rank correlation coefficient (PRCC) to identify critical parameters that have a significant impact on the value of $R_0$. From Figure 3, one can see that the biggest values of the PRCC for $R_0$ were observed for parameters $q$ (the incidence of infected livestock viscera that is not fed to dogs) and $b$ (burial rate for infected livestock carcasses). That means that $q$ and $b$ have the most significant impact on $R_0$. These two parameters were first considered in the CE model, and their values completely depend on health education. This implies that health education is strongly related to $R_0$. Figure 3 also shows that $R_0$ is more sensitive to $\gamma_d$ (deworming recovery rate for infectious dogs) and $\nu$ (vaccination rate of livestock). This is consistent with the Chinese policy of canine deworming as primary and sheep immunization as auxiliary. Regarding the livestock vaccination, the PRCC values for $\nu$ and $\sigma$ inform us that vaccination coverage plays a more significant role in preventing the prevalence of CE. Moreover, $\kappa$ and $c$ are also negatively correlated with $R_0$, that is, the higher the livestock slaughter rate and environmental disinfection rate, the lower the value of $R_0$. In all, Figure 3 shows that health education plays a positive role in reducing the values of $R_0$, especially education on the infected livestock carcass disposal ($b$ and $q$).

To further clarify the effects of $b, q, \nu$, and $\gamma_d$ on $R_0$, we further present the contour plots for $R_0$ from the perspectives of $\gamma_d$ and $\nu$ with different values of $q$ and $b$. From Figure 4(a) and (b) or Figure 4(c) and (d), we have that, once we increase the value of $b$ by 20%, the value of $\gamma_d$ sharply decreases from 0.9267 to 07589 when we fix $\nu = 0.92$ to keep $R_0 = 1$. Similar results can be obtained for $\nu$ if we fix $\gamma_d$. If we want to increase the value $q$ from 0.1 to 0.12, Figure 4(a) and (c) tell us that the dependence on $\gamma_d$ will slightly decrease if we fix $\nu$ to keep $R_0 = 1$. Similarly results can be derived from Figure 4(a) and (c) or Figure 4(b) and (d). These four plots illustrate that increases in $b$ and $q$ are both can independently the dependence on $\gamma_d$ and $\nu$. However, from the perspective of effects, the influence of $b$ is more obvious. That means that deeply burying infected livestock carcasses is the most important behavior to reduce the value of $R_0$. Meanwhile, the time plots of Figure 5 tell us that the trends of infected humans, livestock, and dogs all indicate high correlation with $b$ and $q$. The higher the values of $b$ and $q$, the lower the risk for humans, livestock, and dogs. These results indicate that infected livestock carcass disposal awareness should be stressed during health education.

Finally, time plots for infected humans, livestock, and dogs with different values of $p$ are given in Figure 6. It shows that $I_h$ proportionally decreases with the increase of $p$, while the time series of $I_l$ and $I_d$ are steady without much change. That is because humans do not participate in the spread of CE. In the model, $p$ reflects the actions of washing hands before meals, drinking boiled water, and eating thoroughly cooked meal. That is, if people pay attention to personal hygiene, the spread of CE in humans will be mitigated.

5.   Conclusions

CE, as one of the most important diseases in the world, is mainly distributed in the western part of China, including Ningxia, Tibet, and other western provincial capitals ^[42]. Health education plays a vital role in controlling and preventing CE, which can mitigate and reduce the transmission of CE. This paper presents a deterministic mathematical model that explores the dynamics of a CE model, encompassing the main aspects of health education while also incorporating dog deworming and sheep immunity into the model. The study include a rigorous analysis of the model, with a number of key findings. The dynamical behaviors indicate that the transmission of CE is mainly determined by the basic reproduction number $R_0$. That is, CE will persist when $R_0 > 1$, while it will die out when $R_0 < 1$. With the parameter values in Table 1, model (2.1) can mimic the reported human echinococcosis data from 2004 to 2018 in NHAR (Figure 2). It should be noted that the basic reproduction number in NHAR is $1.0056 > 1$, which means that CE is an endemic disease in NHAR.

For the sensitivity analysis of $R_0$, we analyzed the correlation of various parameters of $R_0$. The strong negative correlation between $R_0$ and the deworming recovery rate for dogs $\gamma_d$ is consistent with the results in ^[30,31]. It is interesting that $R_0$ has strong negative correlations with the burial rate for infected livestock carcasses $b$ and the incidence of infected livestock viscera that is not fed to dogs $q$ (Figures 3 and 4). This indicates that suitable disposal of infected livestock carcasses is very important for the prevention and control of the transmission of CE.

The study presented here is just a primary investigation into the effect of health education on the prevention and control of the spread of CE. The major conclusions are provided as follows.

1) The spread of CE is determined by the basic reproduction number, and the lower the value the better the ability to prevent and control the spread.

2) Health education is an effective means for the prevention of CE.

3) Infected livestock carcass disposal awareness should be stressed in the process of health education.

Various directions can be improved and generalized in the future. As in ^[43], we can further explore cost-effective approaches for CE. We could further explore the most effective health education products and treatment measures for CE, investigate the impact of climate change on the spread of CE, or analyze the geographical distribution of CE in China for our next study.

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 Natural Science Foundation of Ningxia (Grant No. 2023AAC03089), the Alliance of International Science Organizations (Grant No. ANSO-CR-KP-2021-02), and the National Youth Talent Program (Grant No. E1190301).

Conflict of interest

We declare that we have no financial or personal relationships with other people or organizations that could have inappropriately influenced our work.