#### Abstract

Clonorchiasis is the most important food-borne parasitic disease in China. In this paper, a mathematical model of clonorchiasis sinensis is proposed to mimic its transmission dynamics to assess the effects of intervention strategies such as snail control, health education, and chemotherapy. A threshold dynamics in terms of the basic reproductive number has been established; that is, if the disease dies out and the disease-free periodic solution is globally asymptotically stable, and if then the disease breaks out. The effects of different control measures are compared by numerical simulations. The numerical results suggest that it is necessary to strengthen health education and improve faeces management and illustrate that snail control is the most effective way to be implemented in the clonorchiasis sinensis control in Foshan.

#### 1. Introduction

Clonorchiasis sinensis (*C. sinensis*) is a major food-borne parasitosis, which is actively infected in China, the Democratic People’s Republic of Korea, the Republic of Korea, Russia, and Viet Nam [1]. Currently, it is estimated that more than 200 million people are at risk of infection [2], and about 35 million people are infected globally, including 15 million in China [3]. There are two major endemic regions in China—namely, provinces in the southeast, including Guangdong and Guangxi, and provinces in the northeast, such as Heilongjiang and Jilin [4]. Human beings are infected through ingestion of raw or undercooked fish which contains the metacercariae of liver flukes [5–7]. The clonorchiasis infection (*CI*) may cause serious liver and biliary system damage and affect many sectors of human health [4, 8].

Foshan is located in the central of Pearl River Delta in Guangdong. It enjoys subtropical monsoon climate, with warm weather, abundant sunshine, and rainfall. Agriculture is based on rice cultivation and freshwater aquaculture industry. Mulberry fish pond is above 50% of the total farmland area. Local residents like to eat raw fish. Additionally, coupled with misconceptions, such as the belief that consumption of alcohol or spicy food can prevent infection [9], Foshan has been becoming a heavily clonorchiasis-endemic area. Therefore, in order to reduce the CI rate, a number of strategies have been proposed including chemotherapy (morbidity control with praziquantel), information, education, communication, and faeces management (sanitation improvement and animal management). In [10], it is reported that after the implement of integrated control strategies, the CI rate of population in Foshan decreased dramatically from % in 1989 to % in 2000. But the rate shows the resurgence since 2005 owing to the general investigation project eased from 2001 to 2004 and the tradition of consuming raw freshwater fish or shellfish resuscitated (see Figure 1). This shows that health education does play important role in the control of* C. sinensis*.

The amphibious snail is one of intermediate hosts of* C. sinensis*. As far as we know, snail control with molluscicide is one of important control measures for schistosomiasis but has not been widely applied in practice for clonorchiasis control. However, some biologists have tried to study the role of snail-killing in controlling the spread of* C. sinensis*. For example, Yang [11] carried out comprehensive measures in Daliang Village, Yangshan County, Guangdong Province, including health education, chemotherapy, and snail control with molluscicide. The CI rate decreased from % in 1975 to % in 1976 to 0 in 1979-1982. Therefore, it is of great practical significance to assess different tools and strategies for large-scale control of clonorchiasis by using mathematical modelling.

There have been studies of modeling clonorchiasis associated with the human being in China [12, 13]. However, none of the studies were to quantitatively consider the relationship between CI rate and the number of molluscicide to spray when chemotherapy and health education are implemented in combination. Mathematical models can be important for determining the optimal number of molluscicide spraying so as to control the transmission of clonorchiasis. In this paper, a clonorchiasis spread model with an integrated strategy is developed and studied, in which seasonal variation and impulsive control are considered.

This paper is organized as follows. In Section 2, we propose a* C. sinensis* model with pulse snail-killing, health education, and chemotherapy control in periodic environment. Some results are stated which will be essential to our proof. We calculate the basic reproductive number for the model in Section 3. In Section 4, we illustrate that the basic reproductive number serves as a threshold parameter that determines the disease to be extinction or endemic. In Section 5, different control programs are compared and sensitivity analysis is done to evaluate the snails control strategy by numerical simulations. Section 6 gives a brief discussion and future work.

#### 2. Model and Preliminaries

Clonorchiasis belongs to vector-borne disease including two intermediate hosts: snail and fish.* C. sinensis* is transmitted indirectly among the human hosts, first intermediate snails, and second intermediate fishes in the sense that free-swimming stages (miracidia, cercariae) and ingestion stage (metacercariae) are interposed. In addition to human beings, specially, many mammals, such as dogs, cats, pigs, rodents, foxes, and possibly any fish eating mammal, can serve as definitive hosts for* C. sinensis* and humans are the main definitive hosts [14]. In order to simplify the mathematical model, we only consider human hosts here. Figure 2 shows a schematic description of the transmission of clonorchis sinensis in definitive host, snail host, and fish host.

In this section, we mainly formulate an impulsive epidemic model to describe the transmission dynamics of* C. sinensis*.

We firstly formulate a model for the spread of* C. sinensis* incorporating health education and chemotherapy strategies. The total human population at any time , denoted by , is the sum of individual populations in each compartment which includes susceptible , infected , and recovered . We assume that the total human population remains a constant, denoting for all .

Following the idea of Dai and Gao [13], we divide the snail population and fish population into disjoint classes: susceptible and infected , respectively. We suppose that the infection rates of susceptible human, susceptible snail, and susceptible fish are described by respectively, where is per capita (successful) infection rate of human hosts by one fish at time , is per capita (successful) infection rate of snails by one human at time , and is per capita (successful) infection rate of fish by one snail at time .

Some residents would temporarily change their bad eating habit owing to health education. Let denote the proportion of human host at time from susceptible to recovered human hosts owing to health education, is the recovery rate due to chemotherapy, and is the proportion of human host at time of transition from recovered to susceptible human hosts due to the loss of heath awareness and misconceptions. Suppose that sail population and fish population do not result in death and increase at periodic recruitment rates and and also decrease at the natural death rates and , respectively. Incorporating above assumptions, we have the following model, in which seasonal variation and control measures (health education and chemotherapy) are taken into consideration:where all parameters are positive, periodic, andcontinuous functions with period 12 (months).

Since , then (2) can yield

Snail control is conducted with molluscicide in fish ponds and canals. Note that molluscicides are most commonly sprayed to be taken on different dose and different time-interval in a year. The phenomenon exhibits impulsive effects on the transmission of* C. sinensis*. Suppose the number of molluscicide to spray is in one year, is the pulse time, and the elimination rate of snails at time is . Then we have impulsive equations: where , and , ().

The system consisting of (3) and (4) is an impulsive differential system. For simplicity, we will refer to the system as “the model system” in the rest of this paper.

Based on the biological background of the model system, we always assume that all solutions of the model system satisfy the following initial conditions:

It is not difficult to prove that the positive cone of is flow invariant relative to the model system.

Let and denote the density of sail population and fish population at time , respectively. From (3) and (4), we have and

For a continuous, positive periodic function , we set and Define a set where and

Lemma 1. * is a positively invariant set of the model system.*

*Proof. *Let be any positive solution of the model system with initial condition (5).

For system (6), we consider the following auxiliary system:Thus we haveThen it follows from comparison theorem, we have Applying similar method, we can obtain that This completes the proof.

Lemma 2. *Consider the following impulsive differential equation:where and are continuous and positive -periodic functions, and there is a positive integer such that and for all . Then there exists a unique positive periodic solution of system (11)for , which is globally asymptotically stable, where **The proof of Lemma 2 can be seen in the Appendix.**By Lemma 2, it is easy to see that the following conclusion holds true.*

Theorem 3. *The model system always has a unique disease-free periodic solution *

Before proving the main results, we introduce the following notations.

Let be the standard ordered n-dimensional Euclidean space with a norm . For , , we denote if ; if ; and if , respectively.

Let be the evolution operator of the linear -periodic system

Consider the linear impulsive systemwhich satisfies the following three conditions:(a_{1}), , where is a positive real number.(a_{2}), , (), and .(a_{3})There exists such that , ().

DenoteBy the Perron-Frobenius theorem, its spectral radius is the principal eigenvalue of in the sense that it is simple and admits an eigenvector .

The following lemma is useful for our future discussion.

Lemma 4 (see [15]). *Let Then there exists a positive periodic function such that is a solution of system (15).*

#### 3. The Basic Reproductive Number

To use the computation approach of the basic reproductive number (see [15]), we set

and rewrite system (3) as where is the new infection rate and is the decay rate or transfer rate, and

We call the rearranged model system as “new model.” According to Lemma 2, we easily know that the model system has a unique disease-free periodic solution Then is the unique disease-free periodic solution of the new model.

Let , which is equipped with maximum norm . It follows that the new infection matrix at is and the evolution of the initial infective members introduced at is described by Let be the evolution operator of (20) and define a linear operator on by It follows from [15] that the basic reproductive number of the model system is given by .

Following [15], by direct computation, we have andObviously, and , and then the assumptions in [15] hold. By Theorem 3.2 in [15], we can obtain that the following result.

Theorem 5. *For the model system, the following statements are valid:*(i)*.*(ii)*.*(iii)*.*

#### 4. Main Results

In this section, we show that the basic reproductive number is a threshold parameter that determines dynamics of the disease. The first result shows that* C. sinensis* dies out if .

Theorem 6. *The disease-free periodic solution of the model system is asymptotically stable if , and unstable if .*

*Proof. *The linearized system of the new model at the disease-free periodic solution iswhere are defined in (19), (22)-(24), and and are zero matrices. Then the Floquet multipliers of system (25) are the eigenvalues of and By Theorem 5, we have all Floquet multipliers of system (25) are less than 1 provided that . Therefore, the disease-free periodic solution of the model system is asymptotically stable. And if it is unstable. This completes the proof.

Theorem 7. *If , then the disease-free periodic solution of the model system is globally asymptotically stable.*

*Proof. *By Theorem 6, we know that if , then is locally asymptotically stable. Thus we only show that it attracts all nonnegative solution of the model system.

By Theorem 5, we know implies Thus there is a sufficiently small such that whereLet be any solution of the model system. In view of Lemma 1, the first equation of system (3) yields Consider the following comparison system:By [16], we know that the first equation of (29) admits a positive periodic solution which is globally asymptotically stable. Thus , as By the comparison theorem, we have Hence, there exists a sufficiently large and given above , such that , for In the same way, we can also prove that , for

From the model system, we haveBy comparison theorem in impulsive equations and above method, we can obtain for

Let . According to above discussion, we have for From (3), (4), and (31), we have that for , Consider the following linear approximation system:where are defined as (19), (23), and (24).

By Lemma 4, we have that there exists a positive, -period function such that is a solution of system (33), where , . It follows from (26) that Therefore, we have as This implies that the zero solution of system (33) is globally attractive if .

For any nonnegative initial value of system (32), there exists a sufficiently large such that holds. Applying the comparison principle, we have , for all , where is also the solution of system (33). Therefore, we have , and , as By the theory of asymptotically semiflows, it follows thatTherefore the disease-free periodic solution of the model system is globally asymptotically stable. This completes the proof.

The subsequent result shows that the disease is uniformly persistent if .

Define

Let be the Poincaré map associated with the model system, that is, where and is the unique solution of the model system with It is easy to see that Letting , we have

According to Lemma 1, we can easily see that and are positively invariant, and Poincaré map is point dissipative.

Next, we establish the following lemma which will be useful in subsequent main result.

Lemma 8. *If the basic reproductive number , then there exists , such that for any with , we have *

*Proof. *In view of Theorem 5, we know that if and only if , and then there is a sufficiently small such that where is given in (27).

By the continuity of the solutions with respect to the initial values, there exists , such that for any with and , Next, we claim that (39) holds. Suppose the claim is not valid. Then there exists such that Without loss of generality, we assume thatIt follows from (41) and (43) that For any , there exist nonnegative integer and such that . From (44), we have Note that Thus By the first equation of (3) and (46), we have By comparison theorem, we have and as , where is the solution of the following system:Therefore, for above mentioned , there exists sufficiently large such that Using the same method we can get there exists sufficiently large such that From (3), (4), (49), and (50), we obtain for ,Consider the following impulsive linear approximation system:By Lemma 4, we know that there exists a positive, -period function such that is a solution of (52), where and . It follows from (40) that Obviously, we can choose and a proper such that By the comparison theorem we have that for all , Then, we can obtain that , , and , as which is a contradiction. The proof of Lemma 8 is completed.

Theorem 9. *If the basic reproductive number , then there exist constants such that for any solution of the model system with initial value satisfies*

*Proof. *DenoteWe claim thatObviously, . Thus we only need to prove . If it is not true, then there exists a point .

There are six cases to consider: (i) , , , (ii) , , , (iii) , , , (iv) , , , (v) , , , (vi) , , .*Case A.* For case (i), that is, , , , it is easily seen that and , for all . Then from the fourth equation of (3), . Thus for . It follows from the sixth equation of (3) that for . It is easy to obtain that for . This is a contradiction. Using the same method, we can prove the second and the third cases.*Case B.* For case (iv), that is, , , , it is easily seen that , , and for all . Then from the second equation of system (3), . Obviously, for . This is a contradiction. We can also discuss the last two cases using the same method.

From the above discussion, we have that, for any initial , then . Thus

Obviously, Piocar*é* map has a global attractor , is an isolated invariant set in and . Further, is acyclic in and every solution in converges to . By Lemma 8, we know that is weakly uniformly persistent with respect to . According to Zhao [17], we have that is uniformly persistent with respect to . That is, there exist constants such that for any solution of the model system with initial value satisfiesThe proof of Theorem 9 is completed.

#### 5. Numerical Simulations

In this section, we will make numerical simulations in Matlab. Using the theory of impulsive equations and analysis method, dynamic behavior of the model has been studied and a threshold for a disease to be extinct or endemic has been established.

Before carrying out the numerical simulations, we have to estimate the model parameters. Some data are taken from literature [14, 18]. Some data are estimated from the clonorchiasis surveillance data of Foshan city, Guangdong province, China. The values of the model parameters are listed in Table 1.

Set . In view of the clonorchiasis surveillance data, the initial value of the model system is taken as follows: , , , , , .

If we fix and , that is, no snail control strategy is implemented. With numerical simulation, we get . If we fix and