# Modelling the Effects of Snail Control and Health Education in Clonorchiasis Infection in Foshan, China

Authors: Shujing Gao; Ruixia Yuan; Yujiang Liu; Xinzhu Meng
Journal: Complexity
 (2019)

---

## 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 numberR0 has been established; that is, if R0<1, the disease dies out and the disease-free periodic solution is globally asymptotically stable, and if R0>1, 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.


---

## Body

## 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 56.84% in 1989 to 3.47% 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 ofC. sinensis.Figure 1
Positive infection rate in Foshan from 1980 to 2010.The amphibious snail is one of intermediate hosts ofC. 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 ofC. 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 48.2% in 1975 to 2.9% 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 Section2, we propose aC. 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 forC. 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.Figure 2
A transmission diagram of Clonorchis sinensis.In this section, we mainly formulate an impulsive epidemic model to describe the transmission dynamics ofC. sinensis.We firstly formulate a model for the spread ofC. sinensis incorporating health education and chemotherapy strategies. The total human population at any time t, denoted by N(t), is the sum of individual populations in each compartment which includes susceptible Sh(t), infected Ih(t), and recovered Rh(t). We assume that the total human population remains a constant, denoting Nh(t)=Nh for all t≥0.Following the idea of Dai and Gao [13], we divide the snail population and fish population into disjoint classes: susceptible (Ss,Sf) and infected (Is,If), respectively. We suppose that the infection rates of susceptible human, susceptible snail, and susceptible fish are described by (1)β1tShtIft,β2tSstIht,β3tSftIst,respectively, where β1(t) is per capita (successful) infection rate of human hosts by one fish at time t, β2(t) is per capita (successful) infection rate of snails by one human at time t, and β3(t) is per capita (successful) infection rate of fish by one snail at time t.Some residents would temporarily change their bad eating habit owing to health education. Letα(t) denote the proportion of human host at time t from susceptible to recovered human hosts owing to health education, γ(t) is the recovery rate due to chemotherapy, and δ(t) is the proportion of human host at time t 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 Λ2(t) and Λ3(t) and also decrease at the natural death rates μ2(t) and μ3(t), respectively. Incorporating above assumptions, we have the following model, in which seasonal variation and control measures (health education and chemotherapy) are taken into consideration:(2)Sh′t=μ1tNht-β1tShtIft-αtSht-μ1tSht+δtRht,Ih′t=β1tShtIft-γtIht-μ1tIht,Rh′t=αtSht+γtIht-μ1tRht-δtRht,Ss′t=Λ2t-β2tSstIht-μ2tSst,Is′t=β2tSstIht-μ2tIst,Sf′t=Λ3t-β3tSftIst-μ3tSft,If′t=β3tSftIst-μ3tIft,where all parameters are positive, periodic, andcontinuous functions with period 12 (months).SinceRh(t)=Nh-Sh(t)-Ih(t), then (2) can yield(3)Sh′t=μ1t+δtNh-β1tShtIft-αt+μ1t+δtSht-δtIht,Ih′t=β1tShtIft-γtIht-μ1tIht,Ss′t=Λ2t-β2tSstIht-μ2tSst,Is′t=β2tSstIht-μ2tIst,Sf′t=Λ3t-β3tSftIst-μ3tSft,If′t=β3tSftIst-μ3tIft.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 ofC. sinensis. Suppose the number of molluscicide to spray is q in one year, tk(k∈N) is the pulse time, and the elimination rate of snails at time tk is θk. Then we have impulsive equations: (4)Sst+=1-θkSst,Ist+=1-θkIst,fort=tk,where Ss(t+)=limh→0+⁡Ss(t+h),Is(t+)=limh→0+⁡Is(t+h), and θk+q=θk, tk+q=tk+12 (k∈N).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:(5)Si0+≥0,andIi0+≥0fori=h,s,f.It is not difficult to prove that the positive cone ofR+6 is flow invariant relative to the model system.LetNs(t) and Nf(t) denote the density of sail population and fish population at time t, respectively. From (3) and (4), we have (6)Ns′t=Λ2t-μ2tNst,t≠tk,k∈N,Nst+=1-θkNst,t=tk,k∈N,and(7)Nf′t=Λ3t-μ3tNft.For a continuous, positiveω-periodic function f(t), we set fM=supt∈[0,ω]⁡f(t) and fL=inft∈[0,ω]⁡f(t). Define a set (8)Ω=Sht,Iht,Sst,Ist,Sft,Ift∈R+6∣0≤Sht+Iht≤Nh,0≤Sst+Ist≤Ns∗,0≤Sft+Ift≤Nf∗,where Ns∗=Λ2M/μ2L and Nf∗=Λ3M/μ3L.Lemma 1.
Ω is a positively invariant set of the model system.Proof.
Let(Sh(t),Ih(t),Ss(t),Is(t),Sf(t),If(t)) be any positive solution of the model system with initial condition (5).
For system (6), we consider the following auxiliary system:(9)Hs′t=Λ2t-μ2tHst,Hs0+=Ns0+.Thus we have(10)Hs′t≤Λ2M-μ2LHst<0,ifHst>Λ2Mμ2L=Ns∗.Then it follows from comparison theorem, we have limsupt→+∞Ns(t)≤Ns∗. Applying similar method, we can obtain that limsupt→+∞Nf(t)≤Nf∗. This completes the proof.Lemma 2.
Consider the following impulsive differential equation:(11)z′t=at-btzt,t≠tk,zt+=1-θkzt,t=tk,where a(t) and b(t) are continuous and positive ω-periodic functions, and there is a positive integer q such that θk+q=θk and tk+q=tk+ω for all k∈N. Then there exists a unique positive periodic solution of system (11)(12)z∗t=∏l=1i1-θlz0∗e∫nω+t0t-bτdτ+e∫nω+tit-bτdτ∫nω+titase∫nω+tisbτdτds+∑l=1i∏j=li1-θje∫nω+tj-1t-bτdτ∫nω+tj-1nω+tjase∫nω+tj-1sbτdτds,for nω+ti<t≤nω+ti+1,i=0,1,…,q-1, which is globally asymptotically stable, where (13)z0∗=1-∏l=1q1-θle∫nω+t0nω+tq-bτdτ-1∑l=1q∏j=lq1-θje∫nω+tj-1nω+tq-bτdτ∫nω+tj-1nω+tjase∫nω+tj-1sbτdτds.
The proof of Lemma2 can be seen in the Appendix.
By Lemma2, it is easy to see that the following conclusion holds true.Theorem 3.
The model system always has a unique disease-free periodic solution(Sh∗(t),0,Ss∗(t),0,Sf∗(t),0).Before proving the main results, we introduce the following notations.Let(Rn,R+n) be the standard ordered n-dimensional Euclidean space with a norm ·. For ϱ, ν∈Rn, we denote ϱ≥ν if ϱ-ν∈R+n; ϱ>ν if μ-ν∈R+n∖{0}; and ϱ≫ν if ϱ-ν∈Int(R+n), respectively.LetΨA(t,s)(t≥s) be the evolution operator of the linear ω-periodic system (14)x˙t=Atxt,x∈Rn.Consider the linear impulsive system(15)x′t=Atx,t≠tk,k∈N,xt+=Pkxtk,t=tk,k∈N,x0+=x0=x0,t0=0,which satisfies the following three conditions:(a1)
A(·)∈C(R,Rn×n), A(t+ω)=A(t), where ω is a positive real number.(a2)
Pk∈Rn×n, detPk≠0, tk<tk+1 (k∈N), and limk→+∞tk=+∞.(a3)
There existsq∈N+ such that Pk+q=Pk, tk+q=tk+ω (k∈N).Denote(16)ΦAPkω≔∏i=1qPq-i+1ΨAtq-i+1,tq-i.By the Perron-Frobenius theorem, its spectral radius rΦAPk(ω) is the principal eigenvalue of ΦAPk(ω) in the sense that it is simple and admits an eigenvector ν∗≫0.The following lemma is useful for our future discussion.Lemma 4 (see [15]).
Letη=1/ωln⁡rΦAPk(ω). Then there exists a positive ω-periodic function v(t) such that eηtv(t) is a solution of system (15).


## 3. The Basic Reproductive NumberR0

To use the computation approach of the basic reproductive number (see [15]), we set x=x 1 , x 2 , x 3 , x 4 , x 5 , x 6 T = I h , I s , I f , S h , S s , S f T and rewrite system (3) as (17)x′t=Ft,xt-Vt,xt,where F is the new infection rate and V is the decay rate or transfer rate, and (18)Ft,x=β1tx4tx3tβ2tx5tx1tβ3tx6tx2t000,Vt,x=μ1t+γtx1tμ2tx2tμ3tx3t-μ1t+δtNh+β1tx4tx3t+αt+μ1t+δtx4t+δtx1t-Λ2t+β2tx5tx1t+μ2tx5t-Λ3t+β3tx6tx2t+μ3tx6t.We call the rearranged model system as “new model.” According to Lemma2, we easily know that the model system has a unique disease-free periodic solution x~∗(t)=Sh∗t,0,Ss∗t,0,Sf∗t,0. Then x∗(t)=0,0,0,Sh∗t,Ss∗t,Sf∗t is the unique disease-free periodic solution of the new model.LetCω+={ϕ∈CR,R+3∣ϕt+ω=ϕt,forallt∈R}, which is equipped with maximum norm ·∞. It follows that the new infection matrix at x∗(t) is (19)Ft=∂Fit,x∗t∂xj1≤i,j≤3=00β1tSh∗tβ2tSs∗t000β3tSf∗t0,and the evolution of the initial infective members introduced at x∗(t) is described by (20)y˙t=-Vty,t≠tk,k∈N,ytk+=Pkytk,t=tk,k∈N.Let Y(t,s) be the evolution operator of (20) and define a linear operator L on C12+ by (21)Lϕt=∫0+∞Yt,t-aFt-aϕt-ada,∀t∈R.It follows from [15] that the basic reproductive number R0 of the model system is given by R0=r(L).Following [15], by direct computation, we have (22)Pk=10001-θk0001,Qk=10001-θk0001,(23)Mt=∂fit,x∗t∂xj4≤i,j≤6=-μ1t-αt-δt000-μ2t000-μ3t,and(24)Vt=∂Vit,x∗t∂xj1≤i,j≤3=μ1t+γt000μ2t000μ3t.Obviously, rΦMQk(12)<1 and rΦ-VPk(12)<1, and then the assumptions (H1)-(H8) 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)
R0=1⇔rΦ(F-V)Pk(12)=1.(ii)
R0>1⇔rΦ(F-V)Pk(12)>1.(iii)
R0<1⇔rΦ(F-V)Pk(12)<1.


## 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 thatC. sinensis dies out if R0<1.Theorem 6.
The disease-free periodic solutionx~∗(t)=(Sh∗(t),0,Ss∗(t),0,Sf∗(t),0) of the model system is asymptotically stable if R0<1, and unstable if R0>1.Proof.
The linearized system of the new model at the disease-free periodic solutionx∗(t) is(25)x′t=Ft-Vt0AtMtxt,t≠tk,k∈N,xt+=Pk0BkQkxt,t=tk,k∈N,where Ft,Vt,M(t),Pk,Qk are defined in (19), (22)-(24), and A(t) and Bk are zero matrices. Then the Floquet multipliers of system (25) are the eigenvalues of rΦ(F-V)Pk(12) and rΦMQk(12). By Theorem 5, we have all Floquet multipliers of system (25) are less than 1 provided that R0<1. Therefore, the disease-free periodic solution x~∗(t) of the model system is asymptotically stable. And if R0>1 it is unstable. This completes the proof.Theorem 7.
IfR0<1, then the disease-free periodic solution x~∗(t) of the model system is globally asymptotically stable.Proof.
By Theorem6, we know that if R0<1, then x~∗(t) is locally asymptotically stable. Thus we only show that it attracts all nonnegative solution of the model system.
By Theorem5, we know R0<1 implies rΦ(F-V)Pk(12)<1. Thus there is a sufficiently small ε>0 such that (26)rΦF-V+MεPk12<1,where(27)Mε=00β1tεβ2tε000β3tε0.Let (Sht,Iht,Sst,Is(t),Sf(t),If(t)) be any solution of the model system. In view of Lemma 1, the first equation of system (3) yields (28)Sh′t≤μ1t+δtNh-αt+μ1t+δtSht.Consider the following comparison system:(29)S^h′t=μ1t+δtNh-αt+μ1t+δtS^ht,S^h0+=Sh0+.By [16], we know that the first equation of (29) admits a positive periodic solution Sh∗(t) which is globally asymptotically stable. Thus S^h(t)→Sh∗(t), as t→+∞. By the comparison theorem, we have Sh(t)≤S^h(t). Hence, there exists a sufficiently large ξ0>0 and given above ε>0, such that Sh(t)≤Sh∗(t)+ε, for t≥ξ0. In the same way, we can also prove that Sf(t)≤Sf∗(t)+ε, for t≥ξ0′.
From the model system, we have(30)Ss′t≤Λ2t-μ2tSst,t≠tk,k∈Z,Sst+=1-θkSst≤Sst,t=tk,k∈Z.By comparison theorem in impulsive equations and above method, we can obtain Ss(t)≤Ss∗(t)+ε for t≥ξ0′′.
Letξ∗=max⁡{ξ0,ξ0′,ξ0′′}. According to above discussion, we have for t≥ξ∗(31)Sht≤Sh∗t+ε,Sft≤Sf∗t+ε,Sst≤Ss∗t+ε.
From (3), (4), and (31), we have that for t≥ξ∗, (32)Ih′t≤β1tSh∗+εIf-μ1t+γtIh,Is′t≤β2tSs∗+εIh-μ2tIs,If′t≤β3tSf∗+εIs-μ3tIf,t≠tk,k∈N,Ihtk+=Ihtk,Istk+=1-θkIstk,Iftk+=Iftk,t=tk,k∈N.Consider the following linear approximation system:(33)x^′t=Ft-Vt+Mεx^t,t≠tk,k∈N,x^tk+=Pkx^tk,t=tk,k∈N,where F,V,Pk are defined as (19), (23), and (24).
By Lemma4, we have that there exists a positive, 12-period function x^∗(t) such that x^(t)=eηtx^∗(t) is a solution of system (33), where η=1/12ln⁡rΦ(F-V+Mε)Pk(12), x^(t)=(I^h(t),I^s(t),I^f(t)). It follows from (26) that η<0. Therefore, we have x^(t)→0 as t→+∞. This implies that the zero solution of system (33) is globally attractive if R0<1.
For any nonnegative initial valueIh0+,Is0+,If0+T of system (32), there exists a sufficiently large a>0 such that Ih0+,Is0+,If0+T≤ax^(0+) holds. Applying the comparison principle, we have Iht,Ist,IftT≤ax^(t), for all t>ξ0, where ax^(t) is also the solution of system (33). Therefore, we have Ih(t)→0, Is(t)→0 and If(t)→0, as t→+∞. By the theory of asymptotically semiflows, it follows that(34)limt→+∞⁡Sht=Sh∗t,limt→+∞⁡Sst=Ss∗tandlimt→+∞⁡Sft=Sf∗t.Therefore the disease-free periodic solution x∗(t) of the model system is globally asymptotically stable. This completes the proof.The subsequent result shows that the disease is uniformly persistent ifR0>1.Define(35)X≔Sh,Ih,Ss,Is,Sf,If∈R6∣Si≥0,Ii≥0,i=h,s,f,X0≔Sh,Ih,Ss,Is,Sf,If∈X∣Ih>0,Is>0,If>0,∂X0≔X∖X0.LetP:X→X be the Poincaré map associated with the model system, that is, (36)Px0=u12+,x0,∀x0∈X,where x0=(Sh0,Ih0,Ss0,Is0,Sf0,If0) and u(t,x0) is the unique solution of the model system with u(0+,x0)=x0. It is easy to see that (37)Pmx0=u12m+,x0,∀m∈N.Letting U∗=Sh∗0+,0,Ss∗0+,0,Sf∗0+,0, we have (38)PmU∗=u12m+,U∗,∀m∈N.According to Lemma1, we can easily see that X and X0 are positively invariant, and Poincaré map P is point dissipative.Next, we establish the following lemma which will be useful in subsequent main result.Lemma 8.
If the basic reproductive numberR0>1, then there exists ε∗>0, such that for any x0∈X0 with x0-U∗≤ε∗, we have (39)limsupm→∞⁡dPmx0,U∗≥ε∗.Proof.
In view of Theorem5, we know that R0>1 if and only if rΦ(F-V)Pk(12)>1, and then there is a ε>0 sufficiently small such that (40)rΦF-V-MεPk12>1,where Mε is given in (27).
By the continuity of the solutions with respect to the initial values, there existsε∗>0, such that for any x0∈X0 with x0-U∗≤ε∗ and t∈[0,12), (41)ut,x0-ut,U∗≤ε.
Next, we claim that (39) holds. Suppose the claim is not valid. Then there exists x0∈X0 such that (42)limsupm→∞⁡dPmx0,U∗<ε∗.Without loss of generality, we assume that(43)dPmx0,U∗<ε∗,forallm≥0.
It follows from (41) and (43) that (44)ut,Pmx0-ut,U∗<ε,∀m≥0,∀t∈0,12.For any t≥0, there exist nonnegative integer m′ and T1∈[0,12) such that t=12m′+T1. From (44), we have (45)ut,x0-ut,U∗=uT1,Pm′x0-uT1,U∗<ε.Note that Sht,Iht,Sst,Ist,Sft,Ift=ut,x0. Thus (46)Iht<ε,Ist<ε,Ift<εforallt≥0.By the first equation of (3) and (46), we have (47)Sh′t≥μ1t+δtNh-β1tShtε-αt+μ1t+δtSht-δtε.By comparison theorem, we have Sh(t)≥S~h(t) and S~h(t)→Sh∗(t) as ε→0, where S~h(t) is the solution of the following system:(48)S~h′t=μ1t+δtNh-β1tShtε-δtε-αt+μ1t+δtSht,S~h0+=Sh0+.Therefore, for above mentioned ε, there exists T2>0 sufficiently large such that (49)Sht≥S~ht≥Sh∗t-ε,fort≥T2.Using the same method we can get there exists T3≥T2 sufficiently large such that (50)Sft≥S~ft≥Sf∗t-ε,Sst≥S~st≥Ss∗t-ε,fort≥T3.
From (3), (4), (49), and (50), we obtain for t≥T3,(51)Ih′t≥β1tSh∗t-εIft-μ1t+γtIht,Is′t≥β2tSs∗t-εIht-μ2tIst,If′t≥β3tSf∗t-εIst-μ3tIft,t≠tk,k∈N,Ihtk+=Ihtk,Istk+=1-θkIstk,Iftk+=Iftk,t=tk,k∈N.
Consider the following impulsive linear approximation system:(52)x~′t=Ft-Vt-Mεx~t,t≠tk,k∈N,x~tk+=Pkx~tk,t=tk,k∈N.By Lemma 4, we know that there exists a positive, 12-period function x~∗(t)=(I~h∗(t),I~s∗(t),I~f∗(t)) such that x~(t)=eη1tx~∗(t) is a solution of (52), where x~(t)=(I~h(t),I~s(t),I~f(t)) and η1=1/12ln⁡rΦ(F-V-Mε)Pk(12). It follows from (40) that η1>0. Obviously, we can choose 12k~>T3 and a proper b>0 such that (53)I~h12k~+≥bI~h∗0+,I~s12k~+≥bI~s∗0+,andI~f12k~+≥bI~f∗0+.
By the comparison theorem we have that for allt≥12k~, (54)Iht≥I~ht≥beη1t-12k~I~h∗t-12k~,Ist≥I~st≥beη1t-12k~I~s∗t-12k~,Ift≥I~ft≥beη1t-12k~I~f∗t-12k~.Then, we can obtain that Ih(t)→+∞, Is(t)→+∞, and If(t)→+∞, as t→+∞ which is a contradiction. The proof of Lemma 8 is completed.Theorem 9.
If the basic reproductive numberR0>1, then there exist constants δi>0(i=1,2,3) such that for any solution of the model system with initial value (55)Sh0,Ih0,Ss0,Is0,Sf0,If0∈X0satisfies(56)liminft→+∞⁡Iht≥δ1,liminft→+∞⁡Ist≥δ2,andliminft→+∞⁡Ift≥δ3.Proof.
Denote(57)M∂=Sh0,Ih0,Ss0,Is0,Sf0,If0∈∂X0∣HmSh0,Ih0,Ss0,Is0,Sf0,If0∈∂X0,m∈N.
We claim that(58)M∂=Sh,0,Ss,0,Sf,0∈∂X0∣Sh≥0,Ss≥0,Sf≥0.
Obviously,M∂⊇Sh,0,Ss,0,Sf,0∈∂X0∣Sh≥0,Ss≥0,Sf≥0. Thus we only need to prove M∂∖Sh,0,Ss,0,Sf,0∈∂X0∣Sh≥0,Ss≥0,Sf≥0=∅. If it is not true, then there exists a point x0=Sh0,Ih0,Ss0,Is0,Sf0,If0∈M∂∖Sh,0,Ss,0,Sf,0∈∂X0∣Sh≥0,Ss≥0,Sf≥0.
There are six cases to consider: (i)Ih>0, Is=0, If=0, (ii) Ih=0, Is>0, If=0, (iii) Ih=0, Is=0, If>0, (iv) Ih=0, Is>0, If>0, (v) Ih>0, Is=0, If>0, (vi) Ih>0, Is>0, If=0.
Case A. For case (i), that is, Ih>0, Is=0, If=0, it is easily seen that Ih(t)>0 and Ss(t)>0, Sf(t)>0 for all t>0. Then from the fourth equation of (3), dIs(t)/dtt=0=β2(0)Ss(0+)Ih(0+)>0. Thus Is(t)>0 for 0<t≪1. It follows from the sixth equation of (3) that If(t)>0 for 0<t≪1. It is easy to obtain that Sh(t),Ih(t),Ss(t),Is(t),Sf(t),If(t)∉∂X0 for 0<t≪1. This is a contradiction. Using the same method, we can prove the second and the third cases.
Case B. For case (iv), that is, Ih=0, Is>0, If>0, it is easily seen that Is(t)>0, If(t)>0, and Sh(t)>0 for all t>0. Then from the second equation of system (3), dIh(t)/dtt=0=β1(0)Sh(0+)If(0+)>0. Obviously, Sh(t),Ih(t),Ss(t),Is(t),Sf(t),If(t)∉∂X0 for 0<t≪1. 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 initialx0∉{Sh0,0,Ss0,0,Sf0,0∣Sh≥0,Ss≥0,Sf≥0}, then x0∉M∂. Thus M∂=Sh,0,Ss,0,Sf,0∈∂X0∣Sh≥0,Ss≥0,Sf≥0.
Obviously, Piocaré map P has a global attractor U∗, U∗ is an isolated invariant set in X and WsU∗∩X0=⌀. Further, U∗ is acyclic in M∂ and every solution in M∂ converges to U∗. By Lemma 8, we know that P is weakly uniformly persistent with respect to (X0,∂X0). According to Zhao [17], we have that P is uniformly persistent with respect to (X0,∂X0). That is, there exist constants δi>0(i=1,2,3) such that for any solution of the model system with initial value (59)Sh0,Ih0,Ss0,Is0,Sf0,If0∈X0satisfies(60)liminft→+∞⁡Iht≥δ1,liminft→+∞⁡Ist≥δ2,andliminft→+∞⁡Ift≥δ3.The 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.Table 1
The values of parameters in the model system.
Parameter Value Unit Reference μ1(t) 0.0011 m o n t h - 1 [18] β1(t) 7.5 × 1 0 - 11 + 2 × 1 0 - 11 sin ⁡ π t 6 m o n t h - 1 Estimation α(t) 0.0167 + 0.002 sin ⁡ π t 6 m o n t h - 1 Estimation δ(t) 0.02778 + 0.009 sin ⁡ π t 6 m o n t h - 1 Estimation γ(t) 0.076 + 0.006 sin ⁡ π t 6 m o n t h - 1 [14] Λ2(t) 1.3 × 1 0 9 + 1.0 × 1 0 8 sin ⁡ π t 6 m o n t h - 1 [10, 14] μ2(t) 0.01389 + 0.01 sin ⁡ π t 6 m o n t h - 1 [19] β2(t) 3.8 × 1 0 - 8 + 5 × 1 0 - 10 sin ⁡ π t 6 m o n t h - 1 Estimation Λ3(t) 4.6 × 1 0 6 + 2 × 1 0 5 sin ⁡ π t 6 m o n t h - 1 Estimation β3(t) 1.78 × 1 0 - 12 + 5 × 1 0 - 14 sin ⁡ π t 6 m o n t h - 1 Estimation μ 3 ( t ) 0.0208 + 0.101 sin ⁡ π t 6 m o n t h - 1 [10]Setq=2,t1=4,t2=6. In view of the clonorchiasis surveillance data, the initial value of the model system is taken as follows: Sh(0)=2.368×106, Ih(0)=1.6038×106, Ss(0)=7.488×1010, Is(0)=1.872×1010, Sf(0)=1.76×108, If(0)=4.4×107.If we fixθ1=0 and θ2=0, that is, no snail control strategy is implemented. With numerical simulation, we get R0=1.8858>1. If we fix θ1=0.2 and θ2=0, that is, 20% snail-killing once a year is carried out, which results in R0=1.0719>1, these, of course, lead to the persistence of the disease, as clearly indicated by Figures 3 and 4. If we fix θ1=0.2 and θ2=0.1, then R0=0.9140<1. This suggests the extinction of the disease (see Figure 5).Figure 3
This figure shows movement paths ofIh, Is, and If as functions of time t. R0 = 1.8858 > 1, where parameters θ1 = 0 and θ2 = 0. The disease is permanent.Figure 4
This figures show movement paths ofIh, Is, and If as functions of time t. R0 = 1.0719 > 1, where parameters θ1 = 0.2 and θ2 = 0. The disease is permanent.Figure 5
This figure shows movement paths ofIh, Is, and If as functions of time t. R0 = 0.9140 < 1, where parameters θ1 = 0.2 and θ2 = 0.1. The disease will be distinct.Figure6 shows the sensitivity of the basic reproductive number R0 to the elimination rate of snails θ1 and θ2. It is apparent that the elimination rate of snails are very sensitive to R0 when θi<0.2(i=1,2).Figure 6
The graph is a contour plot showing the regions in the(θ1,θ2) plane in which R0 falls in different intervals.Health education, snail control, and feces treatment are three effective strategies to control clonorchiasis. Parametersα (or δ), θi, and β2 reflect the efforts of health education, snail control, and feces treatment, respectively. In Table 2, six projects are studied to determine the impact of control strategies on the basic reproductive number. The first and second projects are only considered the health education strategy. If α is doubled and δ is half, then R0 decreases from 1.8858 to 1.4855. And if α is tripled and δ is reduced by one-third, then R0 decreases from 1.8858 to 1.2192. Combining health education and snail control strategies in the third project, α is doubled and δ is in half. θ1=0.14, by computer simulation. It is shown that R0 decreases to 0.9674. This indicates that the disease will die out. Combining snail control and faeces treatment strategies, the last project shows R0 also becomes less than 1. From Table 2, it is evident that health educations, snail control and faeces treatment are helpful for the disease control. They can all decrease the fraction of infected individuals; however, snail-killing is the most effective way to control the transmission of clonorchiasis.Table 2
Comparison of different control strategies.
Plan Health education efforts Snail controlθ1 Faeces treatment R 0 Extinction Permanence 1 α(t)×2 0 β 2 ( t ) × 1 1.4855 No Yes δ ( t ) ÷ 2 2 α(t)×3 0 β 2 ( t ) × 1 1.2192 No Yes δ ( t ) ÷ 3 3 α(t)×2 0.14 β 2 ( t ) × 1 0.9674 Yes No δ ( t ) ÷ 2 4 α(t)×1 0 β 2 ( t ) ÷ 2 1.4968 No Yes δ ( t ) ÷ 1 5 α(t)×2 0 β 2 ( t ) ÷ 2 1.1790 No Yes δ ( t ) ÷ 2 6 α(t)×1 0.14 β 2 ( t ) ÷ 2 0.9747 Yes No δ ( t ) ÷ 1


## 6. Conclusions and Discussion

Clonorchiasis has been one of significant public health threats in China. Facing up to the epidemic situation, both the central and local governments have been exploring an effective and sustainable strategy on the control of clonorchiasis in endemic areas. Various prevention and control measures have been proposed by many researchers. Chemotherapy with praziquantel is one of vital strategies, but the use of praziquantel for humans and animals has been only temporarily effective. There is no doubt that the chemotherapy-based control strategy would no longer be an ideal measure.Currently, the integrated control strategy mainly focuses on ensuring fish cooked well prior to consumption, health education, and improvement of sanitation to prevent parasite eggs reaching fish habitats. Controlling snail is widely used to control of schistosomiasis [20], but it has not yet been applied to control clonorchiasis in practice, even if it has been theoretically confirmed. In this paper, we proposed a dynamic clonorchiasis model with impulsive snail-killing to study the effects of snail control strategy on the transmission of clonorchiasis. We also derived the basic reproductive number for general epidemic model with seasonality and impulsive control, which extended previous works on such models in that the impulsive control actions can be carried out at several time points within a period. Furthermore, we performed the numerical simulation and sensitivity analysis to recognize the impact of crucial model parameters on R0. We found that controlling snail was the best effective method for eliminating clonorchiasis.As Hepatitis B virus mainly attacks the liver,C. sinensis infection also causes liver diseases, as well as biliary conditions. Both Hepatitis B andC. sinensis are present in almost the same regions in China [8]. Therefore, using HBV/CI coinfection model to assess different tools and strategies for large-scale control of Hepatitis B and clonorchiasis should be an interesting issue which will be considered in the future.



---

*Source: 5878424-2019-07-09.xml*