# Stability Analysis and Optimal Control Strategies of an Echinococcosis Transmission Model

Authors: Run Yang; Jianglin Zhao; Yong Yan
Journal: Computational and Mathematical Methods in Medicine
 (2022)

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## Abstract

This paper presents a deterministic compartmental model for echinococcosis transmission dynamics. The basic reproduction number of the model determines the existence and stability of the disease-free and disease-endemic equilibrium points. We further formulate the optimal control problem and obtain the necessary conditions to minimize the number of infected individuals and the associated costs. Numerical simulations show that optimal control strategies can significantly reduce the number of infected individuals to lower levels. Environmental disinfection may be essential for the elimination of infections. The results of this study will be beneficial for the prevention and control of echinococcosis in the Ganzi Tibetan Autonomous Prefecture and other areas of China.


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## Body

## 1. Introduction

Human echinococcosis is a parasitic zoonosis caused by infection with the larval stage of the tapeworm Echinococcus. Note that more than 1 million people are affected with echinococcosis at any one time, and human echinococcosis is often expensive and complicated to treat and may require extensive surgery and prolonged drug therapy [1]. As a result, human echinococcosis poses a significant burden on patients and health care. The life cycle of Echinococcus consists of three stages: egg, larva, and adult (see [2–6]). Adults reside in the definitive hosts (mainly dogs), produce eggs that are passed in the feces, contaminate the environment (for example, water, dog’s fur, vegetables, grass, and soil), and are immediately infectious. After ingestion by the intermediate host (mainly sheep, goats, and cattle), Echinococcus eggs (EEs) hatch and release six-hooked oncospheres which migrate into various organs (especially the liver and lung) and then develop into a hydatid cyst. The definitive hosts ingest the cyst-containing organs of the infected intermediate hosts and become infected. The protoscolices begin to develop into adult stages. Humans are accidental intermediate hosts because they acquire the infection in the same way as other intermediate hosts but without the biological contribution of spreading the infection to the definitive hosts. For more related knowledge about echinococcosis, please refer to [1, 2].Wang et al. [7] developed a deterministic compartment model of echinococcosis transmission and pointed out that the strict slaughter inspection with regard to meat inspection and offal disposal, dog anthelmintics, and public health education about hygiene and dog contact could effectively reduce the spread of echinococcosis. Wu et al. [8] stressed that human inventions (deworming EEs and killing wild dogs) could be the most effective way to control the spread of echinococcosis. Rong et al. [9] showed that promoting public health education and disposing of stray dogs could significantly help control echinococcosis spreading. Hassan and Munganga [10] emphasized that treating red foxes only or disinfecting the environment alone will not be adequate to eradicate the parasite from the community, and a combination of both control strategies would be more effective in controlling the transmission of the disease in the population. Zhu et al. [11] suggested that the low evacuation rate and high mortality rate of EEs could contribute to a significant reduction in human infection cases. Furthermore, they noted that keeping humans away from EEs and enhancing treatment rates would be highly effective in preventing echinococcosis transmission in humans. Tamarozzi et al. [12] confirmed that environmental contamination, particularly through hand-to-mouth transmission, might be of primary importance from an overall appraisal of published literatures. Craig et al. [13] said that the five key elements, preventing dogs from accessing offal, treating dogs with dewormers, meat inspection, no home slaughter, and health education on hygiene and dog contact, are still valid for reducing the spread of echinococcosis today. Zhao and Yang [14] stated that optimal control strategies aimed at minimizing the number of infected individuals and the associated costs could effectively reduce the transmission of echinococcosis.Hassan and Munganga [10] stated that the joint control is more effective than the single one. Thus, the optimal problem of control measures is a worthwhile discussion since the optimal control strategies could reduce the number of infected individuals at the lowest cost level (see [14–21] for example). Although humans are accidental intermediate hosts and do not participate in the life cycle of Echinococcus, once a person becomes infected with the disease, it will place a significant burden on their health and finances. According to [14], we will consider human infection with echinococcosis in our modeling and discuss optimal control strategies by controlling the intensity of deworming, frequency of environmental disinfection, level of strict slaughter inspection, and frequency of health education.The rest of this paper is organized as follows. In Section2, a dynamical model of echinococcosis transmission with control is given. A mathematical analysis of the model is presented in Section 3. The optimal control problem is formulated, and the necessary conditions are given in Section 4. Numerical simulations are shown to explore the optimal controls in Section 5. A conclusion and discussion are given in Section 6.


## 2. Model Formulation

For the dog population, the definitive hosts are divided into two classes: susceptibleSdt and infected Idt. For the livestock population, the intermediate hosts are decomposed into susceptible Slt and infected Ilt. The definitive hosts are infected by ingesting the cyst-containing organs of the infected intermediate hosts. The intermediate hosts become infected by ingesting EEs from their living environment. Humans act as accidental intermediate hosts, acquiring infection when they ingest EEs. The total human population is separated into susceptible Sht, exposed Eht, and infected Iht. The EEs released from the feces of infected dogs are denoted by Xt. A schematic diagram for the dynamical transmission of echinococcosis is demonstrated in Figure 1. Based on this schematic diagram, it has the following transmission model:
(1)Ṡd=Λd−1−u1tεβdSdIl−μdSd+δdu2tId,İd=1−u1tεβdSdIl−μdId−δdu2tId,Ẋ=γId−μxX−chu3tX,Ṡl=Λl−βlSlX−εSl−μlSl,İl=βlSlX−εIl−μlIl,Ṡh=Λh−1−u4tβhShX−μhSh+δhIh,Ėh=1−u4tβhShX−ωEh−μhEh,İh=ωEh−μhIh−δhIh.Figure 1
Schematic diagram of the transmission dynamics of Echinococcus.In (1), Λd denotes the annual recruitment rate of the susceptible dogs, μd is the natural death rate of the dog population, δd represents the recovery rate of infectious dogs, and u2t∈0,1 is the control on the use of praziquantel (PZQ) dosing for infected dogs. 1−u1tεβdSdIl describes the transmission of echinococcosis between susceptible definitive hosts and infectious intermediate hosts, u1t∈0,1 is the control on the use of very strict slaughter inspection with regard to meat inspection and offal disposal for livestock, and ε is the home slaughter fraction of livestock. In resource-poor pastoral regions, livestock traditionally pervade home slaughter, so the dog infection rate depends on the home slaughter fraction ε of livestock being available for offal of infected livestock. γ denotes the released rate of EEs by infectious definitive hosts, μx accounts for the natural death rate of EEs, ch is the death rate of EEs because of environmental disinfection, and u3t∈0,1 is the control on the use of environmental disinfection for EEs. Λl represents the annual recruitment rate of susceptible intermediate hosts, μl is the natural death rate of livestock, and βlSlX depicts the transmission of EEs to livestock by ingesting EEs in the environment. Λh is the annual recruitment rate of a susceptible human population, μh represents the natural death rate of humans, 1−u4tβhShX describes the transmission of echinococcosis between susceptible humans and an infectious population, and u4t is the control on the use of health education for humans. When ingesting EEs, humans are infected and then undergo an incubation period 1/ω. The infectious humans recover by an infectious period of mean duration 1/δh.


## 3. Model Analysis

When the control variables are considered constant, some mathematical analysis results of model (1) can be obtained.


### 3.1. Positivity and Boundary of Solutions

Theorem 1.
(i)
The solution of model (1) with positive initial conditions is positive for all t>0(ii)
All positive solutions of model (1) with positive initial conditions have the upper boundary in ℝ+7Proof.
(i)
LetSdt,Idt,Xt,Slt,Ilt,Sht,Eht,Iht be a solution of model (1) with positive initial values. Define(2)t1=supt>0:Sdτ>0,Idτ>0,Xτ>0,Slτ>0,Ilτ>0,Shτ>0,Ehτ>0,Ihτ>0,
for allτ∈0,t.
SinceminSd0,Id0,X0,Sl0,Il0,Sh0,Eh0,Ih0>0, then there must be t1>0. If t1<∞, it gives
(3)minSdt1,Idt1,Xt1,Slt1,Ilt1,Sht1,Eht1,Iht1=0,and Sdt>0,Idt>0,Xt>0,Slt>0,Ilt>0,Sht>0,Eht>0,Iht>0 for all t∈0,t1.
On the other hand, the first equation of model (1) could be written as
(4)ddtSdexp∫0t1−u1εβdIl+μdds=Λd+σu2Idexp∫0t1−u1εβdIl+μdds.
Consequently,(5)Sdt1=Sd0exp−∫0t11−u1εβdIl+μddt+exp−∫0t11−u1εβdIl+μddt×∫0t1Λd+σu2Idexp∫0t1−u1εβdIl+μddsdt,
which implies thatSdt1>0. A similar approach could be applied to show that Idt1>0,Slt1>0,Ilt1>0 and Et1>0, which is a contradiction. Therefore, t1=∞.
Hence, all solutions of model (1) with positive initial conditions remain positive when t>0.
(ii)
The first two equations of model (1) could be transformed into(6)dSd+Iddt=Λd−μdSd+Id≤Λd−μdSd+Id.
Thus,limsupt⟶∞Sd+Id≤Λd/μd.
The fourth and fifth equations of model (1) could be transformed into
(7)dSl+Ildt=Λl−μl+εSl+Il≤Λl−μl+εSl+Il,which leads to limsupt⟶∞Sl+Il≤Λl/μl+ε.
The last three equations of model (1) give
(8)dSh+Eh+Ihdt=Λh−μhSh+Eh+Ih≤Λh−μhSh+Eh+Ih,which yields limsupt⟶∞Sh+Eh+Ih≤Λh/μh.
From the third equation of model (1), there is
(9)dXdt=γId−μx+chu3X≤γΛdμd−μx+chu3X.
Hence,limsupt⟶∞X≤γΛd/μdμx+chu3.
Let(10)Γ=Sd,Id,E,Sd,Id,Sh,Ih∈ℝ+7:Sd+Id≤Λdμd,X≤γΛdμdμe+chu3,Sl+Il≤Λlμl+ε,Sh+Eh+Ih≤Λhμh.
Thus, all positive solutions of model (1) with positive initial conditions ultimately have the upper boundary in ℝ+7. The closed set Γ is positively invariant and attracts the solution to model (1).


### 3.2. Equilibrium Points and Stability Analysis

In this section, some mathematical analysis results of model (1) can be obtained when the controls are supposed to be constant.The disease-free equilibrium of model (1) is denoted by
(11)Edfe=Sd0,0,0,Sl0,0,Sh0,0,0=Λdμd,0,0,Λlε+μl,0,Λhμh,0,0.In the next, the next-generation matrix approach [22] will be applied for computing the basic reproduction number R0. The matrix of new infection F and the matrix of transition V are defined as follows:
(12)F=1−u1εβdSdIlγIdβlSlX1−u4βhShX0,V=μd+δdu2Idμx+chu3Xε+μlIlω+μhEh−ωEh+μh+δhIh.Furthermore, the Jacobian matrices ofF and V at the disease-free equilibrium Edfe are, respectively, obtained by
(13)F=001−u1εβdΛdμd00γ00000βlΛlε+μl00001−u4βhΛhμh00000000,V=μd+δdu200000μx+chu300000ε+μl00000ω+μh0000−ωμh+δh.Then, the basic reproduction number that is the largest eigenvalue with a large domain of the next generation matrixFV−1 is given by
(14)R0=R0x·R0d·R0l3,where
(15)R0x=γμx+chu3,R0d=1μd+δdu2·Λdμd·1−u1εβd,R0l=1ε+μl·Λlε+μl·βl.Here,R0x denotes the average number of EEs that might be ingested by the intermediate host livestock and humans, R0d describes the average number of infected dogs by infected livestock, and R0l accounts for the average number of infected livestock by EEs. For more ecological and epidemiological significance in (14), please refer to [7, 8, 22].Assume thatEee=Sd∗,Id∗,X∗,Sl∗,Il∗,Sh∗,Eh∗,Ih∗ should be the endemic equilibrium of model (1). Let the right-hand sides of model (1) vanish. By solving these equations, it gives
(16)Sd∗=μd+δdu2Id∗1−u1εβdIl∗,Id∗=μe+chu3γX∗,Sl∗=ΛlβlX∗+ε+μl,Il∗=βlSl∗X∗ε+μl,X∗=μd+δdu2ε+μl2βlμd+δdu2ε+μl+1−u1εβdΛlR03−1,Sh∗=ω+μhEh∗1−u4βhX∗,Ih∗=ωEh∗μh+δh,Eh∗=1−u4βhX∗μh+δhΛhμhω+μhμh+δh+1−u4βhX∗ω+μh+δh.Then, it is clear that model (1) has a uniquely endemic equilibrium Eee if and only if R0>1.A similar method from [7, 9] is used to obtain the following results. Appendix A gives the detailed proof of Theorem 2. In Appendix B, the proof of Theorem 3 is presented. The proof of Theorem 4 can be displayed in Appendix C.Theorem 2.
The disease-free equilibriumEdfe is locally asymptotically stable if R0<1 and is unstable if R0>1.Theorem 3.
The disease-free equilibriumEdfe is globally asymptotically stable if R0<1.Theorem 4.
The uniquely endemic equilibriumEee of model (1) is globally asymptotically stable when R0>1.


## 3.1. Positivity and Boundary of Solutions

Theorem 1.
(i)
The solution of model (1) with positive initial conditions is positive for all t>0(ii)
All positive solutions of model (1) with positive initial conditions have the upper boundary in ℝ+7Proof.
(i)
LetSdt,Idt,Xt,Slt,Ilt,Sht,Eht,Iht be a solution of model (1) with positive initial values. Define(2)t1=supt>0:Sdτ>0,Idτ>0,Xτ>0,Slτ>0,Ilτ>0,Shτ>0,Ehτ>0,Ihτ>0,
for allτ∈0,t.
SinceminSd0,Id0,X0,Sl0,Il0,Sh0,Eh0,Ih0>0, then there must be t1>0. If t1<∞, it gives
(3)minSdt1,Idt1,Xt1,Slt1,Ilt1,Sht1,Eht1,Iht1=0,and Sdt>0,Idt>0,Xt>0,Slt>0,Ilt>0,Sht>0,Eht>0,Iht>0 for all t∈0,t1.
On the other hand, the first equation of model (1) could be written as
(4)ddtSdexp∫0t1−u1εβdIl+μdds=Λd+σu2Idexp∫0t1−u1εβdIl+μdds.
Consequently,(5)Sdt1=Sd0exp−∫0t11−u1εβdIl+μddt+exp−∫0t11−u1εβdIl+μddt×∫0t1Λd+σu2Idexp∫0t1−u1εβdIl+μddsdt,
which implies thatSdt1>0. A similar approach could be applied to show that Idt1>0,Slt1>0,Ilt1>0 and Et1>0, which is a contradiction. Therefore, t1=∞.
Hence, all solutions of model (1) with positive initial conditions remain positive when t>0.
(ii)
The first two equations of model (1) could be transformed into(6)dSd+Iddt=Λd−μdSd+Id≤Λd−μdSd+Id.
Thus,limsupt⟶∞Sd+Id≤Λd/μd.
The fourth and fifth equations of model (1) could be transformed into
(7)dSl+Ildt=Λl−μl+εSl+Il≤Λl−μl+εSl+Il,which leads to limsupt⟶∞Sl+Il≤Λl/μl+ε.
The last three equations of model (1) give
(8)dSh+Eh+Ihdt=Λh−μhSh+Eh+Ih≤Λh−μhSh+Eh+Ih,which yields limsupt⟶∞Sh+Eh+Ih≤Λh/μh.
From the third equation of model (1), there is
(9)dXdt=γId−μx+chu3X≤γΛdμd−μx+chu3X.
Hence,limsupt⟶∞X≤γΛd/μdμx+chu3.
Let(10)Γ=Sd,Id,E,Sd,Id,Sh,Ih∈ℝ+7:Sd+Id≤Λdμd,X≤γΛdμdμe+chu3,Sl+Il≤Λlμl+ε,Sh+Eh+Ih≤Λhμh.
Thus, all positive solutions of model (1) with positive initial conditions ultimately have the upper boundary in ℝ+7. The closed set Γ is positively invariant and attracts the solution to model (1).


## 3.2. Equilibrium Points and Stability Analysis

In this section, some mathematical analysis results of model (1) can be obtained when the controls are supposed to be constant.The disease-free equilibrium of model (1) is denoted by
(11)Edfe=Sd0,0,0,Sl0,0,Sh0,0,0=Λdμd,0,0,Λlε+μl,0,Λhμh,0,0.In the next, the next-generation matrix approach [22] will be applied for computing the basic reproduction number R0. The matrix of new infection F and the matrix of transition V are defined as follows:
(12)F=1−u1εβdSdIlγIdβlSlX1−u4βhShX0,V=μd+δdu2Idμx+chu3Xε+μlIlω+μhEh−ωEh+μh+δhIh.Furthermore, the Jacobian matrices ofF and V at the disease-free equilibrium Edfe are, respectively, obtained by
(13)F=001−u1εβdΛdμd00γ00000βlΛlε+μl00001−u4βhΛhμh00000000,V=μd+δdu200000μx+chu300000ε+μl00000ω+μh0000−ωμh+δh.Then, the basic reproduction number that is the largest eigenvalue with a large domain of the next generation matrixFV−1 is given by
(14)R0=R0x·R0d·R0l3,where
(15)R0x=γμx+chu3,R0d=1μd+δdu2·Λdμd·1−u1εβd,R0l=1ε+μl·Λlε+μl·βl.Here,R0x denotes the average number of EEs that might be ingested by the intermediate host livestock and humans, R0d describes the average number of infected dogs by infected livestock, and R0l accounts for the average number of infected livestock by EEs. For more ecological and epidemiological significance in (14), please refer to [7, 8, 22].Assume thatEee=Sd∗,Id∗,X∗,Sl∗,Il∗,Sh∗,Eh∗,Ih∗ should be the endemic equilibrium of model (1). Let the right-hand sides of model (1) vanish. By solving these equations, it gives
(16)Sd∗=μd+δdu2Id∗1−u1εβdIl∗,Id∗=μe+chu3γX∗,Sl∗=ΛlβlX∗+ε+μl,Il∗=βlSl∗X∗ε+μl,X∗=μd+δdu2ε+μl2βlμd+δdu2ε+μl+1−u1εβdΛlR03−1,Sh∗=ω+μhEh∗1−u4βhX∗,Ih∗=ωEh∗μh+δh,Eh∗=1−u4βhX∗μh+δhΛhμhω+μhμh+δh+1−u4βhX∗ω+μh+δh.Then, it is clear that model (1) has a uniquely endemic equilibrium Eee if and only if R0>1.A similar method from [7, 9] is used to obtain the following results. Appendix A gives the detailed proof of Theorem 2. In Appendix B, the proof of Theorem 3 is presented. The proof of Theorem 4 can be displayed in Appendix C.Theorem 2.
The disease-free equilibriumEdfe is locally asymptotically stable if R0<1 and is unstable if R0>1.Theorem 3.
The disease-free equilibriumEdfe is globally asymptotically stable if R0<1.Theorem 4.
The uniquely endemic equilibriumEee of model (1) is globally asymptotically stable when R0>1.


## 4. Optimal Control

To obtain the optimal control strategies, an objective functional is defined by(17)Ju=∫0Tgϕ,u,tdt=∫0TId+Il+Ih+12c1u12+c2u22+c3u32+c4u42dt,subject to the state system (1). ϕ=Sd,Id,X,Sl,Il,Sh,Eh,Ih is the solution of model (1) with positive initial values and u=u1t,u2t,u3t,u4t. c1,c2, c3, and c4 represent the weight constants of the control variables u1, u2, u3, and u4, respectively. 1/2c1u12,1/2c2u22, 1/2c3u32, and 1/2c4u42 denote the costs of home slaughter inspection, anthelmintic treatment, environmental disinfection, and health education, respectively.The objective of the optimal control problem (17) is to find a control set that minimizes the infected dogs, the infected livestock, and the infected humans when minimizing the control cost function. Let U=u=u1t,u2t,u3t,u4t:0≤uit≤1,t∈0,T,i=1,2,3,4 be a measurable set. Then, there needs to be the optimal control u∗=u1∗t,u2∗t,u3∗t,u4∗t such that
(18)Ju∗=minJu:u∈U.The necessary conditions that determine the optimal controlu∗ satisfying (5) with constraint model (1) are derived from Pontryagin’s Maximum Principle. Then, the optimal control problem (18) is transformed into minimizing the following Hamiltonian function:
(19)H=gϕ,u,t+∑i=18λifiϕ,u,t,wherefiϕ,u,t,i=1,2,3,4,5,6,7,8, are the right-hand sides of model (1). And λi,i=1,2,3,4,5,6,7,8, are the adjoint variables that satisfy the following costate system:
(20)λ̇1=−∂H∂Sd=1−u1εβdIlλ1−λ2+λ1μd,λ̇2=−∂H∂Id=−1−λ1δdu2+λ2μd+δdu2−λ3γ,λ̇3=−∂H∂X=λ3μx+chu3+λ4−λ5βlSl+λ6−λ71−u4βhSh,λ̇4=−∂H∂Sl=λ4βlX+ε+μl−λ5βlX,λ̇5=−∂H∂Il=−1+1−u1εβdSdλ1−λ2+λ5μl+ε,λ̇6=−∂H∂Sh=1−u4βhXλ6−λ7+λ6μh,λ̇7=−∂H∂Eh=λ7ω+μh−λ8ω,λ̇8=−∂H∂Ih=−1−λ6δh+λ8δh+μh,with boundary conditionsλiT=0,i=1,2,3,4,5,6,7,8. Additionally, the optimality conditions ∂H/∂ui=0,i=1,2,3,4, lead to the optimal controls:
(21)ui∗=min1,max0,uic,i=1,2,3,4,where
(22)u1c=εβdSdIlλ2−λ1c1,u2c=δdIdλ2−λ1c2,u3c=λ3chXc3,u4c=λ7−λ6βhShXc4.


## 5. Numerical Simulations

In this section, the numerical results of different optimal control scenariosu1, u2, u3, and u4 are presented. The numerical solution of the optimality system is solved by the forward-backward sweep method [27]. The ode45 solver in MATLAB is used to solve (1) with initial values Sd0=1.686×105, Id0=4×104, Sl0=3.335×106, Il0=5×105, X0=2×107, Sh0=8.05×105, Eh0=8.064×103, and Ih0=576, where Sd0 can be estimated from [28] and Sl0,Sh0,Ih0 can be estimated from [30]. The other initial values of model (1) are assumed. The costate system (20) with boundary conditions λiT=0,i=1,2,3,4,5,6,7,8, is numerically obtained from the backward Runge-Kutta scheme. The control variables (21) are updated by entering the new state and adjoint values until the current state; the adjoint and control values are negligibly close. It is well established that the cost of anthelmintic treatment is more expensive than that of environmental disinfection, while the cost of slaughter inspection is cheaper than that of environmental disinfection. On the other hand, the cost of health education is cheaper than that of slaughter inspection. Hence, the weighting constants are considered as c1=50, c2=90, c3=70, and c4=60. All other parameters are listed in Table 1. Λl is estimated by using the data from the Statistics Bureau of Ganzi Tibetan Autonomous Prefecture [28]. Λd, ε, δd, and δh are estimated by using the data from Zou [30]. The average life expectancy of people in Ganzi Tibetan Autonomous Prefecture (see [29]) was 72.10 years in 2016. Therefore, the natural death rate μh of humans in Ganzi Tibetan Autonomous Prefecture is estimated as μh=1/72.1≈0.0139. The death rate of echinococcosis eggs due to environmental disinfection cannot be directly acquired. It is instead assumed that the parasite egg mortality rate induced by environmental disinfection should arrive at ten times higher than the natural death rate. The combined employment of two, three, and four control measures will be studied. The following scenarios are considered:
(A)
Scenario one: coupled control measures
(i)
Strategy A: slaughter inspection and anthelmintic treatment (u1,u2)(ii)
Strategy B: slaughter inspection and environmental disinfection (u1,u3)(iii)
Strategy C: slaughter inspection and health education (u1,u4)(iv)
Strategy D: anthelmintic treatment and environmental disinfection (u2,u3)(v)
Strategy E: anthelmintic treatment and health education (u2,u4)(vi)
Strategy F: environmental disinfection and health education (u3,u4)(B)
Scenario two: threefold control measures
(i)
Strategy G: slaughter inspection, anthelmintic treatment, and environmental disinfection (u1,u2,u3)(ii)
Strategy H: slaughter inspection, anthelmintic treatment, and health education (u1,u2,u4)(iii)
Strategy I: slaughter inspection, environmental disinfection, and health education (u1,u3,u4)(iv)
Strategy J: anthelmintic treatment, environmental disinfection, and health education (u2,u3,u4(C)
Scenario three: fourfold control measures
(i)
Strategy K: slaughter inspection, anthelmintic treatment, environmental disinfection, and health education (u1,u2,u3,u4)Table 1
Description of parameters of model (1).
ParametersInterpretationUnitsSourceΛdRecruitment rate of dogs21.1×104 year−1EstimatedβdTransmission rate from livestock to dogs5.8×10−8 year−1[7]μdNatural death rate of dogs0.08 year−1[7]δdRecovery rate of infected dogs0.21 year−1EstimatedγReleased rate from infected dogs9.7 year−1[7]μxDeath rate of EEs1 year−1[8]chDisinfection-induced EE mortality rate10 year−1AssumedΛlRecruitment rate of livestock54.33×104 year−1EstimatedβlInfection rate of livestock by ingesting EEs7.4×10−8 year−1[7]εFraction of home-slaughtered livestock0.189EstimatedμlNatural death rate of livestock0.152 year−1[9]ΛhRecruitment rate of humans1.03×104[29]βhInfection rate of humans by ingesting EEs4.2×10−11 year−1[7]ωReciprocal of human incubation period1/14 year−1[7]μhNatural death rate of humans0.0139 year−1EstimatedδhRecovery rate of humans0.041 year−1Estimatedu1tEffectiveness of home slaughter inspection0–1Assumedu2tEffectiveness of anthelmintic treatment0–1Assumedu3tEffectiveness of environmental disinfection0–1Assumedu4tEffectiveness of health education0–1AssumedFor Strategy A, the slaughter inspection controlu1 and the anthelmintic treatment control u2 are merely carried out while the environmental disinfection control u3 and the health education control u4 are chosen to be ignored. Figure 2(a) shows the paths of optimal controls u1∗ and u2∗. The slaughter inspection (blue dash-dot line in Figure 2(a)) should be executed 100% for 15 years and then decreases gradually to zero. Meanwhile, the anthelmintic treatment (red dotted line in Figure 2(a)) needs to start the 100% use for 10 years and then declines to zero. Figures 2(b)–2(d) illustrates the effect of the optimal controls u1∗ and u2∗. When there is no control (see the blue dashed lines in Figures 2(b)–2(d)), the disease is prevalent. However, when the optimal controls are implemented (see the red dotted lines in Figures 2(b)–2(d)), the number of infected dogs, infected livestock, and infected humans could be significantly minimized to the lower level Id,Il,Ih=61,207,6. For Strategy B, the slaughter inspection control u1 and the environmental disinfection control u3 are merely applied in (17) while the anthelmintic treatment and health education are not considered, i.e., u2=0,u4=0. Figure 3(a) presents the profiles of optimal controls u1∗ and u3∗. The slaughter inspection (blue dash-dot line in Figure 3(a)) is done 100% intensively for 16 years and then decreases gradually till the end of control. Meanwhile, the environmental disinfection control (green dash-dot line in Figure 3(a)) begins with 100% use for 32 years and then declines to zero. Figures 3(b)–3(d) display the effect of u1∗ and u3∗. It is obvious that there is a considerable difference in the number of infected dogs, infected livestock, and infected humans between the controlled cases (see the blue dashed lines in Figures 3(b)–3(d)) and the cases without control (see the red dotted lines in Figures 3(b)–3(d)). The number of infected dogs, infected livestock, and infected humans under Strategy B could drop to the lower level Id,Il,Ih=24,61,6. For Strategy C, u1 and u4 are merely considered in (17) while u2 and u3 are ignored. Figure 4(a) shows the profiles of optimal controls u1∗ and u4∗. The slaughter inspection (blue dash-dot line in Figure 4(a)) is kept at the maximum use of 100% for 34 years and then declines gradually to zero. On the contrary, the health education u4∗ (black dashed line in Figure 4(a)) declines from the maximum use of 46.1% to zero in 16 years. Figures 4(b)–4(d) display that the number of infected dogs, infected livestock, and infected humans could drop to the lower level Id,Il,Ih=319,1070,8. For Strategy D, u2 and u3 are implemented to optimize the objective functional (17) while u1=0 and u4=0. Figure 5(a) shows the paths of u2∗ and u3∗. Both the anthelmintic treatment (red dotted line in Figure 5(a)) and the environmental disinfection (green dash-dot line in Figure 5(a)) should be done 100% intensively for 14 years and then decline gradually to zero. Figures 5(b)–5(d) display that there could exist a considerable significance for reducing the number of infected dogs, infected livestock, and infected humans (blue dashed line) that drops to the lower level Id,Il,Ih=6,16,5. For Strategy E, u2 and u4 are considered while u1=0 and u3=0. Figure 6(a) shows the profiles of u2∗ and u4∗. The control u2∗ (red dotted line in Figure 6(a)) has a 100% use for 25 years and then drops gradually to zero. Meanwhile, the control effort u4∗ (black dashed line in Figure 6(a)) decays from the maximum use of 46.1% to zero in 14 years. Figures 6(b)–6(d) suggest that Strategy E could provide a significant reduction in the number of infected dogs, infected livestock, and infected humans that decreases to the lower level Id,Il,Ih=118,396,7. For Strategy F, u3 and u4 are considered while u1=0 and u2=0. Figure 7(a) presents the paths of u3∗ and u4∗. The control u3∗ (green dash-dot line in Figure 7(a)) needs to perform a 100% use for 39 years and then gradually decreases to zero. Meanwhile, the control u4∗ (black dashed line in Figure 7(a)) drops rapidly from the maximum use of 46.1% to zero in five years. Figures 7(b)–7(d) show that Strategy F could provide a significant reduction in the number of infected dogs, infected livestock, and infected humans that drops to the lower level Id,Il,Ih=35,84,6.Figure 2
Simulation results for Strategy A: (a) depicts the profiles of optimal controlsu1∗ and u2∗; (b–d) represent the number of infected dogs, infected livestock, and infected humans, respectively.
(a)(b)(c)(d)Figure 3
Simulation results for Strategy B: (a) depicts the profiles of optimal controlsu1∗ and u3∗; (b–d) represent the number of infected dogs, infected livestock, and infected humans, respectively.
(a)(b)(c)(d)Figure 4
Simulation results for Strategy C: (a) depicts the profiles of optimal controlsu1∗ and u4∗; (b–d) represent the number of infected dogs, infected livestock, and infected humans, respectively.
(a)(b)(c)(d)Figure 5
Simulation results for Strategy D: (a) depicts the profiles of optimal controlsu2∗ and u3∗; (b–d) represent the number of infected dogs, infected livestock, and infected humans, respectively.
(a)(b)(c)(d)Figure 6
Simulation results for Strategy E: (a) depicts the profiles of optimal controlsu2∗ and u4∗; (b–d) represent the number of infected dogs, infected livestock, and infected humans, respectively.
(a)(b)(c)(d)Figure 7
Simulation results for Strategy F: (a) depicts the profiles of optimal controlsu3∗ and u4∗; (b–d) represent the number of infected dogs, infected livestock, and infected humans, respectively.
(a)(b)(c)(d)For Strategy G, the controlsu1, u2, and u3 are considered while u4=0. Figure 8(a) shows the paths of u1∗, u2∗, and u3∗. The control u1∗ (blue dash-dot line in Figure 8(a)) has a 100% use for 12 years and then decreases gradually to zero. Both the control u2∗ (red dotted line in Figure 8(a)) and the control u3∗ (green dash-dot line in Figure 8(a)) have a 100% use for 10 years and then drop gradually to zero. Figures 8(b)–8(d) suggest that the number of infected dogs, infected livestock, and infected humans under Strategy G (blue dashed line) could be significantly reduced to a lower level Id,Il,Ih=6,15,5 compared to no control (red dotted line). For Strategy H, the controls u1, u2, and u4 are considered while u3=0. Figure 9(a) presents the paths of u1∗, u2∗, and u4∗. The slaughter inspection (blue dash-dot line in Figure 9(a)) should be done 100% for 15 years and then decreases gradually to zero. The anthelmintic treatment u2∗ (red dotted line in Figure 9(a)) has the 100% use for 10 years and then declines to zero. On the contrary, the health education u4∗ (black dashed line in Figure 9(a)) drops rapidly from the maximum use of 46.1% to zero in 11 years. Figures 9(b)–9(d) display that there is a significance for u1∗, u2∗, and u4∗ reducing the number of infected dogs, infected livestock, and infected humans (blue dashed line) that drops to a lower level Id,Il,Ih=62,207,6. For Strategy I, u1, u3, and u4 are implemented in (17) while u2=0. Figure 10(a) shows the paths of u1∗, u3∗, and u4∗. The control u1∗ (blue dash-dot line in Figure 10(a)) should be done 100% intensively for 16 years and then declines gradually to zero, while the control u3∗ (green dash-dot line in Figure 10(a)) has the maximum use (100%) for 32 years before dropping gradually to zero. The control u4∗ drops rapidly from the maximum use (46.1%) to zero in five years. Figures 10(b)–10(d) suggest that there could be a considerable significance for reducing the number of infected dogs, infected livestock, and infected humans (blue dashed line) that decreases to a lower level Id,Il,Ih=24,61,6. For Strategy J, u2, u3, and u4 are considered while u1=0. The optimal controls u2∗, u3∗, and u4∗ are presented in Figure 11(a). Both u2∗ (red dotted line in Figure 11(a)) and u3∗ (green dash-dot line in Figure 11(a)) have a 100% use for 14 years and then drop gradually to zero. Meanwhile, u4∗ (black dashed line in Figure 11(a)) drops rapidly from the maximum use of 46.1% to zero in four years. Figures 11(b)–11(d) show that Strategy J (blue dashed line) could provide a significant reduction in the number of infected dogs, infected livestock, and infected humans that deceases to the lower level Id,Il,Ih=6,16,5.Figure 8
Simulation results for Strategy G: (a) depicts the profiles of optimal controlsu1∗, u2∗, and u3∗; (b–d) represent the number of infected dogs, infected livestock, and infected humans, respectively.
(a)(b)(c)(d)Figure 9
Simulation results for Strategy H: (a) depicts the profiles of optimal controlsu1∗, u2∗, and u4∗; (b–d) represent the number of infected dogs, infected livestock, and infected humans, respectively.
(a)(b)(c)(d)Figure 10
Simulation results for Strategy I: (a) depicts the profiles of optimal controlsu1∗, u3∗, and u4∗; (b–d) represent the number of infected dogs, infected livestock, and infected humans, respectively.
(a)(b)(c)(d)Figure 11
Simulation results for Strategy J: (a) depicts the profiles of optimal controlsu2∗, u3∗, and u4∗; (b–d) represent the number of infected dogs, infected livestock, and infected humans, respectively.
(a)(b)(c)(d)For Strategy K, all the controlsu1,u2,u3, and u4 are considered in (17). The optimal controls u1∗,u2∗,u3∗, and u4∗ are displayed in Figure 12(a). The control u1∗ (blue dash-dot line in Figure 11(a)) starts to have a 100% use for 12 years and then decreases gradually to zero. Both the control u2∗ (red dotted line in Figure 11(a)) and the control u3∗ (green dash-dot line in Figure 12(a)) have a 100% use for 10 years and then decline gradually to zero. Meanwhile, the control u4∗ (black dashed line in Figure 12(a)) drops rapidly from the maximum use of 46.1% to zero in four years. Figures 12(b)–12(d) show that Strategy K (blue dashed line) has a significant reduction in the number of infected dogs, infected livestock, and infected humans that could drop to the lower level Id,Il,Ih=6,15,5.Figure 12
Simulation results for Strategy K: (a) depicts the profiles of optimal controlsu1∗, u2∗, u3∗, and u4∗; (b–d) represent the number of infected dogs, infected livestock, and infected humans, respectively.
(a)(b)(c)(d)


## 6. Conclusion and Discussion

This paper presents and analyzes a deterministic compartmental system for echinococcosis transmission dynamics under the intervention of constant slaughter inspection, anthelmintic treatment, environmental disinfection, and health education. The existence and stability of the disease-free and disease-endemic equilibrium points of the model are discussed. It finds that the basic reproduction number determines entirely whether the disease is extinct or not endemic. In the absence of control measures, the basic reproduction number in Ganzi Tibetan Autonomous Prefecture is estimated to beR0=1.0662>1. This means that echinococcosis is an endemic disease. Craig et al. [13] stated that it is difficult to eliminate the spread of echinococcosis in scattered seminomadic remote communities, even if the Echinococcosis Control Program in Western China is carried out by using PZQ for the dog-dosing frequency monthly. Therefore, comprehensive interventions mainly including slaughter inspection, anthelmintic treatment, environmental disinfection, and health education should be taken into account to control the transmission of echinococcosis. Figures 2–12 have shown that the optimal strategies from Strategy A to Strategy K have a considerable significance in reducing the number of infected dogs, infected livestock, and infected humans. The combined prevention and control measures could eliminate the prevalence of echinococcosis.Note that Strategies D, G, J, and K could reduce the number of infected dogs, infected livestock, and infected humans to a lower level than other strategies. Therefore, anthelmintic treatment and environmental disinfection may play a crucial role in controlling the number of infectious individuals. The anthelmintic treatment against echinococcosis does not eliminate the infection, and most of the time, when the treatments cease, there is a rebound in the infection (see [13]). Environmental disinfection may hence be indispensable for the prevention and control of echinococcosis. However, the importance of environmental disinfection for the prevention and control of echinococcosis is often ignored. Therefore, deworming and environmental disinfection should be the primary consideration in choosing control measures when developing an echinococcosis control and prevention program. The slaughter inspection with regard to meat inspection and offal disposal is aimed at reducing the number of infected dogs. Consequently, the number of EEs naturally decreases when the slaughter inspection is implemented. Thus, infected livestock would be reduced. From this perspective, the slaughter inspection may shorten the control time. The health education is aimed at reducing the possibility of ingestion by humans. The low evacuation rate of EEs would lead to the small possibility of ingestion by humans. Therefore, if the number of EEs drops to a certain level, the health education will become unimportant. That is to say that the health education has effectiveness in a short time for the prevention and control of echinococcosis. Hence, for faster and better prevention and control of echinococcosis, Strategy K may be recommended to be implemented in the real situation. Finally, some parameter values (for example, the death rate of EEs due to environmental disinfection) are not directly available; our model does not necessarily reflect the true picture of the prevalence of echinococcosis in the Ganzi Tibetan Autonomous Prefecture. Nevertheless, our model analysis suggests that environmental disinfection is critical to controlling the spread of echinococcosis and that the optimal integrated control strategy (Strategy K) can control the disease in the shortest possible time.



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*Source: 6154866-2022-05-23.xml*