# Mathematical Modeling of Echinococcosis in Humans, Dogs, and Sheep

Authors: Getachew Bitew Birhan; Justin Manango W. Munganga; Adamu Shitu Hassan
Journal: Journal of Applied Mathematics
 (2020)

---

## Abstract

In this paper, a model for the transmission dynamics of cystic echinococcosis in the dog, sheep, and human populations is developed and analyzed. We first model and analyze the predator-prey interaction model in these populations; then, we propose a mathematical model of the transmission dynamics of cystic echinococcosis. We calculate the basic reproduction numberR0 and prove that the disease-free equilibrium is globally asymptotically stable, and hence, the disease dies out if R0<1. We further show that the endemic equilibrium is globally asymptotically stable, and hence, the disease persists if R0>1. Numerical simulations are performed to illustrate our analytic results. We give sensitivity analysis of the key parameters and give strategies that are helpful to control the transmission of cystic echinococcosis, from which the most sensitive parameter is the transmission rate of Echinococcus’ eggs from the environment to sheep (βes). Thus, the effective controlling strategies are associated with this parameter.


---

## Body

## 1. Introduction

Echinococcosis is a parasitic disease caused by ingesting the eggs of the tapeworm genusEchinococcus through contaminated food and water in the environment. The parasite’s life cycle is maintained by carnivores, which act as definitive hosts (a dog, fox, canine, felid, or hyenid), and by intermediate hosts, which are usually herbivores (e.g., sheep, goats, cattle, camels, and cervids) [1]. Different species of Echinococcus cause different diseases in humans. Of the different forms of echinococcosis occurring in humans, cystic echinococcosis (CE) and alveolar echinococcosis (AE) are of special importance due to their wide geographic distribution and their medical and economic impact [2, 3]. Echinococcus granulosus (E. granulosus) causes cystic echinococcosis while E. multilocularis causes a type of echinococcosis known as alveolar echinococcosis [2–4].Cystic echinococcosis (CE) is a parasitic disease, also called “cystic hydatid disease” caused by the larval stage of small tapeworms (dog tapeworms) known asEchinococcus granulosus [2]. In its transmission dynamics, the domestic dog is the principal definitive host. The parasite is transmitted to dogs when they ingest the organs of infected animals (such as sheep) that contain hydatid cysts. The cysts develop into adult tapeworms in the dog. Infected dogs shed tapeworm eggs in their feces which contaminate the environment. Sheep, cattle, goats, and pigs ingest tapeworm eggs in the contaminated ground; once ingested, the eggs hatch and develop into cysts in the internal organs. The most common mode of transmission to humans is by the accidental consumption of water or food that has been contaminated by the fecal matter of an infected dog. Echinococcus eggs that have been deposited in the soil stay viable for up to a year [5–7].Geographically, the greatest prevalence of cystic echinococcosis in human and animal hosts is found in Australia, some parts of America (especially South America), central Asia, northern and eastern Africa, and the Mediterranean Basin [4, 5, 7]. Cystic echinococcosis typically occurs in poor pastoral regions in which sheep or other livestock are raised and in which dogs are kept, for herding or property guarding, in close proximity to households. Dogs in such regions are frequently fed offal, and, for religious and other reasons, their populations might not be curtailed [8]. The prevalence of the parasite within the offal and the contact rate between sheep and dogs will play an important role in the rate of infection within these populations [9–11].In an ecosystem, predator-prey interaction between species has been understood as a determining factor in the population dynamics. Predation is a biological interaction where a predator feeds on its prey. Predators reduce prey densities while consuming individual prey. In the absence of predators, the population of prey can grow until it approaches the carrying capacity of the environment. In turn, prey densities affect the number of predators because, when prey becomes scarce, predators die of starvation or fail to reproduce [12]. There are considerable and significant efforts to study the population dynamics in predator-prey relationships by using mathematical modeling. Lotka and Volterra initially proposed the prey-predator model [13, 14]. Afterwards, prey-predator models became an important research area in applied mathematics.Mathematical modeling has played a crucial role in understanding the dynamics of an epidemic outbreak and in developing different control measures [15–17]. In the past, various models have been developed to study the transmission dynamics of this parasitic disease, principally in the sheep-dog interaction, and much progress has been made over the last 30 years in modeling [6, 18–23]. To better understand the dynamics of transmission and to propose effective prevention and intervention strategies against cystic echinococcosis outbreak, an advanced study by inclusion of a human transmission component to Echinococcus granulosus is essential. In [24], a mathematical model of the transmission dynamics of alveolar echinococcosis (AE) that took into account the predator-prey interaction between the populations of fox and voles with analysis was presented. Based on the framework of this paper, we have developed a model that helps us to understand the transmission dynamics of cyst echinococcosis. As life cycle of Echinococcus granulosus’ eggs involves dog, sheep, and human populations, the effect of predator-prey interaction between these populations in the transmission of cystic echinococcosis needs to be considered. We formulate and analyze a mathematical model for the transmission dynamics of cystic echinococcosis, which takes into account predator-prey interactions of dog, sheep, and human populations. We consider the populations of dog to be definite host, the populations of sheep and human to be intermediate hosts, and the concentration of parasites in the environment to be the source of infection for intermediate hosts.This paper is organized as follows. In Section2, mathematical models of predator-prey interaction between the populations of sheep, dog, and human with analysis are presented, and cystic echinococcosis with assumptions is given in detail. In Section 3, well-posedness of the system is discussed, and in Section 4, both the disease-free and endemic equilibrium points are determined; local and global stabilities of these equilibrium points with calculation of the basic reproduction number are presented. Moreover, local and global sensitivity analyses, numerical simulations, and controlling strategies are presented in Section 5, and finally, conclusions are drawn in Section 6.


## 2. Mathematical Modeling

### 2.1. Predator-Prey Model

The dynamics of sheep, dog, and human populations can be represented in a predator-prey relationship. We consider the case where dogs feed only on organs of sheep. Without sheep as the source of food, the dog population (Nd) would exponentially decrease and becomes extinct. However, Nd is increased by a rate proportional to the number of encounters between the sheep population (Ns) and dog population (Nd). The growth rate of the dog population due to consumption of sheep is ωNsNd, where ω denotes the conversion efficiency of consumed sheep into the dog reproduction rate. Sheep are the main food source for human. In the absence of sheep, it is assumed that there exists some alternative food source for growth of the human population. With sheep as the source of food, the human population (Nh) is increased by a rate proportional to the number of encounters between the sheep population (Ns) and human population (Nh). The growth rate of the human population due to consumption of sheep is θNsNh, where θ denotes the conversion efficiency of consumed sheep into the human reproduction rate. Without sheep as the source of food, the human population Nh grows exponentially at a per-capita growth rate rh but eventually increases up to the carrying capacity of the environment Kh. As the number of dog and human populations increases, the population decreases in response to increased rates of consumption. The rate of consumption of sheep by dogs and humans is represented by a and c, respectively. However, in the absence of dog and human populations, the sheep population Ns increases logistically. It initially increases exponentially at a per-capita growth rate rs, and it is self-limited by the carrying capacity of the environment Ks. Furthermore, the sheep, dog, and human populations die naturally at rates represented by μs, μd, and μh, respectively. Hence, the predator-prey interaction between the sheep, dog, and human populations is represented by a system of nonlinear ordinary differential equations:(1)dNsdt=rs1−NsKsNs−aNsNd−cNhNs−μsNs,(2)dNddt=ωNsNd−μdNd,(3)dNhdt=rh1−NhKhNh+θNsNh−μhNh,with initial conditions Ns0≥0, Nd0≥0, and Nh0≥0, where rs>0, a>0, ω>0, c>0, μs>0, μd>0, μh>0, θ>0, and rh>0.Theorem 1.
The regionΩ=Ns,Nd,Nh∈ℝ+3:0≤Nst+Ndt+Nht≤η/ε, where 0<ε≤μd and η=Ks/4rsrs+ε2+Kh/4rhrh+ε2, is positively invariant (mathematically well-posed) for the model (1)–(3).Proof.
The proof of this theorem is presented in AppendixA.


#### 2.1.1. Equilibrium Points

Our next result concerns the existence of equilibrium points of the system (1)–(3) that are biologically feasible and determines the conditions for the existence of the equilibrium of the system in the feasible region. At equilibrium, equations of the model (1)–(3) are(4)rs1−NsKsNs−aNsNd−cNhNs−μsNs=0,ωNsNd−μdNd=0,rh1−NhKhNh+θNsNh−μhNh=0.Equilibrium points are as follows:(1)
The trivial fixed pointE0=0,0,0, which represents the extinction of the three populations(2)
E1=Ks1−μs/rs,0,0, which represent the coexistence of the sheep(3)
E2=0,0,Kh1−μh/rh, which represent the coexistence of humans(4)
E3=N~s,N~d,N~h=μd/ω,1/ars1−μd/ωKs−μs,0 and E4=N^s,N^d,N^h=Ksrhrs−μs+cKhμh−rh/rsrh+cθKsKh,0,Khrsrh−μh+θKsrs−μs/rsrh+cθKsKh, which represent the extinction of humans and the extinction of dogs, respectively(5)
Equilibrium point of the coexistence:(5)E5=Ns∗,Nd∗,Nh∗=μdω,1ars1−μdωKs−μs−cKhrhrh−μh+θμdω,Khrhrh−μh+θμdω,which represents the coexistence of all the three populations


#### 2.1.2. Stability of Equilibrium Points

In the next sections, the transmission dynamics of cystic echinococcosis is studied. Since the dog, sheep, and human populations are important for the transmission dynamics of the disease, we are mainly concerned with the equilibrium point of the coexistence. Thus, the detail analysis of the local stability conditions of the equilibrium points is presented in AppendixB, and the stability conditions of E5 are presented in the following theorem.Theorem 2.
(1)
E5 is locally asymptotically stable provided that a1>0, a3>0, and a1a2−a3>0, where(6)a1=rh−μh+rsμdωKs+θμdω,a2=rsμdωKsrh−μh+θμdω+cθμdKhωrhrh−μh+θμdω+μdrs1−μdωKs−μs−cKhrhrh−μh+θμdω,a3=μdrs1−μdωKs−μs−cKhrhrh−μh+θμdω.(2)
The equilibrium pointE5 is globally asymptotically stable in the Int.ℝ+3Proof.
The proof of this theorem is presented in AppendixB.


### 2.2. Modeling the Transmission Dynamics of Cystic Echinococcosis

We model the dynamics of echinococcosis as a compartmental framework. In the human population, there are four classes: the susceptible (Sh), exposed (Eh), infectious (Ih), and removed (Rh) classes. The dog population has three classes: the susceptible (Sd), exposed (Ed), and infectious (Id) classes. The sheep population has also three classes: the susceptible (Ss), exposed (Es), and infectious (Is) classes.In the dynamics of the disease transmission, susceptible sheep are infected by ingesting parasite eggs in the feces of infected definitive hosts (dogs), while humans are infected by accidentally ingestingEchinococcus granulosus’ eggs from the environment. Echinococcus granulosus’ egg contamination rate in the environment by infected dogs is represented by δ. The incorporation of the saturation is relevant since for the disease to be spread from the environment to sheep, there must be a sufficient number (saturation) from the environment that can cause infection. It measures the crowding effect of the parasite in the environment, the inhibition effect of susceptible human and sheep when their number increases, and the probability of contracting the disease by sheep and human populations from the environment. Susceptible humans Sh and susceptible sheep Ss become infected with the disease at rates βehB/χh+B and βesB/χs+B, respectively, where B denotes the concentration of Echinococcus granulosus’ eggs in the environment, βeh denotes the rate of ingestion of Echinococcus’ egg from the environment by humans, βes denotes the rate of ingestion of Echinococcus’ egg from the environment by sheep, and χh and χs are the half-saturation constants of parasite in the environment sufficient to infect humans and sheep, respectively. Susceptible dogs are infected by preying on the infected sheep. The disease transmission rate from sheep to dogs is denoted by βsd.The model is developed based on the following assumptions.(1)
The total population of dogs, sheep, and humans is assumed constant, and at stable equilibrium,E5=Nd∗,Ns∗,Nh∗, so that the birth rate and death rate of each of the populations are equal(2)
Dog, sheep, and human populations are recruited to the susceptible class by birth at ratesμd, μs, and μh, respectively(3)
The infected human population could recover from the disease naturally, and their recovery rate is represented byαh, where as sheep and dogs cannot recover once they are infected. We assume that there is no Echinococcus-induced death. However, dogs, sheep, and humans die naturally at rates μd, μs, and μh, respectivelyAll variables and parameters are introduced in Tables1 and 2, and the flowchart for the transmission dynamics of the disease is shown in Figure 1.Table 1
Definitions of variables.
VariablesDefinitionsSh, Eh, Ih, and RhNumber of susceptible, exposed, infected, and removed humans, respectivelySd, Ed, and IdNumber of susceptible, exposed, and infected dogs, respectivelySs, Es, and IsNumber of susceptible, exposed, and infected sheep, respectivelyBThe concentration ofEchinococcus granulosus’ eggs in the environmentNd∗, Ns∗, and Nh∗Total number of dogs, sheep, and humans, respectivelyTable 2
Descriptions of parameters.
ParametersDescriptionsμdNaturaldeathrate=birthrateofdogμhNaturaldeathrate=birthrateofhumanμsNaturaldeathrate=birthrateofsheepμeNatural death rate ofEchinococcus granulosus’ eggsγd, γs, and γhThe rates at which exposed dogs, sheep, and humans progress to infective classes, respectivelyβsdThe transmission rate from sheep to dogsβesTransmission rate ofEchinococcus’ eggs from the environment to sheepβehTransmission rate ofEchinococcus’ eggs from the environment to humansδEgg contamination rate in the environment by infected dogsχhThe concentration ofEchinococcus granulosus’ eggs at which half of all contacts with humans produce infectionχsThe concentration ofEchinococcus granulosus’ eggs at which half of all contacts with sheep produce infectionαhHuman recovery rateFigure 1
The flow diagram for cystic echinococcosis transmission dynamics.The flowchart in Figure1 demonstrates the interactions between the three populations and the transition of individuals from one compartment to another. The solid arrows show progression of individuals from one compartment to another. The broken line from Is to the line between Sd and Ed tells us that dogs become infected when they feed on organs of an infected sheep. The broken lines from B to the line between Es and Ss and from B to the line between Eh and Sh express the fact that sheep and humans are infected by accidentally ingesting an egg of Echinococcus granulosus.Based on these assumptions and using the transitions among different classes of disease stages given in Figure1, the transmission dynamics of cystic echinococcosis in the populations of dogs, humans, and sheep can be expressed by the following system of ordinary differential equations.(7)dSddt=μdNd∗−βsdIsSd−μdSd,(8)dEddt=βsdIsSd−γ~dEd,(9)dIddt=γdEd−μdId,(10)dBdt=δId−μeB,(11)dShdt=μhNh∗−μhSh−βehBχh+BSh,(12)dEhdt=βehBχh+BSh−γ~hEh,(13)dIhdt=γhEh−q~hIh,(14)dRhdt=αhIh−μhRh,(15)dSsdt=μsNs∗−μsSs−βesBχs+BSs,(16)dEsdt=βesBχs+BSs−γ~sEs,(17)dIsdt=γsEs−μsIs,where γ~d=μd+γd, γ~h=μh+γh, q~h=μh+αh, and γ~s=μs+γs.


## 2.1. Predator-Prey Model

The dynamics of sheep, dog, and human populations can be represented in a predator-prey relationship. We consider the case where dogs feed only on organs of sheep. Without sheep as the source of food, the dog population (Nd) would exponentially decrease and becomes extinct. However, Nd is increased by a rate proportional to the number of encounters between the sheep population (Ns) and dog population (Nd). The growth rate of the dog population due to consumption of sheep is ωNsNd, where ω denotes the conversion efficiency of consumed sheep into the dog reproduction rate. Sheep are the main food source for human. In the absence of sheep, it is assumed that there exists some alternative food source for growth of the human population. With sheep as the source of food, the human population (Nh) is increased by a rate proportional to the number of encounters between the sheep population (Ns) and human population (Nh). The growth rate of the human population due to consumption of sheep is θNsNh, where θ denotes the conversion efficiency of consumed sheep into the human reproduction rate. Without sheep as the source of food, the human population Nh grows exponentially at a per-capita growth rate rh but eventually increases up to the carrying capacity of the environment Kh. As the number of dog and human populations increases, the population decreases in response to increased rates of consumption. The rate of consumption of sheep by dogs and humans is represented by a and c, respectively. However, in the absence of dog and human populations, the sheep population Ns increases logistically. It initially increases exponentially at a per-capita growth rate rs, and it is self-limited by the carrying capacity of the environment Ks. Furthermore, the sheep, dog, and human populations die naturally at rates represented by μs, μd, and μh, respectively. Hence, the predator-prey interaction between the sheep, dog, and human populations is represented by a system of nonlinear ordinary differential equations:(1)dNsdt=rs1−NsKsNs−aNsNd−cNhNs−μsNs,(2)dNddt=ωNsNd−μdNd,(3)dNhdt=rh1−NhKhNh+θNsNh−μhNh,with initial conditions Ns0≥0, Nd0≥0, and Nh0≥0, where rs>0, a>0, ω>0, c>0, μs>0, μd>0, μh>0, θ>0, and rh>0.Theorem 1.
The regionΩ=Ns,Nd,Nh∈ℝ+3:0≤Nst+Ndt+Nht≤η/ε, where 0<ε≤μd and η=Ks/4rsrs+ε2+Kh/4rhrh+ε2, is positively invariant (mathematically well-posed) for the model (1)–(3).Proof.
The proof of this theorem is presented in AppendixA.


### 2.1.1. Equilibrium Points

Our next result concerns the existence of equilibrium points of the system (1)–(3) that are biologically feasible and determines the conditions for the existence of the equilibrium of the system in the feasible region. At equilibrium, equations of the model (1)–(3) are(4)rs1−NsKsNs−aNsNd−cNhNs−μsNs=0,ωNsNd−μdNd=0,rh1−NhKhNh+θNsNh−μhNh=0.Equilibrium points are as follows:(1)
The trivial fixed pointE0=0,0,0, which represents the extinction of the three populations(2)
E1=Ks1−μs/rs,0,0, which represent the coexistence of the sheep(3)
E2=0,0,Kh1−μh/rh, which represent the coexistence of humans(4)
E3=N~s,N~d,N~h=μd/ω,1/ars1−μd/ωKs−μs,0 and E4=N^s,N^d,N^h=Ksrhrs−μs+cKhμh−rh/rsrh+cθKsKh,0,Khrsrh−μh+θKsrs−μs/rsrh+cθKsKh, which represent the extinction of humans and the extinction of dogs, respectively(5)
Equilibrium point of the coexistence:(5)E5=Ns∗,Nd∗,Nh∗=μdω,1ars1−μdωKs−μs−cKhrhrh−μh+θμdω,Khrhrh−μh+θμdω,which represents the coexistence of all the three populations


### 2.1.2. Stability of Equilibrium Points

In the next sections, the transmission dynamics of cystic echinococcosis is studied. Since the dog, sheep, and human populations are important for the transmission dynamics of the disease, we are mainly concerned with the equilibrium point of the coexistence. Thus, the detail analysis of the local stability conditions of the equilibrium points is presented in AppendixB, and the stability conditions of E5 are presented in the following theorem.Theorem 2.
(1)
E5 is locally asymptotically stable provided that a1>0, a3>0, and a1a2−a3>0, where(6)a1=rh−μh+rsμdωKs+θμdω,a2=rsμdωKsrh−μh+θμdω+cθμdKhωrhrh−μh+θμdω+μdrs1−μdωKs−μs−cKhrhrh−μh+θμdω,a3=μdrs1−μdωKs−μs−cKhrhrh−μh+θμdω.(2)
The equilibrium pointE5 is globally asymptotically stable in the Int.ℝ+3Proof.
The proof of this theorem is presented in AppendixB.


## 2.1.1. Equilibrium Points

Our next result concerns the existence of equilibrium points of the system (1)–(3) that are biologically feasible and determines the conditions for the existence of the equilibrium of the system in the feasible region. At equilibrium, equations of the model (1)–(3) are(4)rs1−NsKsNs−aNsNd−cNhNs−μsNs=0,ωNsNd−μdNd=0,rh1−NhKhNh+θNsNh−μhNh=0.Equilibrium points are as follows:(1)
The trivial fixed pointE0=0,0,0, which represents the extinction of the three populations(2)
E1=Ks1−μs/rs,0,0, which represent the coexistence of the sheep(3)
E2=0,0,Kh1−μh/rh, which represent the coexistence of humans(4)
E3=N~s,N~d,N~h=μd/ω,1/ars1−μd/ωKs−μs,0 and E4=N^s,N^d,N^h=Ksrhrs−μs+cKhμh−rh/rsrh+cθKsKh,0,Khrsrh−μh+θKsrs−μs/rsrh+cθKsKh, which represent the extinction of humans and the extinction of dogs, respectively(5)
Equilibrium point of the coexistence:(5)E5=Ns∗,Nd∗,Nh∗=μdω,1ars1−μdωKs−μs−cKhrhrh−μh+θμdω,Khrhrh−μh+θμdω,which represents the coexistence of all the three populations


## 2.1.2. Stability of Equilibrium Points

In the next sections, the transmission dynamics of cystic echinococcosis is studied. Since the dog, sheep, and human populations are important for the transmission dynamics of the disease, we are mainly concerned with the equilibrium point of the coexistence. Thus, the detail analysis of the local stability conditions of the equilibrium points is presented in AppendixB, and the stability conditions of E5 are presented in the following theorem.Theorem 2.
(1)
E5 is locally asymptotically stable provided that a1>0, a3>0, and a1a2−a3>0, where(6)a1=rh−μh+rsμdωKs+θμdω,a2=rsμdωKsrh−μh+θμdω+cθμdKhωrhrh−μh+θμdω+μdrs1−μdωKs−μs−cKhrhrh−μh+θμdω,a3=μdrs1−μdωKs−μs−cKhrhrh−μh+θμdω.(2)
The equilibrium pointE5 is globally asymptotically stable in the Int.ℝ+3Proof.
The proof of this theorem is presented in AppendixB.


## 2.2. Modeling the Transmission Dynamics of Cystic Echinococcosis

We model the dynamics of echinococcosis as a compartmental framework. In the human population, there are four classes: the susceptible (Sh), exposed (Eh), infectious (Ih), and removed (Rh) classes. The dog population has three classes: the susceptible (Sd), exposed (Ed), and infectious (Id) classes. The sheep population has also three classes: the susceptible (Ss), exposed (Es), and infectious (Is) classes.In the dynamics of the disease transmission, susceptible sheep are infected by ingesting parasite eggs in the feces of infected definitive hosts (dogs), while humans are infected by accidentally ingestingEchinococcus granulosus’ eggs from the environment. Echinococcus granulosus’ egg contamination rate in the environment by infected dogs is represented by δ. The incorporation of the saturation is relevant since for the disease to be spread from the environment to sheep, there must be a sufficient number (saturation) from the environment that can cause infection. It measures the crowding effect of the parasite in the environment, the inhibition effect of susceptible human and sheep when their number increases, and the probability of contracting the disease by sheep and human populations from the environment. Susceptible humans Sh and susceptible sheep Ss become infected with the disease at rates βehB/χh+B and βesB/χs+B, respectively, where B denotes the concentration of Echinococcus granulosus’ eggs in the environment, βeh denotes the rate of ingestion of Echinococcus’ egg from the environment by humans, βes denotes the rate of ingestion of Echinococcus’ egg from the environment by sheep, and χh and χs are the half-saturation constants of parasite in the environment sufficient to infect humans and sheep, respectively. Susceptible dogs are infected by preying on the infected sheep. The disease transmission rate from sheep to dogs is denoted by βsd.The model is developed based on the following assumptions.(1)
The total population of dogs, sheep, and humans is assumed constant, and at stable equilibrium,E5=Nd∗,Ns∗,Nh∗, so that the birth rate and death rate of each of the populations are equal(2)
Dog, sheep, and human populations are recruited to the susceptible class by birth at ratesμd, μs, and μh, respectively(3)
The infected human population could recover from the disease naturally, and their recovery rate is represented byαh, where as sheep and dogs cannot recover once they are infected. We assume that there is no Echinococcus-induced death. However, dogs, sheep, and humans die naturally at rates μd, μs, and μh, respectivelyAll variables and parameters are introduced in Tables1 and 2, and the flowchart for the transmission dynamics of the disease is shown in Figure 1.Table 1
Definitions of variables.
VariablesDefinitionsSh, Eh, Ih, and RhNumber of susceptible, exposed, infected, and removed humans, respectivelySd, Ed, and IdNumber of susceptible, exposed, and infected dogs, respectivelySs, Es, and IsNumber of susceptible, exposed, and infected sheep, respectivelyBThe concentration ofEchinococcus granulosus’ eggs in the environmentNd∗, Ns∗, and Nh∗Total number of dogs, sheep, and humans, respectivelyTable 2
Descriptions of parameters.
ParametersDescriptionsμdNaturaldeathrate=birthrateofdogμhNaturaldeathrate=birthrateofhumanμsNaturaldeathrate=birthrateofsheepμeNatural death rate ofEchinococcus granulosus’ eggsγd, γs, and γhThe rates at which exposed dogs, sheep, and humans progress to infective classes, respectivelyβsdThe transmission rate from sheep to dogsβesTransmission rate ofEchinococcus’ eggs from the environment to sheepβehTransmission rate ofEchinococcus’ eggs from the environment to humansδEgg contamination rate in the environment by infected dogsχhThe concentration ofEchinococcus granulosus’ eggs at which half of all contacts with humans produce infectionχsThe concentration ofEchinococcus granulosus’ eggs at which half of all contacts with sheep produce infectionαhHuman recovery rateFigure 1
The flow diagram for cystic echinococcosis transmission dynamics.The flowchart in Figure1 demonstrates the interactions between the three populations and the transition of individuals from one compartment to another. The solid arrows show progression of individuals from one compartment to another. The broken line from Is to the line between Sd and Ed tells us that dogs become infected when they feed on organs of an infected sheep. The broken lines from B to the line between Es and Ss and from B to the line between Eh and Sh express the fact that sheep and humans are infected by accidentally ingesting an egg of Echinococcus granulosus.Based on these assumptions and using the transitions among different classes of disease stages given in Figure1, the transmission dynamics of cystic echinococcosis in the populations of dogs, humans, and sheep can be expressed by the following system of ordinary differential equations.(7)dSddt=μdNd∗−βsdIsSd−μdSd,(8)dEddt=βsdIsSd−γ~dEd,(9)dIddt=γdEd−μdId,(10)dBdt=δId−μeB,(11)dShdt=μhNh∗−μhSh−βehBχh+BSh,(12)dEhdt=βehBχh+BSh−γ~hEh,(13)dIhdt=γhEh−q~hIh,(14)dRhdt=αhIh−μhRh,(15)dSsdt=μsNs∗−μsSs−βesBχs+BSs,(16)dEsdt=βesBχs+BSs−γ~sEs,(17)dIsdt=γsEs−μsIs,where γ~d=μd+γd, γ~h=μh+γh, q~h=μh+αh, and γ~s=μs+γs.


## 3. Well-Posedness of the Solutions

In this section, we test whether the model (7)–(17) is well posed epidemiologically and mathematically in a set or not.Theorem 3.
The regionD=D1×D2×D3×D4⊂ℝ3×ℝ4×ℝ3×ℝ, where D1=Sd,Ed,Id⊂ℝ+3:Sd+Ed+Id≤Nd∗, D2=Sh,Eh,Ih,Rh⊂ℝ+4:Sh+Eh+Ih+Rh≤Nh∗, D3=Ss,Es,Is⊂ℝ+3:Ss+Es+Is≤Ns∗, and D4=B⊂ℝ+:B≤δNd∗/μe, is positively invariant for the model (7)–(17).Proof.
The detailed proof of this theorem is presented in AppendixC.


## 4. Existence and Stability of Equilibria

The equilibrium points of the system (7)–(17) are found by equating each time derivative to zero. Thus,(18)μdNd∗−βsdIsSd−μdSd=0,(19)βsdIsSd−γ~dEd=0,(20)γdEd−μdId=0,(21)δId−μeB=0,(22)μhNh∗−μhSh−βehBχh+BSh=0,(23)βehBχh+BSh−γ~hEh=0,(24)γhEh−q~hIh=0,(25)αhIh−μhRh=0,(26)μsNs∗−μsSs−βesBχs+BSs=0,(27)βesBχs+BSs−γ~sEs=0,(28)γsEs−μsIs=0.From equations (20) and (21), we, respectively, have(29)Id=μeBδ,Ed=μdμeBδγd.From equations (18) and (19) and using (29), we have(30)Sd=Nd∗−μeγ~dBδγd.Substituting (30) in (18) yields(31)Is=1βsdμeμdγ~dBδγdNd∗−μeγ~dB.Similarly, from equations (26)–(28), we obtain(32)Is=γsβesNs∗Bγ~sμsχs+μs+BesB.By equating (31) and (32), we then obtain a quadratic equation:(33)B2μeγ~dμdγ~sμs+βes+γsβesβsdNs∗−BδγdγsβesβsdNs∗Nd∗−μeχsμsμdγ~dγ~s=0.Thus, we have two roots:(34)B=0,B=δγdγsβesβsdNs∗Nd∗−μeχsμsμdγ~dγ~sμeγ~dμdγ~sμs+βes+γsβesβsdNs∗.All the remaining state variablesSh, Eh, Ih, Rh, Ss, and Es obtained from equations (22)–(27) are expressed in terms of B, where B≥0. Thus, we have two cases, and the results are presented in Sections 4.1 and 4.4.


### 4.1. Disease-Free Equilibrium (DFE)

From algebraic computation whenB=0, the system (7)–(17) has the DFE X0=Sd0,Ed0,Id0,B0,Sh0,Eh0,Ih0,Rh0,Ss0,Es0,Is0=Nd∗,0,0,0,Nh∗,0,0,0,Ns∗,0,0.


### 4.2. The Basic Reproduction Number

The basic reproduction number is one of the fundamental concepts in mathematical biology. It is a threshold parameter, intended to quantify the spread of disease by estimating the average number of secondary infections in a wholly susceptible population, giving an indication of the invasion strength of an epidemic. We determine the basic reproduction number using the next-generation matrix (NGM) approach [25]. See Appendix D for the detailed computation. Therefore, the basic reproduction number, denoted by R0, is given by(35)R0=ρFV−1=γsβesNs∗μsγ~sβsdγdNd∗μdγ~dδχsμe.We write it asR0=RdRsRe, where Rd=βsdγdNd∗/μdγ~d corresponds to the average number of dogs in which one infectious sheep causes the disease over its expected infection period in a completely susceptible dog population, Rs=γsβesNs∗/μsγ~s corresponds to the number of sheep in which an infectious dog induces the disease over its expected infection period in a completely susceptible sheep population, and Re=δ/μeχs corresponds to the contribution of the environment to the sheep population as a result of one infectious dog subject during its infectious period.


### 4.3. Stability of the DFE

Theorem 4.
IfR0<1, then the disease-free equilibrium X0 is locally asymptotically stable.Proof.
See Theorem 2 in [25].Theorem 5.
IfR0<1, then the disease-free equilibrium X0 is globally asymptotically stable in D. If R0>1, then the DFE is unstable, the system is persistent, and there is at least one endemic equilibrium in the interior of D.Proof.
To prove the global stability of the disease-free equilibriumX0, we use a matrix-theoretic method as explained in [26].
The disease compartments of the model (7)–(17) can be written as(36)x′=F−Vx−fx,y,where x=Ed,Id,B,Eh,Ih,Es,IsT, y=Sd,Sh,Rh,SsT, F and V are matrices given in (D.2) and (D.3), and(37)fx,y=βsdIsNd∗−Sd00βehBχhNh∗−χhShB+χh0βesBχsNs∗−χsSsB+χs0≥0,since Sd≤Nd∗, Sh≤Nh∗, χh≤χh+B, Ss≤Ns∗, and χs≤χs+B.
MatricesF and V−1 are entry-wise nonnegative with(38)V−1F=000000βsdNd∗γ~d000000γdβsdNd∗μdγ~d000000δγdβsdNd∗μeμdγ~d00βehNh∗χhγ~h000000γhβehNh∗χhγ~hq~h000000βesNs∗χsγ~s000000γsβesNs∗χsγ~sμs0000.
SinceV−1F is a reducible matrix [27], the condition of Theorem 2 in [26] fails. Instead, to establish the global stability of the DFE, we construct a Lyapunov function by using Theorem 1 of [26]. Let WT=w1,w2,w3,w4,w5,w6,w7 be the left eigenvector of V−1F corresponding to the eigenvalue R0. Thus,(39)WTV−1F=R0W.
As a result, we found thatW=0,0,1,0,0,0,0,δγdβsdNd∗/μdμeγ~dR0 is the left eigenvector of V−1F corresponding to the eigenvalue R0. Thus, by Theorem 1 of [26],(40)Q=WTV−1x=δγdEdμdμeγ~d+δIdμdμe+Bμe+γsμsγ~sEs+IsμsδγdβsdNd∗μdμeγ~dR0is a Lyapunov function for the model (7)–(17). Then, differentiating along the solutions of the system (7)–(17) gives(41)Q′=R0−1δγdβsdIsNd∗μdμeγ~dR0+B−δγdβsdIsμdμeγ~dNd∗−Sd+γsγdβesβsdBNd∗μsμeμdχsγ~dR0Ns∗−SsχsB+χs.
SinceSd≤Nd∗, Ss≤Ns∗, and χs≤χs+B, Q′≤0 if R0<1. Furthermore, Q′=0 implies that B=0 and Is=0. Thus, using the results obtained in Section 4, we have Ed=0, Id=0, Eh=0, Ih=0, Rh=0, Es=0, Sd=Nd∗, Sh=Nh∗, and Ss=Ns∗. Hence, the largest invariant set of the model where Q′=0 in intD is the singleton X0. Therefore, by LaSalle’s invariance principle [28], the disease-free equilibrium X0 is globally asymptotically stable if R0<1. Furthermore, from (41), if R0>1, then Q′>0 in D when B>0 and Sd,Ss=Nd∗,Ns∗. Thus, the disease-free equilibrium X0 is unstable, and using Theorem 2 of [26], the system (7)–(17) is uniformly persistent and hence implies that there is at least one endemic equilibrium in the interior of D.


### 4.4. Existence and Stability of the Endemic Equilibrium (EE)

From the result in (34), an endemic equilibrium is obtained when B>0. Thus, the endemic equilibrium point of the system in terms of the reproduction number R0 is given by(42)XE=Sd∗,Ed∗,Id∗,B∗,Sh∗,Eh∗,Ih∗,Rh∗,Ss∗,Es∗,Is∗,where(43)Id∗=μeB∗δ=μeχsμdμsγ~sδμdγ~sμs+βes+γsβsdβesNs∗R02−1,(44)Ed∗=μeχsμd2μsγ~sδγdμdγ~dμs+βes+γsβsdβesNs∗R02−1,(45)Sd∗=Nd∗−μeγ~dBδγd,(46)Sh∗=μhNh∗χh+B∗μhχh+B∗+βehB∗,(47)Eh∗=μhβehNh∗B∗γ~hμhχh+B∗+βehB∗,(48)Ih∗=μhγhβehNh∗B∗q~hγ~hμhχh+B∗+βehB∗,(49)Rh∗=γhαhβehNh∗B∗q~hγ~hμhχh+B∗+βehB∗,(50)Ss∗=μsχs+B∗Ns∗μsχs+B∗+βesB∗,(51)Es∗=μsβesNs∗B∗γ~sμsχs+B∗+βesB∗,(52)Is∗=γsβesNs∗B∗γ~sμsχs+B∗+βesB∗,(53)B∗=χsμdμsγ~sμdγ~sμs+βes+γsβsdβesNs∗R02−1.Theorem 6.
IfR0>1, then an endemic equilibrium XE=Sd∗,Ed∗,Id∗,B∗,Sh∗,Eh∗,Ih∗,Rh∗,Ss∗,Es∗,Is∗ defined by equations (43)–(53) is globally asymptotically stable.Proof.
The detailed proof of this theorem is presented in AppendixE.


## 4.1. Disease-Free Equilibrium (DFE)

From algebraic computation whenB=0, the system (7)–(17) has the DFE X0=Sd0,Ed0,Id0,B0,Sh0,Eh0,Ih0,Rh0,Ss0,Es0,Is0=Nd∗,0,0,0,Nh∗,0,0,0,Ns∗,0,0.


## 4.2. The Basic Reproduction Number

The basic reproduction number is one of the fundamental concepts in mathematical biology. It is a threshold parameter, intended to quantify the spread of disease by estimating the average number of secondary infections in a wholly susceptible population, giving an indication of the invasion strength of an epidemic. We determine the basic reproduction number using the next-generation matrix (NGM) approach [25]. See Appendix D for the detailed computation. Therefore, the basic reproduction number, denoted by R0, is given by(35)R0=ρFV−1=γsβesNs∗μsγ~sβsdγdNd∗μdγ~dδχsμe.We write it asR0=RdRsRe, where Rd=βsdγdNd∗/μdγ~d corresponds to the average number of dogs in which one infectious sheep causes the disease over its expected infection period in a completely susceptible dog population, Rs=γsβesNs∗/μsγ~s corresponds to the number of sheep in which an infectious dog induces the disease over its expected infection period in a completely susceptible sheep population, and Re=δ/μeχs corresponds to the contribution of the environment to the sheep population as a result of one infectious dog subject during its infectious period.


## 4.3. Stability of the DFE

Theorem 4.
IfR0<1, then the disease-free equilibrium X0 is locally asymptotically stable.Proof.
See Theorem 2 in [25].Theorem 5.
IfR0<1, then the disease-free equilibrium X0 is globally asymptotically stable in D. If R0>1, then the DFE is unstable, the system is persistent, and there is at least one endemic equilibrium in the interior of D.Proof.
To prove the global stability of the disease-free equilibriumX0, we use a matrix-theoretic method as explained in [26].
The disease compartments of the model (7)–(17) can be written as(36)x′=F−Vx−fx,y,where x=Ed,Id,B,Eh,Ih,Es,IsT, y=Sd,Sh,Rh,SsT, F and V are matrices given in (D.2) and (D.3), and(37)fx,y=βsdIsNd∗−Sd00βehBχhNh∗−χhShB+χh0βesBχsNs∗−χsSsB+χs0≥0,since Sd≤Nd∗, Sh≤Nh∗, χh≤χh+B, Ss≤Ns∗, and χs≤χs+B.
MatricesF and V−1 are entry-wise nonnegative with(38)V−1F=000000βsdNd∗γ~d000000γdβsdNd∗μdγ~d000000δγdβsdNd∗μeμdγ~d00βehNh∗χhγ~h000000γhβehNh∗χhγ~hq~h000000βesNs∗χsγ~s000000γsβesNs∗χsγ~sμs0000.
SinceV−1F is a reducible matrix [27], the condition of Theorem 2 in [26] fails. Instead, to establish the global stability of the DFE, we construct a Lyapunov function by using Theorem 1 of [26]. Let WT=w1,w2,w3,w4,w5,w6,w7 be the left eigenvector of V−1F corresponding to the eigenvalue R0. Thus,(39)WTV−1F=R0W.
As a result, we found thatW=0,0,1,0,0,0,0,δγdβsdNd∗/μdμeγ~dR0 is the left eigenvector of V−1F corresponding to the eigenvalue R0. Thus, by Theorem 1 of [26],(40)Q=WTV−1x=δγdEdμdμeγ~d+δIdμdμe+Bμe+γsμsγ~sEs+IsμsδγdβsdNd∗μdμeγ~dR0is a Lyapunov function for the model (7)–(17). Then, differentiating along the solutions of the system (7)–(17) gives(41)Q′=R0−1δγdβsdIsNd∗μdμeγ~dR0+B−δγdβsdIsμdμeγ~dNd∗−Sd+γsγdβesβsdBNd∗μsμeμdχsγ~dR0Ns∗−SsχsB+χs.
SinceSd≤Nd∗, Ss≤Ns∗, and χs≤χs+B, Q′≤0 if R0<1. Furthermore, Q′=0 implies that B=0 and Is=0. Thus, using the results obtained in Section 4, we have Ed=0, Id=0, Eh=0, Ih=0, Rh=0, Es=0, Sd=Nd∗, Sh=Nh∗, and Ss=Ns∗. Hence, the largest invariant set of the model where Q′=0 in intD is the singleton X0. Therefore, by LaSalle’s invariance principle [28], the disease-free equilibrium X0 is globally asymptotically stable if R0<1. Furthermore, from (41), if R0>1, then Q′>0 in D when B>0 and Sd,Ss=Nd∗,Ns∗. Thus, the disease-free equilibrium X0 is unstable, and using Theorem 2 of [26], the system (7)–(17) is uniformly persistent and hence implies that there is at least one endemic equilibrium in the interior of D.


## 4.4. Existence and Stability of the Endemic Equilibrium (EE)

From the result in (34), an endemic equilibrium is obtained when B>0. Thus, the endemic equilibrium point of the system in terms of the reproduction number R0 is given by(42)XE=Sd∗,Ed∗,Id∗,B∗,Sh∗,Eh∗,Ih∗,Rh∗,Ss∗,Es∗,Is∗,where(43)Id∗=μeB∗δ=μeχsμdμsγ~sδμdγ~sμs+βes+γsβsdβesNs∗R02−1,(44)Ed∗=μeχsμd2μsγ~sδγdμdγ~dμs+βes+γsβsdβesNs∗R02−1,(45)Sd∗=Nd∗−μeγ~dBδγd,(46)Sh∗=μhNh∗χh+B∗μhχh+B∗+βehB∗,(47)Eh∗=μhβehNh∗B∗γ~hμhχh+B∗+βehB∗,(48)Ih∗=μhγhβehNh∗B∗q~hγ~hμhχh+B∗+βehB∗,(49)Rh∗=γhαhβehNh∗B∗q~hγ~hμhχh+B∗+βehB∗,(50)Ss∗=μsχs+B∗Ns∗μsχs+B∗+βesB∗,(51)Es∗=μsβesNs∗B∗γ~sμsχs+B∗+βesB∗,(52)Is∗=γsβesNs∗B∗γ~sμsχs+B∗+βesB∗,(53)B∗=χsμdμsγ~sμdγ~sμs+βes+γsβsdβesNs∗R02−1.Theorem 6.
IfR0>1, then an endemic equilibrium XE=Sd∗,Ed∗,Id∗,B∗,Sh∗,Eh∗,Ih∗,Rh∗,Ss∗,Es∗,Is∗ defined by equations (43)–(53) is globally asymptotically stable.Proof.
The detailed proof of this theorem is presented in AppendixE.


## 5. Numerics: Elasticity Indices, Numerical Simulations, and Control Strategies

### 5.1. Elasticity Indices

Our analysis in previous sections demonstrated that the quantity ofR0 plays a crucial role in determining the dynamic behavior of our model. In this section, we will identify which parameters have a high impact on the basic reproduction number R0 using data from the literature and assumed (estimated) values as given in Table 3, and we use the result for intervention strategies.Table 3
Parameters, baseline values, and sources.
ParametersBaseline values (Unit=day−1)Sourcesμd0.00088[22]βsd0.00001Assumedγd0.0001Assumedμh0.00004[23]βeh0.0001Assumedγh0.00019[29]αh0.0001Assumedμs0.00114[22]βes0.0001Assumedμe0.0039[22]γs0.0001[30]δ0.003Assumedχh40Assumedχs20AssumedTo determine the parameter that contributes most to the disease transmission, we perform local sensitivity analysis by calculating the normalized sensitivity index (elasticity index) as introduced in [31, 32]. The normalized forward sensitivity index (elasticity index) of a variable (R0) with respect to a parameter p is the ratio of the relative change in the variable to the relative change in the parameter, given by(54)YpR0=∂R0∂p×pR0.Table4 gives the elasticity indices of R0 with respect to the key parameters of the model (7)–(17) at the baseline values indicated in Table 3 and arranged in descending order of magnitudes. The sign of the elasticity index tells whether R0 increases (positive sign) or decreases (negative sign) with the parameter, whereas the magnitude determines the relative importance of the parameter. From the magnitude of the elasticity index, we can notice that four parameters (βes,βsd,δ, and χs) have equal and the greatest influence for the transmission of the disease, followed by γs and γd.Table 4
Elasticity indices ofR0 relative to some model parameters.
ParametersFormulaYpR0=∂R0/∂p×p/R0Baseline valueRangeElasticity indexβes120.00010.00001-0.00020.5βsd120.000010.000001-0.000020.5δ120.0030.0003-0.0060.5χs−122010-40-0.5γsμs2μs+γs0.00010.00001-0.00020.46γdμd2μd+γd0.00010.00001-0.00020.45


### 5.2. Global Sensitivity Analysis

From the local sensitivity analysis, we observed that it is impossible to differentiate explicitly the most influential parameter(s) of the model. In order to determine which parameter(s) among the six is (are) most influential in the dynamics of the disease, global sensitivity analysis is done. We employed the technique of Latin hypercube sampling (LHS) to test the sensitivity of the model to each input parameter, as described and implemented in [33], and partial rank correlation coefficients (PRCC) to assess the significance of each parameter with respect to each metric are used. Latin hypercube sampling is a stratified sampling technique that creates sets of parameters by sampling for each parameter according to a predefined probability distribution. To examine the dependence of R0 on parameter variations, we determine the PRCC values by considering a range of parameters as given in Table 4, with sample size 1000. The result is depicted in Figure 2. The parameter with the PRCC value far away from zero indicates more strongly the parameter influence R0. The negative sign for PRCC indicates inverse proportionality.Figure 2
Global sensitivity analysis displaying the partial rank correlation coefficients (PRCC).From Figure2, it is observed that the transmission rate of Echinococcus’ eggs from the environment to sheep βes, the contamination rate of Echinococcus’ eggs in the environment by infected dogs (δ), and the transmission rate from sheep to dogs (βsd) are the most influential parameters among the six parameters in the disease dynamics.


### 5.3. Numerical Simulations

In this section, numerical simulations to gain insight into some of quantitative features of the model (7)–(17) are presented.For the model (1)–(3), we estimate the parameter values: Ks=2000, Kh=1000, rh=0.01, θ=0.0000001, c=0.000000001, ω=0.0000005, a=0.0000001, μh=0.00004, μd=0.00088, and μs=0.00114. These parameter values result in an asymptotically stable equilibrium point E5=Ns∗,Nd∗,Nh∗=1760,590,1014. Thus, using the total populations Nd∗=590, Nh∗=1014, and Ns∗=1760, an initial concentration of Echinococcus’ eggs B=100, and the baseline parameter values given in Table 3, we obtain a reproductive number R0=0.57<1. Figure 3 depicts the global stability of the disease-free equilibrium as proven in Theorem 5 with different initial conditions. We can notice that all disease compartments Ed∗, Id∗, B∗, Eh∗, Ih∗, Rh∗, Es∗, and Is∗ converge asymptotically to zero, while the noninfected compartments Sd∗, Sh∗, and Ss∗ converge to their respective total populations.Figure 3
Time evolution of the dog, sheep, and human populations with baseline parameter values given in Table3, using different initial conditions, which gives R0=0.57.Figure4 shows the time evolution of human, sheep, and dog populations for the model (7)–(17) with parameter values given in Table 3 by increasing βes=0.0001 to βes=0.001. In this case, the reproductive number is R0=1.8>1 and depicts the global stability of the endemic equilibrium as proven in Theorem 6. It can be noticed that all the compartments of the dog, human, and sheep populations converge asymptotically to their respective endemic equilibrium points irrespective of any initial conditions.Figure 4
Time evolution of the dog, sheep, and human populations with baseline parameter values given in Table3, using different initial conditions, except for βes=0.0001, which gives R0=1.8, and with approximate equilibrium values Sd∗=414, Ed∗=158, Id∗=18, Sh∗=617, Eh∗=69, Ih∗=94, Rh∗=234, Ss∗=1296, Es∗=427, and Is∗=37.


### 5.4. Control Strategies

The global sensitivity analysis presented in Sections5.1 and 5.2 demonstrates that cystic echinococcosis can be controlled by reducing the transmission rate of Echinococcus’ eggs from the environment to sheep βes. In this section, we illustrate the effect of the transmission rate of Echinococcus’ eggs from the environment to sheep βes.The effect of the transmission rate ofEchinococcus’ eggs from the environment to sheep (βes) using baseline parameter values in Table 3, and when βes varied from 0.01 to 0.0001, is displayed in Figure 5. As a result, the infectious sheep, dog, and human populations are, respectively, reduced from 123 to 0, 84 to 0, and 145 to 0, where the basic reproductive number is also reduced from R0=5.72 to R0=0.57. The result shows that increasing the transmission rate of Echinococcus’ eggs from the environment to sheep βes will intensify the transmission of the disease. Thus, to control the disease transmission, we suggest that it is important to plan an intervention strategy to decrease the transmission rate of Echinococcus’ eggs from the environment to sheep βes.Figure 5
The numerical simulations displaying effects of controlling strategies on the cumulative number of infectious dog, human, and sheep populations, using parameter values in Table3, with varying values of βes.


## 5.1. Elasticity Indices

Our analysis in previous sections demonstrated that the quantity ofR0 plays a crucial role in determining the dynamic behavior of our model. In this section, we will identify which parameters have a high impact on the basic reproduction number R0 using data from the literature and assumed (estimated) values as given in Table 3, and we use the result for intervention strategies.Table 3
Parameters, baseline values, and sources.
ParametersBaseline values (Unit=day−1)Sourcesμd0.00088[22]βsd0.00001Assumedγd0.0001Assumedμh0.00004[23]βeh0.0001Assumedγh0.00019[29]αh0.0001Assumedμs0.00114[22]βes0.0001Assumedμe0.0039[22]γs0.0001[30]δ0.003Assumedχh40Assumedχs20AssumedTo determine the parameter that contributes most to the disease transmission, we perform local sensitivity analysis by calculating the normalized sensitivity index (elasticity index) as introduced in [31, 32]. The normalized forward sensitivity index (elasticity index) of a variable (R0) with respect to a parameter p is the ratio of the relative change in the variable to the relative change in the parameter, given by(54)YpR0=∂R0∂p×pR0.Table4 gives the elasticity indices of R0 with respect to the key parameters of the model (7)–(17) at the baseline values indicated in Table 3 and arranged in descending order of magnitudes. The sign of the elasticity index tells whether R0 increases (positive sign) or decreases (negative sign) with the parameter, whereas the magnitude determines the relative importance of the parameter. From the magnitude of the elasticity index, we can notice that four parameters (βes,βsd,δ, and χs) have equal and the greatest influence for the transmission of the disease, followed by γs and γd.Table 4
Elasticity indices ofR0 relative to some model parameters.
ParametersFormulaYpR0=∂R0/∂p×p/R0Baseline valueRangeElasticity indexβes120.00010.00001-0.00020.5βsd120.000010.000001-0.000020.5δ120.0030.0003-0.0060.5χs−122010-40-0.5γsμs2μs+γs0.00010.00001-0.00020.46γdμd2μd+γd0.00010.00001-0.00020.45


## 5.2. Global Sensitivity Analysis

From the local sensitivity analysis, we observed that it is impossible to differentiate explicitly the most influential parameter(s) of the model. In order to determine which parameter(s) among the six is (are) most influential in the dynamics of the disease, global sensitivity analysis is done. We employed the technique of Latin hypercube sampling (LHS) to test the sensitivity of the model to each input parameter, as described and implemented in [33], and partial rank correlation coefficients (PRCC) to assess the significance of each parameter with respect to each metric are used. Latin hypercube sampling is a stratified sampling technique that creates sets of parameters by sampling for each parameter according to a predefined probability distribution. To examine the dependence of R0 on parameter variations, we determine the PRCC values by considering a range of parameters as given in Table 4, with sample size 1000. The result is depicted in Figure 2. The parameter with the PRCC value far away from zero indicates more strongly the parameter influence R0. The negative sign for PRCC indicates inverse proportionality.Figure 2
Global sensitivity analysis displaying the partial rank correlation coefficients (PRCC).From Figure2, it is observed that the transmission rate of Echinococcus’ eggs from the environment to sheep βes, the contamination rate of Echinococcus’ eggs in the environment by infected dogs (δ), and the transmission rate from sheep to dogs (βsd) are the most influential parameters among the six parameters in the disease dynamics.


## 5.3. Numerical Simulations

In this section, numerical simulations to gain insight into some of quantitative features of the model (7)–(17) are presented.For the model (1)–(3), we estimate the parameter values: Ks=2000, Kh=1000, rh=0.01, θ=0.0000001, c=0.000000001, ω=0.0000005, a=0.0000001, μh=0.00004, μd=0.00088, and μs=0.00114. These parameter values result in an asymptotically stable equilibrium point E5=Ns∗,Nd∗,Nh∗=1760,590,1014. Thus, using the total populations Nd∗=590, Nh∗=1014, and Ns∗=1760, an initial concentration of Echinococcus’ eggs B=100, and the baseline parameter values given in Table 3, we obtain a reproductive number R0=0.57<1. Figure 3 depicts the global stability of the disease-free equilibrium as proven in Theorem 5 with different initial conditions. We can notice that all disease compartments Ed∗, Id∗, B∗, Eh∗, Ih∗, Rh∗, Es∗, and Is∗ converge asymptotically to zero, while the noninfected compartments Sd∗, Sh∗, and Ss∗ converge to their respective total populations.Figure 3
Time evolution of the dog, sheep, and human populations with baseline parameter values given in Table3, using different initial conditions, which gives R0=0.57.Figure4 shows the time evolution of human, sheep, and dog populations for the model (7)–(17) with parameter values given in Table 3 by increasing βes=0.0001 to βes=0.001. In this case, the reproductive number is R0=1.8>1 and depicts the global stability of the endemic equilibrium as proven in Theorem 6. It can be noticed that all the compartments of the dog, human, and sheep populations converge asymptotically to their respective endemic equilibrium points irrespective of any initial conditions.Figure 4
Time evolution of the dog, sheep, and human populations with baseline parameter values given in Table3, using different initial conditions, except for βes=0.0001, which gives R0=1.8, and with approximate equilibrium values Sd∗=414, Ed∗=158, Id∗=18, Sh∗=617, Eh∗=69, Ih∗=94, Rh∗=234, Ss∗=1296, Es∗=427, and Is∗=37.


## 5.4. Control Strategies

The global sensitivity analysis presented in Sections5.1 and 5.2 demonstrates that cystic echinococcosis can be controlled by reducing the transmission rate of Echinococcus’ eggs from the environment to sheep βes. In this section, we illustrate the effect of the transmission rate of Echinococcus’ eggs from the environment to sheep βes.The effect of the transmission rate ofEchinococcus’ eggs from the environment to sheep (βes) using baseline parameter values in Table 3, and when βes varied from 0.01 to 0.0001, is displayed in Figure 5. As a result, the infectious sheep, dog, and human populations are, respectively, reduced from 123 to 0, 84 to 0, and 145 to 0, where the basic reproductive number is also reduced from R0=5.72 to R0=0.57. The result shows that increasing the transmission rate of Echinococcus’ eggs from the environment to sheep βes will intensify the transmission of the disease. Thus, to control the disease transmission, we suggest that it is important to plan an intervention strategy to decrease the transmission rate of Echinococcus’ eggs from the environment to sheep βes.Figure 5
The numerical simulations displaying effects of controlling strategies on the cumulative number of infectious dog, human, and sheep populations, using parameter values in Table3, with varying values of βes.


## 6. Conclusion

In this paper, we first derived a mathematical model of the predator-prey interaction of the sheep, dog, and human populations in order to ascertain the conditions for the coexistence of these populations in predator-prey relationship. We then determine the equilibrium points and study the stability of each of these points. We also formulated a compartmental model for transmission dynamics of cystic echinococcosis. We calculated both the disease-free equilibrium (DFE) and the endemic equilibrium (EE) points of the model. To study the behavior of the disease, the basic reproduction numberR0 is derived. We proved that, when R0<1, the DFE is locally asymptotically stable and globally asymptotically stable, which implies that the disease dies out, whereas when R0>1, the EE is globally asymptotically stable, which implies that the disease persists in all the populations. To identify which parameter has a great impact on the disease transmission, we performed both local and global sensitivity analyses on the basic reproduction number, from which we have noticed that the most sensitive parameter is the transmission rate of Echinococcus’ eggs from the environment to sheep (βes). Furthermore, we have also observed that the variation of βes has a significant effect on the number of infectious sheep, dog, and human populations. Thus, the effective controlling strategy that controls the disease transmission is decreasing the transmission rate of Echinococcus’ eggs from the environment to sheep (βes). To this effect, we suggest that the transmission dynamics of the disease by considering different intervention strategies must be studied. Therefore, extension of our work by introducing intervention strategies is an interesting future study.



---

*Source: 8482696-2020-11-02.xml*