Global dynamics of a time-delayed echinococcosis transmission model
- Junli Liu^{1},
- Luju Liu^{2}Email author,
- Xiaomei Feng^{3} and
- Jinqian Feng^{1}
https://doi.org/10.1186/s13662-015-0356-3
© Liu et al.; licensee Springer 2015
Received: 13 October 2014
Accepted: 2 January 2015
Published: 28 March 2015
Abstract
In this paper, we present a time-delayed echinococcosis transmission model to explore effective control and prevention strategies. We first give the basic reproduction number \(R_{0}\). It is shown that if \(R_{0}<1\), the disease-free equilibrium is globally asymptotically stable, and if \(R_{0}>1\), the disease persists. We further show that the endemic equilibrium is globally asymptotically stable for a special case. Numerical simulations are performed to illustrate our analytic results. We give some sensitivity analysis of some parameters and give some useful comments on controlling the transmission of echinococcosis.
Keywords
1 Introduction
Echinococcosis, also called hydatid disease, hydatidosis, or echinococcal disease, is a parasitic disease of tapeworms of the Echinococcus type. The disease occurs in most areas of the world and currently affects about one million people. In some areas of South America, Africa, and Asia up to 10% of certain populations are affected [1]. In 2010, it caused about 1,200 deaths down from 2,000 in 1990 [2]. The economic cost of the disease is estimated to be around \(3\times10^{9}\) USD a year. It can affect both humans and other animals such as pigs, cows, and horses [1]. The most common form found is cystic echinococcosis (also known as unilocular echinococcosis), which is caused by Echinococcus granulosus. The second most common form is alveolar echinococcosis, which is caused by Echinococcus multilocularis.
Like many other parasite infections, the course of Echinococcus infection is complex. The worm has a life cycle that requires definitive hosts and intermediate hosts. Definitive hosts are normally carnivores such as dogs, while intermediate hosts are usually herbivores such as sheep and cattle. Humans function as accidental hosts, because they are usually a dead-end for the parasitic infection cycle.
There are three development stages in the life cycle of Echinococcus, including egg, larva, and adult. An adult worm resides in the small intestine of a definitive host. Afterwards, gravid proglottids release eggs that are passed in the feces of the definitive host. The egg is then ingested by an intermediate host. The egg then hatches in the small intestine of the intermediate host and releases an oncosphere that penetrates the intestinal wall and moves through the circulatory system into different organs, in particular the liver and lungs. Once it has invaded these organs, the oncosphere develops into a cyst. The cyst then slowly enlarges, creating protoscolices and daughter cysts within the cyst. The definitive host then becomes infected after ingesting the cyst-containing organs of the infected intermediate host. After ingestion, the protoscolices attach to the intestine. They then develop into adult worms and the cycle starts all over again.
In China, there are 22 provinces, autonomous regions, and municipalities reported with cystic echinococcosis (CE) which was caused by Echinococcus granulosus and Echinococcus multilocularis [3, 4]. The main endemic areas are in the western and northwestern provinces and autonomous regions: Xinjiang, Gansu, Ningxia, Inner Mongolia, Qinghai, Tibet [5], and Sichuan [6, 7], where extensively developed livestock husbandry maintains stable transmission cycles of Echinococcus granulosus. The number of domestic animals being faced with the infection of echinococcosis is more than 10^{8}, in which the amount of dogs is at least \(5 \times10^{6}\) [8].
Much has been done in terms of modeling and analysis of disease transmission of Echinococcus (see [9–13]). In [14], in order to explore effective control and prevention measures the authors proposed a deterministic model to study the transmission dynamics of echinococcosis in Xinjiang. The results showed that the dynamics of the model was completely determined by the basic reproductive number \(R_{0}\). Du et al. [15] proposed an echinococcosis transmission model with saturation incidence, and they also established a threshold type result, which states that when \(R_{0}<1\), the disease will die out; when \(R_{0}>1\) and the recovery rate of dogs is very small, the disease will persist.
In this paper, we focus on the Echinococcus granulosus, which is the most common cause of human hydatid disease. The egg needs 5 to 6 months to develop into a larva in the intermediate hosts, and protoscoleces may develop into adult worms in about 1.5 to 2 months [16] in the definitive hosts. In view of realistic considerations, we take two time delays into account, to describe the time needed from egg to larva and from larva to adult, respectively. In fact, from the expression of \(R_{0}\) in Section 2, we can see those delays reduce the values of \(R_{0}\). Therefore, the neglect of the delays overestimated the infection risk.
The purpose of this paper is to study the global dynamics of a time-delayed Echinococcus transmission model. In Section 2, we present the model and prove its wellposedness, also we introduce the basic reproduction number \(R_{0}\). In Section 3, we show the global stability of the disease-free equilibrium when \(R_{0}<1\). In Section 4, we show that the disease is uniformly persistent when \(R_{0}>1\). In Section 5, by constructing Lyapunov functionals, we show that the endemic equilibrium is globally asymptotically stable. In Section 6, we perform some sensitivity analysis of several model parameters and give some useful comments on controlling the transmission of echinococcosis.
2 Model formulation
We divide the definitive hosts population (mainly the dogs) into three subclasses: the susceptible population, the exposed population, and the infected population, denoted by \(S_{1}(t)\), \(E_{1}(t)\), and \(I_{1}(t)\), respectively, and \(N_{1}(t)=S_{1}(t)+E_{1}(t)+I_{1}(t)\) is the total number of definitive hosts. The definitive hosts are infected by means of eating infected, cyst-containing organs.
We divide the intermediate hosts population into three subclasses: the susceptible population (\(S_{2}(t)\)), the exposed population (\(E_{2}(t)\)) and the infected population (\(I_{2}(t)\)), and \(N_{2}(t)=S_{2}(t)+E_{2}(t)+I_{2}(t)\) is the total number of intermediate hosts. The intermediate hosts are infected via the ingestion of eggs. Since eggs are released by the infected definitive hosts, we assume that the amount of eggs is proportional to the amount of infected definitive hosts. It follows from [14] that the parameters of the humans do not affect the dynamical behaviors of the echinococcosis model. Hence in the paper we only consider definitive hosts and intermediate hosts in our model.
By a similar proof to Theorem 1 of [14], we can show the following.
Lemma 2.1
The solutions of system (1) with initial conditions (4) and (5) satisfy \(S_{1}(t)>0\), \(E_{1}(t)\geq0\), \(I_{1}(t)\geq0\), \(S_{2}(t)>0\), \(E_{2}(t)\geq0\), \(I_{2}(t)\geq0\) for all \(t>0\).
Lemma 2.2
All solutions of system (1) with initial conditions (4) and (5) ultimately turn into region \(\Omega_{\varepsilon}\) as \(t\rightarrow\infty\).
Proof
Remark 2.1
Lemma 2.2 tells us that all feasible solutions of model (1) enter or remain in the region \(\Omega_{\varepsilon}\) as t becomes large enough. Hence, the dynamics of model (1) can be considered only in \(\Omega_{\varepsilon}\).
Remark 2.2
Near the disease-free equilibrium \(E_{0}\), each infected intermediate host produces \(\frac{\beta_{1}e^{-d_{1}\tau_{1}}}{ d_{2}+\varepsilon_{2}}\) new infected definitive hosts over its expected infectious period, and each definitive host produces \(\frac{\beta_{2}e^{-d_{2}\tau_{2}}}{d_{1}+\sigma}\) new infected intermediate hosts over its expected infectious period. The square root arises from the two ‘generations’ required for an infected definitive host or intermediate host to ‘reproduce’ itself.
3 Global stability of \(E_{0}\)
For the disease-free equilibrium \(E_{0}\), we will show that the disease dies out if \(R_{0}<1\).
Theorem 3.1
The disease-free equilibrium \(E_{0}=(\frac{A_{1}}{d_{1}},0,0,\frac{A_{2}}{d_{2}},0,0)\) is unstable if \(R_{0}>1\), and it is globally asymptotically stable if \(R_{0}<1\).
Proof
Note that if \(R_{0}=\sqrt{\frac{\beta_{1}\beta_{2}e^{-d_{1}\tau_{1}}e^{-d_{2}\tau_{2}}}{ (d_{1}+\sigma)(d_{2}+\varepsilon_{2})}}>1\), then \(f(0)=(d_{1}+\sigma)(d_{2}+\varepsilon_{2})-\beta_{1}\beta_{2}e^{-d_{1}\tau _{1}}e^{-d_{2}\tau_{2}} <0\), and \(f(+\infty)=\infty\). Hence, \(f(\lambda)=0\) has at least one positive root and \(E_{0}\) is unstable.
4 Uniform persistence
Using the method in [17], we now consider the issue of disease persistence.
Theorem 4.1
Proof
Let \(u_{t}\) be the solution of (1), let \(\Phi(t): X\rightarrow X\) be the solution semiflow associated with (1); that is, \(\Phi(t)\phi=u_{t}(\phi)\), \(\phi\in X\), \(t\geq0\). By Lemmas 2.1 and 2.2, the solutions of (1) are ultimately bounded, thus the semiflow \(\Phi(t)\) is point dissipative on X, and \(\Phi(t): X\rightarrow X\) is compact for all \(t>\tau\). By [19], it then follows that \(\Phi(t)\) admits a global attractor, which attracts every bounded set in X.
Since \(\lambda_{1}(\varepsilon)>0\), we have \(\lim_{t\rightarrow\infty} (I_{1}(t),I_{2}(t))=(\infty,\infty)\), a contradiction to (11). Thus (9) holds.
Denote the ω-limit set of the solution of system (1) starting in \(\phi\in X\) by \(\omega(\phi)\).
Claim 2: \(\bigcup_{\phi\in M_{\partial}}\omega(\phi)=E_{0}\).
For any \(\phi\in M_{\partial}\), we have \(I_{1}(t,\phi)\equiv0\) or \(I_{2}(t,\phi)\equiv0\). If \(I_{1}(t,\phi)\equiv0\), then from system (1), we get \(\lim_{t\rightarrow\infty}S_{2}(t,\phi)=\frac{A_{2}}{d_{2}}\), \(\lim_{t\rightarrow\infty}E_{2}(t,\phi)=0\), \(\lim_{t\rightarrow\infty}I_{2}(t,\phi)=0\). By the theory of asymptotically autonomous semiflows [18], it follows that \(\lim_{t\rightarrow\infty}S_{1}(t,\phi)=\frac{A_{1}}{d_{1}}\), \(\lim_{t\rightarrow\infty}E_{1}(t,\phi)=0\). If \(I_{2}(t,\phi)\equiv0\), again from system (1), we get \(\lim_{t\rightarrow\infty}E_{1}(t,\phi)=0\), \(\lim_{t\rightarrow\infty}I_{1}(t,\phi)=0\); furthermore, we obtain \(\lim_{t\rightarrow\infty}S_{1}(t,\phi)=\frac{A_{1}}{d_{1}}\), \(\lim_{t\rightarrow\infty}S_{2}(t,\phi)=\frac{A_{2}}{d_{2}}\), \(\lim_{t\rightarrow\infty}E_{2}(t,\phi)=0\). Therefore, we have \(\bigcup_{\phi\in M_{\partial}}\omega(\phi)=E_{0}\).
5 Global stability of endemic equilibrium
In this section, we will study the global stability of endemic equilibrium of system (1). For simplicity, we assume that \(\varepsilon_{2}=0\), we find that when \(R_{0}>1\), system (1) has one endemic equilibrium; when \(R_{0}\leq1\), there is no endemic equilibrium, system (1) has only the disease-free equilibrium \(E_{0}\).
Theorem 5.1
Assume that \(\varepsilon_{2}=0\). If \(R_{0}>1\), system (1) has a unique endemic (positive) equilibrium \(E^{*}=(S_{1}^{*},E_{1}^{*},I_{1}^{*},S_{2}^{*},E_{2}^{*},I_{2}^{*})\). More specifically,
For the special case \(\varepsilon_{2}=\sigma=0\), that is, the disease-induced death rate of infected livestock population is zero (\(\varepsilon_{2}=0\)), we also assume that the infected dogs will not recover (\(\sigma=0\)). In this case, as regards the stability of the endemic equilibrium, we have the following theorems.
Theorem 5.2
Assume that \(\varepsilon_{2}=\sigma=0\). If \(R_{0}>1\), the endemic equilibrium \(E^{*}=(S_{1}^{*},E_{1}^{*},I_{1}^{*},S_{2}^{*}, E_{2}^{*},I_{2}^{*})\) of system (1) is locally asymptotically stable, where \(E^{*}\) is denoted by (13).
Proof
We can further give the global stability of the endemic equilibrium \(E^{*}\).
Theorem 5.3
Assume that \(\varepsilon_{2}=\sigma=0\). If \(R_{0}>1\), the endemic equilibrium \(E^{*}=(S_{1}^{*},E_{1}^{*},I_{1}^{*},S_{2}^{*}, E_{2}^{*},I_{2}^{*})\) of system (1) is globally asymptotically stable, where \(E^{*}\) is denoted by (13).
Proof
In Theorem 5.2, we have given the local stability of \(E^{*}\). We now prove the global attractivity of \(E^{*}\).
6 Numerical simulations
In this section, we carry out numerical simulations to illustrate our analytic results. Since all the parameters are not easy to find, we will assume some parameters.
In view of [22], we fix \(\beta_{1}=0.185 \mbox{ year}^{-1}\), \(d_{1}=0.04\mbox{ year}^{-1}\), \(d_{2}=0.07\mbox{ year}^{-1}\), we first choose \(\sigma=2\mbox{ year}^{-1}\), \(\varepsilon_{2}=0\mbox{ year}^{-1}\) (see [14]), \(\tau_{1}=\frac{1}{8}\sim\frac{1}{6}\mbox{ year}\), \(\tau_{2}=\frac{5}{12}\sim\frac{1}{2}\mbox{ year}\) (see [16]).
Remark 6.1
Although \(A_{1}\) does not affect the number of \(R_{0}\), reducing \(A_{1}\) can decrease the infection level of \(I_{2}\) (see Figure 4).
Based on the above analysis, we now give some control strategies by adjusting the parameters \(A_{1}\), \(\beta_{1}\), σ, and \(d_{1}\). (1) Decrease \(A_{1}\) by reducing the birth rate of newborn puppies. (2) \(\beta_{1}\) can be reduced by the following measures. Livestock slaughtering regulations and health education should be implemented in endemic areas. Infected offal should be treated harmlessly. Definitive hosts should be barred from slaughter houses. (3) σ can be increased through increasing the frequency of anticestodal drugs (e.g., praziquantel). (4) In order to increase \(d_{1}\), we can kill the infected definitive hosts and the stray dog populations.
Declarations
Acknowledgements
The authors are grateful to both reviewers for their helpful suggestions and comments. This work was supported in part by the National Nature Science Foundation of China (NSFC 11101323, 11101127 and 11302158), the Natural Science Basic Research Plan in Shaanxi Province of China (2014JQ1038), Scientific Research Program Funded by Shaanxi Provincial Education Department (11JK0469), the Research Project at Yuncheng University (SWSX201404).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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