Advanced Nonlinear Dynamics of Population Biology and EpidemiologyView this Special Issue
Research Article | Open Access
Analysis of a Patch Model for the Dynamical Transmission of Echinococcosis
A patch model for echinococcosis due to dogs migration is proposed to explore the effect of dogs migration among patches on the spread of echinococcosis. We firstly define the basic reproduction number . The mathematical results show that the dynamics of the model can be completely determined by . If , the disease-free equilibrium is globally asymptotically stable. When , the model is permanence and endemic equilibrium is globally asymptotically stable. According to the simulations, it is shown that the larger diffusion of dogs from the lower epidemic areas to the higher prevalence areas can intensify the spread of echinococcosis. However, the larger diffusion of dogs from the higher prevalence areas to the lower epidemic areas can reduce the spread and is beneficial for disease control.
Echinococcosis, which is often referred to as hydatid disease, is a parasitic disease that affects both humans and other mammals, such as sheep, dogs, rodents, and horses . The two most clinically relevant species are Echinococcus granulosus and Echinococcus multilocularis, which cause cystic and alveolar echinococcosis respectively. Humans are incidental hosts and, in most cases, do not contribute to continuance of the parasite life cycle, except under unique circumstances .
The prevalent scope of echinococcosis in China is approximately 420 square kilometers, accounting for about 41.7% of the territory. The rate of incidence of echinococcosis has increased in the past decade. The operability of echinococcosis exceeds 10/100000 in each year. High-risk group subject to echinococcosis reaches up to 50 million, and the number of domestic animal amount being faced with the infection of echinococcosis is more than one hundred million, in which the amount of dogs is at least 5 million .
Mathematical modeling has become an important tool in analyzing the epidemiological characteristics of infectious disease and can provide useful control measures. Various models have been used to study different aspects of echinococcosis [4–16]. The models included varied primarily on the basis of six key features that were differentially incorporated in their design . These are the inclusion of a “latent” class (with time delay from host exposure to infectiousness); an age structure for definitive and/or intermediate hosts; the presence of density dependent constraints; accounting for seasonality; stochastic parameters; inclusion of a spatial and risk structures.
In , in order to explore effective control and prevention measures 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 . The model provided an approximate estimate of the basic reproduction number .
Many epidemic models with population dispersal among patches have been proposed and studied (see [19–28]). Wang and Zhao  proposed an epidemic model to describe the dynamics of disease spread among patches due to population dispersal. The effect of population dispersal among patches on the spread of a disease was investigated by Jin and Wang in . To understand the effect of transport-related infection on disease spread, an epidemic model for several regions which are connected by transportation of individuals has been proposed by Cui et al. in . In , an SIS patch model with nonconstant transmission coefficients was formulated to investigate the effect of media coverage and human movement on the spread of infectious diseases among patches. Qiu  developed a mathematical model to explore the effect of host migration between two patches on the spread of a vector-host disease.
To date, few scholars have researched the echinococcosis transmission models with dogs migration among patches. Considering an increasing number of stray dogs, the dispersal is an essential trait for dogs population. Therefore, we expect to explore the effect of dogs migration among patches on the spread of echinococcosis.
The purpose of this paper is to model the transmission dynamics of echinococcosis spread between two patches due to dogs migration and describe the dynamics of the model. The remaining part of this paper is organized as follows. The model is presented in Section 2. The basic properties on the positivity and boundedness of solutions computing the basic reproduction number are in Section 3. In Section 4, we establish the global stability of the disease-free equilibrium for the model. In Section 5, we will apply the theory of permanence to obtain the permanence of the model. The global stability theorem of endemic equilibrium is stated and proved in Section 6. In Section 7, we give some examples to illustrate how the dogs migration affects the dynamics of echinococcosis. A brief discussion is given in Section 8.
2. Model Formulation
In this section, we mainly formulate an epidemic model to describe the transmission dynamics of echinococcosis spread between two discrete patches due to dogs diffusion.
We firstly formulate a model for the spread of echinococcosis in the th patch. It follows from  that the parameters of humans do not affect dynamical behaviors of echinococcosis model. Hence in the paper we only consider dogs, livestock, and Echinococcus eggs in our model. We divide the dogs population in the th patch into two classes: the susceptible population and the infected population denoted by and , respectively. For livestock population, we divide the total livestock population in the th patch into two classes: susceptible and infectious denoted by and , respectively. The density of Echinococcus eggs in the th patch is denoted by . Our assumptions on the dynamical transmission of echinococcosis in the th patch are demonstrated in the flowchart (Figure 1).
If there is no dogs migration among patches, that is, the patches are isolated, we suppose that the echinococcosis dynamics in th patch is governed by All parameters are assumed positive. For the dog population in the th patch, describes the annual recruitment rate; is the natural death rate; denotes the recovery rate of transition from infected to noninfected dogs, including natural recovery rate and recovery due to anthelmintic treatment; describes the transmission of echinococcosis between susceptible dogs and infectious livestock after the ingestion of cyst-containing organs of the infected livestock. For the livestock population in the th patch, is the annual recruitment rate; is the death rate; describes the transmission of echinococcosis to livestock by the ingestion of Echinococcus eggs in the environment. For Echinococcus eggs in the th patch, denotes released rate from infected dogs; is the mortality rate of eggs.
When two patches are connected, we assume that susceptible and infected dogs of every patch leave for patch at a per capita rate . Then the dynamics of echinococcosis is governed by the following model:
3. Basic Properties and Basic Reproduction Number of the Model
In this section, we mainly present the preliminary results and derive reproduction number for model (2). In order to investigate the dynamics of model (2), we begin with stating some results on model (1). Model (1) has been analyzed in . Model (1) admits a disease-free equilibrium and a unique positive equilibrium , where The reproduction number of model (1) is established in , which can be expressed as From Theorems 3 and 5 in , we can obtain the following lemma.
Lemma 1. Considering model (1), one has that(a)if , then disease-free equilibrium is globally asymptotically stable;(b)if , then positive equilibrium is globally asymptotically stable.
In order to obtain our main results, we need the following lemma. Consider the following linear equation: We have the following result on system (6).
Lemma 2. System (6) has a unique equilibrium which is globally stable, where
Proof. The Jacobian matrix of (6) at is By simple calculations, the corresponding characteristic equation is where Therefore, all roots of have negative real parts, and hence is globally stable.
For any , we define region as follows: On the ultimate boundedness of solutions for model (2), we have the following result.
Proof. Let be any solution of model (2) with initial conditions (3) and let , . From model (2) we have and then from Lemma 2 we have . Hence, for any , there is a such that From the third and fourth equations of model (2), we have and therefore, there exists a such that Finally, from the fifth equation of model (2), we have and then there is a such that Let , and then for all we have This completes the proof of Lemma 3.
Simple algebraic calculation shows that model (2) always has a unique disease-free equilibrium . According to the concepts of next generation matrix and reproduction number presented in [29, 30], we define Noting that the disease-free equilibrium of model (2) is , then where Denote . After extensive algebraic calculations, we can obtain whereFrom the proof of Theorem 2 in , it follows that whereand is the maximum real part of the eigenvalues of matrix .
Using Theorem 2 in , we can easily obtain the following stability result.
Theorem 4. For model (2), one has that(a)if , then disease-free equilibrium is locally asymptotically stable;(b)if , then disease-free equilibrium is unstable.
4. Global Stability of the Disease-Free Equilibrium
We start by considering the global stability of disease-free equilibrium when .
Theorem 5. The disease-free equilibrium of model (2) is globally asymptotically stable in if .
Proof. From Theorem 4 we find that disease-free equilibrium is locally asymptotically stable if . In the following we only need to prove the global attractiveness of . From (24) we can see that if , then . Hence, there is a small enough number such that , where and
Let be any solution of model (2) in , then From model (2), it follows that Define an auxiliary linear system: Since system (29) is a linear system, the globally stability of origin is determined by the stability of matrix . Since , then all the eigenvalues of matrix have negative real parts. It then follows that each solution of (29) satisfies By the comparison principle we have Then the limiting system of model (2) is By Lemma 2 we find that there is a unique equilibrium of system (32), which is globally asymptotically stable. Thus, according to the theory of asymptotic autonomous systems , we finally obtain that disease-free equilibrium is globally asymptotically stable for model (2) when . This completes the proof of Theorem 5.
We now turn to the case where . We first establish the permanence for model (2).
satisfies model (2),
In order to prove Theorem 6, it suffices to show that repels uniformly the solutions of .
Firstly, by the form of model (2), it is easy to see that both and are positively invariant. Clearly, is relatively closed in . Furthermore, model (2) is point dissipative (see Lemma 3).
We now show that if , , then Assume It suffices to show that Suppose not, then there exists a such that at least one of , , , , , or is greater than zero. Here we only consider the case , , , , , and , . The other case can be deduced in the same way. Since it follows that there is an small enough such that , , , and , , for all . If , then we have This means that for all . If , it then follows from model (2) that It then follows that there exists an such that By the same way we can obtain that there exists an such that If , then we have This means that for all ; if , it then follows from model (2) that It then follows that there exists an such that By the same way we can obtain that there exists an such that Thus for all we have , , , , and , . This contradicts the assumption that . This proves (35).
From (24) we can see that if , then . Hence, there is a small enough number such that , where and is given by (26). Let and we can see the fact that . Hence we can choose small enough such that Suppose is a solution of model (2) with . We now claim that Suppose, for the sake of contradiction, that there exists a such that , , and , , for all . Then by model (2) we have for . Consider the following auxiliary system: As in our analysis in Lemma 2, system (51) has a unique positive equilibrium which is globally stable. By (48) and comparison principle, there is a such that , , , and for all . Consequently, for , we have Consider an auxiliary system The coefficient matrix of the right hand of (53) is . Since matrix has a positive eigenvalues with a positive eigenvector, it follows from a comparison principle that , , and as , , which leads to a contradiction. This proves (49). Hence . Clearly, every forward orbit in converges to . By Theorem 4.6 in  we are able to conclude that model (2) is uniformly persistent with respect to . Thus, by a well-known result in persistence theory in  we know that model (2) has at least one positive equilibrium . This completes the proof of Theorem 6.
6. Global Stability of
We further have the following result on the stability of the endemic equilibrium.
Theorem 7. If , then model (2) admits a unique equilibrium , which is globally asymptotically stable.
Proof. In Lemma 3, we have proved that and as , . Therefore, in model (2) we can represent and by and , , respectively, and the model (2) will degenerate into the following system with six equations: By Lemma 3, the dynamics of system (54) can be focused on the following region: We will use the theory of cooperate system to prove the global stability of system (54). Therefore, we only verify the assumption in Corollary 3.2  for system (54). Let and then is a continuously differentiable map. Clearly and for all with , . Since for , we have that is cooperative on . For every and , we have Using the same argument, we can show that is strictly sublinear on . By computing , we have whereClearly, is irreducible for . From (24) we can see that if , then . Since , we have . By Corollary 3.2 in , one can conclude that system (54) admits a unique positive equilibrium , which is globally asymptotically stable. According to the theory of asymptotic autonomous systems , we further obtain that endemic equilibrium is globally attractive for model (2).
To complement the mathematical analysis carried out in the previous section, we now investigate some of the numerical properties of the two-patch model (2).
Example 8. Take parameters in model (2) as follows:
, , , , , , , , , , , , , , , , , and . If the two patches are isolated, by simple calculations we have , .
From Lemma 1 we have that the disease will die out in the first patch and will be endemic in the second patch (see Figure 2). From Figure 3 we can easily see that will be larger than 1 under the condition of a larger and a smaller . This means that the larger diffusion of dogs from the lower epidemic areas to the higher prevalence areas can intensify the spread of echinococcosis (see Figure 4). However, when is small and is large, will be smaller than 1. This indicates that the larger diffusion of dogs from the higher prevalence areas to the lower epidemic areas can reduce the spread and is beneficial for disease control (see Figure 5).