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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 703826, 11 pages
Global Dynamic Behavior of a Multigroup Cholera Model with Indirect Transmission
1Department of Mathematics, North University of China, Taiyuan, Shan'xi 030051, China
2Complex Systems Center, Shanxi University, Taiyuan, Shan'xi 030006, China
3Institute of Information Economy, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China
4School of Mathematical Sciences, Fudan University, Shanghai 200433, China
Received 4 October 2013; Accepted 7 November 2013
Academic Editor: Antonia Vecchio
Copyright © 2013 Ming-Tao Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
For a multigroup cholera model with indirect transmission, the infection for a susceptible person is almost invariably transmitted by drinking contaminated water in which pathogens, V. cholerae, are present. The basic reproduction number is identified and global dynamics are completely determined by . It shows that is a globally threshold parameter in the sense that if it is less than one, the disease-free equilibrium is globally asymptotically stable; whereas if it is larger than one, there is a unique endemic equilibrium which is global asymptotically stable. For the proof of global stability with the disease-free equilibrium, we use the comparison principle; and for the endemic equilibrium we use the classical method of Lyapunov function and the graph-theoretic approach.
Cholera, a waterborne gastroenteric infection, remains a significant threat to public health in the developing world. Outbreaks of cholera occur cyclically, usually twice per year in endemic areas, and the intensity of these outbreaks varies over longer periods . Hence, in the last few decades, enormous attention is being paid to the cholera disease and several mathematical dynamic models have been developed to study the transmission of cholera [1–7]. In these papers, they consider the population is uniformly mixed, but many factors can lead to heterogeneity in a host population. So in this paper we divide different population into different groups, which can be divided geographically into communities, cities, and countries, to incorporate differential infectivity of multiple strains of the disease agent.
In the case of cholera, the transmission usually occurs through ingestion of contaminated water or feces rather than through casual human-human contact . Therefore, direct contact of healthy person with an infected person is not a risk for contracting infection, whereas a healthy person may contract infection by drinking contaminated water in which pathogens, V. cholerae, are present . The members of this bacterial genus (V. cholerae) naturally colonize in lakes, rivers, and estuaries. Therefore we consider that cholera transmits to other individuals via bacteria in the aquatic environment and formulates a multi-group epidemic model for cholera. Letbe the total population which is divided into four epidemiological compartments, susceptible compartment, infectious compartment, recovered compartment , and vaccinated compartment. Letbe the density of V. cholerae in the aquatic environment. As a consequence of the increase in the density of virulent V. cholerae in the aquatic environment, humans become infected and begin to shed increasing numbers of bacteria into the aquatic environment, further elevating bacterial density and exacerbating the outbreak . The growth rate of density of bacteria in the aquatic environment is assumed to be proportional to the number of infectious individuals. We assume that the immunity induced by vaccination is perfect; therefore individuals in vaccinated individualscannot be infected. The model is called a multi-group cholera SIRVW epidemic model.
In recent years, multi-group epidemic models have been used to describe the transmission dynamics of many infectious disease in heterogeneous individuals, such as HIV/AIDS , dengue , West-Nile virus , sexually transmitted diseases ; and so on. It is well known that global dynamics of multi-group models with higher dimensions, especially the global stability of the endemic equilibrium, is a very challenging problem. Lajmanovich and York  proved global stability of the unique endemic equilibrium by using a quadratic global Lyapunov function on a class of -group SIS models for gonorrhea; Hethcote  proved global stability of the endemic equilibrium for multi-group SIR model without vital dynamics; Thieme  proved global stability of the endemic equilibrium of multi-group SEIRS model under certain restrictions. However, they only proved global stability of the endemic equilibrium for multi-group model under some special conditions. In 2006, Guo et al.  have first succeeded to establish the complete global dynamics for a multi-group SIR model, by making use of the theory of non-negative matrices, Lyapunov functions and a subtle grouping technique in estimating the derivatives of Lyapunov functions guided by graph theory. By using the results or ideas of , the papers [16, 17] proved the global stability of the endemic equilibrium for multi-group model with nonlinear incidence rates and the papers [18, 19] proved the global stability of the endemic equilibrium for multi-group model with distributed delays.
Distinguishing from these multi-group models with direct transmission from person to person, a multi-group cholera model with indirect transmission from the bacteria of the aquatic environment to person is proposed in this paper. We prove that the disease-free equilibrium is globally asymptotically stable if , while an endemic equilibrium exists uniquely and is globally asymptotically stable if .
The organization of this paper is as follows. In Section 2, we construct a multi-group cholera epidemiological and give some dynamic analysis on the disease-free equilibrium and the endemic equilibrium. An example is given in Section 3 and some conclusions are included in Section 4.
2. Mathematical Modeling and Analysis
For a multi-group epidemic model with cholera, the population of human is divided intodiscrete groups, where. Let , , , and be the numbers of susceptible, infectious, recovered, and vaccinated individuals in group at time, respectively. Letbe the density of bacteria in the aquatic environment in groupat time. Based on the assumptions in Section 1, the disease transmission rate of cholera between compartmentsandis denoted by, which means the susceptible individuals in theth group can contact the bacteria of the aquatic environment in theth () group. So the new infection that occurred in theth group is given by. The recruitment rate of individuals intocompartment with theth group is given by a constant. Within theth group, it is assumed that natural death of human is. A simple immunization policy is considered where the vaccination rate incompartment is given by a constant and the losing immunity rate from vaccination individuals is. We assume that individuals incompartment recover with a rate constant. Incompartment, the brucella shedding rate fromcompartment is, and the decaying rate of brucella is. So a general multi-group SIRVW epidemic model is described by the following system of differential equations:
The parameters , , , , , andare positive for all, which is made for the biological justification. And we assume thatis nonnegative for alland-square matrixis irreducible, which implies that every pair of groups is joined by an infectious path so that the presence of an infectious individual in the first group can cause infection in the second group.
Observe that the variabledoes not appear in the first and last two equations of system (1); this allows us to consider the following reduced system:
For each group, adding the four equations in system (2) gives
then it follows that
Therefore, omega limit sets of system (2) are contained in the following bounded region in the nonnegative cone of:
on the boundary of, where
2.1. The Basic Reproduction Number
According to the next generation matrix formulated in papers [20–22], we define the basic reproduction numberof system (2). In order to formulate, we order the infected variables first by disease state and then by group, that is,
Consider the following auxiliary system:
Follow the recipe from van den Driessche and Watmough  to obtain
We can get the inverse of, which equals
Thus, the next generation matrix is,
So we can calculate the basic reproduction number of system (2),
anddenotes the spectral radius. As we will show,is the key threshold parameters whose values completely characterize the global dynamics of system (2).
2.2. Global Stability of the Disease-Free Equilibrium of System (2)
For the disease-free equilibriumof system (2), we have the following property.
Theorem 1. If, the disease-free equilibriumof system (2) is globally asymptotically stable in the region.
Proof. Let, and defineis an eigenvalue of, sois a simple eigenvalue ofwith a positive eigenvector . By Theorem 2 in , there hold two equivalences:
To prove the locally stability of disease-free equilibrium, we check the hypotheses (A1)–(A5) in . Hypotheses (A1)–(A4) are easily verified, while (A5) is satisfied if all eigenvalues of thematrix
have negative real parts, where,
Calculate the eigenvalues of:
If, thenand, and the disease-free equilibriumof (2) is locally asymptotically stable.
Now we will prove that the disease-free equilibriumof system (2) is globally attractive when. From the third equation of system (2), we have
So we can have that, for a small enough positive number, there exists , such that for all,
Also from the equations of system (2), we have
From system (9) andwith all. Thus, when, we derive
Consider the following auxiliary system
Letbe the matrix defined by
and set. It follows from Theorem 2 in  thatif and only if. Thus, there exists ansmall enough such that. Using the Perron-Frobenius theorem, all eigenvalues of the matrixhave negative real parts when. Therefore it has
which implies that the zero solution of system (24) is globally asymptotically stable. Using the comparison principle of Smith and Waltman , we know that
By the theory of asymptotic autonomous system of Thieme , it is also known that
Sois globally attractive when. It follows that the disease-free equilibriumof (2) is globally asymptotically stable when. This completes the proof.
2.3. The Uniform Persistence and Unique Positive Solution of System (2)
In this section, we give the proof of the uniform persistence and the unique positive solution of system (2). Define
Theorem 2. When, there exists a positive constantsuch that whenfor
Proof. Consider the following system:
Using Corollary 3.2 in Zhao and Jing , it then follows that system (31) has a unique positive equilibrium which is globally asymptotically stable.
As to, choosesmall enough such that, where. Let us consider a perturbed system
From our previous analysis of system (32), we can restrictsmall enough such that (32) admits a unique positive equilibriumwhich is globally asymptotically stable.is continuous in, so we can further restrictsmall enough such that, .
For the sake of contradiction of Theorem 2, there is asuch that, , , for all. Then for, we have
Since the equilibrium (, , ) of (32) is globally asymptotically stable and, . There exists asuch that, for . As a consequence, for , there holds
Consider the following system
Since the matrixhas positive eigenvaluewith a positive eigenvector. It is easy to see that
Using the comparison principle of Smith and Waltman , we also know that
which leads to a contradiction, therefore we claim that
This completes the proof.
We also have the following result of system (2).
Theorem 3. If, then system (2) admits at least one positive equilibrium, and there is a positive constantsuch that every solutionof the system (2) withsatisfies
which implies that system (2) is uniformly persistent.
Proof. Now we prove that system (2) is uniformly persistent with respect to. By the form of (2), it is easy to see that bothandare positively invariant andis relatively closed in. Furthermore system (2) is point dissipative. Let
It is easy to show that
Noting that. We only need to prove. Assume. It suffices to show that, , for all , . Suppose not. Then there exist an , , andsuch that, and we partitioninto two setsandsuch that
is nonempty due to the definition of.is non-empty since. For any and we have that
It follows that there is ansuch that, for, This means thatdoes not belong to for, which contradicts the assumption that . This proves the system (41).
is globally asymptotically stable for system (2). It is clear that there is only an equilibriaumin and by aforementioned claim, it then follows thatis isolated invariant set in,. Clearly, every orbit inconverges to,is acyclic in. Using Theorem 4.6 in Thieme , we conclude that the system (2) is uniformly persistent with respect to. By the result of [27, 28], system (2) has an equilibrium. We further claim that, . Suppose that, ; from of (2), we can get, . It is a contradiction. Then is a componentwise positive equilibrium of system (2). This completes the proof.
The following theorem shows that there exists a unique positive solution for system (2) when.
Theorem 4. If, then there only exists a unique positive equilibrium for system (2).
Proof. Consider the following system:
We have that
Hence, the equilibrium of system (2) is equal to the following system:
Therefore, we only need to prove that (46) has a unique positive equilibrium when. Use the method in  to demonstrate the unique positive equilibrium of (46). First we prove that , , is the only positive solution of (46). Assume that andare two positive solutions of (46), both nonzero. If, thenfor some (). Assume without loss of generality that and moreover thatfor all (). Sinceandare positive solutions of (46), we substitute them into (46). We obtain
Butand; thus from the above equalities we get
This is a contradiction, so there is only one positive solution, , of (46). So when , there only exists a unique positive equilibrium for system (2).
2.4. Global Stability of the Unique Endemic Solution of System (2)
In this section, we prove that the unique endemic equilibrium of system (2) is globally asymptotically stable in. In order to prove global stability of the endemic equilibrium, the Lyapunov function will be used. In the following, we also use a Lyapunov function to prove global stability of the endemic equilibrium.
Theorem 5. If, the unique positive equilibriumof system (2) is globally asymptotically stable in.
Proof. Following  we define
which is a Laplacian matrix whose column sums are zero and which is irreducible. Therefore, it follows from Lemma 2.1 of  that the solution space of linear system
has dimension 1 with a basis
wheredenotes the cofactor of theth diagonal entry of. Note that from (53) we have that
For suchwe define a Lyapunov function
where, , , and. It is easy to see thatfor alland the equalityholds if and only if. The derivative along the trajectories of system (2) is
From system (44), we have
Now we claim that
Appealing to (51), (55) and (59),
From (61) we have
Next we show thatfor allby applying the graph-theoretic approach developed in [29–31]. As in ,denotes the directed graph associated with matrix,presents a subgraph of,denotes the unique elementary cycle of,presents the set of directed arcs in , anddenotes the number of arcs in. Thencan be rewritten as
Note that for each unicycle graph, it is easy to see that
and hencefor each, andif and only if
The equality holds if and only if , , , and for all. Therefore, following from LaSalle’s Invariance Principle , the unique endemic equilibriumof system (2) is globally asymptotically stable. This completes the proof.
3. A Numerical Example
Consider the system (1), when, one has the two-group model as follows:
We can give the basic reproduction number of system (71), which is
Taking, , , , , , , , , , , , , and using Matlab ODE solver, we run numerical simulations for two cases.
If , , , and , we have . Hence the disease-free equilibrium of system (71) is globally asymptotically stable (see Figure 1(a)). If, , , and , we have . Hence the endemic equilibrium of system (71) is globally asymptotically stable (see Figure 1(b)).
Cholera epidemic has become a major health problem for many developing countries. From good understanding of the transmission dynamics of cholera in many emergent epidemic regions, the heterogeneous host population can be divided into several homogeneous groups according to modes of transmission, contact patterns, or geographic distributions. Hence, in this paper, we proposed a multi-group cholera SIRVW epidemiological model. In order to distinguish many multi-group models with direct transmission from person to person, we only considered this multi-group cholera model with indirect transmission from the bacteria of the aquatic environment to person. Firstly, the basic reproduction numberof this model is given. Then, it is found that the model has two non-negative equilibria, the disease-free equilibrium and the endemic equilibrium. The disease-free equilibrium exists without any condition whereas the endemic equilibrium exists provided. Finally, through the analysis of the model it has been found that the global asymptotic behavior of multi-group SIRVW model is completely determined by the size of. That is, the disease-free equilibrium is globally asymptotically stable ifwhile an endemic equilibrium exists uniquely and is globally asymptotically stable if. By running numerical simulations for the cases of two-groups model, we can see that the disease-free equilibrium of system (71) is globally stable whenand the unique endemic equilibrium of system (71) is globally stable when.
This research is supported by the National Natural Science Foundation of China under Grants 11301490, 11301491, 11331009, 11171314, and 11147015, Natural Science Foundation of Shan’Xi Province Grant no. 2012021002-1, the specialized research fund for the doctoral program of higher education preferential development, no. 20121420130001, China Postdoctoral Science Foundation under Grant no. 2012M520814, Shanghai Postdoctoral Science Foundation under Grants no. 13R21410100 and IDRC104519-010.
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