Global Dynamic Behavior of a Multigroup Cholera Model with Indirect Transmission

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 numberR 0 is identified and global dynamics are completely determined byR 0 . It shows thatR 0 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 equilibriumwhich 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.


Introduction
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 [1].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][2][3][4][5][6][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 [1].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 [2].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.Let  be the total population which is divided into four epidemiological compartments, susceptible compartment , infectious compartment , recovered compartment , and vaccinated compartment .Let  be 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 [1].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 individuals  cannot 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 [8], dengue [9], West-Nile virus [10], sexually transmitted diseases [11]; 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 [12] 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 [13] proved global stability of the endemic equilibrium for multi-group SIR model without vital dynamics; Thieme [14] proved global stability of the endemic equilibrium of multigroup 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. [15] 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 [15], 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 multigroup 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 R 0 < 1, while an endemic equilibrium exists uniquely and is globally asymptotically stable if R 0 > 1.
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.

Mathematical Modeling and Analysis
For a multi-group epidemic model with cholera, the population of human is divided into  discrete groups, where  ∈ N. Let   (),   (),   (), and   () be the numbers of susceptible, infectious, recovered, and vaccinated individuals in group  = 1, 2, . . .,  at time , respectively.Let   () be the density of bacteria in the aquatic environment in group  = 1, 2, . . .,  at time .Based on the assumptions in Section 1, the disease transmission rate of cholera between compartments   and   is denoted by   , which means the susceptible individuals in the th group can contact the bacteria of the aquatic environment in the th ( = 1, 2, . . ., ) group.So the new infection that occurred in the th group is given by ∑  =1       .The recruitment rate of individuals into   () compartment with the th group is given by a constant   .Within the th group, it is assumed that natural death of human is   .A simple immunization policy is considered where the vaccination rate in   () compartment is given by a constant   and the losing immunity rate from vaccination individuals is   .We assume that individuals in   () compartment recover with a rate constant   .In   () compartment, the brucella shedding rate from   () compartment 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   ,   ,   ,   ,   , and   are positive for all  = 1, 2, . . ., , which is made for the biological justification.And we assume that   is nonnegative for all ,  = 1, 2, . . .,  and -square matrix  = (  ) 1≤,≤ is 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 variable   does 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 lim Therefore, omega limit sets of system (2) are contained in the following bounded region in the nonnegative cone of R 4 : It can be verified that region  is positively invariant with respect to system (2).System (2) always has a disease-free equilibrium on the boundary of , where 2.1.The Basic Reproduction Number.According to the next generation matrix formulated in papers [20][21][22], we define the basic reproduction number R 0 of system (2).In order to formulate R 0 , 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 [21] to obtain )2×2 .
(10) We can get the inverse of , which equals Thus, the next generation matrix is  −1 , So we can calculate the basic reproduction number of system (2), where and  denotes the spectral radius.As we will show, R 0 is the key threshold parameters whose values completely characterize the global dynamics of system (2).

Global Stability of the Disease-Free Equilibrium of System
(2).For the disease-free equilibrium  0 of system (2), we have the following property.
Theorem 1.If R 0 < 1, the disease-free equilibrium  0 of system (2) is globally asymptotically stable in the region .
So  0 is globally attractive when R 0 < 1.It follows that the disease-free equilibrium  0 of ( 2) is globally asymptotically stable when R 0 < 1.This completes the proof.
Proof.Now we prove that system (2) is uniformly persistent with respect to ( 0 ,  0 ).By the form of (2), it is easy to see that both  and  0 are positively invariant and  0 is relatively closed in .
The following theorem shows that there exists a unique positive solution for system (2) when R 0 > 1.
Theorem 4. If R 0 > 1, then there only exists a unique positive equilibrium  * for system (2).

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  0 .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 R 0 > 1, the unique positive equilibrium  * of system (2) is globally asymptotically stable in  0 .Proof.Following [15] we define which is a Laplacian matrix whose column sums are zero and which is irreducible.Therefore, it follows from Lemma 2.1 of [15] that the solution space of linear system (63) Next we show that   ≤ 0 for all ( 1 ,  1 ,  1 , . . .,   ,   ,   ) ∈  0 by applying the graph-theoretic approach developed in [29][30][31].As in [29],  = () denotes the directed graph associated with matrix B,  presents a subgraph of ,  denotes the unique elementary cycle of , () presents the set of directed arcs in , and  = () denotes the number of arcs in .Then   can be rewritten as )) .
Note that for each unicycle graph , it is easy to see that ∏ (,)∈()  , ≤ 0 for each , and  , = 0 if and only if (S, I, V, W) ≤   ≤ 0.