About this Journal Submit a Manuscript Table of Contents
Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 945767, 13 pages
http://dx.doi.org/10.1155/2013/945767
Research Article

A Schistosomiasis Model with Praziquantel Resistance

1School of Mathematical Sciences, Anhui University, Hefei 230601, China
2College of Science, Beijing University of Civil Engineering and Architecture, Beijing 100044, China

Received 28 August 2012; Accepted 23 October 2012

Academic Editor: Cengiz Çinar

Copyright © 2013 Longxing Qi and Jing-an Cui. 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.

Abstract

A compartmental model is established for schistosomiasis with praziquantel resistance. The model considers the impact of genetic resistance and drug treatment on the transmission of schistosomiasis. We calculate the basic reproductive number and discuss the existence and stability of disease-free equilibrium, boundary equilibrium, and coexistence equilibrium. Our analysis shows that regardless of whether drug treatment leads to the emergence of resistance, once the impact of genetic resistance is larger, the resistant strain will be dominant, which is detrimental to the control of schistosomiasis. In addition, once the proportion of human with drug-resistant strain produced by drug treatment is larger, the number of human and snails with resistant strain is larger. This is not a good result for drug treatment with praziquantel.

1. Introduction

Currently, treatment of human beings infected by schistosomiasis primarily focuses on chemotherapy with praziquantel (PZQ). PZQ appeared as a new schistosomicidal compound during the 1970s [1]. In recent years, PZQ has become the drug of choice in most endemic areas because of its efficacy, its ease of administration, its tolerable side-effects, and its cost [1]. Although the effectiveness of PZQ against schistosomiasis is well documented, the precise mode of action of the drug has not clearly defined [2]. It is reported that the chemotherapy of many helminth infections is complicated by the occurrence of drug resistance and drug tolerance (a natural resistance) to certain anthelmintics [2]. Not surprisingly, recent epidemiological evidence suggests the emergence of PZQ-resistanttolerant schistosomes [1, 3, 4]. Resistance is defined as a genetically transmitted loss of sensitivity in a parasite population that was previously sensitive to a given drug [2]. Tolerance is an innate insusceptibility of a parasite to a drug, with the caveat that the parasite must not have been previously exposed to the drug [2]. The first report of possible PZQ resistance came from an intensive focus in northern Senegal, where the drug had produced very low cure rates (18–39%) [5, 6]. And snails collected in the area carried schistosoma strains. When tested in the laboratory, those snails had a decreased susceptibility to PZQ [7, 8]. Additional PZQ-resistant evidence was collected in Egypt [4, 9]. Preliminary studies have begun on these isolates to identify genetic, physiological, and morphological characteristics associated with PZQ resistance, and some of these may find use as markers for monitoring whether or not resistance is developing in endemic areas, where the drug is used [10].

Many papers have reported that drug treatment results in the emergence of schistosome resistance to PZQ [1113]. Drug treatment can remove drug-susceptible parasites in infected human beings, while resistant parasites survive. However, many investigations find that traits of PZQ resistance of Schistosoma mansoni are dominant inheritance [1416]. The resistant worms can reproduce and pass the resistant genes to the next generation. Furthermore, the resistance of Schistosoma mansoni to PZQ can be expressed in eggs, miracidia, cercariae, adults, and all stages of development [1416]. In other words, some definitive hosts carrying resistant schistosomes can infect snails and make those snails carry resistant schistosomes. On the contrary, some snails carrying resistant schistosomes can infect definitive hosts and also make those definitive hosts carry resistant schistosomes. It goes on, the control of schistosomiasis will face enormous difficulties. Therefore, it is necessary to study the impact of this schistosome genetic resistance on the transmission of schistosomiasis.

In previous schistosomiasis models, resistant problems have been studied by considering that the resistance of schistosomiasis is due to drug treatment [1113]. In [11, 13], the authors proposed a multistrain schistosome model including sensitive and resistant parasite strains. Their goal was to infer the impact of drug treatment on the maintenance of schistosome genetic diversity. In their assumptions, the drug-sensitive parasite strain had an additional per capita death rate, , due to treatment. For a parasite strain that had developed drug resistance with a resistance level (), this treatment-related death rate was assumed to be reduced by the factor to . Their results implied that higher treatment rate could allow for coexistence between sensitive and resistant parasite strains. In [12], the authors formulated a deterministic model with multiple strains of schistosomes in order to explore the role of drug treatment in the maintenance of a polymorphism of parasite strains that differed in their resistance levels. And snails infected by parasite strains were divided into multistrain subclasses according to the different level of parasite strains. Analysis of the model showed that the likelihood that resistant strains would increase in frequency depended on the interplay between their relative fitness, the cost of resistance, and the degree of selection pressure exerted by drug treatments.

Motivated by [1113], we establish a new model considering hosts with sensitive and resistant strains in this paper. Our purpose is mainly to study the impact of drug treatment and genetic resistance on the transmission of schistosomiasis.

Our paper is organized as follows. In Section 2, we establish a mathematical model with praziquantel resistance and obtain basic reproductive number and existence of equilibria. And then the stability of disease-free equilibrium is obtained in Section 3. Section 4 devotes to stability analysis of boundary equilibria. In Section 5, stability analysis of endemic equilibrium is performed.

2. Mathematical Model

According to different level of parasite strains, we divide infected hosts into sensitive and resistant strains. Considering resistance and inheritance of resistance, new resistant strains are composed of two parts. We classify definitive and intermediate hosts as susceptible, sensitive and resistant in the following:(i), the population of susceptible human; (ii), the population of human infected with sensitive parasite strain; (iii), the population of human infected with resistant parasite strain; (iv), the population of susceptible snail host; (v), the population of snail host carrying sensitive parasites; (vi), the population of snail host carrying resistant parasites.

We follow some of the available models for schistosomiasis [1719] and assume that the reproduction rate of hosts is constant, and we ignore the recovery class of host since the life span of infected is short in comparison to that of human [18, 20]. All parameters in the model are assumed to be nonnegative constants:(i), recruitment rate of human; (ii), per capita natural death rate of human; (iii), per capita disease-induced death rate of human; (iv), per capita contact transmission rate from infected snails to susceptible human; (v), recruitment rate of snails; (vi), per capita natural death rate of snails; (vii), per capita disease-induced death rate of snails; (viii), per capita contact transmission rate from infected human to susceptible snails; (ix), the rate of treatment for infected human; (x), the proportion of human with drug-resistant strain produced by treatment; (xi), represents the impact of inheritance and the cost of resistance on transmission rate, we assume that since the transmission rate is reduced due to resistance [12]; (xii), represents the impact of resistance on disease-induced death rate of human; (xiii), represents the impact of resistance on disease-induced death rate of snails.

Then, we have a model with the form: Using standard methods, it is easy to see that disease free equilibrium always exists. Let , , , , , and . According to the concept of next generation matrix [21] and the formula of the basic reproductive number for ODE compartmental models [22], if we let one can calculate that the eigenvalues of the next generation matrix are given by Then, it follows that the basic reproductive number for the system (1) is given by

To obtain other equilibria, we let the right-hand side of (1) equal to zero and obtain If , we have and a formula of as follows:

Note that equals to . Hence, if , there exists a boundary equilibrium with only resistant type, given as , where

If drug treatment does not lead to drug resistance, that is, , then when , we can obtain the other boundary equilibrium with only sensitive type, given as , where

Now, we study existence of coexistence equilibrium for the system (1). From (5)–(10), we obtain Following (6), (7), (9), and (10), we have

In the case that , (15) leads to Note that equals to . It is easy to see that if , (16) cannot hold, which implies that there is not coexistence equilibrium. If , (16) always holds. Hence, if , that is, , (14) represents the existence of coexistence equilibrium in the form of a line.

In the case that , (15) leads to Note that equals to . Following (14) and (17), we can obtain that if and , then the unique coexistence equilibrium is given by , where

Summarizing above analyses, we have the following result.

Lemma 1. The existence of equilibria for the system (1) is as follows:(1)the disease free equilibrium always exists;(2)if , there exists a boundary equilibrium with only resistant type ;(3)if and , there exists a boundary equilibrium with only sensitive type ;(4)if and , there exists coexistence equilibrium in the form of a line (14);(5)if and and , there exists a unique coexistence equilibrium .

The following section shows that the basic reproductive number provides a threshold condition for schistosoma extinction in (1).

3. Stability Analysis of the Disease Free Equilibrium

In this section, we will analyze stability of the disease free equilibrium of the model (1). The stability of the disease free equilibrium determines whether schistosomiasis will be permanent in an uninfected population. The following result shows that schistosome will go extinct if .

Theorem 2. The disease free equilibrium of the system (1) is locally asymptotically stable if and unstable if .

Proof. The Jacobian matrix for the system (1) is given byThen, the eigenvalues of are , and roots of the following equations: Note that equals to and , which leads to and . Hence, if , then all the roots of (20) have negative real parts. Hence, using the Routh-Hurwitz criterion, we can obtain that the disease free equilibrium of the system (1) is locally asymptotically stable if and unstable if .

Now, we turn to the study of the global stability of the disease free equilibrium of the model (1) by using Metzler matrix theory and the technique of Kamgang and Sallet [23].

Consider systems of the following form: where , , and and are . We denote by the state of the system and is a disease free equilibrium on a positively invariant set . Now rewrite (21) as For the system (22), we make the following assumptions.(h1) The system is defined on the positively invariant set of the nonnegative orthant. The system is dissipative on .(h2) The subsystem is globally asymptotically stable at the equilibrium on the canonical projection of on .(h3) The matrix is Metzler and irreducible for any given .(h4) There exists an maximum matrix , then for any such that , .(h5), that is, the greatest real part of eigenvalues of is nonnegative.For convenience, we state two lemmas due to Kamgang and Sallet [23].

Lemma 3. If the above hypotheses, h1–h5, are satisfied, then the disease free equilibrium is globally asymptotically stable in .

Lemma 4. If the same notations and hypotheses in Lemma 3 hold and if, furthermore, we have , the disease free equilibrium is globally asymptotically stable if and only if .

Next, we discuss the global stability of the disease free equilibrium of the system (1) using the above two Lemmas.

From the system (1), we know This proves that the set is a compact positively invariant absorbing set contained in the nonnegative orthant. Thus, the system (1) is dissipative on because the trajectories of (1) are forward bounded. Now, we will study the system (1) on .

We set for system (1) , , and . As in [23], we express the subsystem as and This is a linear system, and its unique equilibrium () (corresponding to the disease free equilibrium of (1)) is globally asymptotically stable, hence the assumptions (h1) and (h2) are satisfied.

The matrix is given by As required by hypothesis h3, for any , the matrix is irreducible.

Now, let us check (h4). There is a maximum which is uniquely realized in if and , which corresponds to the disease free equilibrium. This maximum matrix is then , the subblock of the Jacobian matrix at the disease free equilibrium, corresponding to the matrix . The matrix is given by Therefore, we are in the situation of Lemma 4, where the maximum is attained at the disease free equilibrium.

The hypothesis (h5) requires that . Writing as a block matrix . Since is already a Metzler stable matrix, the condition is equivalent to the condition [23], where Then, Hence, the condition is equivalent to . We have seen that the hypotheses (h1), (h2), (h3), (h4), and (h5) are satisfied. Then, by Lemma 4, we have the following result.

Theorem 5. The disease free equilibrium of the system (1) is globally asymptotically stable if .

4. Stability Analysis of the Boundary Equilibria

In this section, we turn to study stability of the two boundary equilibria. From Lemma 1, we know that if , there exists a boundary equilibrium with only resistant type . Through calculations, we can obtain the characteristic equation as following: Hence, the eigenvalue of are roots of the following equations: Here,

From (12), we can obtain . Then, It is easy to see that . Then, equals to , which implies that the roots of (31) have negative real parts if .

Based on and , we can obtain Here, and if .

If , let and , then and . Note that and , then and . Hence, we can obtain It follows from Routh-Hurwitz criterion that all roots of (32) have negative real parts if . Summering above analyses, we have the following result.

Theorem 6. The boundary equilibrium of the system (1) is locally asymptotically stable if and .

Now, we study the global stability of the boundary equilibrium . Consider the Lyapunov function where and . The Lyapunov derivative is This implies that the sensitive type dies out if . Then the largest compact invariant set of the system (1) in the set is . Using the LaSalle-Lyapunov theorem, we know that all trajectories in eventually tend to as . Then, we only need to study the dynamical behavior of (1) in . At this time, (1) reduces to the following system To show that all trajectories of (39) in the interior of approach the point corresponding to the boundary equilibrium , consider the Lyapunov function where positive constants and are defined in the following. It is easy to see that for , and . Hence, the function is positive definite with respect to the point .

Computing the derivative of along solutions of system (39), we have Substituting , , and into , we obtain Let and . Following from and , we have Hence, for all , It is easy to see that where is an arbitrary positive number. Substituting , , , and into the first equation of system (39), we obtain and then the above formula holds if and only if . Therefore, the only compact invariant subset of the set where is the point , corresponding to the boundary equilibrium . By LaSalle’s Invariance Principle, is globally asymptotically stable if and . Summering above analyses, we have the following result.

Theorem 7. The boundary equilibrium of the system (1) is globally asymptotically stable if .

Now, we turn to the other boundary equilibrium. From Lemma 1, we know that if and , the boundary equilibrium with only sensitive type exists. Through calculations, we can obtain the characteristic equation as following: Hence, the eigenvalue of are roots of the following equations: Here,

From (13), we can obtain that . Then, it is easy to see that the roots of (48) have negative real parts if . Similarly to the case of , using Routh-Hurtwitz criterion, we can obtain that all roots of (49) have negative real parts if . Summering above analysis, we have the following result.

Theorem 8. When , the boundary equilibrium of the system (1) is locally asymptotically stable if and .

Now, we study the global stability of the boundary equilibrium . Consider the Lyapunov function where and . The Lyapunov derivative is This implies that the resistant type dies out if . Then, the largest compact invariant set of the system (1) in the set is . Using the LaSalle-Lyapunov theorem, we know that all trajectories in eventually tend to as . Then, we only need to study the dynamical behavior of (1) in . At this time, (1) reduces to the following system: To show that all trajectories of (53) in the interior of approach the point corresponding to the boundary equilibrium , consider the Lyapunov function where positive constants and are defined in the following. It is easy to see that for and . Hence, the function is positive definite with respect to the point .

Computing the derivative of along solutions of system (39), we have Substituting , , , and into , we obtain Note that for the limiting system , then . Let and . Following from and , we have Hence, for all , It is easy to see that Therefore, the only compact invariant subset of the set where is the point , corresponding to the boundary equilibrium . By LaSalle’s Invariance Principle, is globally asymptotically stable if and . Summering above analysis, we have the following result.

Theorem 9. When , the boundary equilibrium of the system (1) is globally asymptotically stable if .

5. Stability Analysis of the Coexistence Equilibrium

In this section, we turn to study the local stability of the coexistence equilibrium in the limiting system of (1) by using Krasnoselskii sublinearity trick [24], as in [25, 26]. In detail, if is a system of differential equations and is an equilibrium point, then to prove the local asymptotical stability of is to prove that the linearized equation has no solutions of the form with , and, . This implies that the eigenvalues of the characteristic polynomial associated with the linearized equations have negative real part, that is, . Then, the coexistence equilibrium is locally asymptotically stable.

Considering the limiting system where and . In this way, let , . Substituting a solution of the form (60) into the linearized system (61) of the coexistence equilibrium , we obtain the following linear equations: which is equivalent to the system Moving all the negative terms to the left-hand side, after some manipulations we obtain the system where Note that the matrix has nonnegative entries, and satisfies

To show that , we distinguish two cases: and . In the first case, (62) is a homogeneous linear system. Through calculations, we have the determinant of (62) is where