Research Article | Open Access
Analysis of a Simple Vector-Host Epidemic Model with Direct Transmission
Vector-host epidemic models with direct transmission are proposed and analyzed. It is shown that the stability of the equilibria in the proposed models can be controlled by the basic reproduction number of the disease transmission. One model considers that the dynamics of human hosts and vectors are described by SIS and SI model, respectively, where the global asymptotical stability for the equilibria of the model is analyzed by constructing Lyapunov function, respectively. The other model considers that the dynamics of the human hosts and vectors are described by SIRS and SI model, respectively, where the global stability of the disease-free equilibrium and the persistence of the disease in the model are also analyzed, respectively.
Mathematical models of infectious disease have proven to be valuable component for public health planing and responses, as well as an important application of population biology. A simple model may play a significant role in the development of a better understanding of the infectious disease and the various preventive strategies used against it [1–9]. Recently, mathematical models concerning the emergence and reemergence of the vector-host infectious disease have been proposed and analyzed. For example, Esteva and Vargas  have investigated an ordinary differential equation compartmental model for the spread of dengue fever. Their results suggested that the disease can be controlled by the threshold parameter and could persist if and only if exceeds . In , a vector-host epidemic mathematical model with demographic structure has been investigated, where the threshold condition for control of the vector diseases transmission has been obtained and the dynamical behavior of the model is globally performed. Epidemiological models with vector host are numerous in the literature [12–16], and we also refer the reader to [1–3] for a general reference since they are not detailed here.
In the aforementioned modeling work on vector-host disease transmissions, many authors consider that these infections diseases, such as malaria, dengue fever, West Nile virus, and so forth, are transmitted to the human population by insects or vectors (e.g., infected mosquitoes). However, some evidences show that direct transmission is possible through blood transfusion, vertically or through needlestick injury. Such models with direct transmission in addition to vector transmission have been also reported in malaria and in Chagas diseases. For example, in the paper in , an epidemic model of a vector-host disease with direct transmission and the vector-mediated transmission has been investigated. Recently, the paper in  has modeled and analyzed an age-since-infection structured model of Chagas diseases with direct transmission and vector transmission. In this paper, we shall investigate the transmission of a simple vector-host infectious disease by compartmental epidemiological model. A type Ross-MacDonald model for vector-host infectious disease is widely applied, where the host and vector population are divided into the susceptible and infected individuals. Here, we first consider a vector-host epidemic model with direct and vector transmissions. To explore dynamics of the solution of the nonlinear system of differential equations governing the infectious diseases, some mathematical methods and ideas are applied [19–23] in recent years. One of our aims in this paper is to show that the disease-free equilibrium and the endemic equilibrium are, respectively, the global stability by constructing suitable Lyapunov functional. Our results show that the equilibria of the model can be controlled by the basic reproduction number . That is, if is less than one, the disease-free equilibrium is globally asymptotically stable, and in such a case the endemic equilibrium does not exist; if is greater than one, then the disease persists and the unique endemic equilibrium is globally asymptotically stable.
Then, we extend the above model by taking into account that the dynamics of the human hosts and vectors are described by SIRS and SI model, respectively. Mathematical analysis of the dynamical behavior of the equilibria in this model is performed. The global stability of the disease-free equilibrium and the persistence of the disease in the model are obtained, respectively.
The paper is organized as follows. In Section 2, a vector-host epidemic model with direct and vector transmissions is presented, where the dynamics of the human hosts and vectors are described by SIS and SI model, respectively. In Section 3, the global stability of equilibria in the model is investigated by constructing suitable Lyapunov function. In Section 4, an extended vector-host epidemic model with direct and vector transmissions is investigated, where the dynamics of the human hosts and vectors are described by SIRS and SI model, respectively. The paper ends with brief remarks.
2. The Model with Host SIS
In this section, a vector-host epidemic model with direct and vector transmissions is presented and investigated, where the dynamics of the host is described by SIS model. It is assumed that there is no immunity in vector population and host population, and the total host population is partitioned into two distinct epidemiological subclasses which are susceptibles and infectious subclasses, with the sizes denoted by and , respectively, and the total vector population is divided into susceptibles and infectious, with the sizes denoted by and , respectively.
The proposed model satisfies the following assumptions.()Susceptible hosts can be infected via two routes of transmission, that is, directly, through a contact with an infected individual (possibly as a result of blood transfusion), and through being bitten by an infectious vector. Thus, we denote the rate of direct transmission by so that the incidence of new infections via this route is given by a standard incidence rate . We denote the biting rate that a pathogen-carrier vector has to susceptible hosts as , and the incidence of new infections transmitted by the vectors is given again by a standard incidence rate .()It is assumed that the host total population is constant. The birth rate and the per-capita natural mortality rate of host are equal, . is the recovery rate of infective hosts.()The vector total population is constant, and is the per-capita natural mortality rate of vector. The host infectious-to-vector susceptible transmission rate is given by .
The dynamics of this infectious disease in the host and vector populations can be described by the following system of nonlinear differential equations: The feasible region for system (2.1) is (the positive orthant of ). System (2.1) is obviously well-posed. In order to analyze system (2.1), let , , , , and . For the convenience, here, we still write , , , and as , , , and . So system (2.1) can be reduced to the following equations:
3. Stability of the Equilibria of System (2.2)
It is easy to verify that all of the solutions of system (2.2) exist and are nonnegative. Let It can be verified that is positively invariant with respect to (2.2). Direct calculation shows that system (2.2) has always the disease-free equilibrium in . Let . If , then system (2.2) has the unique endemic equilibrium in , where and , which is the unique positive solution of the following equation is given as: where
Remark 3.1. According to Theorem in , is called the basic reproduction number. It represents the average number of people infected directly and indirectly that single infectious host can generate in a totally susceptible population of hosts and vectors.
Now we shall investigate the local geometric properties of the equilibria of the system (2.2). We first give the following results.
Theorem 3.2. If , the disease-free equilibrium of model (2.2) is locally asymptotically stable, and is unstable if .
Proof. Linearizing around the disease-free equilibrium , we obtain the following characteristic equation: Let where Since it follows that from , thus and . So have two negative roots. So, all of the eigenvalues of the characteristic equation (3.5) are negative real parts. Hence, the equilibrium is locally asymptotically stable in the interior of . This completes the proof of Theorem 3.2.
Theorem 3.3. If , the disease-free equilibrium of the model (2.2) is globally asymptotically stable.
Proof. To establish the global stability of the disease-free equilibrium , we construct the following Lyapunov function:
By directly calculating the derivation of along the solution of (2.2), we obtain Using , , and , we have Noting that if and only if , , and , therefore, the largest compact invariant set in is the singleton , where is the disease-free equilibrium in system (2.2). By LaSalle's invariant principle , is globally asymptotically stable in .
This completes the proof of Theorem 3.3.
Now we shall investigate the local geometric properties of the endemic equilibria of system (2.2). We have the following results.
Theorem 3.4. If , the endemic equilibrium of system (2.2) is locally asymptotically stable.
Proof. Since the human and vector populations remain constant in , therefore, letting and , system (2.2) in the invariant set can be written as the equivalent to the following two-dimensional nonlinear system: Thus, the characteristic equation of is Using , it is easy to verify that . Therefore, from (3.12), we obtain that the eigenvalues of have two negative real parts. Therefore is locally asymptotically stable for .
Finally, we shall give the global stability of the endemic equilibrium . We have the following results
Theorem 3.5. If , the endemic equilibrium of the model (2.2) is globally asymptotically stable.
Proof. Let us construct the following Lyapunov function
By directly calculating the derivation of along the solution of (2.2), we have Since the arithmetic mean is greater than or equal to the geometric mean, we have Hence, it follows from (3.15) that we obtain Noting that if and only if , , , and , therefore, the largest compact invariant set in is the singleton , where is the disease-free equilibrium in system (2.2). By LaSalle's invariant principle, is globally asymptotically stable in .
This completes the proof of Theorem 3.5.
Remark 3.6. In this section, by constructing suitable Lyapunov function, it is established in Theorems 3.3 and 3.5 that is a sharp threshold parameter and completely determines the global stability of (2.2) in the feasible region . We can extend model (2.1) to more stage progression compartments model and establish the global stability of the model by constructing Lyapunov functions of the form
4. The Model with Host SIRS
In this section, we shall extend the model (2.1) by considering that the dynamics of the host is described by SIRS model. It is assumed that the host populations are constant that is, (constant). Similar to model (2.2), by using dimensionless, we obtain , and . Thus we consider the following differential equation model: where is the rate at which the host populations acquire immunity. is the per-capita rate of loss of immunity in host populations. The other variables and parameters are the same as those of model (2.1). Let . It is easy to verify that is positively invariant. Now we first investigate the existence of equilibria of (4.1). Letting the equations of system (4.1) with the right-hand side be zero, obviously, is always the disease-free equilibrium of system (4.1), and letting , we can obtain that the unique endemic equilibrium of system (4.1) for , satisfies the following relations: and is the positive solution of the following quadratic polynomial: By linearizing system (4.1) around the disease-free equilibrium of system (4.1) and analyzing the characteristic equation of , it is easy to obtain the following results.
Theorem 4.1. If , the disease-free equilibrium of system (4.1) is locally asymptotically stable, and is unstable if .
Theorem 4.2. If , then the infection-free equilibrium of system (4.1) is globally asymptotically stable in .
Proof. From the last two equations of system (4.1), we have
Let us consider the following equations:
From , we have . It is easy to show that, if for any solutions of (4.5) with nonnegative initial values, we have Let , and If is a solution of system (4.5) with nonnegative initial values , then, by comparison principle, we have , and for all sufficiently large . Hence, we have , and The maximal compact invariant subset in consists of the -axis. From this set, it is easy to obtain that , and for . It follows that all trajectories starting in approach for .
This completes the proof of Theorem 4.2.
Theorem 4.3. If , the disease of system (4.1) is uniformly persistent in .
Proof. Similar to the proof of Theorem in , we choose , , . It is easy to obtain that and , and is an isolated compact invariant set in . Furthermore, letting , thus, is an acyclic isolated covering of .
Now we only need to show that is a weak repeller for . Suppose that there exists a positive orbit of (4.1) such that Since , there exists a small enough such that From (4.1), we choose large enough such that, when , we have
Consider the following matrix defined by Since admits positive off-diagonal element, the Perron-Frobenius Theorem  implies that there is a positive eigenvector for the maximum eigenvalue of . From (4.7), we see that the maximum eigenvalue is positive. Let us consider the following system: Let be a solution of (4.10) through at , where satisfies , and . Since the semiflow of (4.10) is monotone and , it follows that are strictly increasing and as , contradicting the eventual boundedness of positive solutions of system (4.1). Thus, is weak repeller for .
This completes the proof of Theorem 4.3.
Remark 4.4. In this section, although we have not discussed the stability of in model (4.1) (this can be achieved via a tedious process, involving the determination of the signs of the eigenvalues of the corresponding Jacobian), numerical simulation (Figure 1) confirms that the equilibrium () in model (4.1) is stable whenever it exists.
5. The Concluding Remarks
Malaria, dengue fever, and so forht are very sever vector-host disease in some developing countries where hygienic and cultural conditions are inadequate. Despite the improvements in these conditions in the past decades, the endemic levels of these diseases have not tended to decrease; on the contrary, the endemic level in some countries has increased from initial incidence being about 60 per 100,000 yearly to the present 110 per 100,000 . In this paper, mathematical models for a vector-host disease transmission are proposed and analyzed. Our models seem to be quite robust in their qualitative behavior. The constant human recruitment rate and exponential natural death, as well as vector population with asymptotically constant, are incorporated into the model. The basic reproduction numbers of the model (2.1) and the extended models (4.1) are obtained, respectively. The dynamics behavior of the models is determined by their basic reproduction number, respectively. That is, if , the disease-free equilibrium is globally asymptotically stable. If , the disease persists and the unique endemic equilibrium is globally asymptotically stable. Additionally, we show that, if , system (4.1) has a unique positive equilibrium. Numerical simulations suggest that the unique endemic equilibrium is globally asymptotically stable whenever it exists, and we conjecture that the unique positive equilibrium is globally asymptotically stable. The simple model treated in this paper shows that direct transmission rate has played a very important role into the diseases transmission, besides indirect transmission rate, mean duration of host carriers, mean life of vector in the environment, the transmission rate of the host infected to vector susceptible, and so forth.
The authors are very grateful to the anonymous referees for their careful reading, constructive criticisms, helpful comments, and suggestions, which have helped them to improve the presentation of this work significantly. This work is partially supported by the National Natural Science Foundation of China (10971178); University Key Teacher Foundation of Henan Province (2009GGJS-076) and China Postdoctoral Science Foundation (20090460552), Innovative Research Team (in Science and Technology) in University of Henan Province (2010IRTSTHN006), and Natural Science Foundation of Henan Province (102300410022).
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Copyright © 2010 Liming Cai and Xuezhi Li. 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.