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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 293293, 12 pages
http://dx.doi.org/10.1155/2013/293293
Research Article

Stability Analysis of a Vector-Borne Disease with Variable Human Population

1Centre for Advanced Mathematics and Physics, National University of Sciences and Technology, H-12, Islamabad 44000, Pakistan
2Department of Mathematics, Pusan National University, Busan 609-735, Republic of Korea
3National Fisheries Research and Development Institute, Busan 619-705, Republic of Korea

Received 25 November 2012; Revised 1 March 2013; Accepted 2 March 2013

Academic Editor: Ferenc Hartung

Copyright © 2013 Muhammad Ozair 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.

Abstract

A mathematical model of a vector-borne disease involving variable human population is analyzed. The varying population size includes a term for disease-related deaths. Equilibria and stability are determined for the system of ordinary differential equations. If , the disease-“free” equilibrium is globally asymptotically stable and the disease always dies out. If , a unique “endemic” equilibrium is globally asymptotically stable in the interior of feasible region and the disease persists at the “endemic” level. Our theoretical results are sustained by numerical simulations.

1. Introduction

Vector-borne diseases such as malaria, dengue fever, plague, and West Nile fever are infectious diseases caused by the influx of viruses, bacteria, protozoa, or rickettsia which are primarily transmitted by disease-transmitting biological agents, called vectors. A vector-borne disease is transmitted by a pathogenic microorganism from an infected host to another organism results to form from an infection by blood-feeding arthropods [1].

Vector-borne diseases, in particular, mosquito-borne disease, are transmitted to humans by blood-sucker mosquitoes, which have been a big problem for the public health in the world. The literature dealing with the mathematical theory on vector-borne diseases is quite extensive. Many mathematical models concerning the emergence and reemergence of the vector-host infectious disease have been proposed and analyzed in the literature [2, 3].

By direct transmission models, we mean that the infection moves from person to person directly, with no environmental source, intermediate vector, or host. In a vector-host model, direct transmission may take place by transfusion-related transmission, transplantation-related transmission, and needle-stick-related transmission [4]. Some models have been developed to study the dynamics of a vector-borne disease that considers a direct mode of transmission in human host population [57].

Mathematical modeling has proven to play an important role in gaining some insights into the transmission dynamics of infectious diseases and suggest control strategies. Appropriate mathematical models can provide a qualitative assessment for the problem. Some mathematical models discussed in [810] provide, best understanding about the dynamics and control of infectious diseases. Immense literature on the use of mathematical models for communicable diseases is available [11, 12]. The assumption of constant population size in epidemiological models is usually valid when we study the diseases of short duration with limited effects on mortality. It may not be valid when dealing with endemic diseases such as malaria, which has a high mortality rate. Ngwa and Shu [13] assumed density-dependent death rates in both vector and human populations, so that the total populations are varying with time that includes disease-related deaths. Esteva and Vargas [14] analyzed the effect of variable host population size and disease-induced death rate. Recently Ozair et. al analyzed vector-host disease model with nonlinear incidence [15].

In this paper, based on the ideas posed in [6, 14], we develop and analyze a vector-host disease model considering a direct mode of transmission as well as a variable human population. The aim of this paper is to establish stability properties of equilibria and the threshold parameter that completely determines the existence of endemic or disease-free equilibrium. If , the disease-free equilibrium is globally asymptotically stable. If , a unique endemic equilibrium exists and is globally asymptotically stable under parametric restrictions. However, in numerical simulations it is shown that the disease still can be “endemic” even if the conditions are violated.

The rest of the paper is organized as follows. In Section 2, we present a formulation of the extended mathematical model. The dimensionless formulation of proposed model is carried out in Section 3. Section 4 devotes existence and uniqueness of “endemic” equilibria. In Section 5, we use Lyapunov function theory to show global stability of disease-“free" equilibrium and geometric approach to prove global stability of “endemic” equilibrium. Discussions and simulations are done in Section 6.

2. Model Formulation

The human population is partitioned into subclasses of individuals who are susceptible, infectious, and recovered, with sizes denoted by , , and , respectively. The vector population is subdivided into susceptible and infectious vectors, with sizes denoted by and , respectively. The mosquito population does not have an immune class, since their infective period ends with their death. Thus, and are, respectively, the total human and vector populations at time . The model is given by the following system of differential equations:

In model (1), is the recruitment rate of humans into the population which is assumed to be susceptible. Susceptible hosts get infected via two routes of transmission, through a contact with an infected individual and through being bitten by an infectious vector. We denote the infection rate of susceptible individuals which results from effective contact with infectious individuals by and is the infection rate of susceptible humans resulting due to the biting of infected vectors. The incidence of new infections via direct and indirect route of transmission is given by the standard incidence form and , respectively. The term is the natural mortality rate of humans. We assume that infectious individuals acquire permanent immunity by the rate . The infectious humans suffer from disease-induced death at a rate . The recruitment and natural death rate of vector population is assumed to be . The susceptible vectors become infectious as a result of biting effect of infectious humans at a rate , so that the incidence of newly infected vectors is again given by standard incidence form . The total human population is governed by the following equation:

3. Dimensionless Formulation

Denote , , , , and . It is easy to verify that , , , , and satisfy the following system (see the Appendix details for): where solutions are restricted to and . Before analyzing the unnormalized model (1) and (2), we consider the normalized model (3) by scaling, and so we can study the following reduced system that describes the dynamics of the proportion of individuals in each class determining from or from and from , respectively. The correlation between normalized and unnormalized models is explained in the Appendix. Throughout this work, we study the reduced system (4) in the closed, positively invariant set , where denotes the nonnegative cone of with its lower dimensional faces.

4. Existence of Equilibria

We seek the conditions for the existence and stability of the disease-“free” equilibrium (DFE) and the “endemic” proportion equilibrium . Obviously, is the DFE of (4), which exists for all positive parameters. The Jacobian matrix of (4) at an arbitrary point takes the following form:To analyze the stability of DFE, we calculate the characteristic equation of at as follows: where By Routh Hurwitz criteria [16], all roots of (7) have negative real parts if and only if . So, is locally asymptotically stable for . If , the characteristic equation (7) has positive eigenvalue, and is thus unstable. We established the following theorem.

Theorem 1. The disease-free equilibrium is locally asymptotically stable whenever and unstable for .

In order to find the “endemic” equilibrium of (4), we set the right hand side of (4) equal to zero and get where is a positive solution of the equation where From right hand side of (5), we have and second equation of (9) , which means that If , there is no positive , and therefore the only equilibrium point in is . Note that this is a special case of .

Assume that .If , then , we have , and . Further, and . Thus, there exists unique such that (see Figure 1).If , then and , where . We observe that, and . Moreover, and . Therefore, there exists unique such that (see Figure 2).If , then , we have , and still , , . In this case, we can say that there is only one root or  three roots in the interval or . We know that has three real roots if and only if where or

293293.fig.001
Figure 1: .
293293.fig.002
Figure 2: .

If , there is unique such that in the feasible interval.

 If , there are three different real roots for say . Note that, differentiating with respect to , we obtain

The three different real roots for are in the feasible interval if and only if the following inequalities are satisfied: If , there are three real roots for , in which at least two are identical. Similarly, if inequalities (17) are satisfied, then there are three real roots for in the feasible interval, say .

Assume that .If , then and (10) reduces to , which implies that or , which is positive but it lies outside the interval or . If , then , we have , which implies that or is the solution of the equation where , , and . Moreover, and . Therefore, there exists no such that in the interval or .

In summary, regarding the existence and the number of the “endemic” equilibria, we have the following.

Theorem 2. Suppose that . There is always a disease-“free” equilibrium for system (4); if , then there is a unique “endemic” equilibrium with coordinates satisfying (9) and (10) besides the disease-“free” equilibrium.

5. Global Dynamics

5.1. Global Stability of the Disease-“Free” Equilibrium

In this subsection, we analyze the global behavior of the equilibria for system (4). The following theorem provides the global property of the disease-free equilibrium of the system.

Theorem 3. If , then the infection-free equilibrium is globally asymptotically stable in the interior of .

Proof. To establish the global stability of the disease-free equilibrium, we construct the following Lyapunov function: Calculating the time derivative of along (4), we obtain Thus, is negative if and if and only if . Consequently, the largest compact invariant set in , when , is the singelton . Hence, LaSalle’s invariance principle [16] implies that " is globally asymptotically stable in . This completes the proof.

5.2. Global Stability of “Endemic” Equilibrium

Here, we use the geometrical approach of Li and Muldowney to investigate the global stability of the endemic equilibrium in the feasible region . We have omitted the detailed introduction of this approach, and we refer the interested readers to see [17]. We summarize this approach below.

Consider a map from an open set to such that each solution to the differential equation is uniquely determined by the initial value . We have the following assumptions: is simply connected; there exists a compact absorbing set ; (21) has unique equilibrium in .

Let be a nonsingular matrix-valued function which is in and a vector norm on , where .

Let be the Lozinskiĭ measure with respect to the . Define a quantity as where , the matrix is obtained by replacing each entry of by its derivative in the direction of , , and is the second additive compound matrix of the Jacobian matrix of (21). The following result has been established in Li and Muldowney [17].

Theorem 4. Suppose that , , and hold, the unique endemic equilibrium is globally stable in if  .

Obviously is simply connected and is a unique endemic equilibrium for in . To apply the result of the above theorem for global stability of endemic equilibrium , we first prove the uniform persistence of (4) when the threshold parameter , by applying the acyclicity Theorem (see [18]).

Definition 5 (see [19]). The system (4) is uniformly persistent, that is, there exists (independent of initial conditions), such that .
Let be a locally compact metric space with metric and let be a closed nonempty subset of with boundary and interior . Clearly, is a closed subset of . Let be a dynamical system defined on . A set in is said to be invariant if = . Define .

Lemma 6 (see [18]). Assume that has a global attractor; there exists of pair-wise disjoint, compact and isolated invariant set on such that ; no subsets of form a cycle on ;each is also isolated in ; for each , where is stable manifold of . Then is uniformly persistent with respect to .

Proof. We have , , . Obviously . Since is bounded and positively invariant, so there exists a compact set in which all solutions of system (4) initiated in ultimately enter and remain forever. On -axis we have which means as . Thus, is the only omega limit point on , that is, for all . Furthermore, is a covering of , because all solutions initiated on the -axis converge to . Also is isolated and acyclic. This verifies that hypothesis and hold. When , the disease-“free” equilibrium (DFE) is unstable from Theorem 1 and also . Hypothesis and hold. Therefore, there always exists a global attractor due to ultimate boundedness of solutions.

The boundedness of   and the above lemma imply that (4) has a compact absorbing set [19]. Now we shall prove that the quantity . We choose a suitable vector norm in and a matrix-valued function Obviously, is and nonsingular in the interior of . Linearizing system (4) about an endemic equilibrium gives the following Jacobian matrix:The second additive compound matrix of is given by where The matrix can be written in block form as with where Consider the norm in as where denotes the vector in . The Lozinskiĭ, measure with respect to this norm is defined as , where From system (4), we can write Since is a scalar, its Lozinskiĭ measure with respect to any vector norm in will be equal to . Thus and will become Also , and are the operator norms of and which are mapping from , to and from to respectively, and is endowed with the norm. is the Lozinskiĭ measure of matrix with respect to norm in . if . Hence Thus, where . Since (4) is uniformly persistent when , so for such that implies , and for all . Thus, for all , which further implies that . Therefore, all the conditions of Theorem 4 are satisfied. Hence, unique endemic equilibrium is globally stable in .

6. Discussions and Simulations

This paper deals with a vector-host disease model which allows a direct mode of transmission and varying human population. It concerns diseases with long duration and substantial mortality rate (e.g., malaria). Our main results are concerned with the global dynamics of transformed proportionate system. We have constructed Lyapunov function to show the global stability of disease-“free” equilibrium and the geometric approach is used to prove the global stability of “endemic” equilibrium. The epidemiological correlations between the two systems (normalized and unnormalized) have also been discussed. The dynamical behavior of the proportionate model is determined by the basic reproduction number of the disease. The model has a globally asymptotically stable disease-“free” equilibrium whenever (Figures 3 and 4). When , the disease persists at an “endemic” level (Figures 5 and 6) if . Figures 7, 8, 9, and 10 describe numerically “endemic” level of infectious individuals and infectious vectors under the condition . We here question that what are the dynamics of the proportionate system (4) even if the condition is not satisfied? We see numerically that if or then infectious individuals and infectious vectors will also approach to endemic level for different initial conditions (Figures 11, 12, 13, and 14). It is also numerically shown that the same is true for the case or (Figures 15, 16, 17, and 18). This implies that the condition is weak for the global stability of unique “endemic” equilibrium.

293293.fig.003
Figure 3: ; ; ; ; ; ; ; .
293293.fig.004
Figure 4: ; ; ; ; ; ; ; .
293293.fig.005
Figure 5: ; ; ; ; ; ; ; .
293293.fig.006
Figure 6: ; ; ; ; ; ; ;
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Figure 7: ; ; ; ; ; ; , ; .
293293.fig.008
Figure 8: ; ; ; ; ; ; , ; .
293293.fig.009
Figure 9: ; ; ; ; ; ; , ; .
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Figure 10: ; ; ; ; ; ; , ; .
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Figure 11: ; ; ; ; ; ; , ; .
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Figure 12: ; ; ; ; ; ; , ; .
293293.fig.0013
Figure 13: ; ; ; ; ; ; ; .
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Figure 14: ; ; ; ; ; ; ; .
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Figure 15: ; ; ; ; ; ; , ; .
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Figure 16: ; ; ; ; ; ; , ; .
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Figure 17: ; ; ; ; ; ; ; .
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Figure 18: ; ; ; ; ; ; ; .

Appendix

Using the transformation , , , , and for scaling, their differentials: , , , , and , and the system (1) and (2), we obtain the dimensionless form (3). If and , then and so remains fixed at its initial value . In this case, the system (1) becomes the model with constant population whose dynamics are the same as the proportionate system (3). Hence, the solutions with initial conditions tend to if and to if . In the rest of this section, we suppose that . From system (1) and (2), the trivial equilibrium can be easily obtained. Assume that is the endemic equilibrium of system (1) and (2), where . This equilibrium exists if and only if the following equations are satisfied where and . We introduce the parameters From (2) we have for By the definition of , we have following threshold result.

Theorem A.1. The total population for the system (1) decreases to zero if and increases to if as . The asymptotic rate of decrease is if , and the asymptotic rate of increase is when .

Theorem A.2. Suppose , for , tend to if and tend to if .

Proof. Since as , so in the limiting case the proportion of infectious mosquitoes is related to the proportion of infectious humans as thus, the equation for has limiting form which shows that decreases exponentially if and increases exponentially if .
The solution is given by From the exponential nature of , it follows that declines exponentially if and grows exponentially if .
Suppose , then corresponding to and the differential equation for will have the form which means that is bounded for all , the equilibria have one eigenvalue zero, and the other eigenvalues have negative real parts. Therefore, each orbit approaches an equilibrium point.
If , the disease becomes “endemic.” From the global stability of and the equation we observe that approaches to or if or . From the global stability of , we have converges to some as approaches to . Since ,,, so we have ,, . All the above discussion is summarized in Table 1.

tab1
Table 1: Asymptotic behavior with threshold criteria.

Acknowledgment

This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (MEST) (2012-000599).

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