BioMed Research International

Volume 2013 (2013), Article ID 710917, 9 pages

http://dx.doi.org/10.1155/2013/710917

Research Article

Global Stability of Vector-Host Disease with Variable Population Size

Muhammad Altaf Khan,¹ Saeed Islam,¹ Sher Afzal Khan,² and Gul Zaman³

¹Department of Mathematics, Abdul Wali Khan University, Mardan, Khyber Pakhtunkhwa, Pakistan

²Department of Computer Sciences, Abdul Wali Khan University, Mardan, Khyber Pakhtunkhwa, Pakistan

³Department of Mathematics, University of Malakand, Dir, Pakistan

Received 26 March 2013; Accepted 22 May 2013

Academic Editor: Ali Khraibi

Copyright © 2013 Muhammad Altaf Khan 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.

The paper presents the vector-host disease with a variability in population. We assume, the disease is fatal and for some cases the infected individuals become susceptible. We first show the local and global stability of the disease-free equilibrium, for the case when . We also show that for , the disease free-equilibrium of the model is both locally as well as globally stable. For , there exists a unique positive endemic equilibrium. For , the disease persistence occurs. The endemic equilibrium is locally as well as globally asymptotically stable for . Numerical results are presented for the justifications of theoratical results.

1. Introduction

Mathematical modeling for disease transmission in host population is of great practical value in predicting and controlling disease spread (West Nile virus in North America in the 1990s, Avian influenza worldwide in the 2000s, SARS in Asia in 2003, etc.). The battle between infectious diseases and humans was heavily lopsided for much of the history. Since the pioneering work of Edward Jenner (a doctor, who worked in Gloucestershire, UK, noticed that individuals who had contracted cowpox rarely caught smallpox) on smallpox [1], the process of protecting individuals from infection by vaccination has become a routine, with substantial historical success in reducing both morbidity and mortality (see [2, 3] and references cited therein). Typically, after the initial infection, the host remains in a latent stage for a period of time before becoming infectious. For some diseases, the latent period is neither short nor negligible compared with the infectious period (scarlet fever: 1-2 days versus 14–21 days [4]; measles: 4–12 days versus 17–31 days [5]), leptospirosis, 2–12 days.

In this paper, we consider an epidemic model of vector-host population. The disease spread due to vector, for example, leptospirosis, dengue, malaria, west Nile virus, and so forth, is considered. We assume that the individuals after some time become susceptible again. Therefore, the term is added in the model. The model consists of the interaction of human and vector. The human population is divided in three subclasses, that is, susceptible human , infected human , and recovered human . The total population size of human is shown by and . The vector population is divided in two subclasses, susceptible vector and . The total size of the vector population is denoted by , with . The disease spread from vector, like leptospirosis, effects humans as well as cattle [9]. The human are infected by means of drinking water contaminated by dead rats or by infectious cattle while drinking water. This infection can also spread through the urine of infected human. Those who work in the fields, like marshy places, rice planters, going in dirty water, those who swimming in water are mostly infected. Weil's first time describes leptospirosis as a unique disease process in 1886, while 30 years before Inada and his colleagues identified the causal organism. The symptoms of leptospirosis are high fever, headache, chills, muscle aches, conjunctivitis (red eyes), diarrhea, vomiting, and kidney or liver problems (which may also include jaundice), anemia, and, sometimes, rash. Symptoms may last from a few days and up to several weeks. Deaths from this disease may occur but they are rare. For some cases, the infections can be mild and without obvious symptoms [10–14].

Many works have been done on vector-host models, as in [15–17]. Reference [15] presented a mathematical model of vector host in which the population dynamics of an SIR vector transmitted disease with two pathogen strains. They discussed the stability of the vector-host model and also presented the numerical simulation of their model. Reference [16], they presented the vector-host model of dengue disease; they analyzed the dengue model and presented their stability and numerical results for vector host dengue model. Reference [17], they presented a mathematical model in the form of demographic stochasticity and heterogeneity in transmission of infection dynamics of host-vector disease systems. Mathematical and theoretical discussion is presented in the paper. For more discussion, we refer the readers to the previously mentioned articles. In our models, we have presented the vector-host model with their stability analysis. We obtain, if , it recover the community. However, for the disease remains in the community. We present the global stability of the model and also we present in a good way the numerical simulation of the proposed model, choosing the different values for the parameters.

Many models have been proposed to represent the dynamics of both human and vector population [18–20]. Pongsuumpun et al. [21] developed mathematical models to study the behavior of leptospirosis disease. They represent the rate of change for both rats and human population. The human population are further divided into two main groups juveniles and adults. Triampo et al. [7] considered a deterministic model for the transmission of leptospirosis disease [7]. In their work, they considered a number of leptospirosis infections in Thailand and shown the numerical simulations. Zaman [6] considered the real data presented in [7] to study the dynamical behavior and role of optimal control theory. The dynamical interaction including local and global stability of leptospirosis infected vector and human population which can be found in Zaman et al. [22]. In their work they also presented the bifurcation analysis and presented the numerical simulations for different values of infection rate.

The structure of the paper is organized as follows. Section 2 is devoted to the formulation of the mathematic model and reducing it to the normalized model. In Section 3, we present the infection-free equilibrium, the basic reproduction and the local and global stability of infection equilibrium. In Section 4, we present the disease persistence and existence of the endemic equilibrium. In Section 5, we show the local as well as the global stability of the endemic equilibrium for the reproduction number . The numerical results, conclusion and references are presented in Section 6.

2. Mathematical Model

In this section, a vector-host epidemic model with direct transmission is presented. The host population at time is divided into susceptible , infected, and recovered individuals. The vector population at time is divided into susceptible and infected vector population . The total population of humans is denoted by , and the total population of the vector is denoted by . Thus, and . The mathematical representation of the model which consists of the system of nonlinear differential equations with five state variables is given by

Here, is the recruitment rate of human population; susceptible human can be infected by two ways of transmission, that is, directly, or through infected individuals; , are the mediate transmission coefficients. is the natural mortality rate for humans; is the recovery rate for humans from the infections. We assumed that the disease may be fatal to some infectious hosts, so disease-related death rate from infected class occurs at human populations at . The immune human once again susceptible at constant rate , for some disease like dengue, the chances for susceptibility are less compared to dengue, West Nile virus, malaria, and so forth. is the recruitment rate for vector population. The death rate of vector , is the disease carrying the vector to the host per unit time:

2.1. Normalized Model

For the normalization of the model, we let , , , , and . It is easy to verify that , , , , and satisfy the following system of differential equations:

With restriction, , and . In the first equation of the normalized model, we substituted . So in the normalized system the does not appear. We reduced to the normalized model (3), and we will study the reduced model:

We determine and from and , respectively. For reduced system (4), the feasible region is

With the nonnegative initial conditions values of , the system is positively invariant, and the proof is easy.

3. Infection-Free Equilibrium and Basic Reproduction Number

The basic reproduction for the reduced system (4) is given by

The disease eradicated from the population by two ways, first with the varying size in population and , and the second one is , for detail see [23, 24]. We are thus inspired to seek the conditions for infection-free and endemic equilibrium. The infection-free equilibrium point for model (4) is and for endemic equilibrium . The infection-free equilibrium is obtained by setting the left side of the reduced model (4), we obtain . Obviously the infection-free equilibrium belongs to of reduced model (4), which exists for all positive parameters. Next, we prove the infection-free local asymptotical stability of model (4) at the arbitrary point .

Theorem 1. The infection-free equilibrium of reduced model (4) is stable locally asymptotically stable for when and unstable for .

Proof. The Jacobean matrix of the reduced model about the equilibrium point is given by

where .

The characteristics equation of the Jacobian matrix is obtained by

The eigenvalue has a negative real part, and the rest of the two eigenvalues is calculated by Routh-Harwitz-Criteria. We write

when , then Routh-Hurtwiz Criteria are satisfied if . The infection-free equilibrium is locally asymptotically stable.

Next, we show the global asymptotical stability of infection-free equilibrium, by defining the Lyapunove function.

Theorem 2. If the threshold quantity , the infection-free equilibrium of the reduced model (4) is globally asymptotically stable and is an unstable infection-free equilibrium for system (4), when .

Proof. To show the global stability of infection-free equilibrium of reduced model (4), we define the Lyapunove function in the following:

Taking the time derivative of (10), along the solution of system (4), we obtain

Using and simplifying, we get

When , the infection free-equilibrium is globally asymptotically stable, and is negative. becomes zero when is zero and vice versa. By the Lasalle invariant principle [25], which implies that the infection-free equilibrium at the point is globally asymptotically stable in .

4. Disease Persistence

In this section, we study the uniform persistence of the reduced system (4). The disease persistence occurs for the case when the threshold parameter , by applying the acyclicity Theorem [26].

Definition 3. The reduced model (4) is called uniformly persistence if there exists a constant such that any solution with satisfies

Let be a locally compact metric space with metric , and let be a closed nonempty subset of with boundary and interior of . Obviously, is the closed subset of . Suppose that be a dynamical system defined on . A subset in is said to be invariant if . Define .

Lemma 4. Assume that

   has a global attractor; there exists an of pair-wise disjoint, compact, and isolated invariant set on such that;  ;no subset of forming a cycle on ; each of is also isolated in ;   for every , where is the stable manifold of . Then is uniformly persistent with respect to [26].

By the application of Lemma 4 to our model, suppose that

from (5),

Clearly, .

Hypotheisis and hold, for (4), reducing to , when , then . When , the infection-free equilibrium is unstable. Also, . and are satisfied. Due to the boundedness the reduced system (4) always admits a global attractor, so is satisfied. We now state the above discussion in the form of the following result.

Theorem 5. For , the reduced system (4), is uniformly persistent.

4.1. Existence of the Endemic Equilibrium

We have proved in Section 4 the local asymptotical stability of infection-free equilibrium when . In such case, when the infection-free equilibrium is locally asymptotically stable for is , the disease dies out and no endemic equilibrium exists. From epidemiological point of view, it is important to show the existence of endemic equilibrium when .

Let belong to which is an endemic equilibrium. From reduced system (4), its coordinates should satisfy

with , , and . By adding the system (16), and solve for , we obtain

This gives the range for in the following:

From (18), note that the disease-related death , less than the , the birth rate , the rate at which the human become susceptible , the sum of , and the contact rate coefficient , and the less value of (natural death rate of vector) will lie in the interval . Now, further eliminate and from (10), then satisfies

Further simplification gives

where

and the equilibria of the reduced system (4) is given by

The positive endemic equilibrium depends upon and when we get a positive endemic equilibrium point. We now state the above in the following result.

Theorem 6. When , a unique positive endemic equilibrium exists for reduced system (4), In other case the existence of disease-free equilibrium.

5. Global Stability of Endemic Equilibrium

Theorem 7. For , the reduced model (4), about the endemic equilibrium point , is globally asymptotically stable, and unstable for .

Proof. To prove that the reduced model (4) is globally asymptotically stable, we obtain the Jacobian matrix about which is given by

The second additive compound matrix for is given in the following. Also see, the Appendix for the second additive compound matrix.

where

where

And is

And is

So we write

where

Suppose that the vector in and its norm will be defined as

Suppose that represents Lozinski measure with the previously defined norm. So, as described in [27], we choose

where

and are the matrix norm with respect to vector , and represents the Lozinski measure with respect to this norm, then

Use

From system (4) and equation (2),

Using

From the third equation of system (4),

So,

then

implies that . Thus, the result [28] implies that the positive equilibrium point of is globally asymptotically stable.

6. Numerical Simulations and Conclusion

In this section, we discuss the numerical simulation of the reduced model (4), by using Runge-Kutta order four scheme. The model for different parameters and their numerical results are presented in Figures 1, 2, 3, 4, 5, 6, 7, and 8. The parameters and their values are presented in Table 1. Figure 1 represents the population dynamics of model (4). Varying the parameters in Figures 2 to 6, we obtained different results. Increasing , the number of infected human increases. The variability of the population effects the individuals numbers (infected individuals). For changing the value of , we get different results in Figures 7 and 8. In this work, we have presented a mathematical model of vector-host disease like (leptospirosis, West Nile virus, dengue, etc.), which spreads through the vector, has been presented. The system is stable locally as well as globally about the disease-free equilibrium , when reproduction number , and the unstable equilibrium occurs for . When the reproduction number , there exists persistence. The disease permanently exists in the community if . Then, we obtained the global stability of endemic equilibrium. The numerical simulations were presented for the illustration of theoretical results.

Table 1: Parameter values used in the numerical simulations of the model.

Notation
Parameter description
Value
Reference

Recruitment rate for human

[6]

Proportionality constant

[7]

Natural death rate of human

[6]

Natural death rate of vector

[6]

Death rate due to disease at human class

[8]

Recovery rate of the infection

[8]

Birth rate of vector

Assumed

Transmission between and

Assumed

Transmission between and

Assumed

Transmission coefficient between and

Assumed

Natural death rate of vector

[6]

Figure 1: The plot shows the human population.

Figure 2: The plot shows the human population.

Figure 3: The plot shows the human population.

Figure 4: The plot shows the human population.

Figure 5: The plot shows the human population.

Figure 6: The plot shows the human population.

Figure 7: The plot shows the human population.

Figure 8: The plot shows the human population.

Appendix

Consider the following:

Acknowledgment

The authors would like to thank the handling Editor Ali Khraibi and reviewers for their careful reading of the manuscripts and for their valuable comments and suggestions which improved our work.

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