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.


Introduction
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 10 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 R 0 and could persist if and only if R 0 exceeds 1.In 11 , 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 17 , an epidemic model of a vector-host disease with direct transmission and the vector-mediated transmission has been investigated.Recently, the paper in 18 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 R 0 .That is, if R 0 is less than one, the disease-free equilibrium is globally asymptotically stable, and in such a case the endemic equilibrium does not exist; if R 0 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.

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 N H t is partitioned into two distinct epidemiological subclasses which are susceptibles and infectious subclasses, with the sizes denoted by S H t and I H t , respectively, and the total vector population N V t is divided into susceptibles and infectious, with the sizes denoted by S V t and I V t , respectively.
The proposed model satisfies the following assumptions.
H 1 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 β 1 so that the incidence of new infections via this route is given by a standard incidence rate β 1 S H t I H t /N H .We denote the biting rate that a pathogen-carrier vector has to susceptible hosts as β 2 , and the incidence of new infections transmitted by the vectors is given again by a standard incidence rate H 2 It is assumed that the host total population N H is constant.The birth rate and the per-capita natural mortality rate of host are equal, μ. φ is the recovery rate of infective hosts.
H 3 The vector total population N V is constant, and η 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:

2.1
The feasible region for system 2.1 is R 4 the positive orthant of R 4 .System 2.1 is obviously well-posed.In order to analyze system 2.1 , let For the convenience, here, we still write s h , i h , s v , and i v as S H , I H , S V , and I V .So system 2.1 can be reduced to the following equations: 2.2

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 Γ is positively invariant with respect to 2.2 .Direct calculation shows that system 2.2 has always the disease-free equilibrium where

3.4
Remark 3.1.According to Theorem 2 in 24 , R 0 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.

is locally asymptotically stable, and is unstable
Proof.Linearizing around the disease-free equilibrium E 0 , we obtain the following characteristic equation: where

3.7
Since it follows that μ φ > β 1 from R 0 < 1, thus A 0 > 0 and A 1 > 0. So f λ have two negative roots.So, all of the eigenvalues of the characteristic equation 3.5 are negative real parts.Hence, the equilibrium E 0 is locally asymptotically stable in the interior of Γ.This completes the proof of Theorem 3.2.
2 is globally asymptotically stable.
Proof.To establish the global stability of the disease-free equilibrium E 0 , we construct the following Lyapunov function:

3.8
By directly calculating the derivation of L along the solution of 2.2 , we obtain 3.9 Using

3.10
Noting that dL/dt 0 if and only if S H S * H , S V S * V , and I H I V 0, therefore, the largest compact invariant set in { S H , I H , S V , I V ∈ Γ : dL/dt 0} is the singleton {E 0 }, where E 0 is the disease-free equilibrium in system 2.2 .By LaSalle's invariant principle 19 , E 0 is globally asymptotically stable in Γ.
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 R 0 > 1, the endemic equilibrium E * of system 2.2 is locally asymptotically stable.
Proof.Since the human and vector populations remain constant in Γ, therefore, letting S H 1 − I H and S V 1 − I V , system 2.2 in the invariant set Γ can be written as the equivalent to the following two-dimensional nonlinear system:

3.11
Thus, the characteristic equation of H , it is easy to verify that B 1 > 0. Therefore, from 3.12 , we obtain that the eigenvalues of J E * have two negative real parts.Therefore E * is locally asymptotically stable for R 0 > 1.
Finally, we shall give the global stability of the endemic equilibrium E * .We have the following results Theorem 3.5.If R 0 > 1, the endemic equilibrium E * of the model 2.2 is globally asymptotically stable.
Proof.Let us construct the following Lyapunov function

3.14
By directly calculating the derivation of V t along the solution of 2.2 , we have

3.15
Since the arithmetic mean is greater than or equal to the geometric mean, we have

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, S H I H R H N H constant .Similar to model 2.2 , by using dimensionless, we obtain S H I H R H 1, and S V I V 1. Thus we consider the following differential equation model: where γ is the rate at which the host populations acquire immunity.δ 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.
It is easy to verify that Ω 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, E 0 1, 0, 0 is always the disease-free equilibrium of system 4.1 , and letting R 0 ββ 2 m/ μ φ γ η β 1 / μ φ γ , we can obtain that the unique endemic equilibrium of system 4.1 E * S * H , I * H , I * V for R 0 > 1, S * H , I * V satisfies the following relations: and I * H is the positive solution of the following quadratic polynomial:

4.3
By linearizing system 4.1 around the disease-free equilibrium E 0 of system 4.1 and analyzing the characteristic equation of E 0 , it is easy to obtain the following results.
Theorem 4.1.If R 0 < 1, the disease-free equilibrium E 0 of system 4.1 is locally asymptotically stable, and is unstable if R 0 > 1.
Theorem 4.2.If R 0 < 1, then the infection-free equilibrium E 0 of system 4.1 is globally asymptotically stable in Ω.
Proof.From the last two equations of system 4.1 , we have

4.4
Let us consider the following equations:

4.5
From R 0 < 1, we have ββ 2 m β 1 η < μ φ γ η.It is easy to show that, if ββ 2 m β 1 η < μ φ γ η for any solutions of 4.5 with nonnegative initial values, we have lim t → ∞ Z i t 0, i 1, 2. Let 0 < I H 0 ≤ Z 1 0 , and 0 < I V 0 ≤ Z 2 0 .If Z 1 t , Z 2 t is a solution of system 4.5 with nonnegative initial values Z 1 0 , Z 1 0 , then, by comparison principle, we have I H t ≤ Z 1 t , and I V t ≤ Z 2 t for all sufficiently large t.Hence, we have lim t → ∞ I H t 0, and lim t → ∞ I V t 0. The maximal compact invariant subset in { S H , I H , I V ∈ Ω : I H I V 0} consists of the S H -axis. From this set, it is easy to obtain that S H → 1, I H 0, and I V 0 for t ≥ 0. It follows that all trajectories starting in Ω approach E 0 for R 0 < 1.
This completes the proof of Theorem 4.2.
Proof.Similar to the proof of Theorem 3.4 in 23 , we choose X Ω, X 1 int Ω, X 2 bd Ω .It is easy to obtain that Y 2 { S, 0, 0 : 0 < S ≤ 1} and Ω 2 y∈Y 2 ω y {E 0 }, and {E 0 } is an isolated compact invariant set in X.Furthermore, letting M {E 0 }, thus, M is an acyclic isolated covering of Ω 2 .Now we only need to show that {E 0 } is a weak repeller for X 1 .Suppose that there exists a positive orbit S H , I H , I V of 4.1 such that lim Since R 0 > 1, there exists a small enough ε > 0 such that From 4.1 , we choose t 0 > 0 large enough such that, when t ≥ t 0 , we have

4.8
Consider the following matrix M ε defined by Since M ε admits positive off-diagonal element, the Perron-Frobenius Theorem 19 implies that there is a positive eigenvector v v 1 , v 2 for the maximum eigenvalue λ * of M ε .From 4.7 , we see that the maximum eigenvalue λ * is positive.Let us consider the following system:

4.10
Let u t u 1 t , u 2 t be a solution of 4.10 through lv 1 , lv 2 at t t 0 , where l > 0 satisfies lv 1 < I H t 0 , and lv 2 < I V t 0 .Since the semiflow of 4.10 is monotone and M ε v > 0, it follows that u i t are strictly increasing and u i t → ∞ as t → ∞, contradicting the eventual boundedness of positive solutions of system 4.1 .Thus, E 0 is weak repeller for X 1 .
This completes the proof of Theorem 4.3.
Remark 4.4.In this section, although we have not discussed the stability of E * 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 E * in model 4.1 is stable whenever it exists.

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 25 .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 R 0 R 0 ≤ 1, the disease-free equilibrium is globally asymptotically stable.If R 0 > 1, the disease persists and the unique endemic equilibrium is globally asymptotically stable.Additionally, we show that, if R 0 > 1, 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.
is the disease-free equilibrium in system 2.2 .By LaSalle's invariant principle, E * is globally asymptotically stable in Γ.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 R 0 is a sharp threshold parameter and completely determines the global stability of 2.2 in the feasible region Γ.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 W x 1 , x 2 , . . ., x n * }, where E * n i 1 Variation of S H , I H , and I V with time for the parameter values μ 0.00042, β 1 0.00004, β 2 0.00002, β 0.00003, α 0.01, m 0.005, γ 0.0012, and δ 0.00002 when R 0 17.0124.