Global Stability of Vector-Host Disease with Variable Population Size

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 𝑅₀<1. We also show that for 𝑅₀<1, the disease free-equilibrium of the model is both locally as well as globally stable. For 𝑅₀>1, there exists a unique positive endemic equilibrium. For 𝑅₀>1, the disease persistence occurs. The endemic equilibrium is locally as well as globally asymptotically stable for 𝑅₀>1. 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 𝜆_ℎ 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 𝑆_ℎ(𝑡), infected human 𝐼_ℎ(𝑡), and recovered human 𝑅_ℎ(𝑡). The total population size of human is shown by 𝑁₁ and 𝑁₁=𝑆_ℎ(𝑡)+𝐼_ℎ(𝑡)+𝑅_ℎ(𝑡). The vector population is divided in two subclasses, susceptible vector 𝑆_𝑣(𝑡) and 𝐼_𝑣(𝑡). The total size of the vector population is denoted by 𝑁₂, with 𝑁₂=𝑆_𝑣(𝑡)+𝐼_𝑣(𝑡). 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 𝑅₀≤1, it recover the community. However, for 𝑅₀>1 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 𝑅₀>1. 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 𝑡 is divided into susceptible 𝑆_ℎ(𝑡), 𝐼_ℎ(𝑡) infected, and recovered 𝑅_ℎ(𝑡) individuals. The vector population at time 𝑡 is divided into susceptible 𝑆_𝑣(𝑡) and infected vector population 𝐼_𝑣(𝑡). The total population of humans is denoted by 𝑁₁, and the total population of the vector is denoted by 𝑁₂. Thus, 𝑁₁(𝑡)=𝑆_ℎ(𝑡)+𝐼_ℎ(𝑡)+𝑅_ℎ(𝑡) and 𝑁₂(𝑡)=𝑆_𝑣(𝑡)+𝐼_𝑣(𝑡). The mathematical representation of the model which consists of the system of nonlinear differential equations with five state variables is given by 𝑑𝑆_ℎ𝑑𝑡=Λ_ℎ𝑁₁−𝜇_ℎ𝑆_ℎ−𝛽₂𝑆_ℎ𝐼_𝑣𝑁₂−𝛽₁𝑆_ℎ𝐼_ℎ𝑁₁+𝜆_ℎ𝑅_ℎ,𝑑𝐼_ℎ=𝛽𝑑𝑡²𝑆_ℎ𝐼_𝑣𝑁₂+𝛽₁𝑆_ℎ𝐼_ℎ𝑁₁−𝜇_ℎ𝐼_ℎ−𝛿_ℎ𝐼_ℎ−𝛾_ℎ𝐼_ℎ,𝑑𝑅_ℎ𝑑𝑡=𝛾_ℎ𝐼_ℎ−𝜇_ℎ𝑅_ℎ−𝜆_ℎ𝑅_ℎ,𝑑𝑆_𝑣𝑑𝑡=Λ_𝑣𝑁₂−𝛾_𝑣𝑆_𝑣−𝛽₃𝑆_𝑣𝐼_ℎ𝑁₁,𝑑𝐼_𝑣=𝛽𝑑𝑡³𝑆_𝑣𝐼_ℎ𝑁₁−𝛾_𝑣𝐼_𝑣. (1) Here, Λ_ℎ is the recruitment rate of human population; susceptible human can be infected by two ways of transmission, that is, directly, or through infected individuals; 𝛽₁, 𝛽₂ are the mediate transmission coefficients. 𝜇_ℎ is the natural mortality rate for humans; 𝛾_ℎ 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 𝛿_ℎ. The immune human once again susceptible at constant rate 𝜆_ℎ, for some disease like dengue, the chances for susceptibility are less compared to dengue, West Nile virus, malaria, and so forth. Λ_𝑣 is the recruitment rate for vector population. The death rate of vector 𝛾_𝑣, 𝛽₃ is the disease carrying the vector to the host per unit time: 𝑑𝑁₁𝑑𝑡=Λ_ℎ𝑁₁−𝜇_ℎ𝑁₁−𝛿_ℎ𝐼_ℎ. (2)

2.1. Normalized Model

For the normalization of the model, we let 𝑆_ℎ=𝑆_ℎ/𝑁₁, 𝐼_ℎ̂=𝐼_ℎ/𝑁₁, 𝑅_ℎ=𝑅_ℎ/𝑁₁, 𝑆_𝑣=𝑆_𝑣/𝑁₂, and 𝐼_𝑣̂=𝐼_𝑣̂/𝑁₂. It is easy to verify that 𝑆_ℎ, 𝐼_ℎ̂, 𝑅_ℎ, 𝑆_𝑣, and 𝐼_𝑣̂ satisfy the following system of differential equations: 𝑑𝑆_ℎ𝑑𝑡=Λ_ℎ1−𝑆_ℎ+𝛿_ℎ𝑆_ℎ𝐼_ℎ̂−𝛽₂𝑆_ℎ𝐼_𝑣̂−𝛽₁𝑆_ℎ𝐼_ℎ̂+𝜆_ℎ1−𝑆_ℎ−𝐼_ℎ̂,𝑑𝐼_ℎ̂𝑑𝑡=𝛽₂𝑆_ℎ𝐼_𝑣̂+𝛽₁𝑆_ℎ𝐼_ℎ̂−Λ_ℎ+𝛿_ℎ+𝛾_ℎ𝐼_ℎ̂+𝛿_ℎ𝐼²_ℎ̂,𝑑𝑅_ℎ𝑑𝑡=𝛾_ℎ𝐼_ℎ̂−Λ_ℎ+𝜆_ℎ𝑅_ℎ+𝛿_ℎ𝑅_ℎ𝐼_ℎ̂,𝑑𝑆_𝑣𝑑𝑡=Λ_𝑣1−𝑆_𝑣−𝛽₃𝑆_𝑣𝐼_ℎ̂,𝑑𝐼_𝑣𝑑𝑡=𝛽₃𝑆_𝑣𝐼_ℎ̂−𝛾_𝑣𝐼_𝑣̂. (3) With restriction, 𝑆_ℎ+𝐼_ℎ̂+𝑅_ℎ=𝑁₁=1, 𝑆_𝑣+𝐼_𝑣̂=𝑁₂=1 and 𝐼_𝑣̂=1−𝑆_𝑣. In the first equation of the normalized model, we substituted 𝑅_ℎ=1−𝑆_ℎ−𝐼_ℎ̂. So in the normalized system the 𝑅_ℎ does not appear. We reduced to the normalized model (3), and we will study the reduced model: 𝑑𝑆_ℎ𝑑𝑡=Λ_ℎ1−𝑆_ℎ−𝛽₂𝑆_ℎ𝐼_𝑣̂+𝛿_ℎ𝑆_ℎ𝐼_ℎ̂−𝛽₁𝑆_ℎ𝐼_ℎ̂+𝜆_ℎ1−𝑆_ℎ−𝐼_ℎ̂,𝑑𝐼_ℎ̂𝑑𝑡=𝛽₂𝑆_ℎ𝐼_𝑣̂+𝛽₁𝑆_ℎ𝐼_ℎ̂−Λ_ℎ+𝛿_ℎ+𝛾_ℎ𝐼_ℎ̂+𝛿_ℎ𝐼²_ℎ̂𝑑𝐼_𝑣𝑑𝑡=𝛽₃1−𝐼_𝑣̂𝐼_ℎ̂−𝛾_𝑣𝐼_𝑣̂. (4) We determine 𝑆_𝑣 and 𝑅_ℎ from 𝑆_𝑣=1−𝐼_𝑣̂ and 𝑅_ℎ=1−𝑆_ℎ−𝐼_ℎ̂, respectively. For reduced system (4), the feasible region is Ω∶=𝑆_ℎ,𝐼_ℎ̂,𝐼_𝑣̂∈𝐑^𝟑₊∣0≤𝑆_ℎ+𝐼_ℎ̂+𝐼_𝑣̂≤1. (5) With the nonnegative initial conditions values of Ω, 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 𝑅₀=𝛽₁Λ_ℎ+𝛿_ℎ+𝛾_ℎ+𝛽₂𝛽₃𝛾_𝑣Λ_ℎ+𝛿_ℎ+𝛾_ℎ. (6) The disease eradicated from the population by two ways, first with the varying size in population and 𝐼_ℎ̂→0, and the second one is 𝐼_ℎ→0, 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 𝐸_𝑜=(𝑆_ℎ=𝑆⁰_ℎ,0,0) and for endemic equilibrium 𝐸^∗=(𝑆^∗_ℎ,𝐼^∗_ℎ̂,𝑆^∗_𝑣). The infection-free equilibrium is obtained by setting the left side of the reduced model (4), we obtain 𝑆⁰_ℎ=1. Obviously the infection-free equilibrium 𝐸_𝑜=(𝑆_ℎ=𝑆⁰_ℎ=1,0,0) belongs to Ω 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 𝐸₁=(𝑆_ℎ,𝐼_ℎ̂,𝐼_𝑣̂).

Theorem 1. The infection-free equilibrium of reduced model (4) is stable locally asymptotically stable for 𝑅₀<1 when 𝛾_𝑣+𝐶₁>𝛽₁ and unstable for 𝑅₀≥1.

Proof. The Jacobean matrix of the reduced model about the equilibrium point 𝐸₀ is given by𝐽𝐸⁰=−𝛽₂𝐼_𝑣̂−𝛽₁𝐼_ℎ̂−Λ_ℎ+𝛿𝐼_ℎ̂𝛿_ℎ−𝛽₁𝑆_ℎ−𝜆_ℎ−𝛽₂𝑆_ℎ𝛽₂𝐼_𝑣̂+𝛽₁𝐼_ℎ̂𝛽₁𝑆_ℎ−𝐶₁+2𝛿_ℎ𝐼_ℎ̂𝛽₂𝑆_ℎ0𝛽₃1−𝐼_𝑣̂−𝛽₃𝐼_ℎ̂−𝛾_𝑣, (7)where 𝐶₁=Λ_ℎ+𝛾_ℎ+𝛿_ℎ.
The characteristics equation of the Jacobian matrix 𝐽(𝐸⁰) is obtained by −Λ_ℎ−𝜆𝛽₁−𝐶₁−𝜆𝛾_𝑣−𝜆−𝛽₂𝛽₃=0. (8) The eigenvalue −𝜆_ℎ has a negative real part, and the rest of the two eigenvalues is calculated by Routh-Harwitz-Criteria. We write 𝜆²+𝜆𝛾_𝑣+𝐶₁−𝛽₁+𝛾_𝑣𝐶₁1−𝑅₀=0, (9) when 𝑅₀<1, then Routh-Hurtwiz Criteria are satisfied if 𝛾_𝑣+𝐶₁>𝛽₁. 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 𝑅₀≤1, the infection-free equilibrium of the reduced model (4) is globally asymptotically stable and is an unstable infection-free equilibrium for system (4), when 𝑅₀>1.

Proof. To show the global stability of infection-free equilibrium of reduced model (4), we define the Lyapunove function in the following: 𝑃(𝑡)=𝛾_𝑣𝐼_ℎ̂+𝛽₂𝐼_𝑣̂. (10) Taking the time derivative of (10), along the solution of system (4), we obtain 𝑃^′(𝑡)=𝛾_𝑣𝛽₂𝑆_ℎ𝐼_𝑣̂+𝛽₁𝑆_ℎ𝐼_ℎ̂−Λ_ℎ+𝛿_ℎ+𝛾_ℎ𝐼_ℎ̂+𝛿_ℎ𝐼̂_ℎ²+𝛽₂𝛽₃1−𝐼_𝑣̂𝐼_ℎ̂−𝛾_𝑣𝐼_𝑣̂. (11) Using 𝑆_ℎ=1−𝐼_ℎ̂ and simplifying, we get 𝑃^′(𝑡)=−𝛾_𝑣𝛽₂𝐼_𝑣̂𝐼_ℎ̂−𝛾_𝑣𝛽₁−𝛿_ℎ𝐼̂_ℎ²−𝛽₂𝛽₃𝐼_ℎ̂𝐼_𝑣̂−𝛾_𝑣Λ_ℎ+𝛾_ℎ+𝛿_ℎ1−𝑅₀𝐼_ℎ̂. (12) When 𝑅₀≤1, the infection free-equilibrium is globally asymptotically stable, and 𝑃^′(𝑡) is negative. 𝑃^′(𝑡) becomes zero when 𝐼_ℎ̂ is zero and vice versa. By the Lasalle invariant principle [25], which implies that the infection-free equilibrium at the point 𝐸₀ is globally asymptotically stable in Ω.

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 𝑅₀>1, by applying the acyclicity Theorem [26].

Definition 3. The reduced model (4) is called uniformly persistence if there exists a constant 𝑐∈(0,1) such that any solution (𝑆_ℎ,𝐼_ℎ̂,𝐼_𝑣̂) with (𝑆_ℎ(0),𝐼_ℎ̂(0),𝐼_𝑣̂(0))∈Ω satisfies minliminf_𝑡→∞𝑆_ℎ(𝑡),liminf_𝑡→∞𝐼_ℎ̂(𝑡),liminf_𝑡→∞𝐼_𝑣̂(𝑡)≥𝑐. (13) Let 𝑋 be a locally compact metric space with metric 𝑑, and let 𝐶 be a closed nonempty subset of 𝑋 with boundary 𝜕Ω and interior of Ω^𝑜. Obviously, 𝜕Ω is the closed subset of Ω. Suppose that 𝜙_𝑡(𝑥) be a dynamical system defined on Ω. A subset 𝐵 in 𝑋 is said to be invariant if 𝜙(𝐵,𝑡)=𝐵. Define 𝑇_𝜕∶={𝑥∈𝜕Ω∶𝜙_𝑡(𝑥)∈𝜕Ω,forall𝑡≥0}.

Lemma 4. Assume that(𝐻₁)??𝜙(𝑡) has a global attractor;(𝐻₂) there exists an 𝑁=𝑁₁,…,𝑁_𝑘 of pair-wise disjoint, compact, and isolated invariant set on 𝜕Ω such that;(𝑎₁)??𝑉_{𝑥∈𝜕Ω}⊂𝑉^𝑘_𝑗=1𝑁_𝑗;(𝑎₂)no subset of 𝑁 forming a cycle on 𝜕Ω;(𝑎₃) each of 𝑁_𝑗 is also isolated in 𝜕Ω;(𝑎₄)??𝑊^𝑠(𝑁_𝑗)∩Ω^𝑜=𝜙 for every 1≤𝑗≤𝑘, where 𝑊^𝑠(𝑁_𝑗) is the stable manifold of 𝑁_𝑗. Then 𝜙(𝑡) is uniformly persistent with respect to Ω^𝑜 [26].

By the application of Lemma 4 to our model, suppose that Ω∶=𝑆_ℎ,𝐼_ℎ̂,𝐼_𝑣̂∈𝐑^𝟑₊∣0≤𝑆_ℎ+𝐼_ℎ̂+𝐼_𝑣̂≤1, (14) from (5), Ω^𝑜∶=𝑆_ℎ,𝐼_ℎ̂,𝐼_𝑣̂∈𝐸,𝐼_ℎ̂,𝐼_𝑣̂Ω>0,𝜕Ω=Ω^𝑜. (15) Clearly, 𝑁_𝜕=𝜕Ω.

Hypotheisis (𝑎₁) and (𝑎₂) hold, for (4), reducing to 𝑆^′_ℎ=(Λ_ℎ+𝜆_ℎ)−(Λ_ℎ+𝜆_ℎ)𝑆_ℎ, when 𝑡_∞, then 𝑆^′_ℎ(𝑡)=1. When 𝑅_𝑜>1, the infection-free equilibrium is unstable. Also, 𝑊^𝑠(𝑁)=𝜕Ω. (𝑎₃) and (𝑎₄) are satisfied. Due to the boundedness the reduced system (4) always admits a global attractor, so 𝐻₁ is satisfied. We now state the above discussion in the form of the following result.

Theorem 5. For 𝑅₀>1, 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 𝑅₀<1. In such case, when the infection-free equilibrium is locally asymptotically stable for is 𝑅₀<1, 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 𝑅₀>1.

Let 𝐸^∗=(𝑆^∗_ℎ,𝐼^∗_ℎ̂,𝐼^∗_𝑣̂) belong to Ω which is an endemic equilibrium. From reduced system (4), its coordinates should satisfy Λ_ℎ1−𝑆^∗_ℎ−𝛽₂𝑆^∗_ℎ𝐼^∗_𝑣̂+𝛿_ℎ𝑆_ℎ𝐼^∗_ℎ̂−𝛽₁𝑆^∗_ℎ𝐼^∗_ℎ̂+𝜆_ℎ1−𝑆^∗_ℎ−𝐼^∗_ℎ̂𝛽=0,₂𝑆^∗_ℎ𝐼^∗_𝑣̂+𝛽₁𝑆^∗_ℎ𝐼^∗_ℎ̂−Λ_ℎ+𝛿_ℎ+𝛾_ℎ𝐼^∗_ℎ̂+𝛿_ℎ𝐼̂_ℎ^∗2𝛽=0,₃1−𝐼^∗_𝑣̂𝐼^∗_ℎ̂−𝛾_𝑣𝐼^∗_𝑣̂=0, (16) with 𝑆^∗_ℎ>0, 𝐼^∗_ℎ̂>0, and 𝐼^∗_𝑣̂>0. By adding the system (16), and solve for 𝐼^∗_ℎ̂, we obtain Λ_ℎ+𝜆_ℎ−𝛿_ℎ𝐼^∗_ℎ̂1−𝑆^∗_ℎ−𝐼^∗_ℎ̂=−𝛾_𝑣+𝛾_ℎ+𝛽₃𝐼^∗_ℎ̂−𝛾_𝑣1−𝐼^∗_𝑣̂. (17) This gives the range for 𝐼^∗_ℎ̂ in the following: 0<𝐼^∗_ℎ̂<1,minΛ_ℎ+𝜆_ℎ𝛿_ℎ,𝛽1,min³𝛾_𝑣. (18) From (18), note that the disease-related death 𝛿, less than the (Λ_ℎ+𝜆_ℎ), the birth rate Λ, the rate at which the human become susceptible 𝜆_ℎ, the sum of (Λ_ℎ+𝜆_ℎ), and the contact rate coefficient 𝛽₃, and the less value of 𝛾_𝑣 (natural death rate of vector) will lie in the interval (0,1). Now, further eliminate 𝑆^∗_ℎ and 𝐼^∗_𝑣̂ from (10), then 𝐼^∗_ℎ̂ satisfies Λ_ℎ+𝜆_ℎ−𝛾_ℎ𝐼^∗_ℎ̂−𝜆_ℎ+𝛿_ℎ+𝛾_ℎ−𝛿_ℎ𝐼^∗_ℎ̂𝐼^∗_ℎ̂=Λ_ℎ+𝜆_ℎ−𝛿_ℎ𝐼^∗_ℎ̂Λ_ℎ+𝛿_ℎ+𝛾_ℎ−𝛿_ℎ𝐼^∗_ℎ̂𝛽₂𝛽₃+𝛽₁𝛾_𝑣+𝛽₃𝐼^∗_ℎ̂𝛾_𝑣+𝛽₃𝐼^∗_ℎ̂. (19) Further simplification gives 𝑓𝐼^∗_ℎ̂=𝐼̂_ℎ^∗3+𝐵₁𝐼̂_ℎ^∗2+𝐵₂𝐼^∗_ℎ̂+𝐵₃, (20) where 𝐵₁=𝛽₁𝛾_𝑣𝛿_ℎ+𝛽₃𝛿_ℎ𝛽₂+Λ_ℎ+𝛾_ℎ+𝛽₃𝛽₁2𝛾_ℎ+𝜆_ℎ+𝛿_ℎ−Λ_ℎ+𝜆_ℎΛ_ℎ+𝛾_ℎ𝛽₁𝛽₃𝛿_ℎ,𝐵₂=𝛽₁𝛾_𝑣2𝛾_ℎ+𝜆−ℎ+𝛿_ℎ+𝛽₁𝛽₃Λ_ℎ+𝜆_ℎ+Λ_ℎ+𝜆_ℎ−𝛿_ℎ𝛿_ℎ𝛾_𝑣+𝛾_𝑣Λ_ℎ+𝛿_ℎ+𝛾_ℎ𝛽₁𝛽₃𝛿_ℎ,𝐵₃=Λ_ℎ+𝛾_ℎ+𝛿_ℎΛ_ℎ+𝜆_ℎ𝛾_𝑣𝑅₀−1𝛽₁𝛽₃𝛿_ℎ, (21) and the equilibria of the reduced system (4) is given by 𝑆^∗_ℎ=Λ_ℎ+𝜆_ℎ−𝜆_ℎ𝐼^∗_ℎ̂𝛽₃𝐼^∗_ℎ̂+𝛾_𝑣𝛽₂𝛽₃𝐼^∗_ℎ̂+Λ_ℎ+𝜆_ℎ+𝐼^∗_ℎ̂𝛽₁−𝛿_ℎ𝛽₃𝐼_ℎ+𝛾_𝑣,𝐼^∗_𝑣=𝛽₃𝐼_ℎ𝛽₃𝐼_ℎ+𝛾_𝑣. (22) The positive endemic equilibrium depends upon (Λ_ℎ+𝜆_ℎ)>𝜆_ℎ𝐼^∗_ℎ̂ and 𝛽₁≥𝛿_ℎ when 𝑅₀>1 we get a positive endemic equilibrium point. We now state the above in the following result.

Theorem 6. When 𝑅₀>1, 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 𝑅₀>1, the reduced model (4), about the endemic equilibrium point 𝐸^∗, is globally asymptotically stable, and unstable for 𝑅₀>1.

Proof. To prove that the reduced model (4) is globally asymptotically stable, we obtain the Jacobian matrix 𝐽^∗ about 𝐸^∗ which is given by𝐽^∗𝐸^∗=−Λ_ℎ−𝛽₂𝐼_𝑣̂+𝛿_ℎ𝐼_ℎ̂−𝛽₁𝐼_ℎ̂−𝜆_ℎ𝛿_ℎ𝑆_ℎ−𝛽₁𝑆_ℎ−𝜆_ℎ−𝛽₂𝑆_ℎ𝛽₂𝐼_𝑣̂+𝛽₁𝐼_ℎ̂𝛽₁𝑆_ℎ−Λ_ℎ+𝛿_ℎ+𝛾_ℎ+2𝛿_ℎ𝐼_ℎ̂𝛽₂𝑆_ℎ0𝛽₃1−𝐼_𝑣̂−𝛽₃𝐼_ℎ̂−𝛾_𝑣. (23)
The second additive compound matrix for 𝐽^∗(𝐸^∗) is given in the following. Also see, the Appendix for the second additive compound matrix.𝐽^[2]𝐸^∗=𝐴₁₁𝛽₂𝑆_ℎ𝛽₂𝑆_ℎ𝛽₃1−𝐼_𝑣̂𝐴₂₂−𝛽₁𝑆_ℎ+𝛿_ℎ𝑆_ℎ−𝜆_ℎ0𝛽₂𝐼_𝑣̂+𝛽₁𝐼_ℎ̂𝐴₃₃, (24) where 𝐴₁₁=−Λ_ℎ−𝛽₂𝐼_𝑣̂+𝛿_ℎ𝐼_ℎ̂−𝛽₁𝐼_ℎ̂−𝜆_ℎ+𝛽₁𝑆_ℎ−Λ_ℎ+𝛿_ℎ+𝛾_ℎ+2𝛿_ℎ𝐼_ℎ̂,𝐴₂₂=−Λ_ℎ−𝛽₂𝐼_𝑣̂+𝛿_ℎ𝐼_ℎ̂−𝛽₁𝐼_ℎ̂−𝜆_ℎ−𝛽₃𝐼_ℎ̂−𝛾_𝑣,𝐴₃₃=𝛽₁𝑆_ℎ−Λ_ℎ+𝛿_ℎ+𝛾_ℎ+2𝛿_ℎ𝐼_ℎ̂−𝛽₃𝐼_ℎ̂−𝛾_𝑣,𝑃=𝑃𝑆_ℎ,𝐼_ℎ̂,𝐼_𝑣̂𝐼̂=diag1,^𝑣𝐼_ℎ̂,𝐼_𝑣̂𝐼_ℎ̂, (25) where 𝑃⁻¹𝐼̂=diag1,^ℎ𝐼_𝑣̂,𝐼_ℎ̂𝐼_𝑣̂,𝑃_𝑓𝐼̂=diag0,^𝑣𝐼^′_ℎ̂−𝐼^′_𝑣̂𝐼_ℎ̂𝐼̂_ℎ²,𝐼_𝑣̂𝐼^′_ℎ̂−𝐼^′_𝑣̂𝐼_ℎ̂𝐼̂_ℎ². (26) And 𝑃_𝑓𝑃⁻¹ is 𝑃_𝑓𝑃⁻¹𝐼̂=diag0,^′ℎ𝐼_ℎ̂−𝐼^′_𝑣̂𝐼_𝑣̂,𝐼^′_ℎ̂𝐼_ℎ̂−𝐼^′_𝑣̂𝐼_𝑣̂. (27) And 𝑃_𝑓𝐽^[2]𝑃⁻¹ is 𝑃_𝑓𝐽^[2]𝑃⁻¹=𝐽^[2]=𝐴₁₁𝛽₂𝑆_ℎ𝛽₂𝑆_ℎ𝛽₃1−𝐼_𝑣̂𝐴₂₂−𝛽₁𝑆_ℎ+𝛿_ℎ𝑆_ℎ−𝜆_ℎ0𝛽₂𝐼_𝑣̂+𝛽₁𝐼_ℎ̂𝐴₃₃. (28) So we write 𝐵=𝑃_𝑓𝑃⁻¹+𝑃_𝑓𝐽^[2]𝑃⁻¹=𝐵₁₁𝐵₁₂𝐵₂₁𝐵₂₂, (29) where 𝐵₁₁=−Λ_ℎ−𝛽₂𝐼_𝑣̂+𝛿_ℎ𝐼_ℎ̂−𝛽₁𝐼_ℎ̂−𝜆_ℎ+𝛽₁𝑆_ℎ−Λ_ℎ+𝛿_ℎ+𝛾_ℎ+2𝛿_ℎ𝐼_ℎ̂,𝐵₁₂=𝛽₂𝑆_ℎ,𝛽₂𝑆_ℎ,𝐵₂₁=𝛽₃1−𝐼_𝑣̂,0^𝑇,𝐵₂₂=𝐴₂₂+𝐼^′_ℎ̂𝐼_ℎ̂−𝐼^′_𝑣̂𝐼_𝑣̂−𝛽₁𝑆_ℎ+𝛿_ℎ𝑆_ℎ−𝜆_ℎ0𝐼^′_ℎ̂𝐼_ℎ̂−𝐼^′_𝑣̂𝐼_𝑣̂+𝐴₃₃. (30) Suppose that the vector (𝑢,𝑣,̂𝑤) in 𝑅³ and its norm ‖⋅‖ will be defined as ‖(𝑢,𝑣,̂𝑤)‖=max{|𝑢|,|𝑣|̂+|𝑤|}. (31) Suppose that 𝜇𝐵 represents Lozinski measure with the previously defined norm. So, as described in [27], we choose 𝜇𝐵≤sup𝑔₁,𝑔₂, (32) where 𝑔₁=𝜇₁𝐵₁₁+𝐵₁₂,𝑔₂=𝐵₂₁+𝜇₂𝐵₂₂. (33)|𝐵₂₁| and |𝐵₁₂| are the matrix norm with respect to vector ℓ, and 𝜇₁ represents the Lozinski measure with respect to this ℓ norm, then 𝜇₁𝐵₁₁=−Λ_ℎ−𝛽₂𝐼_𝑣̂+𝛿_ℎ𝐼_ℎ̂−𝛽₁𝐼_ℎ̂−𝜆_ℎ+𝛽₁𝑆_ℎ−Λ_ℎ+𝛿_ℎ+𝛾_ℎ+2𝛿_ℎ𝐼_ℎ̂,𝐵₂₁=𝛽₃1−𝐼_𝑣̂,𝐵₁₂=max𝛽₂𝑆_ℎ,𝛽₂𝑆_ℎ=𝛽₂𝑆_ℎ,𝜇₁𝐵₂₂𝐼̂=max^′ℎ𝐼_ℎ̂−𝐼^′_𝑣̂𝐼_𝑣̂+𝐴₂₂,𝐼^′_ℎ̂𝐼_ℎ̂−𝐼^′_𝑣̂𝐼_𝑣̂+𝐴₃₃∴𝑔₁=𝜇₁𝐵₁₁+𝐵₁₂=−Λ_ℎ+𝜆_ℎ−𝛽₂𝐼_𝑣̂−𝛽₁𝐼_ℎ̂+𝛿_ℎ𝐼_ℎ̂+𝛿_ℎ𝐼_ℎ̂+𝛽₂𝑆_ℎ+2𝛿_ℎ𝐼_ℎ̂−Λ_ℎ+𝛾_ℎ+𝛾_ℎ=−Λ_ℎ+𝜆_ℎ−𝛽₂𝐼_𝑣̂−𝛽₁𝐼_ℎ̂+𝛽₂𝑆_ℎ+2𝛿_ℎ𝐼_ℎ̂+𝐼^′_ℎ̂𝐼_ℎ̂−𝛽₂𝑆_ℎ𝐼_𝑣̂𝐼_ℎ̂−𝛽₁𝑆_ℎ. (34) Use 𝐼^′_ℎ̂𝐼_ℎ̂−𝛽₂𝑆_ℎ𝐼_𝑣̂𝐼_ℎ̂−𝛽₁𝑆_ℎ=−Λ_ℎ+𝛾_ℎ+𝛾_ℎ+𝛿_ℎ𝐼_ℎ̂. (35)
From system (4) and equation (2), 𝑔₁≤𝐼^′_ℎ̂𝐼_ℎ̂−Λ_ℎ+𝜆_ℎ−𝐼_ℎ̂𝛽₁−2𝛿_ℎ,𝑔₂=𝐵₂₁+𝜇₁𝐵₂₂,=𝛽₃1−𝐼_𝑣̂+𝐼^′_ℎ̂𝐼_ℎ̂−𝐼^′_𝑣̂𝐼_𝑣̂+𝐴₂₂+𝐴₃₃=𝛽₃1−𝐼_𝑣̂+𝐼^′_ℎ̂𝐼_ℎ̂−𝐼^′_𝑣̂𝐼_𝑣̂−Λ_ℎ−𝛽₂𝐼_𝑣̂+𝛿_ℎ𝐼_ℎ̂−𝛽₁𝐼_ℎ̂−𝜆_ℎ−𝛽₃𝐼_ℎ̂+𝛽₁𝑆_ℎ−Λ_ℎ+𝛿_ℎ+𝛾_ℎ+2𝛿_ℎ𝐼_ℎ̂−𝛽₃𝐼_ℎ̂−𝛾_𝑣. (36)
Using 𝐼^′_𝑣̂𝐼_𝑣=𝛽₃𝑆_ℎ𝐼_ℎ̂𝐼_𝑣−𝛾_𝑣. (37) From the third equation of system (4), 𝑔₂≤𝐼^′_ℎ̂𝐼_ℎ̂−Λ_ℎ+𝜆_ℎ−𝛽₁−2𝛿_ℎ𝐼_ℎ̂. (38) So, 𝜇₁𝐵≤sup𝑔₁,𝑔₂≤𝐼^′_ℎ̂𝐼_ℎ̂−𝛽,where𝛽=𝛽₁−2𝛿_ℎ, (39) then 1𝑡^𝑡₀𝜇₁1𝐵𝑑𝑠≤𝑡^𝑡₀𝐼^′_ℎ̂𝐼_ℎ̂1−𝛽𝑑𝑠=𝑡𝐼̂ln^′ℎ(𝑡)𝐼^′_ℎ̂(0)−𝛽 (40) implies that 𝑞≤−𝛽/2<0. Thus, the result [28] implies that the positive equilibrium point of 𝐸^∗ 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 𝛽₃, the number of infected human increases. The variability of the population effects the individuals numbers (infected individuals). For changing the value of 𝛽₁, 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 𝑆^𝑜_ℎ=1,0,0, when reproduction number 𝑅₀<1, and the unstable equilibrium occurs for 𝑅₀≥1. When the reproduction number 𝑅₀>1, there exists persistence. The disease permanently exists in the community if 𝑅₀>1. 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.

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: 𝐽^[2]=𝑎₁₁+𝑎₂₂𝑎₂₃−𝑎₁₃𝑎₃₂𝑎₁₁+𝑎₃₃𝑎₁₂−𝑎₃₁𝑎₂₁𝑎₂₂+𝑎₃₃. (A.1)

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.