Stability and Hopf Bifurcation of a Vector-Borne Disease Model with Saturated Infection Rate and Reinfection

This paper established a delayed vector-borne disease model with saturated infection rate and cure rate. First of all, according to the basic reproductive number R 0, we determined the disease-free equilibrium E 0 and the endemic equilibrium E 1. Through the analysis of the characteristic equation, we consider the stability of two equilibriums. Furthermore, the effect on the stability of the endemic equilibrium E 1 by delay was studied, the existence of Hopf bifurcations of this system in E 1 was analyzed, and the length of delay to preserve stability was estimated. The direction and stability of the Hopf bifurcation were also been determined. Finally, we performed some numerical simulation to illustrate our main results.


Introduction
Malaria is a vector-borne infectious disease [1], caused by parasites. It is popular in 102 countries and regions, especially in some countries in Africa, southeast Asia, and South America. In the 30s of this century, malaria spread throughout the country. Clinical symptoms and signs of this disease, such as typical periodic onset of malaria, secondary anemia, and spleen, can cause serious consequences, including dangerous malaria, malarial kidney disease, and black urine fever. e main way of transmission of malaria is the bite of an infected female anopheline mosquito. e mosquitoes would also be infected when uninfected mosquitoes bite infected people, and this transmission process has an incubation period [2]. e important feature of malaria is that the recovered immune system may establish immune memory for such antigens. It is this characteristic that greatly reduces the spread of malaria [3,4]. Immune process is slow and, however, takes years or even decades [5]. As time goes by, the immune system gradually weakens, and at this time, reinfection likely occurs; therefore, considering the function of delay and immune system is necessary in the study of malaria.
For the vector-borne diseases such as malaria, a large number of mathematical models have been created [2,6,7,8,9], most of which consider the local immunity and delay of the spread of malaria in the crowd. Different time delay has been used to describe the latent period in the course of disease transmission [7,8,9]. Local stability conditions for the equilibrium of a model with two time delays have been considered by Wan and Cui [8]. e global stability of the equilibrium has been studied for a vectorborne disease model with distributed delay by Cai et al. [10].
Based on the above model, this paper considers a delayed vector-borne model with saturated infection rate and partial immunity to reinfection. We prove that the stability of this system can be changed by time delay and produce Hopf bifurcation, calculating the length of delay to preserve stability. Using the center manifold theorem [11] and norm theory, we determine the stability and bifurcation direction.

Model Formulation
N 1 (t) represented as the host population at time t is divided into three subclasses: the susceptible S(t), the infected I(t), and the recovered R(t). N 2 (t) represented as the vector population at time t is divided into two subclasses: the susceptible T(t) and the infected V(t). e Hopf bifurcation was determined in a model with direct infection and delay by Wei et al. [9]. e mathematical formulation still needs improvements. We consider an improved model as follows: where Λ 1 and Λ 2 represent the recruitment rate of the host population and vector population, respectively. b represents the average number of bites per mosquito per day. e incidence rate bβ 1 S(t)V(t)/1 + αV(t) is the number of infections of the susceptible host caused by the infected vector, and α is the inhibitory effect rate caused by the infected vector. μ 1 and μ 2 represent the death rates of the host population and vector population, respectively. β 1 is the infection rate from vector to human. σ(0 ≤ σ ≤ 1) represents the degree of partial protection for recovered people given by a primary infection, where σ � 0 represents complete protection and σ � 1 represents no protection. c is the per capita recovery rate of the infected host population. β 2 represents the infection rate from human to vector. τ is the time delay, representing the incubation period in the vector population; that is to say, a susceptible vector that bites an infective host at time t − τ will become infective at time t. e model (1) meets the initial conditions: + with the topology of uniform convergence. e norm is defined as follows: Based on the fundamental theory of functional differential equations [12], it is easy to show that the solution of the model (1) with the initial condition (2) is unique and is nonnegative for all t ≥ 0.

Stability of Equilibrium and Hopf Bifurcation
In this section, we study the stability of equilibrium and the existence of Hopf bifurcation of system (7). e characteristic equation of the linear approximate equation of the system (7) at equilibrium E � (S, I, V) is where Q � −σbβ 1 ((Λ 1 /μ 1 ) − S − I).

e Local and Global Stability of the Disease-Free Equilibrium.
At the disease-free equilibrium E 0 , equation (13) can be expressed as follows: Obviously, equation (14) has a negative real root λ 1 � −μ 1 . To discuss the rest of the characteristic roots of (14), we consider the following equation: When τ � 0, equation (15) is equivalent to By the using Routh-Hurwitz criterion, (16) has two eigenvalues with negative real parts if R 0 < 1.
When τ > 0, then the roots of (15) can enter the right-half plane in the complex plane by crossing the imaginary axis as the delay τ increases.
Let λ � ωi(ω > 0) be a purely imaginary root of equation (15), then separating the real and imaginary parts yields Squaring and taking the sum of (17) yields Equation (18) has no roots if R 0 < 1. erefore, we conclude that all eigenvalues of equation (14) have negative real parts.
which implies that By the continuity of f(λ) and zero point theorem, f(λ) � 0 has at least one positive root. So, the disease-free equilibrium E 0 is unstable. Based on the results, we can draw the conclusion. Theorem 3.1. For any τ, the virus-free equilibrium E 0 of the system (7) is locally asymptotically stable if R 0 < 1, and it is unstable if R 0 > 1.
In fact, using a similar approach to the literature [14], we can know that E 0 is globally asymptotically stable if R 0 < 1. A detailed proof is given below.
For a continuous and bounded function f(t), we define (21) For system (7), any solution with the initial conditions is (S(t), I(t), V(t)), and we have

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By the fluctuation lemma [15], we know that there is a sequence t n ; when t n ⟶ ∞, we have S(t n ) ⟶ S ∞ and S ′ (t n ) ⟶ 0(n ⟶ ∞). Substituting t n into the first equation of (7) yields Let us take the limits on both sides: Similarly, Combining (24) and (25), we know that at is to say, V ∞ � 0, which implies that lim t⟶∞ V(t) � 0. In the same way by using (25), we have lim t⟶∞ I(t) � 0. According to the limit theorem [13], we have lim t⟶∞ S(t) � Λ 1 /μ 1 . Combined with the local asymptotic stability of E 0 , we can get the following theorem: Theorem 3.2. For any τ, the virus-free equilibrium E 0 of system (7) is globally asymptotically stable if R 0 < 1. (13), the characteristic equation of linear approximate equation of the system (7) at the endemic equilibrium E 1 is

e Local Stability of the Endemic Equilibrium. From
When τ � 0, equation (27) is equivalent to where

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Notice that It follows that By using the Routh-Hurwitz criterion, equation (29) only has eigenvalues with negative real parts if R 0 > 1. We can obtain the following theorem: Theorem 3.3. For τ � 0, the endemic equilibrium E 1 of system (7) is locally asymptotically stable if R 0 > 1.

Hopf Bifurcation.
In this subsection, we devote to investigating the stability of the endemic equilibrium and the existence of Hopf bifurcation.

Estimation of the Length of Delay to Preserve Stability
In this section, we use a Nyquist criterion [17] to calculate the length of delay to preserve stability.
Consider the system (7) and the space of the real continuous functions that is defined in [−τ, +∞] and satisfied the initial conditions (2) (46) Linearization system (7) at the endemic equilibrium E 1 is expressed as follows: By taking the Laplace transformation for (47), we can obtain where Similarly, where M 2 � 0 −τ e −st Z(t)dt. us, (48) can be written as where

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(53)

e inverse Laplace transformation of L[X(t)], L[Y(t)], and L[Z(t)] will have terms which exponentially increase with time if L[X(t)], L[Y(t)], and L[Z(t)] have poles with positive real parts. us, E 1 is locally asymptotically stable if and only if all the poles of L[X(t)], L[Y(t)], and L[Z(t)]
have negative real parts.
By the method of [17] and the Nyquist criterion, the local asymptotic stability of E 1 needs to satisfy the following two conditions: where where μ 0 is the smallest positive root of (54). us (54) and (55) can be written as In order to estimate the length of delay to preserve stability, under the premise of ensuring stability, the following conditions need to be satisfied: If (58) and (59) are satisfied simultaneously, they are sufficient conditions to guarantee stability. Our aim is to find an upper bound μ + to μ 0 independent of τ and then to estimate τ so that (59) holds true for all values of 0 ≤ μ ≤ μ + and in particular at μ � μ 0 .

Direction and Stability of the Hopf Bifurcation
We have obtained the conditions under which the Hopf bifurcation occurs at E 1 of the system (7). is section will use the normal form theory and the center manifold theory to give the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions of system (7). We suppose that system (7) undergoes Hopf bifurcation at E 1 for . Let ± iω be a pair of conjugate pure virtual roots at E 1 when τ � τ. Define (67) us, system (7) is equivalent to the following functional differential equation where where

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Applying the Riesz representation theorem, there exists a 3 × 3 matrix-valued function η(·, μ) : where δ is the Dirac delta function, meeting δ(θ) � 0(θ ≠ 0) and us, (68) becomes In order to construct coordinates to describe the integral manifold near the origin, we need to define inner product and the adjoint operator A * � A * (0) of A as follows: where ψ ∈ C([0, 1], R 3 ) and η(θ) � η(θ, 0). Form the discussion in Section 3, we know that ± iωτ are eigenvalues of A(0). us they are also eigenvalues of A * . Define q(θ) � (1, q 1 , q 2 ) T e iθωτ and q * (s) � D(1, q * 1 , q * 2 ) T e −isωτ to be the eigenvectors of A(0) and A * corresponding to the eigenvalues iωτ and −iωτ, then We can calculate that According to (75), we know that where By using (74) and (80), we can know where η(θ) � η(θ, 0). By (89), when θ � 0, at leads to where Since we have We can calculate Computational and Mathematical Methods in Medicine

Conclusions
In this paper, we discuss the dynamics of the vector-borne disease model with delay-saturated infection rate and reinfection. By calculation, we have the basic reproductive number R 0 . rough R 0 , we determined the existence of disease-free equilibrium E 0 and the endemic equilibrium E 1 . According to the characteristic equation of the equilibrium points and using the Routh-Hurwitz criterion, we obtained that if R 0 < 1 the disease-free equilibrium will be stable, and the endemic equilibrium is locally asymptotically stable if R 0 > 1 and in the absence of time delay. Furthermore, by the fluctuation lemma and the limit theory, we analyzed the global stability of the disease-free equilibrium. We find that the time delay does not affect the Computational and Mathematical Methods in Medicine stability of the boundary equilibrium but can change the stability of E 1 and lead to the occurrence of Hopf bifurcation. en by using the Nyquist criterion, we get the maximum length of delay to preserve stability. Next, we found that the conditions for determining the direction and stability of bifurcating periodic solutions. Finally, the correctness of the main conclusion is verified by numerical simulation.
Data Availability e data we selected is only to verify the correctness of the results. ese data are not real data.

Conflicts of Interest
e authors declare that they have no conflicts of interest.