Stability and Hopf Bifurcation Analysis of a Vector-Borne Disease Model with Two Delays and Reinfection

In this paper, a vector-borne disease model with two delays and reinfection is established and considered. First of all, the existence of the equilibrium of the system, under different cases of two delays, is discussed through analyzing the corresponding characteristic equation of the linear system. Some conditions that the system undergoes Hopf bifurcation at the endemic equilibrium are obtained. Furthermore, by employing the normal form method and the center manifold theorem for delay differential equations, some explicit formulas used to describe the properties of bifurcating periodic solutions are derived. Finally, the numerical examples and simulations are presented to verify our theoretical conclusions. Meanwhile, the influences of the degree of partial protection for recovered people acquired by a primary infection on the endemic equilibrium and the critical values of the two delays are analyzed.


Introduction
Vector-borne diseases (VBDs) are one of the complex infectious diseases that endanger human beings. Vectors are living biological agents, such as ticks, mosquitoes, and fleas, with the ability to transmit parasites, bacteria, or viruses between people or from animals to people. It is reported that VBDs cause more than 1 billion infections and 1 million deaths worldwide every year [1]. Malaria is the most prevalent parasitic vectorborne disease caused by plasmodium parasitizing the human body [2][3][4][5]. Plasmodium enters stem cells through the blood to parasitize and reproduce. It then invades red blood cells to reproduce after maturity, causing red blood cells to burst in batches and attack. The source of malaria infection is malaria patients and those with Plasmodium. The natural transmission medium of malaria is female mosquitoes of the genus Anopheles, which is transmitted by biting the human body, and a few by blood transfusion and vertical transmission, with Anopheles gambiae being a major carrier of the disease [2,5,6]. Different types of malaria have different incubation periods, some of which are about 7-12 days, and some of which are more than 6 months. The morbidity and mortality of malaria are high [7]. According to the latest WHO reports, in 2018, there were estimated 228 million malaria cases worldwide, of which the death toll was about 405,000. It has led to great global economic and social losses, especially in the tropical and subtropical regions on five continents [3].
The population is generally susceptible to malaria. Although there is a certain degree of immunity after several infections, its acquisition process is so slow that it may take years or decades to develop, and it gradually weakens over time [5,7,8]. The remaining plasmodium may escape from the immune function due to the antigenic variation and reproduce again (recrudescence); even if the parasite has been eliminated by human immunity or drugs after the initial onset of malaria, the possibility of relapse is not excluded with time. That is, reinfection likely occurs [6,7]. It is shown that time delay is of great significance in many biological modelling, and its change may affect the dynamic behavior of the system [4][5][6][9][10][11][12][13][14][15][16][17][18][19], such as stability, uniqueness, and oscillation of solution. So, it is instructive to consider a mathematical model with time delays to research the influence of immunity on disease control of malaria transmission.
Considering the incubation period of virus transmission in the vector population, some vector-borne epidemic models with a delay were considered [4][5][6]. In [5], Xu and Zhou proposed a delayed vector-borne epidemic model and reinfection, investigated its existence and stability of equilibrium, and analyzed its dynamical behavior. It suggested that there are two effective preventive measures to reduce infections: one is to minimize vector to human contacts and the other is to use insecticides to control vector. According to [4,5], a vector-borne disease model with delay-saturated infection rate and cure rate was given by [6]. The existence and local stability of the epidemic equilibrium were discussed, and the length of the delay of the system preserving stability was estimated. This paper develops an improved vector-borne disease model with two delays and reinfection to consider the time required for the malaria virus to spread to the host population and vector population.
The remainder of this paper is organized as follows. In Section 2, an improved vector-borne disease model is formulated. In Section 3, the stability of the equilibrium and the existence of local Hopf bifurcation are discussed. In Section 4, some explicit formulas determining properties bifurcating periodic solutions are obtained by employing the normal form method and the center manifold theorem for the delay differential equations developed by Hassard et al. [20]. Some numerical examples and simulations are performed in Section 5 to demonstrate the main theoretical results, and the conclusions of this paper are summarized in Section 6.

Model Description
Generally, in the mathematical model of a vector-transmitted disease, the host population size at time t, denoted by N 1 ðtÞ, is divided into three subclasses: susceptible, infected, and recovered, with numbers denoted by SðtÞ, IðtÞ, and RðtÞ, respectively. The vector population size at time t, given by N 2 ðtÞ, is partitioned into two subclasses: susceptible vectors MðtÞ and infectious vectors VðtÞ. Based on the models [4][5][6], we consider an improved vector-borne disease model with two delays and reinfection as follows: where Λ 1 and Λ 2 are the recruitment rates of the hosts and vectors, respectively. b is the average number of bites per mosquito per day. β 1 and β 2 represent the infection rates from vector to human and human to vector, respectively. μ 1 and μ 2 are the natural death rates of the hosts and vectors, respectively. σð0 ≤ σ ≤ 1Þ is the degree of partial protection of individuals that recovered from primary infection. γ is the recovery rate per capita of the infective host population. τ 1 and τ 2 are two delays, in which τ 1 represents the incubation period of the host population, and τ 2 denotes the incubation period of the vector population. The term bβ 1 Sðt − τ 1 ÞVðt − τ 1 Þ represents the incidence number of the susceptible host infections caused by the infective vector at time t − τ 1 . bβ 2 Mðt − τ 2 ÞIðt − τ 2 Þ denotes the number of infections of the susceptible vectors that bite the infected host at time t − τ 2 and become infective at time t. According to [21], system (1) is equivalent to the following model: Notice that N i ′ðtÞ = Λ i − μ i N i ðtÞ, i = 1, 2, and its solution is Obviously, lim and any ε > 0, 0 ≤ S + I ≤ N 1 ðtÞ < ðΛ 1 /μ 1 Þ + ε holds by limit theorem, which follows that 0 ≤ S + I ≤ Λ 1 /μ 1 . Similarly, 0 ≤ V ≤ Λ 2 /μ 2 also holds. Thus, all the solutions of system (2) enter into the region: For the existence of the equilibrium of system (2), according to [5], one can have the following result.  Computational and Mathematical Methods in Medicine Lemma 1. For system (2), E 0 = ðΛ 1 /μ 1 , 0, 0Þ is a disease-free equilibrium if R 0 ≤ 1, and E * = ðS * , I * , V * Þ is an endemic equilibrium if R 0 > 1, where R 0 = ðb 2 Λ 1 Λ 2 β 1 β 2 Þ/ðμ 1 μ 2 2 ðμ 1 + γÞÞ is the basic reproductive number, S * = ðΛ 1 μ 2 ðbβ 2 I * + μ 2 ÞÞ/ðb 2 Λ 2 β 1 β 2 I * + μ 1 μ 2 ðbβ 2 I * + μ 2 ÞÞ, V * = ðbΛ 2 β 2 I * Þ/ðμ 2 ðbβ 2 I * + μ 2 ÞÞ, and I * satisfy the following equation: where The disease-free equilibrium E 0 denotes no infection, and the endemic equilibrium E * represents that the disease will exist and persist. The basic reproductive number R 0 describes the expected number of secondary infections, which is mainly sensitive to parameters b and β 1 but not affected by parameter σ [5]. The process of deriving R 0 using the nextgeneration method [22][23][24] is presented in Appendix A.
Hence, according to the Routh-Hurwitz criterion, the following conclusion can be drawn.
The characteristic equation (7) becomes where (11), replacing it into equation (11) and separating the real and imaginary parts, then, we can obtain It follows that  Computational and Mathematical Methods in Medicine It is easy to get that the two roots of f ′ ðzÞ = 3z 2 + 2p 2 z Þ. According to [25,26], we can get the conditions that equation (14) exists positive roots.
By Lemma 3, it is easy to see that the stability of E * will not change when τ changes if (ii) is set up. However, the stability may change when τ crosses through some critical values if equation (14) has a positive root.
According to [27], and applying Lemma 3 and the above transversal condition to system (2), the following theorem is obtained.
Then, for every fixed ω

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Let τ * 2 = min fτ 0 2i ji = 1, 2,⋯,kg, when τ 2 = τ * 2 , then equation (7) has a pair of purely imaginary roots ±iω * 2 . Next, we also check the transversality condition. By taking the differentiation of (20) with respect to τ 2 and further calculating, we can obtain where Thus, we can get where

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Therefore, when (H3): P 31 Q 31 + P 32 Q 32 ≠ 0 holds, then Re f½dλ/dτ 2 −1 i.e., the transversality condition is satisfied. According to the above discussions and based on [27], we can get the following result.

Properties of Hopf Bifurcation
From the analysis in the last section, we can see that for some critical values of delays, system (2) can occur in a series of periodic solutions at the equilibrium. On the basis of Theorem 5, we will employ the normal form method and the center manifold theorem introduced by Hassard et al. [20] to provide the properties of bifurcating periodic solutions in this section. Without loss of generality, we assume that τ * 1 < τ * 2 , τ * 1 ∈ ð0, τ 10 Þ, τ 2 = τ * 2 + μ, u 1 ðtÞ = SðτtÞ − S * , u 2 ðtÞ = Ið τtÞ − I * , and u 3 ðtÞ = VðτtÞ − V * . Then, equation (2) can be rewritten into the following functional differential equation in the Banach Space C = Cð½−1, 0, ℝ 3 Þ.
For φ ∈ Cð½0, 1, ðℝ 3 Þ * Þ, we define and the bilinear inner product where ηðθÞ = ηðθ, 0Þ. Let A = Að0Þ, then A and A * are a pair of adjoint operators. From the discussions in Section 3, we know that ±iω * 2 τ * 2 are a pair of eigenvalues of Að0Þ, it follows that they are also a pair of eigenvalues of A * . Next, we calculate the eigenvectors of Að0Þ and A * with respect to iω * 2 τ * 2 and −iω * 2 τ * 2 , respectively.
Next, using the methods given in [20], we can calculate some explicit expressions as follows that are used to determine the qualities of bifurcating periodic solution. where 11 Computational and Mathematical Methods in Medicine Δ 11 , Δ 12 , and Δ 13 are the third-order determinants obtained by substituting the first, second, and third columns of Δ 1 by vector ðP 1 , P 2 , P 3 Þ T , respectively. Δ 21 , Δ 22 , and Δ 23 are also third-order determinants obtained by replacing the first, second, and third columns of Δ 2 by vector ðQ 1 , Q 2 , Q 3 Þ T , respectively, where Therefore, based on [20], we can obtain the main results in the present section as follows.
Besides, the influences of the parameter σ on the endemic equilibrium E * and the critical values τ 10 and τ * 2 of the two delays τ 1 and τ 2 are shown in Table 1. When σ decreases, I * and V * also decrease, but S * , τ 10 , and τ * 2 increase, which suggests that the degree of partial protection for recovered people acquired by a primary infection can change the densities of distributions of susceptible and infective people and can also affect on the stability of positive equilibrium. It is noteworthy that it can not change the direction of Hopf bifurcation and the stability of bifurcating periodic solutions.

Conclusions
In this paper, we considered a vector-borne disease model with two delays and reinfection based on the models proposed by [4][5][6]. Some theoretical analyses of this model were implemented. From the characteristic equations, firstly, we discussed the conditions of the endemic equilibrium E * , which is locally asymptotically stable in the model without delays by the Routh-Hurwitz criterion. Then, by setting τ 2 = 0, regarding τ 1 as a bifurcating parameter, and setting τ 1 in its stable interval, regarding τ 2 as a parameter, respectively, we employed the Hopf bifurcation theorem to analyze the conditions for the stability of equilibrium and existence of Hopf bifurcation. Furthermore, when τ 1 is in its stable interval, for parameter τ 2 , the directions and stability of Hopf bifurcation were investigated, and some explicit formulas were derived by using the normal form theory and center manifold theorem. In the end, our numerical examples and simulations verified the correctness of theoretical conclusions. As the facts show, the disease will be persistent under the set of parameters, and the number of the host population and vector population will fluctuate. Also, when both delays exceed their critical values, the stability of the system and the properties and behaviors of the bifurcating periodic solutions may lead to change. The oscillatory behavior and phenomenon that an increased partial immunity may give rise to instability will prompt medical workers engaged in infectious diseases to remain vigilant against the virus even if they have noticed that the number of infected populations decreases. The qualitative analysis and numerical simulation used in the research of dynamic behavior of system can provide theoretical basis for revealing the development process, transmission pattern, and change trend of diseases and give important reference to the analysis, prediction, and control of human infectious diseases.