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.

1. 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 vector-borne disease caused by plasmodium parasitizing the human body [25]. 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 [46, 919], 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 [46]. 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.

2. Model Description

Generally, in the mathematical model of a vector-transmitted disease, the host population size at time , denoted by , is divided into three subclasses: susceptible, infected, and recovered, with numbers denoted by , , and , respectively. The vector population size at time , given by , is partitioned into two subclasses: susceptible vectors and infectious vectors . Based on the models [46], we consider an improved vector-borne disease model with two delays and reinfection as follows: where and are the recruitment rates of the hosts and vectors, respectively. is the average number of bites per mosquito per day. and represent the infection rates from vector to human and human to vector, respectively. and are the natural death rates of the hosts and vectors, respectively. is the degree of partial protection of individuals that recovered from primary infection. is the recovery rate per capita of the infective host population. and are two delays, in which represents the incubation period of the host population, and denotes the incubation period of the vector population. The term represents the incidence number of the susceptible host infections caused by the infective vector at time . denotes the number of infections of the susceptible vectors that bite the infected host at time and become infective at time .

According to [21], system (1) is equivalent to the following model:

Notice that , , and its solution is .

Obviously, . So, for and any , holds by limit theorem, which follows that . Similarly, 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.

Lemma 1. For system (2), is a disease-free equilibrium if , and is an endemic equilibrium if , where is the basic reproductive number, , , and satisfy the following equation: where

The disease-free equilibrium denotes no infection, and the endemic equilibrium represents that the disease will exist and persist. The basic reproductive number describes the expected number of secondary infections, which is mainly sensitive to parameters and but not affected by parameter [5]. The process of deriving using the next-generation method [2224] is presented in Appendix A.

3. Stability of Endemic Equilibrium and Hopf Bifurcation

The Jacobian matrix of the linear system (2) at is and the characteristic equation is where

Case (1). .

Characteristic equation (7) becomes where , , and .

When the following condition (H1) holds, all roots of equation (9) have negative real parts.

Hence, according to the Routh-Hurwitz criterion, the following conclusion can be drawn.

Theorem 2. If (H1) holds, the endemic equilibrium is locally asymptotically stable when .

Case (2). .

The characteristic equation (7) becomes where .

Suppose is a root of equation (11), replacing it into equation (11) and separating the real and imaginary parts, then, we can obtain

It follows that

Let , , , and , then, equation (13) is equivalent to

It is easy to get that the two roots of are and .

According to [25, 26], we can get the conditions that equation (14) exists positive roots.

Lemma 3. For equation (14), (i)If , then equation (14) has at least one positive root(ii)If and , then equation (14) has no positive roots(iii)If and , then equation (14) has positive roots if and only if and

By Lemma 3, it is easy to see that the stability of will not change when changes if (ii) is set up. However, the stability may change when crosses through some critical values if equation (14) has a positive root.

Assume that , , and are three positive roots of equation (14), then, there are three positive numbers , , and . It follows from equation (12) that

Denote where , , then, is a pair of purely imaginary roots of equation (11) for . Let , be the that corresponds to the minimum , and be the root of equation (11) near satisfying and .

Furthermore, we consider the transversality condition. By differentiating both sides of equation (11) with respect to , we get

It follows that


The transversality condition is satisfied when .

According to [27], and applying Lemma 3 and the above transversal condition to system (2), the following theorem is obtained.

Theorem 4. For system (2), when , (i)if and hold, then, the endemic equilibrium is locally asymptotically stable for all (ii)if or , , , and , then, the endemic equilibrium is locally asymptotically stable for (iii)if the conditions of (ii) and hold, then, it undergoes a Hopf bifurcation at when

Case (3). .

Fixing in its stable interval and regarding as a parameter, we analyze the roots of characteristic equation (7). Let be a characteristic root of equation (7). By performing some calculations as those in case (2), we can obtain where

Furthermore, we suppose that

(H2): equation (22) has finite positive roots .

Then, for every fixed , there exists a sequence of critical values such that equation (22) holds, where

Let , when , then equation (7) has a pair of purely imaginary roots .

Next, we also check the transversality condition. By taking the differentiation of (20) with respect to and further calculating, we can obtain where

Thus, we can get where

Therefore, when

(H3): holds, then , i.e., the transversality condition is satisfied.

According to the above discussions and based on [27], we can get the following result.

Theorem 5. Let , if (H2) and (H3) hold, then, the endemic equilibrium is locally asymptotically stable for and is unstable for . System (2) undergoes Hopf bifurcation at the endemic equilibrium for .

4. 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 , , , , , and . Then, equation (2) can be rewritten into the following functional differential equation in the Banach Space . where , , , and are defined as follows: where

Based on the Riesz representation theorem, we know that there exists a function of bounded variation components , , such that

In fact, we can take

Define and by

Then, system (29) can be further represented as where .

For , we define and the bilinear inner product where . Let , then and are a pair of adjoint operators. From the discussions in Section 3, we know that are a pair of eigenvalues of , it follows that they are also a pair of eigenvalues of . Next, we calculate the eigenvectors of and with respect to and , respectively.

Suppose that is an eigenfunction of corresponding to . Then, by the definition of , we can take where

Similarly, is the eigenfunction of corresponding to , where

From equation (38), we can get

Thus, one can choose as

which satisfies .

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

, , and are the third-order determinants obtained by substituting the first, second, and third columns of by vector , respectively. , , and are also third-order determinants obtained by replacing the first, second, and third columns of by vector , respectively, where