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

Zhang, Yanxia; Li, Long; Huang, Junjian; Liu, Yanjun

doi:https://doi.org/10.1155/2021/6648959

Computational and Mathematical Methods in Medicine

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Volume 2021 | Article ID 6648959 | https://doi.org/10.1155/2021/6648959

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

Yanxia Zhang,¹Long Li,¹Junjian Huang,²and Yanjun Liu³

Academic Editor: Alireza Jolfaei

Received30 Nov 2020

Revised09 May 2021

Accepted20 May 2021

Published08 Jun 2021

Abstract

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 [2–5]. 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 [4–6, 9–19], 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 [4–6]. 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.

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 [4–6], 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 [22–24] 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

Thus,

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

Thus, the following quantities can be calculated:

Therefore, based on [20], we can obtain the main results in the present section as follows.

Theorem 6. For system (2), the periodic solution is supercritical (resp., subcritical) if (resp., ). The bifurcating periodic solutions are orbitally asymptotically stable with an asymptotical phase (resp., unstable) if (resp., ). The period of the bifurcating periodic solutions increases (resp., decreases) if (resp. ).

5. Numerical Examples and Simulations

We select the parameters to simulate the above theoretical results as follows:

By calculations, we can obtain that the endemic equilibrium of system (2) is . Under the above parameters, the condition (H1) is satisfied. According to Theorem 2, we know that the endemic equilibrium of system (2) is asymptotically stable when (see Figure 1). When , , we get , , , and ; from Theorem 4, the endemic equilibrium is locally asymptotically stable for and is unstable for , and a Hopf bifurcation occurs at (see Figures 2–4). When , fixing that belongs to its stable interval , then, by calculation, we obtain that a positive root of equation (22) is and , which follow from Theorem 5 that the endemic equilibrium is locally asymptotically stable for and is unstable for , a Hopf bifurcation exits at (see Figures 5–7).

The numerical results show that , , , and , which imply that, for system (2), its periodic solution is subcritical at and unstable, and the period of the bifurcating periodic solutions increases (see Figure 8). Also, when the two delays are greater than their critical values, the system will produce irregular periodic behaviors, which is depicted in Figure 9.

Besides, the influences of the parameter on the endemic equilibrium and the critical values and of the two delays and are shown in Table 1. When decreases, and also decrease, but , , and 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.

6. Conclusions

In this paper, we considered a vector-borne disease model with two delays and reinfection based on the models proposed by [4–6]. Some theoretical analyses of this model were implemented. From the characteristic equations, firstly, we discussed the conditions of the endemic equilibrium , which is locally asymptotically stable in the model without delays by the Routh-Hurwitz criterion. Then, by setting , regarding as a bifurcating parameter, and setting in its stable interval, regarding 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 is in its stable interval, for parameter , 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.

Appendix

A. The Derivation Process of Based on the Next-Generation Method

is the dominant eigenvalue of the next generation matrix whose entries are interpreted as the number of secondary infections produced by infected vectors and hosts in the process of their infection. Here, the notation we use follows [22–24]. For system (2), we calculate

Thus, . Note that we use as in [5], though it is called . Either way, its threshold behavior remains unchanged.

B. The Derivation Process of Coefficients , , , and of Equation (47)

Using the algorithms and marks presented in [20], when , the projection system of system (2) on the center manifold can be calculated. For the solution of equation (29), we define , which follows from equations (35)–(39) that where .

Let then, on the center manifold , we can have where

and are local coordinates on in the directions of and . From equations (39) and (B.2), we can see

Therefore, we can obtain

Furthermore, we can get as follows:

Substituting , , , , , , and into the above expression of and comparing the coefficients yield the coefficients , , , and of equation (47).

Obviously, , , and can be determined directly by substituting system parameters, but and in need to be calculated through some methods [25, 26].

By equations (36) and (B.2), we can get where

Thus, we have

From equation (B.4), we can get

Therefore, we can obtain

For , we have

By comparing coefficients of (B.9) with (B.13), one can get

From equations (B.12) and (B.14) and the definition of , we can obtain

Since , then according to (B.16), we can get where .

Similarly, where .

Next, we seek the appropriate constant vectors and in expressions (B.17) and (B.19). From equation (B.12) and the definition of , we can have where . Thus, where , , and and , , and .

Note that

.

Substituting expressions (39) and (B.22) into equation (B.20), we can have which leads to , where

It follows that where ,

Similarly, substituting expressions (B.19) and (B.23) into equation (B.21), we can have which leads to , where

It follows that where ,

Data Availability

The data included in this paper is only to demonstrate the validity of the results.

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Acknowledgments

This work was supported in part by the Fundamental Research Funds for the Central Universities, China (SWU020005), in part by the Natural Science Foundation Project of Chongqing CSTC (cstc2018jcyjAX0810), in part by the Science and Technology Research Program of Chongqing Municipal Education Commission (KJQN201801605), and in part by CQUE (KY201708B and KY20200146).

References

K. S. Moore, “Vector-borne diseases: an ongoing threat,” The journal Nurse Practitioners, vol. 15, no. 6, pp. 449–457, 2019.
View at: Publisher Site | Google Scholar
F. Catteruccia, “Malaria-carrying mosquitoes get a leg up on insecticides,” Nature, vol. 577, no. 7790, pp. 319-320, 2020.
View at: Publisher Site | Google Scholar
J. Sachs and P. Malaney, “The economic and social burden of malaria,” Nature, vol. 415, no. 6872, pp. 680–685, 2002.
View at: Publisher Site | Google Scholar
H.-M. Wei, X.-Z. Li, and M. Martcheva, “An epidemic model of a vector-borne disease with direct transmission and time delay,” Journal of Mathematical Analysis and Application, vol. 342, no. 2, pp. 895–908, 2008.
View at: Publisher Site | Google Scholar
J. Xu and Y. Zhou, “Hopf bifurcation and its stability for a vector-borne disease model with delay and reinfection,” Applied Mathematical Modeling, vol. 40, no. 3, pp. 1685–1702, 2016.
View at: Publisher Site | Google Scholar
Z. Hu, S. Yin, and H. Wang, “Stability and Hopf bifurcation of a vector-borne disease model with saturated infection rate and reinfection,” Computational and Mathematical Methods in Medicine, vol. 2019, Article ID 1352698, 17 pages, 2019.
View at: Google Scholar
A. M. Niger and A. B. Gumel, “Mathematical analysis of the role of repeated exposure on malaria transmission dynamics,” Differential Equations and Dynamical Systems, vol. 16, no. 3, pp. 251–287, 2008.
View at: Publisher Site | Google Scholar
L. Hviid, “Naturally acquired immunity to Plasmodium falciparum malaria in Africa,” Acta Tropica, vol. 95, no. 3, pp. 270–275, 2005.
View at: Publisher Site | Google Scholar
Y. Xue and T. Li, “Stability and Hopf bifurcation for a delayed SIR epidemic model with logistic growth,” Abstract and Applied Analysis, vol. 2013, Article ID 916130, 11 pages, 2013.
View at: Publisher Site | Google Scholar
Z. Zhang, S. Kundu, J. P. Tripathi, and S. Bugalia, “Stability and Hopf bifurcation analysis of an SVEIR epidemic model with vaccination and multiple time delays,” Chaos Solitons & Fractals, vol. 131, article 109483, 2020.
View at: Publisher Site | Google Scholar
H. Wan and J. A. Cui, “A malaria model with two delays,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 601265, 8 pages, 2013.
View at: Publisher Site | Google Scholar
J. L. Liu and T. L. Zhang, “Stability and Hopf bifurcation analysis of a plant virus propagation model with two delays,” Discrete Dynamics in Nature and Society, vol. 2018, Article ID 7126135, 12 pages, 2018.
View at: Google Scholar
L. Li, C. H. Zhang, and X. P. Yan, “Stability and Hopf bifurcation analysis for a two-enterprise interaction model with delays,” Communications in Nonlinear Science and Numerical Simulation, vol. 30, no. 1–3, pp. 70–83, 2016.
View at: Publisher Site | Google Scholar
C. Xu and M. Liao, “Bifurcation analysis of an autonomous epidemic predator-prey model with delay,” Annali di Matematica Pura ed Applicata, vol. 193, no. 1, pp. 23–38, 2014.
View at: Publisher Site | Google Scholar
C. Xu and P. Li, “Oscillations for a delayed predator-prey model with Hassell–Varley-type functional response,” Comptes Rendus Biologies, vol. 338, no. 4, pp. 227–240, 2015.
View at: Publisher Site | Google Scholar
S. Khajanchi, M. Perc, and D. Ghosh, “The influence of time delay in a chaotic cancer model,” Chaos, vol. 28, article 103101, 2018.
View at: Google Scholar
C. Xu, M. Liao, and P. Li, “Bifurcation control of a fractional-order delayed competition and cooperation model of two enterprises,” Science China Technological Sciences, vol. 62, no. 12, pp. 2130–2143, 2019.
View at: Publisher Site | Google Scholar
R. Kumar, A. K. Sharma, and K. Agnihotri, “Hopf bifurcation analysis in a multiple delayed innovation diffusion model with Holling II functional response,” Mathematical Methods in the Applied Sciences, vol. 43, no. 4, pp. 2056–2075, 2020.
View at: Publisher Site | Google Scholar
Z. Zhang and H. Yang, “Stability and bifurcation in a stage-structured predator–prey system with Holling-II functional response and multiple delays,” International Journal of Computer Mathematics, vol. 92, no. 3, pp. 542–561, 2015.
View at: Publisher Site | Google Scholar
B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, “Theory and applications of Hopf bifurcation,” London Mathematical Society Lecture Note Series, vol. 41, 1981.
View at: Google Scholar
C. Castillochavez and H. R. Thieme, “Asymptotically autonomous epidemic models,” in Mathematical Population Dynamics: Analysis of Heterogeneity, Theory of Epidemics, vol. 1, Wuerz, Winnipeg, Canada, 1994.
View at: Google Scholar
M. Jackson and B. M. Chen-Charpentier, “Modeling plant virus propagation with delays,” Journal of Computational and Applied Mathematics, vol. 309, no. 47, pp. 611–621, 2017.
View at: Publisher Site | Google Scholar
P. Driessche and J. Watmough, “Further notes on the basic reproduction number,” in Mathematical Epidemiology Lecture Notes in Mathematics, vol. 1945, Springer, Berlin, 2008.
View at: Google Scholar
J. M. Heffernan, R. J. Smith, and L. M. Wahl, “Perspectives on the basic reproductive ratio,” Journal of the Royal Society Interface, vol. 2, no. 4, pp. 281–293, 2005.
View at: Publisher Site | Google Scholar
Y. Song and S. Yuan, “Bifurcation analysis in a predator-prey system with time delay,” Nonlinear Analysis: Real World Applications, vol. 7, no. 2, pp. 265–284, 2006.
View at: Publisher Site | Google Scholar
Y. Song and J. Wei, “Bifurcation analysis for Chen’s system with delayed feedback and its application to control of chaos,” Chaos, Solitons & Fractals, vol. 22, no. 1, pp. 75–91, 2004.
View at: Publisher Site | Google Scholar
J. Hale, “Theory and application of Hopf bifurcation,” in London Math Society Lecture Note Series, Cambridge University Press, Cambridge, UK, 1977.
View at: Google Scholar

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Copyright © 2021 Yanxia Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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