Stability and Hopf Bifurcation Analysis of a Vector-Borne Disease with Time Delay

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 804204, 8 pages http://dx.doi.org/10.1155/2014/804204 Research Article Stability and Hopf Bifurcation Analysis of a Vector-Borne Disease with Time Delay Yuanyuan Chen1 and Ya-Qing Bi2 1 2 Department of Applied Mathematics, Zhongyuan University of Technology, Zhengzhou, Henan 450007, China Department of Library, Chongqing Normal University, Chongqing 401331, China Correspondence should be addressed to Yuanyuan Chen; Received 3 March 2014; Accepted 27 April 2014; Published 20 May 2014 Academic Editor: Qiu-Ming Luo Copyright © 2014 Y. Chen and Y.-Q. Bi. 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. A delay-differential modelling of vector-borne is investigated. Its dynamics are studied in terms of local analysis and Hopf bifurcation theory, and its linear stability and Hopf bifurcation are demonstrated by studying the characteristic equation. The stability and direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold argument. 1. Introduction Vector-borne diseases are an important public health problem. Vector-borne diseases are infectious diseases caused by virus, bacteria, and so on which are primarily transmitted by disease biological agents, called vector carrying the disease. Malaria is the most prevalent vector-borne disease, which is transmitted to the human host through a bite by an infected mosquito. It can lead to serous affecting the brain, lungs, kidneys, and other organs, and it caused the greatest number of deaths. Approximately, 40 percent of the world’s population is at risk, and 2 million deaths per year can be attributed to malaria, half of those in children under 5 years old. Especially in Africa, more than one million children mostly under 5 years die each year. No effective vaccines are available for the disease. In many years, the effective way to prevent the malaria and other mosquito-borne disease is to control mosquito. Several theoretical studies have proposed vector-borne models. Reference [1] used a mathematical model to show that bringing a mosquito population below a certain threshold was sufficient to eliminate malaria. Reference [2] studied both a baseline ODE version of the model and a model with a discrete time delay and gave the conditions under which equilibrium is globally stable and the disease dies out. Reference [3] showed that reducing the number of mosquitoes is an inefficient control strategy that would have little effect on the epidemiology of malaria in areas of intense transmission. Reference [4] used a mathematical model to evaluate the impact from the programs of selective mass drug administration and vector control through mosquito nets. References [5, 6] models took into account the acquired immunity to malaria depends on exposure (i.e., that immunity is boosted by additional infections). For a long time, it has been recognized that delay may have very complicated impact on the dynamics of a system. Delay can cause the loss of stability and can bifurcate various periodic solutions. Recently, there has been extensive work dealing with time delay systems (see, e.g., [7–11]). As far as we know, there are few works on the delayed vector-borne system, let alone the existence of Hopf bifurcation, and the stability and direction of bifurcating periodic solutions. In this paper, we focus on investigating these problems. This paper is organized as follows. In Section 2, we provide a vector-borne model and analyze the property of the nonnegative equilibria. In Section 3, we get the existence of the Hopf bifurcation. In Section 4, the stability and direction of periodic solutions bifurcating from the Hopf bifurcation are determined by using the normal form theory and center manifold argument introduced by Hassard et al. [12]. 2 Journal of Applied Mathematics 2. Property of the Nonnegative Equilibria We can describe the dynamics of the disease in the host population as follows: 𝑆 ̇ (𝑡) = 𝑏1 − 𝜆 1 𝑆 (𝑡) 𝐼 (𝑡) − 𝜆 2 𝑆 (𝑡) 𝑉 (𝑡) − 𝜇1 𝑆 (𝑡) , 𝐼 ̇ (𝑡) = 𝜆 1 𝑆 (𝑡) 𝐼 (𝑡) + 𝜆 2 𝑆 (𝑡) 𝑉 (𝑡) − 𝛾𝐼 (𝑡) − 𝜇1 𝐼 (𝑡) , (1) Here 𝑆(𝑡), 𝐼(𝑡), and 𝑅(𝑡) represent the population density of susceptible, infectious, and recovered at time 𝑡, respectively. The total host population size at time 𝑡 is given by 𝑁1 (𝑡). The host population dies at a natural rate 𝜇1 , and the host grows with intrinsic growth rate 𝑏1 . 𝜆 1 is the rate of direct transmission, while 𝜆 2 is the biting rate of a pathogencarrier vector. The host recovers at the rate 𝛾. The recovered individuals are assumed to acquire permanent immunity. The system that describes the dynamics of the vector is given by 𝑉̇ (𝑡) = 𝜆 3 𝑀 (𝑡) 𝐼 (𝑡) − 𝜇2 𝑉 (𝑡) . (2) 𝑉̇ (𝑡) = 𝜆 3 𝑀 (𝑡) 𝐼 (𝑡 − 𝜏) − 𝜇2 𝑉 (𝑡) . (3) The systems (1) and (3) satisfy the initial conditions: 𝑆(𝜃) = 𝑆0 , 𝐼(𝜃) = 𝐼0 , 𝑅(𝜃) = 𝑅0 , 𝑀(𝜃) = 𝑀0 , 𝑉(𝜃) = 𝑉0 , and 𝜃 ∈ [−𝜏, 0). The total host population size 𝑁1 (𝑡) can be determined by 𝑁1 (𝑡) = 𝑆(𝑡) + 𝐼(𝑡) + 𝑅(𝑡) or 𝑁̇ 1 (𝑡) = 𝑏1 − 𝜇1 𝑁1 (𝑡) . The initial condition of system (6) is (𝑆 (𝜃) , 𝐼 (𝜃) , 𝑉 (𝜃) ∈ 𝐶+ = 𝐶 ((−𝜏, 0] , 𝑅+3 )) , 𝑆0 , 𝐼0 , 𝑉0 > 0. (7) System (6) has two equilibria 𝐸10 = (𝑏1 /𝜇1 , 0, 0) and 𝐸∗ = (𝑆∗ , 𝐼∗ , 𝑉∗ ), where 𝑏1 − (𝛾 + 𝜇1 ) 𝐼∗ , 𝜇1 𝑉∗ = 𝜆 3 𝑏2 𝐼∗ . 𝜇2 (𝜇2 + 𝜆 3 𝐼∗ ) (8) 𝐼∗ is determined by the following equation: 𝑏1 − (𝛾 + 𝜇1 ) 𝐼 𝜆 2 𝜆 3 𝑏2 (𝜆 1 + ) = 𝛾 + 𝜇1 . 𝜇1 𝜇2 (𝜆 3 𝐼 + 𝜇2 ) (9) Equation (9) has a unique positive root, when (𝛾 + 𝜇1 )𝜇1 𝜇22 < 𝑏1 (𝜆 1 𝜇22 + 𝜆 2 𝜆 3 𝑏2 ). For the equilibrium 𝐸10 , the characteristic equation is (𝜇1 + 𝜆) (𝛾 + 𝜇1 − 𝜆 1 𝑏1 + 𝜆) (𝜇2 + 𝜆) = 0. 𝜇1 (10) We can easily get the following theorem by some calculation. Theorem 1. 𝐸10 is asymptotically stable if 𝜆 1 𝑏1 /𝜇1 −𝛾−𝜇1 < 0; it is unstable if 𝜆 1 𝑏1 /𝜇1 − 𝛾 − 𝜇1 > 0. 3. Existence of Hopf Bifurcation We study 𝐸∗ under the condition (𝛾 + 𝜇1 )𝜇1 𝜇22 < 𝑏1 (𝜆 1 𝜇22 + 𝜆 2 𝜆 3 𝑏2 ). The characteristic equation of 𝐸∗ is (4) The total number of vectors 𝑁2 (𝑡) can be determined by 𝑁2 (𝑡) = 𝑀(𝑡) + 𝑉(𝑡) or 𝑁2 (𝑡) = 𝑏2 − 𝜇2 𝑁2 (𝑡) . (6) 𝑏 𝑉̇ (𝑡) = 𝜆 3 ( 2 − 𝑉 (𝑡)) 𝐼 (𝑡 − 𝜏) − 𝜇2 𝑉 (𝑡) . 𝜇2 𝑆∗ = Here, 𝑉(𝑡) is the number of vectors at time 𝑡 carrying the pathogen at time 𝑡, and 𝑀(𝑡) represents the population density of pathogen-free vector at time 𝑡. The total vectors population size at time 𝑡 is given by 𝑁2 (𝑡). 𝑏2 and 𝜇2 are the birth rate and death rate of vector population, respectively. Suspectable vectors start carrying the pathogen after getting into contact with an infective host at a rate 𝜆 3 . We assume that the vectors carry the microparasite for life once they became carrier of it. The time delay 𝜏 > 0 is introduced in the system (2) to describe the dynamics of the vector. At time 𝑡, the susceptible vectors bite the host 𝜏 time ago, and the vector became infecti (...truncated)