Computational and Mathematical Methods in Medicine

Volume 2019, Article ID 3859815, 10 pages

https://doi.org/10.1155/2019/3859815

## State-Dependent Pulse Vaccination and Therapeutic Strategy in an SI Epidemic Model with Nonlinear Incidence Rate

^{1}College of Mathematics and Information Science, Anshan Normal University, Anshan 114016, China^{2}College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China^{3}State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China^{4}Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

Correspondence should be addressed to Tongqian Zhang; nc.ude.tsuds@naiqgnotgnahz

Received 26 July 2018; Revised 18 November 2018; Accepted 22 November 2018; Published 6 February 2019

Guest Editor: Ke-jun Dong

Copyright © 2019 Kaiyuan Liu 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.

#### Abstract

In this paper, the state-dependent pulse vaccination and therapeutic strategy are considered in the control of the disease. A pulse system is built to model this process based on an SI ordinary differential equation model. At first, for the system neglecting the impulse effect, we give the classification of singular points. Then for the pulse system, by using the theory of the semicontinuous dynamic system, the dynamics is analyzed. Our analysis shows that the pulse system exhibits rich dynamics and the system has a unique order-1 homoclinic cycle, and by choosing *p* as the control parameter, the order-1 homoclinic cycle disappears and bifurcates an orbitally asymptotical stable order-1 periodic solution when *p* changes. Numerical simulations by maple 18 are carried out to illustrate the theoretical results.

#### 1. Introduction

Infectious diseases are caused by various pathogens that can be transmitted from person to person, animal to animal, or human to animal. The ever-changing changes in the pathogens of ancient infectious diseases and the emergence of new pathogens have brought new challenges to the discovery, diagnosis, and prevention of infectious diseases. According to the 2016 report of the World Health Organization [1], about 36.7 million people have been infected with HIV/AIDS, 1.0 million people died of HIV/AIDS, and more than 18 million people worldwide living with HIV are receiving antiviral drugs. And tuberculosis is currently the biggest “killer” caused by a single infectious pathogen after AIDS in the world. As of the end of 2016, there were 10.4 million new tuberculosis cases [2]. Therefore, the control and elimination of infectious diseases has attracted wide attention of people. Various dynamic models have been proposed by mathematicians to investigate the spread and evolution of infectious diseases [3–14]. In particular, mathematical models of differential equations have been extensively investigated, and among them, the most classical well-known model is SIR model [15] or SIS model [16], which have been widely investigated [17–23].

It is well known that vaccination is mostly a medical behavior that can evoke the individual’s natural defense mechanism to prevent possible future diseases. This kind of vaccination is known as prophylactic vaccination. Diphtheria, whooping cough, polio, tetanus, herpes, rubella, and mumps are the most common types of vaccines. There are many types of vaccination, the two common types are continuous vaccination and pulsed vaccination. Continuous vaccination is when people are vaccinated at birth to protect themselves from illness, while pulsed vaccination is when people are vaccinated at a fixed period of time in all age groups which was firstly investigated by Agur et al in [24]. Pulse vaccination strategy (PVS) has been studied by many scholars [25, 26]. For example, Lu et al. [27] studied the pulse epidemic model with bilinear incidence and compared the effectiveness of the continuous and pulsed vaccination strategies. Liu et al. [28] investigated the SIR epidemic model with the saturated transmission rate. However, the strategy is taken at certain fixed times and does not depend on the status of infectious diseases. In general, taking into account the limited medical resources and costs, vaccines to susceptible people according to the number of susceptible people or infected people are more reasonable than continuous vaccination and fixed time pulse vaccination. This control strategy relies on the individual (or susceptible individuals) of the infection state and is called a state-dependent pulse vaccination strategy. Based on this idea, Tang et al. [29], Nie et al. [30], Guo et al. [31], and Qin et al. [32] have considered a state-dependent pulse strategy in SIR model and SIRS model. In fact, using state-dependent feedback control strategies to simulate real-world problems is more reasonable. Therefore, the impulsive state feedback control is also widely applied to the population dynamics model [33–46], chemostat model [47], and turbidostat model [48].

Firstly, we consider an epidemic model with nonlinear incidence rate described by the ordinary differential equations as follows:which is a special case in the study of Liu et al. [49] and and represents the number of susceptible and infected individuals at time respectively. *θ* is the birth rate, *β* is the contact rate, and *γ* is the natural death rate.

Motivated by the studies of Tang et al. [29], Nie et al. [30], and Zhang et al. [50], we consider state-dependent pulse vaccination and treatment strategy in model (1) and get the following model:where When the amount of infected reaches the hazardous threshold value vaccination and treatment are taken into account, and the number of susceptible and infected suddenly turn to and , respectively, where denote the vaccination rate of susceptible individuals and treatment rate of infected individuals, respectively. By the scaling,then, model (2) transforms into the following form:where In the following, according to the actual condition, we always suppose that , and based on practical significance, our research scope is limited to the first quadrant, i.e., .

The purpose of this paper is to study the dynamic behavior under the effect of state-dependent pulse vaccination and treatment strategy. This article is organized as follows. In Section 2, we introduce some definitions and notations of the geometric theory of semicontinuous dynamic systems, which will be useful for the latter discussion. In Section 3, we qualitatively analyze the dynamics of model (3). In Section 4, the existence of the homoclinic cycle is studied by using the geometrical theory of semicontinuous dynamical systems. At last, we present some numerical simulations.

#### 2. Preliminaries

In this section, we introduce some notations, definitions, and lemmas of the geometric theory of semicontinuous dynamic system, which will be useful for the following discussions. The following definitions and lemmas of semicontinuous dynamic system come from the studies of Chen et al. [51] and Wei and Chen [36].

*Definition 1. *Consider the following state-dependent impulsive differential systemThe solution mapping of system (4) is called the semicontinuous dynamical system denoted by where is the semicontinuous dynamical system mapping with initial point the sets *M* and *N* are called the impulse set and phase set, which are lines or curves on The continuous function is called impulse mapping.

*Remark 1. *System (4) constitutes a semicontinuous dynamic system where

*Definition 2. *If there exists a point and such that and , then is called order-1 periodic solution.

*Definition 3. *The trajectory combining with impulse line is called the order-1 cycle. If the order-1 cycle has a singularity, then the order-1 cycle is called the order-1 singular cycle. Furthermore, if the order-1 cycle only has a saddle, then the order-1 singular cycle is called the order-1 homoclinic cycle.

*Definition 4. *We assume that G is a bounded closed simple connected region, which has the following properties:(i)Impulse set *M* is a simple connected bounded closed straight line segments or curve segments which do not contain closed branch(ii)The boundaries , , and of region G are nontangent arcs of semicontinuous dynamical system (4). The boundary is the impulse set of system (4), and its phase set satisfies ;(iii)The orientation of the vector fields of semicontinuous dynamical system (4) on the , , and points of the internal of region G. There are no equilibriums on the boundaries and also in the internal of region G of semicontinuous dynamical system (4).Then region G is called Bendixson’s region of semicontinuous dynamical system (4).

Lemma 1. *(Bendixson theorem of semicontinuous dynamical system.) If region G is Bendixson’s region of semicontinuous dynamical system (4), then there exists at least an order-1 periodic solution in the internal of region G (Figure 1).*