Acta Biotheoretica

, Volume 63, Issue 1, pp 1–21

Mathematical Analysis of a Chlamydia Epidemic Model with Pulse Vaccination Strategy

Authors

    • Institute of MathematicsNational Autonomous University of Mexico
    • Department of MathematicsIndian Institute of Engineering Science and Technology
Regular Article

DOI: 10.1007/s10441-014-9234-8

Cite this article as:
Samanta, G.P. Acta Biotheor (2015) 63: 1. doi:10.1007/s10441-014-9234-8

Abstract

In this paper, we have considered a dynamical model of Chlamydia disease with varying total population size, bilinear incidence rate and pulse vaccination strategy. We have defined two positive numbers \(R_{0}\) and \(R_{1}(\le R_{0})\). It is proved that there exists an infection-free periodic solution which is globally attractive if \(R_{0}<1\) and the disease is permanent if \(R_{1}>1.\) The important mathematical findings for the dynamical behaviour of the Chlamydia disease model are also numerically verified using MATLAB. Finally epidemiological implications of our analytical findings are addressed critically.

Keywords

Chlamydia trachomatisPulse vaccinationPermanenceExtinctionGlobal stability

1 Introduction

Infectious diseases have tremendous influence on human life and are usually caused by pathogenic microorganisms, such as bacteria, viruses, parasites, or fungi. The diseases can be spread directly or indirectly. Controlling infectious diseases has been an increasingly complex and significant issue in recent years. Sexually transmitted infections (STIs) remain a major public health challenge globally and are among the most common infections in the United States. Chlamydia, caused by the bacterium Chlamydia trachomatis, is one of the most important sexually-transmitted infections worldwide. It is a major and the commonest sexually-transmitted bacterial disease in European countries (Fenton and Lowndes 2004), Manavi (2006) and the United States (Miller et al. 2004; Schillinger et al. 2005). It was estimated that around 92 million Chlamydia infections occurred worldwide in 1999, affecting more women (50 million) than men (42 million) (Averting HIV and AIDS 2010; World Health Organization 2001). In 2007, 1,108,374 Chlamydia diagnoses were reported in the United States, up from 1,030,911 in 2006. The 2007 total represents the largest number of cases ever reported to the Centers for Disease Control and Prevention (CDC) for any condition (Mushayabasa 2012).

Chlamydia trachomatis is sexually-transmitted (via vaginal, oral and anal sex) and it is also transmitted through other modes, such as vertically (from mother to child) (Centers for Disease Control and Prevention 2008a). This disease causes numerous irreversible complications such as chronic pelvic pain, infertility in females and pelvic inflammatory disease. Chlamydia can be easily treated and cured, if diagnosed effectively, with antibiotics. But usually occurs without symptoms and often goes undiagnosed. Untreated, it can cause severe health consequences for women and up to 40 % of females with untreated Chlamydia infections develop pelvic inflammatory disease (PID), a condition which can lead to such long-term complications as infertility, ectopic pregnancy and chronic pelvic pain (Hillis and Wasserheit 1996). In pregnant women, it may lead to premature delivery and babies born to infected mothers can get infections in their eyes, called conjunctivitis or pinkeye, as well as pneumonia. Complications from Chlamydia among men are relatively uncommon, but may include epididymitis and urethritis which can cause pain, fever and in rare cases, sterility. Men with Chlamydia symptoms might have a discharge from the penis and a burning sensation when urinating which may range from clear to pussy. Men might also have burning and itching around the opening of the penis or pain and swelling in the testicles/scrotum, or both which can be a sign of epididymitis, an inflammation of a part of the male reproductive system located in the testicles. Both PID and epididymitis can result in infertility. In addition, investigations indicate that the presence of Chlamydia infection increases the risk of HIV transmission (Fleming and Wasserheit 1999; Kahn et al. 2005). Chlamydia is also known as a “silent” disease since 75 % of infected women and at least 50 % of infected men have no symptoms. If symptoms do occur, they usually appear after 1 to 3 weeks of exposure (Adetunde et al. 2009; Centers for Disease Control and Prevention 2008a; Sharomi and Gumel 2009). Women, especially young and minority women, are hit hardest by Chlamydia trachomatis and it is found that women are most severely impacted by the long-term consequences of untreated Chlamydia. The reported Chlamydia case rate for females in 2007 was almost three times higher than for males (543.6 and 190.0 per 100,000 population respectively). Young females 15 to 19 years of age had the highest Chlamydia rate followed by females 20 to 24 years of age (Mushayabasa 2012). Some women may still have no signs or symptoms, which is the cause that it is often transmitted from one sexual partner to another without either knowing (Adetunde et al. 2009). The annual cost of treatment of Chlamydia and its consequences in the United States is more than \(\$2\) billion (Adetunde et al. 2009; Centers for Disease Control and Prevention 2008b). Chlamydia trachomatis causes more cases of sexually-transmitted diseases (STDs) than any other bacterial pathogen and so making it a major public health problem throughout the world (World Health Organization 2010). Another important fact about Chlamydia is that infected individuals can acquire re-infection while recovering from the disease (Centers for Disease Control and Prevention 2008a) and often arises in situations where infected individuals have multiple sex partners. Although Chlamydia-related mortality is negligible in comparison to other STDs such as HIV/AIDS, the aforementioned Chlamydia-associated irreversible complications makes Chlamydia a disease of major public health significance (Centers for Disease Control and Prevention 2008a; World Health Organization 2008, 2010).

Human immune system is a collection of organs, special cells and substances that help to protect from infections and some other diseases. Immune system cells and the substances they make travel through the body to protect it from pathogens (germs) causing infections. Pathogens (viruses, bacteria, parasites) are foreign armies as they are not normally found in the body. They try to invade human body to use its resources to serve their own purposes and so they can hurt the body in the process. In fact, people often use the word foreign to describe invading germs or other substances not normally found in the body. The immune system is acting as body’s defense force. It helps keep invading germs out, or helps kill them if they do get into the body. The immune system basically works by keeping track of all of the substances normally found in the body and any new substance in the body that the immune system does not recognize raises an alarm to attack it. Substances that cause an immune system response are known as antigens. The immune response can lead to destruction of anything containing the antigen, such as pathogens. Pathogens (viruses, bacteria, parasites) have substances on their outer surfaces, such as certain proteins, that are not normally found in the human body. The immune system identifies these foreign substances as antigens.

Our body can become immune to pathogens (bacteria, viruses and other germs) in two ways:(i) by getting a disease (called natural immunity), (ii) through vaccines (called vaccine induced immunity). Natural immunity develops after any one has been exposed to a certain pathogen. Our immune system puts into play a complex array of defenses to keep invading germs out, or to help kill them if they do get into the body. Exposure to a foreign invader stimulates production of certain white blood cells in our body called B-cells and these cells produce plasma cells, which in turn produce antibodies designed specifically to attack that particular invader. These antibodies circulate in our body fluids and next time that invader enters our body, the antibodies will recognize it and kill it. Once the body has produced a particular antibody, it rapidly produces more antibodies if needed. In addition to the work of B-cells, other white blood cells known as macrophages confront and kill foreign invaders. If our body encounters a pathogen that it has never been exposed to before, information about the germ is relayed to white blood cells called helper T-cells. These cells help for production of other infection fighting cells, including memory T-cells. Once we have been exposed to a specific virus or bacterium, the next time antibodies and memory T-cells go to fight. They immediately react to the foreign invader, attacking it before disease can develop. Our body immune system can recognize and effectively combat thousands of different pathogens.

Vaccine-induced immunity results after a vaccine is given. The vaccine triggers immune system’s infection-attacking ability and memory without exposure to the actual disease-producing pathogens. A vaccine consists of a killed or weakened form or derivative of the infectious germ. When given to a healthy person, the vaccine triggers an immune response and makes the body to think that it is being invaded by a specific organism. Then the immune system goes to work to destroy the invader and prevent it from infection. If we are exposed to a disease for which we have been vaccinated, the invading germs are met by antibodies that will kill them. The immunity one develops following vaccination is similar to the immunity acquired from natural infection. Several doses of a vaccine may be required for a full immune response. Some people may fail to achieve full immunity to the first doses of a vaccine but respond to later doses. The immune response may decrease over time, one may need another dose of a vaccine (booster shot) to restore or increase immunity.

The pulse vaccination strategy (PVS) consists of repeated application of vaccine at discrete time with equal interval in a population in contrast to the traditional constant vaccination (Gakkhar and Negi 2008; Zhou and Liu 2003). Compared to the proportional vaccination models, the study of pulse vaccination models is in its infancy (Zhou and Liu 2003). At each vaccination time a constant fraction of the susceptible population is vaccinated successfully. Since 1993, attempts have been made to develop mathematical theory to control infectious diseases using pulse vaccination (Agur et al. 1993; Gakkhar and Negi 2008). Nokes and Swinton (1995) discussed the control of childhood viral infections by pulse vaccination strategy. Stone et al. (2000) presented a theoretical examination of the pulse vaccination strategy in the \(SIR\) epidemic model and d’Onofrio (2002a, b) analyzed the use of pulse vaccination policy to eradicate infectious disease for \(SIR\) and \(SEIR\) epidemic models. Different types of vaccination policies and strategies combining pulse vaccination policy, treatment, pre-outbreak vaccination or isolation have already been introduced by several researchers (Babiuk et al. 2002; d’Onofrio 2005; Gao et al. 2006, 2007; Gjorrgjieva et al. 2005; Hui and Chen 2004; Meng et al. 2007; Tang et al. 2005; Wei and Chen 2008; Zhang and Teng 2008).

Mathematical epidemiology is the study of the spread of diseases, in space and time, with the objective to identify factors that are responsible for or contributing to their occurrence. Mathematical models are becoming important tools in analyzing the spread and control of infectious diseases. Epidemic models of ordinary differential equations have been studied by a number of researchers (Anderson and May 1992; Brauer and Castillo-Chavez 2001; Cai et al. 2009; Capasso 1993; Cooke and van Den Driessche 1996; Diekmann and Heesterbeek 2000; Hethcote and van Den Driessche 1991; Kermack and Mckendrick 1927; Ma et al. 2004; Mena-Lorca and Hethcote 1992; Naresh et al. 2006; Ruan and Wang 2003; Thieme 2003; Wang 2002). The basic and important objectives for these models are the existence of the threshold values which distinguish whether the infectious disease will be going to extinct, the local and global stability of the disease-free equilibrium and the endemic equilibrium, the existence of periodic solutions and the persistence of the disease. Stability, persistence and permanence in population biology have been studied by many researchers (Takeuchi et al. 2006a, b). Hence, as a part of population biology, permanence of disease plays an important role in mathematical epidemiology.

Although Chlamydia is a disease of significant public health importance, not much has been analyzed in terms of using mathematical modelling to gain insight into its transmission dynamics at population level (Sharomi and Gumel 2009). In this paper we have used the Kermack-McKendrick compartmental modelling framework, which entails sub-dividing the entire high-risk human population into mutually-exclusive epidemiological compartments (based on disease status), to gain insights into the qualitative features of Chlamydia trachomatis in a human population (with the aim of finding effective ways to control its spread). The main feature of this paper is to introduce valid pulse vaccination strategy. We have introduced two threshold values \(R_{0}\) and \(R_{1}(\le R_{0})\) and further obtained that the disease will be going to extinct when \(R_{0}<1\) and the disease will be permanent when \(R_{1}>1\). The important mathematical findings for the dynamical behaviour of the Chlamydia disease model are numerically verified using MATLAB and also epidemiological implications of our analytical findings are addressed critically in the Sect. 5. The aim of the analysis of this model is to trace the parameters of interest for further study, with a view to informing and assisting policy-maker in targeting prevention and treatment resources for maximum effectiveness.

2 Model Derivation and Preliminaries

In the following, we consider a dynamical model of Chlamydia disease that spread by Chlamydia trachomatis (a type of bacteria) with pulse vaccination strategy (PVS) which satisfies the following assumptions:

The underlying high-risk human population is split up into five mutually-exclusive classes (compartments), namely, susceptible (\(S\)), infective in asymptomatic phase (showing no symptoms of Chlamydia) (\(I_{A}\)), infective in symptomatic phase (showing symptoms of Chlamydia) (\(I_{S}\)), naturally recovered (infectious people who have cleared (or recovered from) Chlamydia infection naturally) (\(R\)) and vaccinated individuals (\(V\)). Here, it is assumed that the recovered individuals acquire the permanent immunity but the vaccinated acquire temporary immunity. So, the natural immunity is permanent but the vaccine-induced immunity is temporary.

The susceptible population increases by the recruitment through new sexually-active individuals, migration and the vaccinated individuals return to the susceptible class (due to immunity waning) and decreases due to direct contact with infected individuals, natural death and pulse vaccination strategy.

Standard epidemiological models use a bilinear incidence rate \(\beta SI\) based on the law of mass action (Anderson and May 1979, 1992) and it is reasonable when the mixing of susceptible with infective is considered to be homogeneous.

The infected class is increased by infection of susceptible and vaccinated individuals. A fraction of the infectious individuals will start to show symptoms of Chlamydia infection (and move to the class \(I_{S}\)), while the remaining fraction will not (but still remain capable of infecting others and move to the class \(I_{A}\)). A fraction of the asymptomatically infectious individuals eventually show disease symptoms and a fraction recovers naturally. The infected class is decreased through natural recovery from infection, by disease-related death and by natural death. It is assumed that natural recovery is possible from both types of infection (Regan et al. 2008).

Thus, the following dynamical model of Chlamydia disease that spread by Chlamydia trachomatis with bilinear incidence and pulse vaccination strategy is formulated:
$$\begin{aligned} \displaystyle \frac{dS(t)}{dt}&= \Lambda -\beta _{1}S(t)I_{A}(t)-\beta _{2}S(t)I_{S}(t) -\mu S(t)+\alpha V(t),\quad t\ne nT, \nonumber \\ \displaystyle \frac{dI_{A}(t)}{dt}&= \rho S(t)\left\{ \beta _{1}I_{A}(t)+\beta _{2}I_{S}(t)\right\} +\rho \sigma V(t)\left\{ \beta _{1}I_{A}(t)+\beta _{2}I_{S}(t)\right\} \nonumber \\&\quad -(r_{1}+d_{1}+\mu )I_{A}(t),\quad t\ne nT, \nonumber \\ \displaystyle \frac{dI_{S}(t)}{dt}&= (1-\rho )S(t)\left\{ \beta _{1}I_{A}(t)+\beta _{2}I_{S}(t)\right\} \nonumber \\&\quad +(1-\rho )\sigma V(t)\left\{ \beta _{1}I_{A}(t)+\beta _{2}I_{S}(t)\right\} \nonumber \\&\quad +(1-k)r_{1}I_{A}(t)-(r_{2}+d_{2}+\mu )I_{S}(t),\quad t\ne nT, \nonumber \\ \displaystyle \frac{dR(t)}{dt}&= kr_{1}I_{A}(t)+r_{2}I_{S}(t)-\mu R(t),\quad t\ne nT, \nonumber \\ \displaystyle \frac{dV(t)}{dt}&= -\sigma V(t)\left\{ \beta _{1}I_{A}(t)+\beta _{2}I_{S}(t)\right\} -(\mu +\alpha )V(t), \quad t\ne nT,\nonumber \\ \displaystyle S(t^{+})&= (1-p)S(t),\quad t=nT, n=1,2,\ldots \nonumber \\ \displaystyle I_{A}(t^{+})&= I_{A}(t),\quad t=nT, n=1,2,\ldots \nonumber \\ \displaystyle I_{S}(t^{+})&= I_{S}(t), \quad t=nT, n=1,2,\ldots \nonumber \\ \displaystyle R(t^{+})&= R(t),\quad t=nT, n=1,2,\ldots \nonumber \\ \displaystyle V(t^{+})&= V(t)+pS(t),\quad t=nT, n=1,2,\ldots , \end{aligned}$$
(2.1)
where all coefficients are nonnegative constants. Here \(S(t)\) denotes the number of susceptible, \(I_{A}(t)\) denotes the number of infective in asymptomatic phase, \(I_{S}(t)\) denotes the number of infective in symptomatic phase, \(R(t)\) denotes the number of recovered individuals and \(V(t)\) denotes the number of vaccinated individuals. The pulse vaccination does not give life-long immunity, there is an immunity waning for the vaccination with the per capita immunity waning rate \(\alpha \), and return to the susceptible class. The influx of susceptible comes from two sources: a constant recruitment \(\Lambda \) and vaccinated hosts \(\alpha V\). The parameters \(\beta _{1},\beta _{2},\sigma ,\mu ,\rho ,d_{1},d_{2},r_{1},r_{2},k,p\) are:
  • \(\beta _{1}\): The coefficient of transmission (Chlamydia infection) rate from infective in asymptomatic phase to susceptible humans and the rate of transmission of infection is of the form: \(\beta _{1}S(t)I_{A}(t).\)

  • \(\sigma \beta _{1}\): The coefficient of transmission (Chlamydia infection) rate from infective in asymptomatic phase to vaccinated individuals and the rate of transmission of infection is of the form: \(\sigma \beta _{1}V(t)I_{A}(t).\)

  • \(\beta _{2}\): The coefficient of transmission (Chlamydia infection) rate from infective in symptomatic phase to susceptible humans and the rate of transmission of infection is of the form: \(\beta _{2}S(t)I_{S}(t).\)

  • \(\sigma \beta _{2}\): The coefficient of transmission (Chlamydia infection) rate from infective in symptomatic phase to vaccinated individuals and the rate of transmission of infection is of the form: \(\sigma \beta _{2}V(t)I_{S}(t).\)

Here \(\beta _{1}>0,\ \beta _{2}>0\) and \(0\le \sigma \le 1\). The fraction \(\sigma \) reflects the effect of reducing the infection rate due to vaccination: \(\sigma =0\) indicates that the vaccine is completely effective in preventing infection, while \(\sigma =1\) indicates that the vaccine is utterly ineffective and so the vaccination need not to be considered when \(\sigma =1\).
  • \(\mu \): The coefficient of natural death rate of all epidemiological human classes.

  • \(d_{1}\): The coefficient of additional disease-related death rate of infective in asymptomatic phase.

  • \(d_{2}\): The coefficient of additional disease-related death rate of infective in symptomatic phase.

  • \((1-\rho )\): The fraction of the infectious individuals that will start to show symptoms of Chlamydia infection and move to the class \(I_{S}\). The remaining fraction \(\rho \ (0<\rho <1)\) will not start to show symptoms of Chlamydia infection (but still remain capable of infecting others) and move to the class \(I_{A}\).

  • \((1-k)r_{1}\): The rate at which the asymptomatically infectious individuals eventually show disease symptoms (move to the class \(I_{S}\)) and recover naturally at the rate \(kr_{1}\ (0<k<1)\) (move to the class \(R\)).

  • \(r_{2}\): The rate at which the infectious individuals showing symptoms of Chlamydia (in symptomatic phase) clear infections naturally and move to the class \(R\).

  • \(p (0<p<1)\): The fraction of susceptible who are vaccinated successfully at discrete time \(t=T,2T,3T,\ldots \), which is called impulsive vaccination rate.

The initial condition of (2.1) is given as
$$\begin{aligned}&S(0^{+})=S^{0},\ I_{A}(0^{+})=I_{A}^{0},\ I_{S}(0^{+})=I_{S}^{0},\ R(0^{+})=R^{0},\ V(0^{+})=V^{0},\ \mathrm{where}\nonumber \\&\displaystyle (S^{0},I_{A}^{0},I_{S}^{0},R^{0},V^{0})\in {\mathbb {R}}^{5}_{+}\equiv \left\{ (x_{1},x_{2},x_{3},x_{4},x_{5})\in {\mathbb {R}}^{5}:x_{i}\ge 0,\ i=1,2,3,4,5\right\} .\nonumber \\ \end{aligned}$$
(2.2)
The total human population size \(N(t)=S(t)+I_{A}(t)+I_{S}(t)+R(t)+V(t)\) can be determined by the following differential equation:
$$\begin{aligned} \frac{dN(t)}{dt}=\Lambda -\mu N(t)-d_{1}I_{A}(t)-d_{2}I_{S}(t), \end{aligned}$$
(2.3)
which is derived by adding first five equations of system (2.1). Therefore,
$$\begin{aligned}&\Lambda -(\mu +d_{1}+d_{2})N(t)\le \frac{dN(t)}{dt}\le \Lambda -\mu N(t) \nonumber \\&\Rightarrow \frac{\Lambda }{\mu +d_{1}+d_{2}}\le \liminf _{t\rightarrow \infty }N(t)\le \limsup _{t\rightarrow \infty }N(t)\le \frac{\Lambda }{\mu }. \end{aligned}$$
(2.4)
There exists a unique solution of (2.1) with initial condition (2.2) since the right hand sides of (2.1) and the pulse are smooth functions (Bainov and Simeonov 1993, 1995; Lakshmikantham et al. 1989).
From biological considerations, we analyze system (2.1) and (2.2) in the closed set:
$$\begin{aligned} G=\left\{ (S(t),I_{A}(t),I_{S}(t),R(t),V(t))\in {\mathbb {R}}^{5}_{+}:0\le S+I_{A}+I_{S}+R+V \le \frac{\Lambda }{\mu }\right\} \end{aligned}$$
(2.5)
and it can be verified that \(G\) is positively invariant with respect to (2.1) and (2.2).

Before starting our main results, we give the following two lemmas which will be essential for study.

Lemma 2.1

Consider the following impulsive differential equation:
$$\begin{aligned}&\displaystyle \frac{du(t)}{dt}=a-bu(t),t\ne nT,\nonumber \\&\displaystyle u(t^{+})=(1-p)u(t),\quad t=nT, n=1,2,\ldots \end{aligned}$$
(2.6)
where\(a>0, \ b>0,\ 0<p<1.\)Then there exists a unique positive periodic solution of system (2.6):
$$\begin{aligned}&\displaystyle \widetilde{u}_{e}(t)=\frac{a}{b}+\left( u^{*}-\frac{a}{b}\right) e^{-b(t-nT)}, \ nT<t\le (n+1)T, \\&\displaystyle where \ u^{*}=\frac{a(1-p)\left( 1-e^{-bT}\right) }{b\left\{ 1-(1-p)e^{-bT}\right\} }, \end{aligned}$$
and\(\widetilde{u}_{e}(t)\)is globally asymptotically stable.

Proof

From the first equation of system (2.6) we get,
$$\begin{aligned} \frac{d}{dt}\left( e^{bt}u(t)\right) =ae^{bt}.\ \hbox {Integrating between pulses:}\ \int _{nT}^{t}d\left( e^{bt}u(t)\right) =\int _{nT}^{t}ae^{bt}dt\\ \Rightarrow u(t)=\frac{a}{b}+\left\{ u(nT)-\frac{a}{b}\right\} e^{-b(t-nT)}, \ nT<t\le (n+1)T, \end{aligned}$$
where \(u(nT)\) is the initial value at time \(nT\). Using the second equation of system (2.6) we have the following stroboscopic map:
$$\begin{aligned} u((n+1)T)=(1-p)\left[ \frac{a}{b}+\left\{ u(nT)-\frac{a}{b}\right\} e^{-bT}\right] =f(u(nT)), \end{aligned}$$
(2.7)
where \(f(u)=(1-p)\left\{ \frac{a}{b}+\left( u-\frac{a}{b}\right) e^{-bT}\right\} \). Solving the following equation:
$$\begin{aligned}&u=(1-p)\left\{ \frac{a}{b}+\left( u-\frac{a}{b}\right) e^{-bT}\right\} ,\ \hbox {we get,}\ u^{*}=\frac{a(1-p)\left( 1-e^{-bT}\right) }{b\left\{ 1-(1-p)e^{-bT}\right\} }. \end{aligned}$$
Since \(\mid f'(u) \mid =(1-p)e^{-bT}<1,\) as \(0<p<1\) and \(b>0\), the system (2.7) has a unique positive equilibrium \(u^{*}=\frac{a(1-p)\left( 1-e^{-bT}\right) }{b\left\{ 1-(1-p)e^{-bT}\right\} }\) which is globally asymptotically stable. Hence the corresponding periodic solution of system (2.6)
$$\begin{aligned} \widetilde{u}_{e}(t)=\frac{a}{b}+\left( u^{*}-\frac{a}{b}\right) e^{-b(t-nT)}, \ nT<t\le (n+1)T, \ \hbox {where} \ u^{*}=\frac{a(1-p)\left( 1-e^{-bT}\right) }{b\left\{ 1-(1-p)e^{-bT}\right\} } \end{aligned}$$
is globally asymptotically stable. This completes the proof. \(\square \)

Lemma 2.2

Lakshmikantham et al. (1989) Assume that the time sequence\(\{t_{j}\}\)satisfies\(0\le t_{0}<t_{1}<t_{2}<t_{3}<\ldots <t_{j}<t_{j+1}<\ldots \ \hbox {with}\ \displaystyle \lim _{j\rightarrow \infty }t_{j}=\infty \). Let\(f(t,x): {\mathbb {R}}_{+}\rightarrow {\mathbb {R}}\)be quasi-monotone nondecreasing in x for each t and\(\phi _{k}(u)\in C[{\mathbb {R}},{\mathbb {R}}]\)is nondecreasing in u for\(k=1,2,3,\ldots \)Suppose that\(u(t),v(t)\in PC([t_{0},\infty ],{\mathbb {R}})\)satisfy
$$\begin{aligned}&\displaystyle D^{+}u(t)\le f(t,u(t)),\ u(t_{k}^{+})\le \phi _{k}(u(t_{k})),\ t\ge t_{0},\ k=1,2,\ldots \\&\displaystyle D^{+}v(t)\ge f(t,v(t)),\ v(t_{k}^{+})\ge \phi _{k}(v(t_{k})),\ t\ge t_{0},\ k=1,2,\ldots . \end{aligned}$$
Then\(u(t_{0})\le v(t_{0})\)implies\(u(t)\le v(t)\)for\(t\ge t_{0}\).

3 Global Stability of the Disease-Free Periodic Solution

In this section, we discuss the existence of the disease-free periodic solution of system (2.1), in which infectious individuals (both in symptomatic and asymptomatic phases) are completely absent, that is, \(I_{A}(t)=0,\ \forall t\ge 0\) and \(I_{S}(t)=0,\ \forall t\ge 0.\) Under this circumstances, system (2.1) reduces to the following impulsive system:
$$\begin{aligned}&\displaystyle \frac{dS(t)}{dt}=\Lambda -\mu S(t)+\alpha V(t),\quad t\ne nT, \nonumber \\&\displaystyle \frac{dR(t)}{dt}=-\mu R(t),\quad t\ne nT,\nonumber \\&\displaystyle \frac{dV(t)}{dt}=-(\mu +\alpha )V(t),\quad t\ne nT,\nonumber \\&\displaystyle S(t^{+})=(1-p)S(t),\quad t=nT, n=1,2,\ldots \nonumber \\&\displaystyle R(t^{+})=R(t),\quad t=nT, n=1,2,\ldots \nonumber \\&\displaystyle V(t^{+})=V(t)+pS(t),\quad t=nT, n=1,2,\ldots . \end{aligned}$$
(3.1)
\(\displaystyle \hbox {From these equations, we have}\ \frac{dN(t)}{dt}=\Lambda -\mu N(t)\Rightarrow \lim _{t\rightarrow \infty }N(t)=\frac{\Lambda }{\mu }.\)

Further, from the second and fifth equations of system (3.1) it follows that \(\displaystyle \lim _{t\rightarrow \infty }R(t)=0.\)

Consider the following limit system of system (3.1) as per the previous discussions:
$$\begin{aligned} V(t)&= \frac{\Lambda }{\mu }-S(t), \nonumber \\ \frac{dS(t)}{dt}&= (\mu + \alpha )\left\{ \frac{\Lambda }{\mu }-S(t)\right\} ,\quad t\ne nT,\nonumber \\ S(t^{+})&= (1-p)S(t),\quad t=nT, n=1,2,\ldots . \end{aligned}$$
(3.2)
Using Lemma 2.1, the periodic solution of system (3.2) is given below:
$$\begin{aligned} \widetilde{S}_{e}(t)&= \frac{\Lambda }{\mu }+\left( S^{*}-\frac{\Lambda }{\mu }\right) e^{-(\mu + \alpha )(t-nT)}, \ nT<t\le (n+1)T, \end{aligned}$$
(3.3)
where
$$\begin{aligned} \ S^{*}=\frac{\Lambda (1-p)\left( 1-e^{-(\mu + \alpha )T}\right) }{\mu \left\{ 1-(1-p)e^{-(\mu + \alpha )T}\right\} } \end{aligned}$$
and \(\widetilde{S}_{e}(t)\) is globally asymptotically stable. Denote \(\displaystyle \widetilde{V}_{e}(t)=\frac{\Lambda }{\mu }-\widetilde{S}_{e}(t)\) and so \(\widetilde{S}_{e}(t)\) and \(\widetilde{V}_{e}(t)\) are independent of initial values. These indicate that the susceptible and vaccinated will approach to \(\widetilde{S}_{e}(t)\) and \(\widetilde{V}_{e}(t)\) respectively when the infective are very few.
Let us define the basic reproduction number as
$$\begin{aligned} R_{0}&= \frac{1}{\theta T}\int _{0}^{T}\left\{ \beta \widetilde{S}_{e}(t)+\sigma \beta \widetilde{V}_{e}(t)\right\} dt \nonumber \\&\Rightarrow R_{0}=\frac{1}{\theta T}\left[ \frac{\beta \Lambda }{\mu }T+\beta (1-\sigma )\left\{ \left( S^{*}-\frac{\Lambda }{\mu }\right) \frac{1}{\mu + \alpha }(1-e^{-(\mu + \alpha )T})\right\} \right] , \end{aligned}$$
(3.4)
where \(\beta =\max \{\beta _{1},\beta _{2}\}\ \hbox {and}\ \theta =\min \{kr_{1}+d_{1}+\mu , r_{2}+d_{2}+\mu \}>0\).

The biological interpretation of \(R_{0}\) is the maximum average number of the secondary infections produced by a typical infective during the entire period of Chlamydia infectiousness.

Theorem 3.1

If\(R_{0}<1\), then the disease-free periodic solution\(\left( \widetilde{S}_{e}(t),0,0,0,\widetilde{V}_{e}(t)\right) \)of system (2.1) with initial conditions (2.2) is globally asymptotically stable.

Proof

Since \(R_{0}<1\), we can choose \(\epsilon >0\) small enough such that
$$\begin{aligned}&\displaystyle \int _{0}^{T}\left\{ \beta \widetilde{S}_{e}(t)+\sigma \beta \widetilde{V}_{e}(t)+(\beta +\sigma \beta )\epsilon -\theta \right\} dt<0, \end{aligned}$$
(3.5)
where \(\beta =\max \{\beta _{1},\beta _{2}\}\ \hbox {and}\ \theta =\min \{kr_{1}+d_{1}+\mu , r_{2}+d_{2}+\mu \}>0\).
From the first and fifth equations of (2.1), it follows that
$$\begin{aligned} \displaystyle \frac{dS(t)}{dt}&= \Lambda -\beta _{1}SI_{A}-\beta _{2}SI_{S} -\mu S+\alpha (N-S-I_{A}-I_{S}-R) \nonumber \\ \displaystyle&\le \Lambda +\frac{\alpha \Lambda }{\mu }-(\mu +\alpha )S \ \hbox {and}\ \frac{dV(t)}{dt}\le -(\mu +\alpha )V. \end{aligned}$$
(3.6)
So, we consider the following comparison impulsive differential system:
$$\begin{aligned}&\displaystyle \frac{dz_{1}(t)}{dt}=\Lambda +\frac{\alpha \Lambda }{\mu }-(\mu +\alpha )z_{1}(t),\quad t\ne nT,\nonumber \\&\displaystyle \frac{dz_{2}(t)}{dt}=-(\mu +\alpha )z_{2}(t), \quad t\ne nT,\nonumber \\&\displaystyle z_{1}(t^{+})=(1-p)z_{1}(t), \quad t=nT, n=1,2,\ldots \nonumber \\&\displaystyle z_{2}(t^{+})=z_{2}(t)+pz_{1}(t), \quad t=nT, n=1,2,\ldots . \end{aligned}$$
(3.7)
Observe that \(\displaystyle \frac{d}{dt}(z_{1}(t)+z_{2}(t))=\Lambda +\frac{\alpha \Lambda }{\mu }-(\mu +\alpha )(z_{1}(t)+z_{2}(t)).\) So, by using Lemma 2.1, we see that the periodic solution of system (3.7) is \((\widetilde{S}_{e}(t),\widetilde{V}_{e}(t))\) which is globally asymptotically stable.
Let \((S(t),I_{A}(t),I_{S}(t),R(t),V(t))\) be the solution of system (2.1) with initial conditions (2.2) and \(S(0^{+})=S^{0}>0,\ V(0^{+})=V^{0}\ge 0\); \((z_{1}(t)+z_{2}(t))\) be the solution of system (3.7) with initial conditions \(z_{1}(0^{+})=S^{0},\ z_{2}(0^{+})=V^{0}\). By Lemma 2.2, there exists an integer \(n_{1}>0\) such that
$$\begin{aligned} S(t)\le z_{1}(t)<\widetilde{S}_{e}(t)+\epsilon ,\ V(t)\le z_{2}(t)<\widetilde{V}_{e}(t)+\epsilon ,\ \forall t\in (nT,(n+1)T],\ n\ge n_{1}. \end{aligned}$$
(3.8)
Moreover, from the second and third equations of system (2.1) and (3.8), we get
$$\begin{aligned}&\frac{d}{dt}(I_{A}(t)+I_{S}(t))\le \left\{ \beta S(t)+\sigma \beta V(t)-\theta \right\} (I_{A}(t)+I_{S}(t)) \nonumber \\&\quad \le \left\{ \beta \widetilde{S}_{e}(t)+\sigma \beta \widetilde{V}_{e}(t)+(\beta +\sigma \beta )\epsilon -\theta \right\} (I_{A}(t)+I_{S}(t)), \end{aligned}$$
(3.9)
where \(\beta =\max \{\beta _{1},\beta _{2}\}\ \hbox {and}\ \theta =\min \{kr_{1}+d_{1}+\mu , r_{2}+d_{2}+\mu \}>0, \hbox {for all}\ t \ge nT\ \hbox {and}\ n>n_{1}\).
Using (3.5) and since \(I_{A}(t)\ge 0,\ I_{S}(t)\ge 0\), we have
$$\begin{aligned} \lim _{t\rightarrow \infty }\left\{ I_{A}(t)+I_{S}(t)\right\} =0 \Rightarrow \lim _{t\rightarrow \infty }I_{A}(t)=\lim _{t\rightarrow \infty }I_{S}(t)=0. \end{aligned}$$
(3.10)
Hence for any \(\epsilon _{1}>0\) (sufficiently small), there exists a positive integer \(n_{2}\ge n_{1}\), such that \(\displaystyle 0<I_{A}(t),I_{S}(t)<\epsilon _{1},\ \forall t>n_{2}T.\) From system (2.1), we have
$$\begin{aligned}&\displaystyle \frac{dN(t)}{dt}>\Lambda -\mu N(t)-(d_{1}+d_{2})\epsilon _{1},\ \forall t> n_{2}T, \nonumber \\&\displaystyle \frac{dR(t)}{dt}<(kr_{1}+r_{2})\epsilon _{1}-\mu R(t),\ \forall t>n_{2}T. \end{aligned}$$
(3.11)
So, by the comparison theorem, there exists an integer \(n_{3}\ge n_{2}\) such that
$$\begin{aligned}&\displaystyle N(t)\ge \frac{\Lambda -(d_{1}+d_{2})\epsilon _{1}}{\mu }-\epsilon _{1},\ R(t)\le \frac{(kr_{1}+r_{2})\epsilon _{1}}{\mu }+\epsilon _{1},\ \forall t>n_{3}T \nonumber \\&\displaystyle \Rightarrow \lim _{t\rightarrow \infty }N(t)=\frac{\Lambda }{\mu },\ \lim _{t\rightarrow \infty }R(t)=0, \end{aligned}$$
(3.12)
as \(\epsilon _{1}>0\ \hbox {is arbitrarily small and}\ \limsup _{t\rightarrow \infty }N(t)\le \frac{\Lambda }{\mu }\).
Therefore, from the first equation of system (2.1), we get
$$\begin{aligned} \frac{dS(t)}{dt}&= \Lambda -\beta _{1}SI_{A}-\beta _{2}SI_{S}-\mu S+\alpha (N-S-I_{A}-I_{S}-R) \nonumber \\&\ge \Lambda +\frac{\alpha \Lambda }{\mu }-K\epsilon _{1}-(\mu +\alpha )S,\ \forall t>n_{3}T, \end{aligned}$$
(3.13)
where \(K=2\beta \frac{\Lambda }{\mu }+\alpha \left\{ \frac{1}{\mu }(d_{1}+d_{2}+kr_{1}+r_{2})+4\right\} \) and \(\beta =\max \{\beta _{1},\beta _{2}\}\).
Let us consider the following comparison impulsive differential system:
$$\begin{aligned}&\displaystyle \frac{dz_{3}(t)}{dt}=\left( \Lambda +\frac{\alpha \Lambda }{\mu }-K\epsilon _{1}\right) -(\mu +\alpha )z_{3}(t),\ t\ne nT,\nonumber \\&\displaystyle z_{3}(t^{+})=(1-p)z_{3}(t),\ t=nT, n=1,2,\ldots . \end{aligned}$$
(3.14)
By Lemma 2.1, we know that the periodic solution of system (3.14) is
$$\begin{aligned}&\displaystyle \widetilde{z}_{3e}(t)=\Phi +\left( z_{3}^{*}-\Phi \right) e^{-(\mu +\alpha )(t-nT)}, \ nT<t\le (n+1)T, \end{aligned}$$
(3.15)
where \(\Phi =\frac{\Lambda +\frac{\alpha \Lambda }{\mu }-K\epsilon _{1}}{\mu +\alpha }\ \hbox {and} \ z_{3}^{*}=\Phi \frac{(1-p)\left( 1-e^{-(\mu +\alpha )T}\right) }{\left\{ 1-(1-p)e^{-(\mu +\alpha )T}\right\} }\), which is globally asymptotically stable.
By the comparison theorem for impulsive differential equation (Lakshmikantham et al. 1989), there exists an integer \(n_{4}>n_{3}\) such that
$$\begin{aligned} S(t)>\widetilde{z}_{3e}(t)-\epsilon _{1},\ nT<t\le (n+1)T, \ n>n_{4}. \end{aligned}$$
(3.16)
Making \(\epsilon \rightarrow 0\) and \(\epsilon _{1}\rightarrow 0\), it follows from (3.8), (3.10), (3.12) and (3.16) that
$$\begin{aligned} \lim _{t\rightarrow \infty }S(t)=\widetilde{S}_{e}(t)\ \hbox {and} \ \lim _{t\rightarrow \infty }V(t)=\widetilde{V}_{e}(t). \end{aligned}$$
(3.17)
Therefore, from (3.10), (3.12) and (3.17) we conclude that if \(R_{0}<1\), then the disease-free periodic periodic solution \(\left( \widetilde{S}_{e}(t),0,0,0,\widetilde{V}_{e}(t)\right) \) of system (2.1) with initial conditions (2.2) is globally asymptotically stable. This completes the proof. \(\square \)
Let us find the critical vaccination proportion, i.e., the value \(p=p^{\star }\) such that \(R_{0}(p^{\star })=1\), where \(R_{0}(p^{\star })\) is the value for \(R_{0}\) in which \(p\) is replaced by \(p^{\star }\).
$$\begin{aligned}&\displaystyle \therefore \; R_{0}(p^{\star })=1\Rightarrow p^{\star }=\frac{T(\mu +\alpha )\left( \beta \frac{\Lambda }{\mu }-\theta \right) }{\beta (1-\sigma )\frac{\Lambda }{\mu }-e^{-(\mu +\alpha )T}\left\{ \frac{T(\mu +\alpha )\left( \beta \frac{\Lambda }{\mu }-\theta \right) }{1-e^{-(\mu +\alpha )T}}\right\} }, \end{aligned}$$
(3.18)
where \(\beta =\max \{\beta _{1},\beta _{2}\}\ \hbox {and}\ \theta =\min \{kr_{1}+d_{1}+\mu , r_{2}+d_{2}+\mu \}>0\).

From (3.3) and (3.4), it is easily seen that \(\displaystyle \frac{dR_{0}}{dp}<0\) and so \(R_{0}<1\) only when \(p>p^{\star }\). Using Theorem 3.1, we have the following result:

Corollary 3.1

The disease-free periodic solution\(\left( \widetilde{S}_{e}(t),0,0,0,\widetilde{V}_{e}(t)\right) \)of system (2.1) with initial condition (2.2) is globally asymptotically stable when\(p>p^{\star }\).

So, the Chlamydia disease will disappear, i.e., the infectious population extinct if the pulse vaccination rate is larger than \(p^{\star }\). Thus we theoretically prove that the pulse vaccination is an effective strategy in preventing the Chlamydia disease.

4 Permanence of the Disease

In this section, we wish to discuss the permanence of the disease of the system (2.1), this means that the long-term survival (i.e., will not extinct as time goes) of the total numbers of infectious population (\(I_{A}(t)+I_{S}(t)\)) of system (2.1) with initial condition (2.2). It demonstrates how the disease will be permanent (i.e., will not vanish as time goes) under some conditions.

Definition

The disease in system (2.1) is said to be permanent, i.e., the long-term survival (will not extinct as time goes) of the total numbers of infectious population (\(I_{A}(t)+I_{S}(t)\)) of system (2.1) with initial condition (2.2), if there exists a positive constant \(m\) such that \(\displaystyle \liminf _{t\rightarrow \infty }\left\{ I_{A}(t)+I_{S}(t)\right\} \ge m\) holds for any solution \((S(t),I_{A}(t),I_{S}(t),R(t),V(t))\) of (2.1) with initial condition (2.2).

Theorem 4.1

If\(R_{1}>1\), then the Chlamydia disease in system (2.1) is permanent, where
$$\begin{aligned} \displaystyle R_{1}&= \frac{1}{\theta ' T}\int _{0}^{T}\left\{ \beta ' \widetilde{S}_{e}(t)+\sigma \beta ' \widetilde{V}_{e}(t)\right\} dt \nonumber \\ \displaystyle&\Rightarrow R_{1}=\frac{1}{\theta ' T}\left[ \frac{\beta ' \Lambda }{\mu }T+\beta '(1-\sigma )\left\{ \left( S^{*}-\frac{\Lambda }{\mu }\right) \frac{1}{\mu + \alpha }(1-e^{-(\mu + \alpha )T})\right\} \right] ; \nonumber \\ \displaystyle \beta '&= \min \{\beta _{1},\beta _{2}\}\ \hbox {and}\ \theta '=\max \{kr_{1}+d_{1}+\mu , r_{2}+d_{2}+\mu \}>0. \end{aligned}$$
(4.1)

Proof

We first prove that,
$$\begin{aligned} \limsup _{t\rightarrow \infty }\{I_{A}(t)+I_{S}(t)\}\ge \epsilon , \end{aligned}$$
(4.2)
where \(\epsilon >0\) is small enough. Suppose (4.2) is not true, then there exists a \(T_{1}\ge 0\), such that \(\displaystyle 0<I_{A}(t)+I_{S}(t)<\epsilon ,\ \forall t\ge T_{1}.\) From system (2.1), we have
$$\begin{aligned}&\displaystyle \frac{dN(t)}{dt}>\Lambda -\mu N(t)-(d_{1}+d_{2})\epsilon ,\ \forall t\ge T_{1}, \nonumber \\&\displaystyle \frac{dR(t)}{dt}<(kr_{1}+r_{2})\epsilon -\mu R(t),\ \forall t\ge T_{1}. \end{aligned}$$
(4.3)
So, by the comparison theorem, there exists a \(T_{2}\ge T_{1}\) such that
$$\begin{aligned} N(t)\ge \frac{\Lambda -(d_{1}+d_{2})\epsilon }{\mu }-\epsilon ,\ R(t)\le \frac{(kr_{1}+r_{2})\epsilon }{\mu }+\epsilon ,\ \forall t\ge T_{2}. \end{aligned}$$
(4.4)
Therefore, from the first equation of system (2.1), we get
$$\begin{aligned}&\displaystyle \frac{dS(t)}{dt}=\Lambda -\beta _{1}SI_{A}-\beta _{2}SI_{S}-\mu S+\alpha (N-S-I_{A}-I_{S}-R) \nonumber \\&\displaystyle \ge \Lambda +\frac{\alpha \Lambda }{\mu }-K_{1}\epsilon -(\mu +\alpha )S,\ \forall t\ge T_{2}, \nonumber \\ \end{aligned}$$
(4.5)
where \(K_{1}=2\beta \frac{\Lambda }{\mu }+\alpha \left\{ \frac{1}{\mu }(d_{1}+d_{2}+kr_{1}+r_{2})+4\right\} \ \hbox {and}\ \beta =\max \{\beta _{1},\beta _{2}\}\).
Let us consider the following comparison impulsive differential system:
$$\begin{aligned}&\displaystyle \frac{dz_{4}(t)}{dt}=\left( \Lambda +\frac{\alpha \Lambda }{\mu }-K_{1}\epsilon \right) -(\mu +\alpha )z_{4}(t),\ t\ne nT,\nonumber \\&\displaystyle z_{4}(t^{+})=(1-p)z_{4}(t),\ t=nT, n=1,2,\ldots . \end{aligned}$$
(4.6)
By Lemma 2.1, we know that the periodic solution of system (4.6) is
$$\begin{aligned}&\displaystyle \widetilde{z}_{4e}(t)=\Phi +\left( z_{4}^{*}-\Phi \right) e^{-(\mu +\alpha )(t-nT)}, \ nT<t\le (n+1)T, \end{aligned}$$
(4.7)
where \(\Phi =\frac{\Lambda +\frac{\alpha \Lambda }{\mu }-K_{1}\epsilon }{\mu +\alpha }\ \hbox {and} \ z_{4}^{*}=\Phi \frac{(1-p)\left( 1-e^{-(\mu +\alpha )T}\right) }{\left\{ 1-(1-p)e^{-(\mu +\alpha )T}\right\} }\), which is globally asymptotically stable.
By the comparison theorem for impulsive differential equation (Lakshmikantham et al. 1989), there exists an integer \(n_{1}\) such that
$$\begin{aligned} S(t)>\widetilde{z}_{4e}(t)-\epsilon ,\ nT<t\le (n+1)T, \ \forall n\ge n_{1}. \end{aligned}$$
(4.8)
On the other hand, using (3.6), the following system:
$$\begin{aligned}&\displaystyle \frac{dS(t)}{dt}\le \left( \Lambda +\frac{\alpha \Lambda }{\mu }\right) -(\mu +\alpha )S(t),\ t\ne nT,\nonumber \\&\displaystyle S(t^{+})=(1-p)S(t),\ t=nT, n=1,2,\ldots . \end{aligned}$$
(4.9)
implies that there exists an integer \(n_{2}\) such that
$$\begin{aligned} S(t)<\widetilde{S}_{e}(t)+\epsilon ,\ nT<t\le (n+1)T, \ n\ge n_{2}. \end{aligned}$$
(4.10)
Let us choose \(T_{3}=\max \{T_{2},n_{1}T,n_{2}T\}\), than
$$\begin{aligned}&\displaystyle V(t)=N(t)-S(t)-I_{A}(t)-I_{S}(t)-R(t)\ge \widetilde{V}_{e}(t)-K_{2}\epsilon , \end{aligned}$$
(4.11)
where \(K_{2}=\frac{1}{\mu }(d_{1}+d_{2}+kr_{1}+r_{2})+5,\ \forall t\ge T_{3}\).
From the second and third equations of system (2.1), (4.8) and (4.11), we get
$$\begin{aligned} \frac{d}{dt}(I_{A}(t)+I_{S}(t))&\ge \left\{ \beta ' S(t)+\sigma \beta ' V(t)-\theta '\right\} (I_{A}(t)+I_{S}(t)) \nonumber \\&\ge \left\{ \beta ' \widetilde{z}_{4e}(t)+\sigma \beta ' \widetilde{V}_{e}(t)-\beta '(1+\sigma K_{2})\epsilon -\theta '\right\} (I_{A}(t)+I_{S}(t)), \end{aligned}$$
(4.12)
where \(\beta '=\min \{\beta _{1},\beta _{2}\}\ \hbox {and}\ \theta '=\max \{kr_{1}+d_{1}+\mu , r_{2}+d_{2}+\mu \}>0, \hbox {for all}\ t \ge T_{3}\).
Integrating (4.12) from \(T_{3}\) to \(t\), we get
$$\begin{aligned} I(t)\ge I(T_{3})\exp \left[ \int _{T_{3}}^{t}\left\{ \beta ' \widetilde{z}_{4e}(t)+\sigma \beta ' \widetilde{V}_{e}(t)-\beta '(1+\sigma K_{2})\epsilon -\theta '\right\} dt\right] . \end{aligned}$$
(4.13)
From (3.3) and (4.7), it follows that \(\widetilde{z}_{4e}(t)\) uniformly tends to \(\widetilde{S}_{e}(t)\) as \(\epsilon \rightarrow 0^{+}\).
Since \(R_{1}>1\), we can choose \(\epsilon >0\) small enough such that
$$\begin{aligned} \int _{0}^{T}\left\{ \beta ' \widetilde{z}_{4e}(t)+\sigma \beta ' \widetilde{V}_{e}(t)-\beta '(1+\sigma K_{2})\epsilon -\theta '\right\} dt>0. \end{aligned}$$
(4.14)
From (4.13) and (4.14), it follows that \(\{I_{A}(t)+I_{S}(t)\}\rightarrow \infty \ \hbox {as}\ t\rightarrow \infty \). This is a contradiction and hence the result (4.2) is true.
In the following, we will prove that there is a constant \(m>0\) such that
$$\begin{aligned} \liminf _{t\rightarrow \infty }I(t)\ge m,\ \hbox {where}\ I(t)=I_{A}(t)+I_{S}(t), \end{aligned}$$
(4.15)
for any solution \((S(t),I_{A}(t),I_{S}(t),R(t),V(t))\) of system (2.1) with initial condition \(S(0^{+})>0,\ I_{A}(0^{+})>0,\ I_{S}(0^{+})>0,\ R(0^{+})\ge 0,\ V(0^{+})\ge 0\).
In fact, from (4.14), we can obtain two positive constants \(P_{1}\) and \(P_{2}\) such that
$$\begin{aligned} \int _{t}^{t+\eta }\left\{ \beta ' \widetilde{z}_{4e}(t)+\sigma \beta ' \widetilde{V}_{e}(t)-\beta '(1+\sigma K_{2})\epsilon -\theta '\right\} dt>P_{2},\ \hbox {for all}\ t\ge 0 \ \hbox {and}\ \eta \ge P_{1}. \end{aligned}$$
(4.16)
If (4.15) is not true, then there is a sequence of initial values \(X_{n}=(S_{n},I_{An},I_{Sn},R_{n},V_{n})\in {\mathbb {R}}^{5}_{+}\ (n=1,2,\ldots )\) such that
$$\begin{aligned} \liminf _{t\rightarrow \infty }I(t,X_{n})< \frac{\epsilon }{n^{2}},\ n=1,2,\ldots \end{aligned}$$
(4.17)
From (4.2), for every \(n\) there are two time sequences \(\{t_{j}^{(n)}\}\) and \(\{s_{j}^{(n)}\}\) satisfying
$$\begin{aligned}&\displaystyle 0<s_{1}^{(n)}<t_{1}^{(n)}<s_{2}^{(n)}<t_{2}^{(n)}<\ldots <s_{j}^{(n)}<t_{j}^{(n)}<\ldots \ \hbox {and}\ \lim _{j\rightarrow \infty }s_{j}^{(n)}=\infty , \end{aligned}$$
(4.18)
such that \(I(s_{j}^{(n)},X_{n})=\frac{\epsilon }{n},\ I(t_{j}^{(n)},X_{n})=\frac{\epsilon }{n^{2}} \,\hbox {and}\ \frac{\epsilon }{n^{2}}<I(t,X_{n})<\frac{\epsilon }{n},\ \forall t\in (s_{j}^{(n)},t_{j}^{(n)})\). From the second and third equations of system (2.1), we get
$$\begin{aligned}&\displaystyle \frac{d}{dt}I(t,X_{n})\ge -\theta ' I(t,X_{n})\Rightarrow I(t_{j}^{(n)},X_{n})\ge I(s_{j}^{(n)},X_{n})\exp \{-\theta '( t_{j}^{(n)}-s_{j}^{(n)})\} \nonumber \\&\displaystyle \Rightarrow \frac{\epsilon }{n^{2}}\ge \frac{\epsilon }{n}\exp \{-\theta '( t_{j}^{(n)}-s_{j}^{(n)})\}\Rightarrow t_{j}^{(n)}-s_{j}^{(n)}\ge \frac{\ln n}{\theta '}\rightarrow \infty \ \hbox {as}\ t\rightarrow \infty , \end{aligned}$$
(4.19)
where \(\theta '=\max \{kr_{1}+d_{1}+\mu , r_{2}+d_{2}+\mu \}>0\).

From (4.8) and (4.11), we conclude that there exists a \(T^{\star }(>0)\) which is independent of any \(n\) and \(j\) such that \(S(t)\ge \widetilde{z}_{4e}(t)-\epsilon \) and \(V(t)\ge \widetilde{V}_{e}(t)-K_{2}\epsilon \) for all \(t\in (s_{j}^{(n)}+T^{\star },t_{j}^{(n)})\), provided \(t_{j}^{(n)}-s_{j}^{(n)}> T^{\star }\). Let us choose \(n\) large enough to make \(t_{j}^{(n)}-s_{j}^{(n)}> T^{\star }+P_{1}\).

So, finally we have
$$\begin{aligned} \frac{\epsilon }{n^{2}}&= I(t_{j}^{(n)},X_{n}) \nonumber \\&\ge I(s_{j}^{(n)}+T^{\star },X_{n})\exp \left[ \int _{s_{j}^{(n)}+T^{\star }}^{t_{j}^{(n)}}\left\{ \beta ' S(t)+\sigma \beta ' V(t)-\theta '\right\} dt\right] \nonumber \\&\ge \frac{\epsilon }{n^{2}} \exp \left[ \int _{s_{j}^{(n)}+T^{\star }}^{t_{j}^{(n)}}\left\{ \beta ' \widetilde{z}_{4e}(t)+\sigma \beta ' \widetilde{V}_{e}(t)-\beta '(1+\sigma K_{2})\epsilon -\theta '\right\} dt\right] >\frac{\epsilon }{n^{2}}. \end{aligned}$$
(4.20)
This lead to a contradiction and so we finally prove that the inequality (4.15) is true. Hence if \(R_{1}>1\), then the Chlamydia disease in system (2.1) is permanent. This completes the proof. \(\square \)
Let us find the critical vaccination proportion, i.e., the value \(p=p_{\star }\) such that \(R_{1}(p_{\star })=1\), where \(R_{1}(p_{\star })\) is the value for \(R_{1}\) in which \(p\) is replaced by \(p_{\star }\).
$$\begin{aligned}&\displaystyle R_{1}(p_{\star })=1\Rightarrow p_{\star }=\frac{T(\mu +\alpha )\left( \beta '\frac{\Lambda }{\mu }-\theta '\right) }{\beta '(1-\sigma )\frac{\Lambda }{\mu }-e^{-(\mu +\alpha )T}\left\{ \frac{T(\mu +\alpha )\left( \beta '\frac{\Lambda }{\mu }-\theta '\right) }{1-e^{-(\mu +\alpha )T}}\right\} }, \end{aligned}$$
(4.21)
where \(\beta '=\min \{\beta _{1},\beta _{2}\}\ \hbox {and}\ \theta '=\max \{kr_{1}+d_{1}+\mu , r_{2}+d_{2}+\mu \}>0\).

From (3.3) and (4.1), it is easily seen that \(\displaystyle \frac{dR_{1}}{dp}<0\) and so \(R_{1}>1\) only when \(p<p_{\star }\). Using Theorem 4.1, we have the following result:

Corollary 4.1

The Chlamydia disease in system (2.1) is permanent when\(p<p_{\star }.\)

It indicates that a small pulse vaccination rate will give rise the permanence of the disease, that is to say, the disease spreads around and generates an endemic ultimately.

Note 1

The biological interpretation of \(R_{1}\) is the minimum average number of the secondary infections produced by a typical infective during the entire period of Chlamydia infectiousness.

Note 2

From (3.4) and (4.1) we observe that \(R_{0}=R_{1}\) and so \(p^{\star }=p_{\star }\) when \(\beta _{1}=\beta _{2}\) and \(kr_{1}+d_{1}+\mu =r_{2}+d_{2}+\mu \). While natural recovery happens at the same per-capita recovery rate from both types of infections, the identical coefficients of transmission (Chlamydia infection) rates occur from infective of both types of infections and disease-related death rates are same from both types of infections, then \(R_{0}=R_{1}\) (is called the basic reproductive number).

Note 3

From (3.4) and (4.1) we also observe that \(R_{0}\ge R_{1}\) and so eradication of the Chlamydia disease need to make the vaccination rate at least attain the critical value \(p^{\star }\) given by (3.18).

Note 4

Unfortunately, we cannot establish the comparison of \(R_{0}\) and \(R_{1}\). When \(R_{1}\le 1\le R_{0}\), the dynamical behaviour of model (2.1) and (2.2) has not been clear. These works will be left as a part of our future consideration.

5 Numerical Simulations and Biological Interpretations

We first consider the case when \(R_{0}=0.6296<1\) using the parameter values given in Table 1. Using these parameter values, the movement paths of \(S(t), I_A(t), I_S(t), R(t)\) and \(V(t)\) are presented in Fig. 1a. This figure shows that the Chlamydia disease dies out when \(R_0<1,\) which supports our analytical result given in Theorem 3.1. Its epidemiological implication is that the infectious population vanishes, i.e., the disease dies out when \(R_0<1\) (see Fig. 1a). In Fig. 1b, \(R_{1}=3.0243>1\) using the parameter values given in Table 1. Using these parameter values, the movement paths of \(S(t), I_A(t), I_S(t), R(t)\) and \(V(t)\) are presented in Fig. 1b. This figure shows that the disease will be permanent when \(R_1>1,\) which supports our analytical result given in Theorem 4.1.

We also consider the case when \(R_{0}=1.2660>1\) and \(R_1=0.2978<1\) with parameter values given in Table 1. Using these parameter values, the movement paths of \(S(t), I_A(t), I_S(t), R(t)\) and \(V(t)\) are presented in Fig. 2a. This figure shows that the Chlamydia disease dies out. For \(R_0=1.8365>1\) and \(R_1=0.4155<1\) with parameter values are given in Table 1, the movement paths of \(S(t), I_A(t), I_S(t), R(t)\) and \(V(t)\) are presented in Fig. 2b. This figure shows that the disease is still permanent.
Table 1

Parameter values for Figures 1(a, b), 2(a, b)

Parameter

Value for Fig. 1a

Value for Fig. 1b

Value for Fig. 2a

Value for Fig. 2b

\(\Lambda \)

0.1

0.1

0.2

0.2

\(\beta _1\)

0.02

0.04

0.04

0.04

\(\beta _2\)

0.01

0.02

0.02

0.02

\(\sigma \)

0.6

0.4

0.4

0.4

\(\mu \)

0.01

0.01

0.01

0.01

\(\alpha \)

0.2

0.2

0.2

0.2

\(\rho \)

0.4

0.4

0.4

0.4

\(r_1\)

0.4

0.04

0.04

0.004

\(r_2\)

0.2

0.02

0.02

0.002

\(d_1\)

0.2

0.02

0.9

0.09

\(d_2\)

0.15

0.015

0.4

0.07

\(k\)

0.1

0.1

0.1

0.1

\(p\)

0.8

0.8

0.8

0.1

\(T\)

5

5

5

5

https://static-content.springer.com/image/art%3A10.1007%2Fs10441-014-9234-8/MediaObjects/10441_2014_9234_Fig1_HTML.gif
Fig. 1

Movement paths of \(S(t), I_A(t), I_S(t), R(t)\) and \(V(t)\)a for \(R_{0}=0.6296<1\) with parameter values given in Table 1, b for \(R_{1}=3.0243>1\) with parameter values given in Table 1

https://static-content.springer.com/image/art%3A10.1007%2Fs10441-014-9234-8/MediaObjects/10441_2014_9234_Fig2_HTML.gif
Fig. 2

Movement paths of \(S(t), I_A(t), I_S(t), R(t)\) and \(V(t)\)a for \(R_{0}=1.2660>1\) and \(R_1=0.2978<1\) with parameter values given in Table 1, b for \(R_0=1.8365>1\) and \(R_1=0.4155<1\) with parameter values given in Table 1

Remark

When \(R_{1}\le 1\le R_{0}\), the dynamical behaviour of model (2.1) and (2.2) has not been clear. So, in this case the fate of the disease is not clear.

6 Conclusions

In this paper we have considered a dynamical model of Chlamydia diseases with bilinear incidence rate and pulse vaccination strategy. The entire high-risk human population is split up into five mutually-exclusive epidemiological compartments (based on disease status), namely, susceptible (\(S\)), infective in asymptomatic phase (showing no symptoms of Chlamydia) (\(I_{A}\)), infective in symptomatic phase (showing symptoms of Chlamydia) (\(I_{S}\)), naturally recovered (infectious people who have cleared (or recovered from) Chlamydia infection naturally) (\(R\)) and vaccinated individuals (\(V\)). It is assumed that the recovered individuals acquire the permanent immunity but the vaccinated acquire temporary immunity. So, the natural immunity is permanent but the vaccine-induced immunity is temporary. The susceptible population increases by the recruitment through new sexually-active individuals, migration and vaccinated hosts and decreases due to direct contact with infected individuals, natural death and pulse vaccination strategy. The infected class is increased by infection of susceptible. A fraction of the infectious individuals will start to show symptoms of Chlamydia infection (and move to the class \(I_{S}\)), while the remaining fraction will not (but still remain capable of infecting others and move to the class \(I_{A}\)). A fraction of the asymptomatically infectious individuals eventually show disease symptoms and a fraction recovers naturally. The infected class is decreased through natural recovery from infection, by disease-related death and by natural death. It is assumed that natural recovery is possible from both types of infection (Regan et al. 2008). The most basic and important questions to ask for the systems in the theory of mathematical epidemiology are the persistence, extinctions, the existence of periodic solutions, global stability, etc. Here, we have established some sufficient conditions on the permanence and extinction of the disease by using inequality analytical techniques. We have introduced two threshold values \(R_{0}\) and \(R_{1}(\le R_{0})\) and further obtained that the disease will be going to extinct when \(R_{0}<1\) and the disease will be permanent when \(R_{1}>1\). If natural recovery happens at the same per-capita recovery rate from both types of infection, the identical coefficient of transmission (Chlamydia infection) rate occurs from infective from both types of infection and disease-related death rate are same from both types of infection, then \(R_{0}=R_{1}\) (is called the basic reproductive number). The important mathematical findings for the dynamical behaviour of the Chlamydia disease model are also numerically verified using MATLAB. It is observed that when \(R_{1}\le 1\le R_{0}\), the dynamical behaviour of the infectious disease is not clear. The aim of the analysis of this model is to trace the parameters of interest for further study, with a view to informing and assisting policy-maker in targeting prevention and treatment resources for maximum effectiveness.

Acknowledgments

The author is grateful to the anonymous referees and the Editor-in-Chief (Dr. Diedel Kornet, Ph.D.) for their careful reading, valuable comments and helpful suggestions, which have helped him to improve the presentation of this work significantly. He likes to thank TWAS, UNESCO and National Autonomous University of Mexico (UNAM) for financial support. He is grateful to Prof. Javier Bracho Carpizo, Prof. Marcelo Aguilar and Prof. Ricardo Gomez Aiza, Institute of Mathematics, National Autonomous University of Mexico for their helps and encouragements.

Copyright information

© Springer Science+Business Media Dordrecht 2014