Global results for a cholera model with imperfect vaccination

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Abstract

In this paper, we consider a cholera model with imperfect vaccination. We can calculate the infection threshold (i.e., the basic reproduction number Rv) for the cholera model. The disease-free equilibrium of the system is globally asymptotically stable when Rv1. If Rv>1, the disease persists and the unique endemic equilibrium is globally asymptotically stable in the interior of the feasible region under some conditions, which is obtained by compound matrices and geometric approaches. We also perform the effect of vaccination on the disease transmission and prevalence.

Introduction

Cholera is a serious infectious disease caused by the bacteria Vibrio cholera, which affects the intestinal system of the body. Vibrio cholera 01 and 0139 serotypes have been considered the etiologic agents of epidemic cholera [23]. Vibrio cholera is typically found in water environments such as freshwater lakes and rivers. Vibrio cholera accumulates and begins to produce toxins when an adequate quantity of the bacteria has passed into the stomach by eating food or drinking water. It's the toxin that causes the symptoms of the cholera. The main symptoms are profuse watery diarrhea and vomiting.

Cholera has killed millions of people since it emerged out of the filthy water and living conditions of Calcutta India in the early 1800's. Since then, there have been a total of seven cholera pandemic. The first cholera pandemic of 1817–1823 spread from India to Southeast Asia, Central Asia, the Middle East and Russia leaving hundreds of thousands of people dead in its wake. The seventh cholera pandemic, which began in 1961 in Celebes, Indonesia, spread to Bangladesh in 1963 and the following year reached India.

Now the control of deadly outbreaks remains a challenge. The number of cholera cases reported to World Health Organization (WHO) continues to rise. From 2004 to 2008, cases increased by 24% compared with the period from 2000 to 2004. For 2008 alone, a total of 190,130 cases were notified from 56 countries, including 5143 deaths. Many more cases were unaccounted for due to limitations in surveillance systems and fear of trade and travel sanctions. The true burden of the disease is estimated to be 3–5 million cases and 100,000–120,000 deaths annually [29]. In December 2006, a large outbreak hit the Republic of Congo, affecting 7098 people including 101 deaths over a period of 5 months [30]. Between 1 January and 31 December 2006, a total of 14,297 cases including 254 deaths were reported in the United Republic of Tanzania [30]. Since mid-August 2008 and as of 30 July 2009, 98,592 cases including 4288 deaths have been reported from all 10 provinces in Zimbabwe [30]. In developing countries, cholera is prevalent in areas that do not enjoy sanitary living conditions because of poverty and a lack of resources. Cholera in 2007–2009 has been reported by the WHO as occurring mainly in Africa and Asia [31] (See Fig. 1).

Mathematical biology is an exciting and fast growing field. Most of the current topics of mathematical biology consist of the formulation and analysis of various mathematical models, often in the forms of difference equations or differential equations. In recent years, models in the area of mathematical biology are found in many references, for example, prey–predator models [16], [22], [32], epidemic models [8], [9], eco-epidemiological models [33], [34], etc.

Recently, a number of mathematical models have been developed to help in understanding the dynamics of cholera outbreaks. In 1979, Capasso and Paveri-Fontana [1] presented a mathematical model for the 1973 cholera epidemic in the port city of Bari (a city in Italy). In Capasso's version, two equations describe the dynamics of infected people in the community and the dynamics of the aquatic population of pathogenic bacteria. In 2001, Codeco [2] extended the model of Capasso and Paveri-Fontana. He added an equation for the dynamics of the susceptible population. And he studied the role of the aquatic reservoir in the endemic and epidemic dynamics of cholera. In [21], Pascual et al. generalized Codeco's model by including a fourth equation for the volume of water in which the formative live following Codeco [2]. In 2009, Richard I. Joh et al. considered the dynamics of infectious diseases for which the primary mode of transmission is indirect and mediated by contact with a contaminated reservoir [11]. In [18], Rachael L. Miller Neilan et al. formulated a mathematical model to include essential components such as a hyperinfectious, short-lived bacterial state, a separate class for mild human infections, and waning disease immunity. In [24], Senelani D. Hove-Musekwa et al. presented a deterministic model for cholera in a community which is rigorously analyzed in order to determine the effects of malnutrition in the spread of the disease. To the best of our knowledge, these studies do not explicitly consider a deterministic compartmental model with vaccination.

Cholera vaccine is a vaccine used against cholera. Cholera vaccine is a suspension of two strains of killed cholera bacteria in saline solution. Phenol is added as a preservative. Cholera vaccine is about 50% effective in preventing disease. Therefore, there are some argue that a more effective vaccination is needed. However, the cholera vaccine does provide immunization at some level, and has been proven to be effective at helping people resist a cholera infection when visiting foreign nations. That is to say, persons who are at an increased for cholera infection (for example, travelers to a cholera endemic country, persons living in condition where sanitation is poor and the risk of cholera is high, refugees of a camp where the possible outbreak of cholera is high, etc.) should be vaccinated. There are two types of cholera vaccines: parenteral vaccines and oral vaccines. Parenteral vaccine is prepared by growing phenol-killed strains of V. cholera on trypticase soy agar and harvested with isotonic sodium chloride solution. To date, three oral cholera vaccines (i.e., WC/rBS vaccine, Variant WC/rBS vaccine and CVD 103-HgR vaccine) are available, which have been shown to be safe, immunogenic and effective.

In this paper, we will present and analyze a cholera model with vaccination. We consider the total human population sizes at time t denoted by N(t), which including susceptible individuals S(t), vaccinated individuals V(t), infected individuals I(t) and recovered individuals R(t). The pathogen population at time t, is given by B(t). The model assumes that at any moment in time, new recruits (including newborns, travel, etc.) enter the susceptible population at a constant rate A (the same assumptions can be found in [4], [17], [20], [24]), with a fraction ρ of the susceptible recruited individuals taken to be under vaccination control and enter the vaccinated individuals. We assume that susceptible people becomes infected at a rate βλ(B), where β is the rate of contact with untreated water and λ(B) is the probability of such person to catch cholera. And λ(B) depends on the concentration of Vibrio cholera, B, which is given by the dose response function B/(K+B), where K is the concentration of V. cholera in water that yields 50% chance of catching cholera [2]. The source of infection in through oral ingestion of faecal contaminated water or fool. We also assume a constant recovery rate, α. The rate at which the susceptible population is vaccinated is ϕ, and the rate at which the vaccine wears off is θ. The vaccine has the effect of reducing infection by a factor of σ so that σ=0 means that the vaccine is completely effective in preventing infection, while σ=1 means that the vaccine is utterly ineffective. We assume that there can be disease related death and define d to be the rate of disease-related death, while μ1 is the rate of natural death that is not related to the disease (the same assumptions also can be found in [4], [17], [20], [24]). Infected people contribute to the concentration of vibrios at a rate η. The pathogen population is generated at a rate μ^ and the cholera pathogen has a natural death rate μˇ in the aquatic environment, which in this case, is the set of untreated water consumed by the population. According to Islam [10], we know that V. cholera population decay does not necessarily imply death but also the transition towards a non-culturable state. Hence, we assume μˇ>μ^.

From the above assumption, we will build the following model:dS(t)dt=(1ρ)AβS(t)B(t)K+B(t)ϕS(t)+θV(t)μ1S(t),dV(t)dt=ρA+ϕS(t)σβV(t)B(t)K+B(t)θV(t)μ1V(t),dI(t)dt=βS(t)B(t)K+B(t)+σβV(t)B(t)K+B(t)(d+α+μ1)I(t),dR(t)dt=αI(t)μ1R(t),dB(t)dt=μ^B(t)+ηI(t)μˇB(t).

Since the first three and last equations in Eq. (1.1) are independent of the variable R, it suffices to consider the following reduced model:dS(t)dt=(1ρ)AβS(t)B(t)K+B(t)ϕS(t)+θV(t)μ1S(t),dV(t)dt=ρA+ϕS(t)σβV(t)B(t)K+B(t)θV(t)μ1V(t),dI(t)dt=βS(t)B(t)K+B(t)+σβV(t)B(t)K+B(t)(d+α+μ1)I(t),dB(t)dt=ηI(t)μ2B(t),where μ2=μˇμ^>0.

All parameters are assumed positive. The initial conditions of the system (1.2) are assumed as following:S(0)0,V(0)0,I(0)0,B(0)0.

The paper is organized as follows. In Section 2, we present some preliminaries, such as the positivity, the boundedness of solutions. In Section 3, we firstly calculate the basic reproduction number. Then we obtain the local and global stability of the disease-free equilibrium. In Section 4, we present the persistence of the system (1.2). We get the local and global stability of the endemic equilibrium in Section 5. In Section 6, we analyze the effect of vaccination. The paper ends with a discussion.

Section snippets

Positivity and boundedness of solutions

It is important to prove that the solutions of the system (1.2) are positive with the positive initial conditions (1.3) for the epidemiological meaning.

Theorem 2.1

The solutions (S(t),V(t),I(t),B(t)) of the model (1.2) are positive for all t>0 with initial conditions (1.3).

Proof

Let t1=sup{t>0|S>0,V>0,I>0,B>0}. Thus t1>0. It follows from the firth three equations of the system (1.2) that dS(t)dt(1ρ)AβB(t)K+B(t)+ϕ+μ1S(t),which can be re-written as ddtS(t)exp0tβB(ς)K+B(ς)dς+(ϕ+μ1)t(1ρ)Aexp0tβB(ς)K+B(ς)dς+(ϕ+

Stability of the disease-free equilibrium

For all infectious disease, the basic reproduction number, sometimes called basic reproductive rate or basic reproductive ratio, is one of the most useful threshold parameters which characterize mathematical problems concerning infections diseases. This metric is useful because it helps determine whether or not an infectious disease will spread through a population. In this section, we will calculate the basic reproduction number of system (1.2). Moreover, we will also obtain the local and

Persistence

The epidemiological implication of Theorem 3.2 is that the infected fraction (i.e., I and B) goes to zero in time when Rv1; that is, the cholera eventually disappears from the population. In this section we will present the persistence of system (1.2). The disease is endemic if the infected fraction remains above a certain positive level for sufficiently large time. This definition of endemicity has been characterized using the notion of uniform persistence in several epidemiological models

Global stability of the endemic equilibrium

In this section, we will discuss the local and global stability of the endemic equilibrium. The endemic equilibrium P(S,V,I,B) of system (1.2) can be deduced by the following system: (1ρ)AβSBK+BϕS+θVμ1S=0,ρA+ϕSσβVBK+BθVμ1V=0,βSBK+B+σβVBK+B(d+α+μ1)I=0,ηIμ2B=0,which gives f(I)=A1I2+A2I+A3=0,where A1=μ2(μ1+d+α)(μ12+μ1θ+ϕμ1+μ1βσ+μ1β+β2σ+βθ+βϕσ),A2=Kμ2(μ1+d+α)(2μ12+2μ1ϕ+βμ1+2μ1θ+μ1σβ+βθ+βϕσ)Aβη(σβ+μ1σρ+ϕσ+μ1+θρμ1),A3=Kμ1μ2(μ1+d+α)(μ1+ϕ+θ)(1Rv).

Obviously, A1>0

Effect of vaccination

The reproduction number of the disease in the absence of vaccination is obtained by letting ρ=ϕ=θ=0, and is given by R0=ηβA/Kμ1μ2(μ1+d+α). One can see that the reproduction number in the presence of vaccination Rv is a decreasing function of the vaccination rate ρ and ϕ, and a increasing function of the wane rate θ, Thus, the higher the vaccination rate, the smaller the reproduction number. And the lower the wane rate, the smaller the reproduction number. We can see that limϕ,θ0,ρ1Rv=σR0.

Discussion

In this paper, we model the use of vaccination as a public health strategy for the control of cholera. The results of this model show that it is possible to force Rv under 1, and thus it may be possible to eradicate cholera. In fact, we have obtained the prominent parameter, the basic reproduction number Rv, by using the next generation matrix method. And we have shown that the system always has a disease-free equilibrium and a unique endemic equilibrium when Rv>1. It has been proved that the

Acknowledgment

This work is supported by the National Natural Science Foundation of China (No. 11071011), Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality (No. PHR201107123), Mathematics Tianyuan Funds of NSFC (No. 11026133) and Innovative Program for University Postgraduate Students in Jiangsu Province of China (No. CX10B_387Z), and Outstanding Doctoral Thesis Cultivation Foundation of Nanjing Normal University (No.

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