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ISRN Biomathematics
VolumeΒ 2012Β (2012), Article IDΒ 621939, 13 pages
http://dx.doi.org/10.5402/2012/621939
Research Article

Qualitative Analysis of a Cholera Bacteriophage Model

1Department of Public Health and Center for Infectious Disease Epidemiology Research, China Medical University, 91 Hsueh-Shih Road, Taichung 40402, Taiwan
2Department of Mathematics, Vivekananda College Thakurpukur, Kolkata 700 063, India

Received 22 January 2012; Accepted 19 February 2012

Academic Editors: A.Β MacKenzie and M.Β SantillΓ‘n

Copyright Β© 2012 Prasenjit Das and Debasis Mukherjee. 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

Cholera still remains as a severe global threat and is currently spreading in Africa and other parts of the world. The role of lytic bacteriophage as an intervention of cholera outbreaks is investigated using a mathematical model. Dynamics of cholera is discussed on basis of the basic reproduction number 𝑅0. Conditions of Hopf bifurcation are also derived for a positive net growth rate of Vibrio cholerae. Stability analysis and numerical simulations suggest that bacteriophage may contribute to lessening the severity of cholera epidemics by reducing the number of Vibrio cholerae in the environment. Hence with the presence of phage virus, cholera is self-limiting in nature. By using phage as a biological control agent in endemic areas, one may also influence the temporal dynamics of cholera epidemics while reducing the excessive use of chemicals. We also performed stochastic analysis which suggests that the model system is globally asymptotically stable in probability when the strengths of white noise are less than some specific quantities.

1. Introduction

A highly pathogenic gram-negative bacterium Vibrio cholerae O1 (classical or EI Tor) is the causative agent of the waterborne diarrheal disease, cholera. Ingestion of contaminated water or feces is mainly responsible for cholera transmission rather than casual human to human contact. In spite of the recent progress of medical sciences, cholera still remains as a severe global threat in view of morbidity or mortality and is currently spreading in countries such as Zimbabwe and Mozambique in Africa [1] and other parts of the world. In 2005, for example, 52 nations reported 131943 cholera cases and 2272 deaths despite the paucity of data reporting. The case-fatality rate (CFR) is still high and a number of countries presented a CRF above 5%. Among the vulnerable groups living in high-risk area, CFR is as high as 40% [2]. Moreover, the emergence of pathogenic bacteria, including Vibrio cholerae, resistant to most, if not all, currently available antimicrobial agents has become a most critical dilemma in clinical medicine, in particular because of the concomitant increase in immunosuppressed patients. The development of alternative anti-infection strategy for cholera has become one of the highest priorities of global public health concerns.

In early part of 20th century, bacteriophage, which is a virus that invades bacterial cells and cause bacteria to self-destruct, was used for the prophylaxis and treatment of cholera before the invention and widespread use of antibiotics. More recently, much attention has been focused on bacteriophages that are reemerging as an alternative to antibiotics for treating infection, because specific phages can be used to attack specific bacteria and bacteria are less likely to develop resistance to them [3]. However, there has been much controversy about the real effect of the bacteriophages in cholera treatment [4–7]. Some recent studies [8–10] suggest that phage predation of Vibrio cholerae is an important factor that can influence the seasonal cholera epidemics in Bangladesh. Phage or bacteriophage has recently been found in the environment to inversely correlate with the abundance of toxigenic Vibrio cholerae in water samples and the incidence rate of cholera. Consequently, decrease of the environmental Vibrio cholerae population coincides with the plateau of environmental phage. These studies further suggest that vibriophages may also be used as biological control agents in cholera endemic areas.

Study by [8] also indicates that cholera is characterized by two annual peaks, one with a large peak in the autumn and a smaller one in the spring, in endemic regions (in particular, Ganges delta region of Bangladesh and India). Koelle et al. [11] explain the interannual disease cycle by highlighting the effect of climate variability and temporary immunity. They assert that acquired immunity to reinfection with EI Tor, from previous Classical and EI Tor infections, is long lasting and the degree of immunity starts to wane 3 years after infection, although partial immunity may last for up to 10 years. Moreover, complete cross-immunity is conferred by classical infection for more than 6 years whereas EI Tor reinfection does not occur within 10 years after infection. Seasonal changes in water temperature may also play an important role in transmission of cholera [12–15]. The timing of epidemics varied from one area to the next although outbreaks are seasonal and climatic conditions, such as temperature, precipitation, and humidity, also influence the severity of cholera outbreaks [16]. On the other hand, it is well known that in a temperate region the climatic factors such as temperature or rainfall may not substantially change during the course of epidemic [8]. Hence, climatic factors cannot adequately explain the rapid collapse of epidemic, once it is initiated. Furthermore, in an endemic region such as Bangladesh, cholera epidemics reoccur regularly. So the argument that development of adequate immunity by the population leads to the collapse of the epidemic may not also be fully justified here [8].

This work is aimed to study the impact of bacteriophage in the environment while the cholera epidemic is in progress. More precisely, the focus of our paper is to investigate the role of lytic bacteriophage in the cyclic behaviour of cholera outbreaks.

2. Mathematical Model

We formulate a mathematical model to study the dynamics of cholera in terms of interplay between Vibrio cholerae and bacteriophage. The cholera bacteriophage model flow diagram is given in Figure 1.

621939.fig.001
Figure 1: Model Flow Diagram.

Our proposed mathematical model is as follows:𝑑𝑆𝑑𝑑=π‘Žβˆ’π›Όπ‘‰π‘†π‘˜+π‘‰βˆ’πœ‡1𝑆+π‘ŸπΌ,𝑑𝐼=π‘‘π‘‘π›Όπ‘‰π‘†βˆ’ξ€·π‘˜+π‘‰π‘Ÿ+πœ€+πœ‡1𝐼,𝑑𝑉𝑑𝑑=𝑔𝑉+π›ΏπΌβˆ’π›Ύπ‘‰π‘ƒβˆ’πœ‡2𝑉,𝑑𝑃𝑑𝑑=π›½π›Ύπ‘‰π‘ƒβˆ’πœ‡3𝑃,(1) where 𝑆 is the number of Susceptible, 𝐼 is the number of Infective with Vibrio cholerae and Phage, 𝑉 is the concentration of toxigenic Vibrio cholerae in water, and 𝑃 is the phage density at any time 𝑑. Further the parameters π‘Ž, 𝛼, π‘Ÿ, and πœ€ represent the constant immigration rate, the rate of exposure to contaminated water, the recovery rate, and the disease induced death rate, respectively. The parameter π‘˜ is the concentration of Vibrio cholerae in water that yields 50% chance of catching cholera, 𝛿 is the concentration of each infected population of Vibrio cholerae in the aquatic environment that is, each infected persons may contribute a lot of Vibrio cholera either by vomiting or through passing stool in the aquatic environment, 𝛾 is phage adsorption rate, 𝑔 is the growth rate of Vibrio cholerae, and 𝛽 is the number of phage produced per infected bacterium. Moreover, πœ‡1, πœ‡2 and πœ‡3, respectively, denote the death rate of human, Vibrio cholerae, and bacteriophage. All the parameters are assumed to be positive.

In this paper we first analyze the equilibrium points of the system and discuss the effect of reproduction number on cholera dynamics in Section 3. Then we consider the case πœ‡2>𝑔 in Section 4 and study the system in terms of stability analysis using reproduction number. In this section boundedness of the system is derived. The local stability analysis is carried out for the disease-free, phage-free, and endemic equilibrium. Persistent aspects of the population are presented with the survival of bacteria. The global analyses of the disease-free, phage-free, and endemic equilibria are also carried out. Next by considering the case 𝑔>πœ‡2, we derive the local stability conditions for the disease-free and endemic equilibrium. Hopf-bifurcation of the system is also investigated. The case 𝑔=πœ‡2 is highlighted also in this section. Stochastic analysis is performed in Section 5. Discussion and numerical simulations follow in Section 6.

3. Equilibria and Basic Reproduction Number

The system has the following equilibrium points.(i)Disease-free equilibrium: 𝐸0=(π‘Ž/πœ‡1,0,0,0).(ii)Phage-free equilibrium: 𝐸1=(𝑆1,𝐼1,𝑉1,0), where 𝑆1=ξ€·π‘Ÿ+πœ€+πœ‡1ξ€·πœ‡ξ€Έξ€½π‘Žπ›Ώ+π‘˜1πœ‡+πœ€ξ€Έξ€·2βˆ’π‘”ξ€Έξ€Ύπ›Ώξ€½ξ€·π›Όπœ€+π‘Ÿ+𝛼+πœ€+πœ‡1ξ€Έπœ‡1ξ€Ύ,𝐼1=π‘Žπ›Όπ›Ώβˆ’π‘˜πœ‡1ξ€·πœ‡2βˆ’π‘”ξ€Έξ€·π‘Ÿ+πœ€+πœ‡1ξ€Έπ›Ώξ€½ξ€·π›Όπœ€+π‘Ÿ+𝛼+πœ€+πœ‡1ξ€Έπœ‡1ξ€Ύ,𝑉1=π‘Žπ›Όπ›Ώβˆ’π‘˜πœ‡1ξ€·πœ‡2βˆ’π‘”ξ€Έξ€·π‘Ÿ+πœ€+πœ‡1ξ€Έξ€·πœ‡2ξ€·βˆ’π‘”ξ€Έξ€½π›Όπœ€+π‘Ÿ+𝛼+πœ€+πœ‡1ξ€Έπœ‡1ξ€Ύ,(2) provided that 0<πœ‡2βˆ’π‘”<π‘Žπ›Όπ›Ώ/π‘˜πœ‡1(π‘Ÿ+πœ€+πœ‡1).(iii) Phage-cholera equilibrium: 𝐸2=(𝑆2,𝐼2,𝑉2,𝑃2), where 𝑆2=π‘Žξ€·π‘Ÿ+πœ€+πœ‡1ξ€Έξ€·π‘˜π›½π›Ύ+πœ‡3ξ€Έπ›Όπœ‡3ξ€·πœ‡1ξ€Έ+πœ€+πœ‡1ξ€·π‘˜π›½π›Ύ+πœ‡3πœ‡ξ€Έξ€·1ξ€Έ,𝐼+π‘Ÿ+πœ€2=π‘Žπ›Όπœ‡3π›Όπœ‡3ξ€·πœ‡1ξ€Έ+πœ€+πœ‡1ξ€·π‘˜π›½π›Ύ+πœ‡3πœ‡ξ€Έξ€·1ξ€Έ,𝑉+π‘Ÿ+πœ€2=πœ‡3,𝑃𝛽𝛾2=π‘Žπ›Όπ›½π›Ώπ›Όπœ‡3ξ€·πœ‡1ξ€Έ+πœ€+πœ‡1ξ€·π‘˜π›½π›Ύ+πœ‡3πœ‡ξ€Έξ€·1ξ€Έβˆ’1+π‘Ÿ+πœ€π›Ύξ€·πœ‡2ξ€Έ,βˆ’π‘”(3) provided that 0<πœ‡2βˆ’π‘”<π‘Žπ›Όπ›½π›Ώπ›Ύπ›Όπœ‡3ξ€·πœ‡1ξ€Έ+πœ€+πœ‡1ξ€·π‘˜π›½π›Ύ+πœ‡3πœ‡ξ€Έξ€·1ξ€Έ+π‘Ÿ+πœ€(4) or πœ‡2βˆ’π‘”β‰€0.(5)

The basic reproduction number 𝑅0 is the number of secondary infections produced by an infective newcomer (index case) in a disease-free population. We calculate 𝑅0 by using the next generation operator method [17, 18]. Denoting 𝐅 and 𝐕, respectively, as two matrices for the new infection generation terms and the remaining transition terms, we getβŽ›βŽœβŽœβŽœβŽ0𝐅=π‘Žπ›Όπ‘˜πœ‡1βŽžβŽŸβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽ00,𝐕=π‘Ÿ+πœ€+πœ‡10βˆ’π›Ώπœ‡2βŽžβŽŸβŽŸβŽ βˆ’π‘”.(6) By definition, 𝑅0ξ€·=πœŒπ…π•βˆ’πŸξ€Έ=π‘Žπ›Όπ›Ώπ‘˜πœ‡1ξ€·π‘Ÿ+πœ€+πœ‡1πœ‡ξ€Έξ€·2ξ€Έβˆ’π‘”,(7) where 𝜌 is the spectral radius.

We can see that 𝑑𝑅0/π‘‘π‘Ž=𝛼𝛿/π‘˜πœ‡1(π‘Ÿ+πœ€+πœ‡1)(πœ‡2βˆ’π‘”)>0. Therefore 𝑅0 increases as π‘Ž increases. Again 𝑅0 increases rapidly even for relatively small value of π‘Ž, when rate of exposure to the contaminated water is also increased. Hence cholera may spread faster to attain the epidemic level when the rate of exposure to contaminated water increased albeit with small number of immigration.

When 𝑅0<1, then disease-free equilibrium is stable and there exists no meaningful endemic equilibrium. On the other hand, if 𝑅0>1, then the disease-free equilibrium is unstable while endemic equilibrium point is an attractor. Hence a forward transcritical bifurcation occurs at 𝑅0=1 in the equilibria (see Figure 2), since the endemic disease prevalence is an increasing function of 𝑅0.

621939.fig.002
Figure 2: Transcritical bifurcation occurs at 𝑅0=1.

From the previous analysis we observe that 𝐸1 exists only when 𝑅0>1 and 𝐸2 exists only when 𝑅0>𝜏 with 𝜏=1+πœ‡3{π‘Ÿπœ‡1+(πœ€+πœ‡1)(𝛼+πœ‡1)}/π‘˜π›½π›Ύπœ‡1(π‘Ÿ+πœ€+πœ‡1). Consequently, the equilibrium points in the system, on the basis of 𝑅0, are graphically presented in Figure 3.

621939.fig.003
Figure 3: Scale of 𝑅0.

We now present dynamical behaviour of the system for three cases πœ‡2>𝑔, 𝑔>πœ‡2, and 𝑔=πœ‡2 in the following section.

4. Stability Analysis

Case I (πœ‡2>𝑔). If πœ‡2>𝑔, then there exist the three aforementioned equilibrium points 𝐸0, 𝐸1, and 𝐸2 in the system. We now discuss the dynamical behaviour of these equilibria.

Theorem 1. Assume that πœ‡2βˆ’π‘”>0 and 𝛿<πœ€+πœ‡1. Then all solutions of system (1) which starts in β„œ4+ are bounded.

Proof. Let us define the function π‘Š=𝑆+𝐼+𝑉+𝑃/𝛽. Then the time derivative along the solution of system (1) is Μ‡π‘Š=π‘Žβˆ’πœ‡1π‘†βˆ’(πœ€+πœ‡1βˆ’π›Ώ)πΌβˆ’(πœ‡2βˆ’π‘”)π‘‰βˆ’(πœ‡3/𝛽)𝑃. For each 𝜎>0, the inequality Μ‡π‘Š+πœ†π‘Šβ‰€π‘Ž is satisfied, where 𝜎=min{πœ‡1,πœ€+πœ‡1βˆ’π›Ώ,πœ‡2βˆ’π‘”,πœ‡3}. Applying differential inequality argument, we obtain 0β‰€π‘Šβ‰€π‘Ž/𝜎. Hence system (1) is bounded and the solution of the system enter into the region 𝐡={(𝑆,𝐼,𝑉,𝑃)βˆˆβ„œ4+∢0β‰€π‘Šβ‰€π‘Ž/𝜎}.

Here we present the stability behaviour of the disease-free, phage-free, and endemic equilibrium points, respectively.

Theorem 2. If πœ‡2>𝑔, then the disease-free equilibrium point 𝐸0 is locally asymptotically stable if 𝑅0<1, neutrally stable if 𝑅0=1, and unstable if 𝑅0>1.

Proof. The proof follows from the characteristic equation (πœ†+πœ‡1)(πœ†+πœ‡3){(πœ†+π‘Ÿ+πœ€+πœ‡1)(πœ†+πœ‡2βˆ’π‘”)βˆ’π‘Žπ›Όπ›Ώ/π‘˜πœ‡1}=0 of the Jacobian matrix of system (1) at the disease-free equilibrium point 𝐸0.

Theorem 3. Phage-free equilibrium 𝐸1, whenever exists, is locally asymptotically stable if 𝑅0<𝜏 and is unstable if 𝑅0>𝜏.

Proof. The characteristic equation of the Jacobian matrix of system (1) at 𝐸1=(𝑆1,𝐼1,𝑉1,0) is given by (𝛽𝛾𝑉1βˆ’πœ‡3βˆ’πœ†)(πœ†3+π΄πœ†2+π΅πœ†+𝐢)=0, where 𝐴=πœ‡2βˆ’π‘”+π‘Ÿ+πœ€+2πœ‡1+𝛼𝑉1π‘˜+𝑉1,ξ€·πœ‡π΅=2ξ€Έξ‚΅βˆ’π‘”π‘Ÿ+πœ€+2πœ‡1+𝛼𝑉1π‘˜+𝑉1ξ‚Ά+𝛼𝑉1ξ€·πœ€+πœ‡1ξ€Έπ‘˜+𝑉1+πœ‡1ξ€·π‘Ÿ+πœ€+πœ‡1ξ€Έβˆ’π›Όπ›Ώπ‘˜π‘†1ξ€·π‘˜+𝑉1ξ€Έ2,ξ€·πœ‡πΆ=2ξ€Έβˆ’π‘”π›Όπ‘‰1ξ€·πœ€+πœ‡1ξ€Έπ‘˜+𝑉1+πœ‡1ξƒ―ξ€·πœ‡2βˆ’π‘”ξ€Έξ€·π‘Ÿ+πœ€+πœ‡1ξ€Έβˆ’π›Όπ›Ώπ‘˜π‘†1ξ€·π‘˜+𝑉1ξ€Έ2ξƒ°.(8) Again, 𝐴, 𝐡, 𝐢, and (π΄π΅βˆ’πΆ) are all positive if (πœ‡2βˆ’π‘”)(π‘Ÿ+πœ€+πœ‡1)βˆ’π›Όπ›Ώπ‘˜π‘†1/(π‘˜+𝑉1)2>0 since πœ‡2βˆ’π‘”>0. Furthermore, the existence of 𝐸1 implies (πœ‡2βˆ’π‘”)(π‘Ÿ+πœ€+πœ‡1)βˆ’π›Όπ›Ώπ‘˜π‘†1/(π‘˜+𝑉1)2>0.
Hence, 𝐸1 is locally asymptotically stable if πœ‡3>𝛽𝛾𝑉1 and unstable if πœ‡3<𝛽𝛾𝑉1, whenever it exists. Moreover, πœ‡3>𝛽𝛾𝑉1⇒𝑅0<𝜏.

Theorem 4. Endemic equilibrium point 𝐸2 is locally asymptotically stable if and only if 𝐴4<(𝐴1𝐴2βˆ’π΄3)𝐴3/𝐴21 holds.

Proof. The characteristic equation of the Jacobian matrix of system (1) at 𝐸2=(𝑆2,𝐼2,𝑉2,𝑃2) is given by πœ†4+𝐴1πœ†3+𝐴2πœ†2+𝐴3πœ†+𝐴4=0, where 𝐴1=π‘Ÿ+πœ€+2πœ‡1+πœ‡2βˆ’π‘”+𝛾𝑃2+𝛼𝑉2π‘˜+𝑉2,𝐴2=𝛼𝑉2π‘˜+𝑉2ξ€·πœ€+πœ‡1ξ€Έ+πœ‡1ξ€·π‘Ÿ+πœ€+πœ‡1ξ€Έ+ξ‚΅π‘Ÿ+πœ€+2πœ‡1+𝛼𝑉2π‘˜+𝑉2ξ‚Άξ€·πœ‡2βˆ’π‘”+𝛾𝑃2ξ€Έ+𝛽𝛾2𝑃2𝑉2βˆ’π›Όπ›Ώπ‘˜π‘†2ξ€·π‘˜+𝑉2ξ€Έ2,𝐴3=𝛼𝑉2π‘˜+𝑉2ξ€·πœ€+πœ‡1ξ€Έ+πœ‡1ξ€·π‘Ÿ+πœ€+πœ‡1ξ€Έξ‚ΌΓ—ξ€·πœ‡2βˆ’π‘”+𝛾𝑃2ξ€Έ+𝛽𝛾2𝑃2𝑉2Γ—ξ‚΅π‘Ÿ+πœ€+2πœ‡1+𝛼𝑉2π‘˜+𝑉2ξ‚Άβˆ’π›Όπ›Ώπ‘˜πœ‡1𝑆2ξ€·π‘˜+𝑉2ξ€Έ2,𝐴4=𝛼𝑉2π‘˜+𝑉2ξ€·πœ€+πœ‡1ξ€Έ+πœ‡1ξ€·π‘Ÿ+πœ€+πœ‡1𝛽𝛾2𝑃2𝑉2.(9) Hence the result follows by applying Routh-Hurwitz criterion.

Now we present the persistence aspects of the population.

Theorem 5. The population 𝑆(𝑑) is uniformly persistent whenever 𝑆(0)>0.

Proof. From the first equation of system (1), we get 𝑑𝑆/𝑑𝑑β‰₯π‘Žβˆ’(𝛼+πœ‡1)𝑆, since 𝑉/(π‘˜+𝑉)<1⇒𝑑𝑆/𝑑𝑑+(𝛼+πœ‡1)𝑆β‰₯π‘Ž. Applying differential inequality argument, we obtain limπ‘‘β†’βˆžinf𝑆(𝑑)β‰₯π‘Ž/(𝛼+πœ‡1).

Theorem 6. The population 𝐼(𝑑) is strongly persistent whenever 𝐼(0)>0 and is uniformly persistent when 𝑉(𝑑) is uniformly persistent.

Proof. From the last equation of system (1) we get ̇𝑃<0 when 𝑉<πœ‡3/𝛽𝛾. In this case, 𝑃→0 as π‘‘β†’βˆž which is impossible from biological point of view. Hence, 𝑉>πœ‡3/𝛽𝛾. Now from the second equation of the system, we get 𝐼(𝑑)β‰₯𝛼𝑉𝑆(π‘˜+𝑉)π‘Ÿ+πœ€+πœ‡1ξ€Έβ‰₯π›Όπ‘‰π‘Žξ€·(π‘˜+𝑉)π‘Ÿ+πœ€+πœ‡1ξ€ΈπœŽβ‰₯π›Όπœ‡3π‘Žξ€·π‘˜π›½π›Ύ+πœ‡3ξ€Έξ€·π‘Ÿ+πœ€+πœ‡1ξ€ΈπœŽ.(10) Moreover, if 𝐼(𝑑)<𝛼𝑉𝑆/(π‘˜+𝑉)(π‘Ÿ+πœ€+πœ‡1), then, 𝑑𝐼/𝑑𝑑β‰₯0 and hence 𝐼(𝑑) is increasing. So 𝐼(𝑑) becomes unbounded and hence we have 𝐼(𝑑)β‰₯𝛼𝑉𝑆/(π‘˜+𝑉)(π‘Ÿ+πœ€+πœ‡1).

We now present the global stability result for the disease-free, phage-free, and endemic equilibrium point.

Theorem 7. The disease-free equilibrium point 𝐸0 is globally asymptotically stable if 𝑅0<1 when all solutions of system (1) which starts in 𝑅4+ are bounded.

Proof. From system (1) we get βŽ›βŽœβŽœβŽπ‘‘πΌ(𝑑)𝑑𝑑𝑑𝑉(𝑑)βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βˆ’βŽ›βŽœβŽœβŽœβŽ0𝑑𝑑=(πΉβˆ’π‘‰)𝐼(𝑑)𝑉(𝑑)π‘Žπ›Όπ‘˜πœ‡1βˆ’π›Όπ‘†(𝑑)βŽžβŽŸβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ π‘˜+𝑉(𝑑)0𝛾𝑃(𝑑)𝐼(𝑑)𝑉(𝑑).(11) Since π‘Žπœ‡1>(π‘˜π‘†(𝑑))(π‘˜+𝑉(𝑑))βˆ€π‘‘β‰₯0in𝑅4+,βŽ›βŽœβŽœβŽπ‘‘πΌ(𝑑)𝑑𝑑𝑑𝑉(𝑑)βŽžβŽŸβŽŸβŽ β‰€βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ .𝑑𝑑(πΉβˆ’π‘‰)𝐼(𝑑)𝑉(𝑑)(12) Using the fact that eigenvalues of πΉβˆ’π‘‰ all have negative real parts when 𝑅0<1 [18] it follows that linearized differential inequality of system (12) is stable whenever 𝑅0<1. Consequently, (𝐼(𝑑),𝑉(𝑑))β†’(0,0) as π‘‘β†’βˆž. It follows by comparison theorem [19, 20] that (𝐼(𝑑),𝑉(𝑑))β†’(0,0). Substituting 𝐼=𝑉=0 in the first and fourth equations of model system (1), we obtain 𝑆(𝑑)β†’π‘Ž/πœ‡1 and 𝑃(𝑑)β†’0 as π‘‘β†’βˆž. Thus (𝑆(𝑑),𝐼(𝑑),𝑉(𝑑),𝑃(𝑑))β†’(π‘Ž/πœ‡1,0,0,0) as π‘‘β†’βˆžif 𝑅0<1, and subsequently 𝐸0 is globally asymptotically stable if 𝑅0<1.

Theorem 8. Assume that πœ‡2βˆ’π‘”>0 and 𝛿<πœ€+πœ‡1. 𝐸1 is globally asymptotically stable, if 𝑀 is positive definite where 𝑀=[π‘Žπ‘–π‘—]4Γ—4.

Proof. Define 1π‘ˆ(𝑆,𝐼,𝑉,𝑃)=2ξ€·π‘†βˆ’π‘†1ξ€Έ2+12ξ€·πΌβˆ’πΌ1ξ€Έ2+12ξ‚΅π‘‰βˆ’π‘‰1+𝑃𝛽2.(13) The time derivative of π‘Š along the solution of system (1) is Μ‡ξ€·π‘ˆ=π‘†βˆ’π‘†1̇𝑆+πΌβˆ’πΌ1̇𝐼+π‘‰βˆ’π‘‰1+𝑃𝛽̇̇𝑃𝑉+π›½ξ‚ΆβŸΉΜ‡π‘ˆ=βˆ’πœ‡1ξ€·π‘†βˆ’π‘†1ξ€Έ2βˆ’ξ€·π‘Ÿ+πœ€+πœ‡1ξ€Έξ€·πΌβˆ’πΌ1ξ€Έ2βˆ’ξ€·πœ‡2βˆ’π‘”ξ€Έξ€·π‘‰βˆ’π‘‰1ξ€Έ2βˆ’πœ‡3𝛽2𝑃2ξ€·+π‘Ÿπ‘†βˆ’π‘†1ξ€Έξ€·πΌβˆ’πΌ1ξ€Έξ€·+π›ΏπΌβˆ’πΌ1ξ€Έξ€·π‘‰βˆ’π‘‰1ξ€Έβˆ’πœ‡2βˆ’π‘”+πœ‡3π›½π‘ƒξ€·π‘‰βˆ’π‘‰1ξ€Έ+π›Ώπ›½π‘ƒξ€·πΌβˆ’πΌ1ξ€Έξ‚΅βˆ’π›Όπ‘‰π‘†βˆ’π‘‰π‘˜+𝑉1𝑆1π‘˜+𝑉1ξ‚Άξ€½ξ€·π‘†βˆ’π‘†1ξ€Έβˆ’ξ€·πΌβˆ’πΌ1βŸΉΜ‡ξ€Έξ€Ύπ‘ˆ=βˆ’πœ‡1ξ€·π‘†βˆ’π‘†1ξ€Έ2βˆ’ξ€·π‘Ÿ+πœ€+πœ‡1ξ€Έξ€·πΌβˆ’πΌ1ξ€Έ2βˆ’ξ€·πœ‡2βˆ’π‘”ξ€Έξ€·π‘‰βˆ’π‘‰1ξ€Έ2βˆ’πœ‡3𝛽2𝑃2ξ€·+π‘Ÿπ‘†βˆ’π‘†1ξ€Έξ€·πΌβˆ’πΌ1ξ€Έξ€·+π›ΏπΌβˆ’πΌ1ξ€Έξ€·π‘‰βˆ’π‘‰1ξ€Έβˆ’πœ‡2βˆ’π‘”+πœ‡3π›½π‘ƒξ€·π‘‰βˆ’π‘‰1ξ€Έ+π›Ώπ›½π‘ƒξ€·πΌβˆ’πΌ1ξ€Έξ‚΅βˆ’π›Όπ‘‰π‘†βˆ’π‘˜+𝑉𝑉𝑆1+π‘˜+𝑉𝑉𝑆1βˆ’π‘‰π‘˜+𝑉1𝑆1π‘˜+𝑉1ξ‚ΆΓ—ξ€½ξ€·π‘†βˆ’π‘†1ξ€Έβˆ’ξ€·πΌβˆ’πΌ1βŸΉΜ‡ξ€·πœ‡ξ€Έξ€Ύπ‘ˆβ‰€βˆ’1+π›Όξ€Έξ€·π‘†βˆ’π‘†1ξ€Έ2βˆ’ξ€·π‘Ÿ+πœ€+πœ‡1ξ€Έξ€·πΌβˆ’πΌ1ξ€Έ2βˆ’ξ€·πœ‡2βˆ’π‘”ξ€Έξ€·π‘‰βˆ’π‘‰1ξ€Έ2βˆ’πœ‡3𝛽2𝑃2||+(π‘Ÿ+𝛼)π‘†βˆ’π‘†1||||πΌβˆ’πΌ1||+𝛿+𝛼𝑆1π‘˜+𝑉1ξ‚Ά||πΌβˆ’πΌ1||||||+πœ‡π‘‰βˆ’π‘‰2βˆ’π‘”+πœ‡3𝛽𝑃||||+π›Ώπ‘‰βˆ’π‘‰π›½π‘ƒ||πΌβˆ’πΌ1||+𝛼𝑆1π‘˜+𝑉1||π‘†βˆ’π‘†1||||||π‘‰βˆ’π‘‰=βˆ’π‘Ž11ξ€·π‘†βˆ’π‘†1ξ€Έ2βˆ’π‘Ž22ξ€·πΌβˆ’πΌ1ξ€Έ2βˆ’π‘Ž33ξ€·π‘‰βˆ’π‘‰1ξ€Έ2βˆ’π‘Ž44𝑃2+2π‘Ž12||π‘†βˆ’π‘†1||Γ—||πΌβˆ’πΌ1||+2π‘Ž13||π‘†βˆ’π‘†1||||||π‘‰βˆ’π‘‰+2π‘Ž23||πΌβˆ’πΌ1||Γ—||||π‘‰βˆ’π‘‰+2π‘Ž24𝑃||πΌβˆ’πΌ1||+2π‘Ž34𝑃||||π‘‰βˆ’π‘‰=βˆ’π‘‹π‘‡π‘€π‘‹,(14) where 𝑋𝑇={|π‘†βˆ’π‘†1|,|πΌβˆ’πΌ1|,|π‘‰βˆ’π‘‰1|,𝑃}and𝑀=[π‘Žπ‘–π‘—]4Γ—4. Elements of the matrix 𝑀 are given by π‘Ž11=𝛼+πœ‡1,π‘Ž22=π‘Ÿ+πœ€+πœ‡1,π‘Ž33=πœ‡2π‘Žβˆ’π‘”,44=πœ‡3𝛽2,π‘Ž12=π‘Ž21=βˆ’(π‘Ÿ+𝛼)2,π‘Ž13=π‘Ž31=βˆ’π›Όπ‘†12ξ€·π‘˜+𝑉1ξ€Έ,π‘Ž14=π‘Ž41π‘Ž=0,23=π‘Ž321=βˆ’2𝛿+𝛼𝑆1π‘˜+𝑉1ξ‚Ά,π‘Ž24=π‘Ž42𝛿=βˆ’2𝛽,π‘Ž34=π‘Ž43πœ‡=βˆ’2βˆ’π‘”+πœ‡3.2𝛽(15) Here, 𝑀 is positive definite if ||||||π‘Ž11π‘Ž12π‘Ž21π‘Ž22|||||||||||||||π‘Ž>0,11π‘Ž12π‘Ž13π‘Ž21π‘Ž22π‘Ž23π‘Ž31π‘Ž32π‘Ž33|||||||||||||||||||||π‘Ž>0,11π‘Ž12π‘Ž130π‘Ž21π‘Ž22π‘Ž23π‘Ž24π‘Ž31π‘Ž32π‘Ž33π‘Ž340π‘Ž42π‘Ž43π‘Ž44||||||||||||>0holdsimultaneously.(16) Further since 𝐡 is a global attractor, we may restrict our attention to solutions initiating in 𝐡. From the aforementioned inequalities, the right-hand side of (14), which is considered as a quadratic form in the variables |π‘†βˆ’π‘†1|,|πΌβˆ’πΌ1|,and|π‘‰βˆ’π‘‰1|,𝑃 is negative definite for (𝑆,𝐼,𝑉,𝑃)∈𝐡. Hence (𝑑/𝑑𝑑)π‘ˆ(𝑆,𝐼,𝑉,𝑃) is negative definite about 𝐸1 and consequently π‘ˆ(𝑆,𝐼,𝑉,𝑃) is a Lyapunov function for (𝑆,𝐼,𝑉,𝑃)∈𝐡. This completes the proof.

Theorem 9. Assume that πœ‡2βˆ’π‘”>0 and 𝛿<πœ€+πœ‡1. 𝐸2 is globally asymptotically stable, if 𝑁 is positive definite, whereβŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽπ‘=𝛼+πœ‡βˆ’(π‘Ÿ+𝛼)2βˆ’π›Όπ‘†22ξ€·π‘˜+𝑉2ξ€Έ0βˆ’(π‘Ÿ+𝛼)2π‘Ÿ+πœ€+πœ‡1βˆ’12𝛿+𝛼𝑆2π‘˜+𝑉2ξ‚Άβˆ’π›Ώβˆ’2𝛽𝛼𝑆22ξ€·π‘˜+𝑉2ξ€Έβˆ’12𝛿+𝛼𝑆2π‘˜+𝑉2ξ‚Άπœ‡2πœ‡βˆ’π‘”βˆ’2βˆ’π‘”+πœ‡3𝛿2𝛽0βˆ’βˆ’πœ‡2𝛽2βˆ’π‘”+πœ‡3πœ‡2𝛽3𝛽2⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(17)

Proof. Theorem 9 can be proved in a similar fashion as the proof of Theorem 8 by considering the positive definite function: 1πœ“(𝑆,𝐼,𝑉,𝑃)=2ξ€·π‘†βˆ’π‘†2ξ€Έ+12ξ€·πΌβˆ’πΌ2ξ€Έ2+12ξ‚΅π‘‰βˆ’π‘‰2+π‘ƒπ›½βˆ’π‘ƒ2𝛽2.(18)

Case II (𝑔>πœ‡2). If 𝑔>πœ‡2, then there exist two equilibrium points 𝐸0 and 𝐸2 of the system, the local stability of which is discussed in the following theorems.

Theorem 10. The disease-free equilibrium 𝐸0 is unstable when 𝑔>πœ‡2 or if 𝑅0=1 and 𝑔>πœ‡2+π‘Ÿ+πœ€+πœ‡1 hold simultaneously.

The proof is obvious from the characteristic equation of the Jacobian matrix of system (1) at the disease-free equilibrium point 𝐸0.

Theorem 11. 𝐸2 is locally asymptotically stable if and only if 𝐷𝑖>0(𝑖=1,2,3,4) for πœ‡2<𝑔, where 𝐷1=𝐴1, 𝐷2=𝐴1𝐴2βˆ’π΄3, 𝐷3=𝐴1(𝐴2𝐴3βˆ’π΄1𝐴4)βˆ’π΄23, and 𝐷4=𝐴4𝐷3.

The result follows from Routh-Hurwitz criterion.

In this case 𝐸0 is unstable while 𝐸2 is stable.

Now we present Hopf bifurcation criterion for system (1). Since 𝛽 is a parameter, we show that a classical Hopf bifurcation occurs for (1) at a critical value 𝛽=𝛽𝑐. For simplicity of notation, we assume that πΈβˆ—(π‘†βˆ—(𝛽),πΌβˆ—(𝛽),π‘‰βˆ—(𝛽),π‘ƒβˆ—(𝛽)) is the positive equilibrium of the system and let πœ†4+𝐴1(𝛽)πœ†3+𝐴2(𝛽)πœ†2+𝐴3(𝛽)πœ†+𝐴4(𝛽)=0(19) be the characteristic equation of the variational system associated with system (1) about it. Let πœ†π‘–(𝛽),𝑖=1,2,3,4 be the roots of (19) with Reπœ†1(𝛽)≀Reπœ†2(𝛽)≀Reπœ†3(𝛽)≀Reπœ†4(𝛽),𝑓(𝛽)=𝐴1(𝛽)𝐴2(𝛽)𝐴3(𝛽)βˆ’π΄21(𝛽)𝐴4(𝛽)βˆ’π΄23(𝛽).(20)

Now we state the following theorem.

Theorem 12. If there exists 𝛽=𝛽𝑐 such that (i)   𝑔>πœ‡2, (ii)𝑓(𝛽𝑐)=0, π‘“ξ…ž(𝛽𝑐)>0, and (iii)   𝐴3(𝛽𝑐)>0, then positive equilibrium πΈβˆ—(π‘†βˆ—(𝛽),πΌβˆ—(𝛽),π‘‰βˆ—(𝛽),π‘ƒβˆ—(𝛽)) is locally stable if 𝛽>𝛽𝑐, π›½βˆ’π›½π‘β‰ͺ1 but unstable for 𝛽<𝛽𝑐, π›½π‘βˆ’π›½β‰ͺ1,and a Hopf bifurcation of periodic solution occurs at 𝛽=𝛽𝑐.

Proof. Here 𝑓′(𝛽𝑐)>0 indicates that 𝑓(𝛽) is a monotonic increasing function in the neighborhood of 𝛽=𝛽𝑐. This, together with 𝐴3(𝛽𝑐)>0, implies that 𝑓(𝛽𝑐)<0, 𝐴1(𝛽)𝐴2(𝛽)βˆ’π΄3(𝛽)<0 for 𝛽>𝛽𝑐, π›½βˆ’π›½π‘β‰ͺ1. Thus we have Reπœ†3(𝛽𝑐)<0 and Reπœ†4(𝛽)<0, that is, the positive equilibrium πΈβˆ— is locally stable.
Again monotonic increasing function 𝑓(𝛽) in the neighborhood of 𝛽=𝛽𝑐 also implies that 𝑓(𝛽𝑐)>0 for 𝛽<𝛽𝑐, π›½π‘βˆ’π›½β‰ͺ1. Hence we have Reπœ†4(𝛽)>0, that is, πΈβˆ— is unstable.
It is easy to see that 𝑓(𝛽𝑐)=𝐴1(𝛽𝑐)𝐴2(𝛽𝑐)𝐴3(𝛽𝑐)βˆ’π΄21(𝛽𝑐)𝐴4(𝛽𝑐)βˆ’π΄23(𝛽𝑐)=0; 𝐴1(𝛽𝑐)>0 implies that 𝐴2(𝛽𝑐)𝐴3(𝛽𝑐)βˆ’π΄1(𝛽𝑐)𝐴4(𝛽𝑐)>0.
Since𝑓′(𝛽𝑐)>0, then by a theorem of [21] we have a simple Hopf bifurcation of periodic solution occurs at 𝛽=𝛽𝑐.

One can check the conditions of Theorems 11 and 12 by a given set of parameters. If for a particular set of parameters the conditions of the previous two theorems are not satisfied, then also one can predict the dynamical behavior of the system.

Further, 𝑔>πœ‡2⇒𝑅0<0. In this scenario growth rate of the Vibrio cholerae dominates its death rate and biologically this is the worst situation where disease outbreaks cannot be controlled. Moreover in this situation occurrence of disease might be periodical.

Case III (𝑔=πœ‡2). If 𝑔=πœ‡2, then we obtain the two aforementioned equilibrium points 𝐸0 and 𝐸2 of the system. The local stability conditions of these equilibria are stated here in without proof as it follows from Routh-Hurwitz criterion.

Theorem 13. For πœ‡2=𝑔, 𝐸0 is neutrally stable when 𝑅0=1 and unstable otherwise. Further 𝐸2 coexists with 𝐸0 and is locally asymptotically stable when 𝐴4<(𝐴1𝐴2βˆ’π΄3)𝐴3/𝐴21 holds.

5. Stochastic Analysis

Stochastic perturbations can be introduced in some of the model parameters [22–24]. In this study, we allow stochastic perturbations of the variables 𝑆(𝑑),𝐼(𝑑),𝑉(𝑑), and 𝑃(𝑑) around their values at the positive equilibrium 𝐸2, when it is feasible and locally asymptotically stable. Local stability of 𝐸2 is implied by the existence condition of 𝐸2 with 0<πœ‡2βˆ’π‘”<π‘Žπ›Όπ›½π›Ώπ›Ύ/(π›Όπœ‡3(πœ‡1+πœ€)+πœ‡1(π‘˜π›½π›Ύ+πœ‡3)(πœ‡1+π‘Ÿ+πœ€)) or πœ‡2βˆ’π‘”β‰€0. Hence we assume that stochastic perturbations of the variables around their values at 𝐸2 are of white noise type, which are proportional to the distances of 𝑆(𝑑),𝐼(𝑑),𝑉(𝑑), and 𝑃(𝑑) from values 𝑆2,𝐼2,𝑉2, and 𝑃2.

Subseqsuently we have𝑑𝑆=π‘Žβˆ’π›Όπ‘‰π‘†π‘˜+π‘‰βˆ’πœ‡1𝑆+π‘ŸπΌπ‘‘π‘‘+𝜎1ξ€·π‘†βˆ’π‘†2ξ€Έπ‘‘πœ‰1𝑑,𝑑𝐼=π›Όπ‘‰π‘†βˆ’ξ€·π‘˜+π‘‰π‘Ÿ+πœ€+πœ‡1𝐼+𝜎2ξ€·πΌβˆ’πΌ2ξ€Έπ‘‘πœ‰2𝑑,=𝑑𝑉𝑔𝑉+π›ΏπΌβˆ’π›Ύπ‘‰π‘ƒβˆ’πœ‡2𝑉+𝜎3ξ€·π‘‰βˆ’π‘‰2ξ€Έπ‘‘πœ‰3𝑑,𝑑𝑃=π›½π›Ύπ‘‰π‘ƒβˆ’πœ‡3𝑃+𝜎4ξ€·π‘ƒβˆ’π‘ƒ4ξ€Έπ‘‘πœ‰4𝑑.(21) Here πœŽπ‘—,𝑗=1, 2, 3, 4, are real constants, and πœ‰π‘—π‘‘=πœ‰π‘—(𝑑),𝑗=1, 2, 3, 4 are independent from each other as standard Wiener processes [25]. We study the robustness of dynamical behaviour of model (1) with respect to stochasticity by investigating the asymptotic stochastic stability behaviour of the equilibrium 𝐸2 for system (21) and comparing the results with those obtained for (1). We will consider (21) as the Ito stochastic differential system.

5.1. Stochastic Stability of the Positive Equilibrium

The stochastic differential system (21) can be centered at its positive equilibrium 𝐸2 by the changes of the variables𝑒1=π‘†βˆ’π‘†2,𝑒2=πΌβˆ’πΌ2,𝑒3=π‘‰βˆ’π‘‰2,𝑒4=π‘ƒβˆ’π‘ƒ2.(22) The linearized SDEs around 𝐸2 take the form𝑑𝑒(𝑑)=𝑓(𝑒(𝑑))𝑑𝑑+𝑔(𝑒(𝑑))π‘‘πœ‰(𝑑),(23) where 𝑒𝑒(𝑑)=col1(𝑑),𝑒2(𝑑),𝑒3(𝑑),𝑒4ξ€Έ(𝑑),(24)βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£βˆ’ξ‚΅π‘“(𝑒(𝑑))=𝛼𝑉2π‘˜+𝑉2+πœ‡1ξ‚Άπ‘Ÿβˆ’π‘˜π›Όπ‘†2ξ€·π‘˜+𝑉2ξ€Έ20𝛼𝑉2π‘˜+𝑉2βˆ’ξ€·π‘Ÿ+πœ€+πœ‡1ξ€Έπ‘˜π›Όπ‘†2ξ€·π‘˜+𝑉2ξ€Έ200π›Ώπ‘”βˆ’π›Ύπ‘ƒ2βˆ’πœ‡2βˆ’π›Ύπ‘‰200𝛽𝛾𝑃2𝛽𝛾𝑉2βˆ’πœ‡3⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦π‘’(𝑑),(25)βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£πœŽπ‘”(𝑒)=1𝑒10000𝜎2𝑒20000𝜎3𝑒30000𝜎2𝑒2⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦.(26) In (23) the positive equilibrium 𝐸2 corresponds to the trivial solution, 𝑒(𝑑)=0.

Let π‘ˆ=(𝑑β‰₯𝑑0)×𝑅𝑛,𝑑0βˆˆπ‘…+, and consider a twice continuously differentiable function, π‘Š. With reference to [26], the following theorem holds. Note that for (23),πΏπ‘Š(𝑑,𝑒)=πœ•π‘Š(𝑑,𝑒)πœ•π‘‘+𝑓𝑇(𝑒)πœ•π‘Š(𝑑,𝑒)+1πœ•π‘’2ξ‚Έπ‘”π‘‡π‘Ÿπ‘‡πœ•(𝑒)2π‘Š(𝑑,𝑒)πœ•π‘’2ξ‚Ή,𝑔(𝑒)(27) whereπœ•π‘Šξ‚΅πœ•π‘’=Colπœ•π‘Šπœ•π‘’1,πœ•π‘Šπœ•π‘’2,πœ•π‘Šπœ•π‘’3,πœ•π‘Šπœ•π‘’4ξ‚Ά,πœ•2π‘Š(𝑑,𝑒)πœ•π‘’2=ξ‚΅πœ•2π‘Šπœ•π‘’π‘—πœ•π‘’π‘–ξ‚Ά,𝑖,𝑗=1,2,3,4,(28)

and β€œπ‘‡β€ represents transposition.

Theorem 14. Assume that there exists a function π‘Š(𝑑,𝑒)∈𝐢02(π‘ˆ) satisfying the following inequalities: 𝐾1|𝑒|π‘β‰€π‘Š(𝑑,𝑒)≀𝐾2|𝑒|𝑝,(29)πΏπ‘Š(𝑑,𝑒)β‰€βˆ’πΎ3|𝑒|𝑝,(30) where 𝐾𝑖and p are positive constants. Then the trivial solution of (23) is exponentially p-stable for 𝑑β‰₯0.

It is important to note that, if 𝑝=2 in (29)-(30), then the trivial solution of (23) is globally asymptotically stable in probability. For definition of mean square stability, we again refer to [26].

Theorem 15. Let 0<πœ‡2βˆ’π‘”<π‘Žπ›Όπ›½π›Ώπ›Ύ/(π›Όπœ‡3(πœ‡1+πœ€)+πœ‡1(π‘˜π›½π›Ύ+πœ‡3)(πœ‡1+π‘Ÿ+πœ€)) or πœ‡2βˆ’π‘”β‰€0. Further assume that 𝜎21<4(πœ‡1βˆ’π‘Ÿ),𝜎22<4(πœ€+πœ‡1βˆ’π›Όπ‘‰2/(π‘˜+𝑉2)),𝜎23<4(βˆ’π‘”+𝛾𝑃2+πœ‡2), and 𝜎24<4(βˆ’π›½π›Ύπ‘‰2+πœ‡3)   hold. Then the zero solution of (23) is asymptotically mean square stable.

Proof. Let us consider the Lyapunov function π‘Š(𝑒)=(1/2)(𝑒21+𝑒22+𝑒23+𝑀𝑒24), where 𝑀>0 is a constant to be chosen later. It is easy to check that inequality (29) holds true with 𝑝=2. Moreover, ξ‚Έ2ξ‚΅πΏπ‘Š(𝑒)=βˆ’π›Όπ‘‰2π‘˜+𝑉2+πœ‡1ξ‚Άβˆ’12𝜎21𝑒21βˆ’ξ‚ƒ2ξ€·π‘Ÿ+πœ€+πœ‡1ξ€Έβˆ’12𝜎22𝑒22βˆ’ξ‚ƒ2ξ€·βˆ’π‘”+𝛾𝑃2+πœ‡3ξ€Έβˆ’12𝜎23𝑒232ξ€·βˆ’π‘€βˆ’π›½π›Ύπ‘‰2+πœ‡4ξ€Έβˆ’12𝜎24𝑒24ξ‚΅+2π‘Ÿ+𝛼𝑉2π‘˜+𝑉2𝑒1𝑒2βˆ’2π›Όπ‘˜π‘†2ξ€·π‘˜+𝑉2ξ€Έ2𝑒1𝑒3ξ€·+2βˆ’π›Ύπ‘‰2+𝑀𝛽𝛾𝑃2𝑒3𝑒4.(31) Choosing 𝑀=𝑉2/𝛽𝑃2 and using 𝑒21+𝑒22β‰₯2𝑒1𝑒2 from (31), we obtain 2ξ€·πœ‡πΏπ‘Š(𝑒)β‰€βˆ’1ξ€Έβˆ’1βˆ’π‘Ÿ2𝜎21𝑒21βˆ’ξ‚Έ2ξ‚΅πœ€+πœ‡1βˆ’π›Όπ‘‰2π‘˜+𝑉2ξ‚Άβˆ’12𝜎22𝑒22βˆ’ξ‚ƒ2ξ€·βˆ’π‘”+𝛾𝑃2+πœ‡3ξ€Έβˆ’12𝜎23𝑒232ξ€·βˆ’π‘€βˆ’π›½π›Ύπ‘‰2+πœ‡4ξ€Έβˆ’12𝜎24𝑒24β‰€βˆ’π‘Œπ‘‡π‘π‘Œ,(32) where 𝑍=[π‘Žπ‘–π‘—]4Γ—4 and π‘Œ=(𝑒1,𝑒2,𝑒3,𝑒4)𝑇. Here π‘Ž11ξ€·πœ‡=21ξ€Έβˆ’1βˆ’π‘Ÿ2𝜎21,π‘Ž22ξ‚΅=2πœ€+πœ‡1βˆ’π›Όπ‘‰2π‘˜+𝑉2ξ‚Άβˆ’12𝜎22,π‘Ž33ξ€·=2βˆ’π‘”+𝛾𝑃2+πœ‡3ξ€Έβˆ’12𝜎23,π‘Ž44ξ€·=2βˆ’π›½π›Ύπ‘‰2+πœ‡4ξ€Έβˆ’12𝜎24,π‘Ž12=π‘Ž21=0,π‘Ž13=π‘Ž31π‘Ž=0,14=π‘Ž41=0,π‘Ž23=π‘Ž32π‘Ž=0,24=π‘Ž42=0,π‘Ž34=π‘Ž43=0.(33)𝑍 is positive definite if 𝜎21<4(πœ‡1+𝛼𝑉2/(π‘˜+𝑉2)),𝜎22<4(π‘Ÿ+πœ€+πœ‡1),𝜎23<4(βˆ’π‘”+𝛾𝑃2+πœ‡2), and 𝜎24<4(βˆ’π›½π›Ύπ‘‰2+πœ‡3) hold.
Hence inequality (32) can be written as πΏπ‘Š(𝑑,𝑒(𝑑))β‰€βˆ’πœ†|𝑒|2, where πœ† is the minimum of the eigenvalues of  𝑍. Thus we conclude that the conditions for mean square stability of the trivial solution of (23) are satisfied.

6. Discussions and Numerical Simulations

We now present some cases which occur in this system with their biological interpretations.

Case I (0<𝑅0<1). In this situation disease-free equilibrium is also globally asymptotically stable. It is unstable when 𝑅0>1, that is, when endemic equilibrium exists. Therefore reproduction number plays a vital role for the initiation of cholera in the community.

Case II (1<𝑅0<𝜏). In this case phage-free equilibrium is stable where as disease-free equilibrium is unstable when death rate of phage is above some threshold value 𝛽𝛾𝑉1. Otherwise when 𝑅0>𝜏, 𝐸1 is unstable and switches to 𝐸2. In this case, death rate of phage population controls the dynamics of the disease system and also influences its severity.

Case III (𝜏<𝑅0). In this case, we observe that the disease-free equilibrium and the phage-free equilibrium both are unstable, while the endemic equilibrium is locally stable. Furthermore, with the choice of the following parameters: π‘Ž = 5 dayβˆ’1, π‘˜ = 1000000 cells/liter, 𝛼 = 1 dayβˆ’1, πœ‡1 = 0.0000548 dayβˆ’1, π‘Ÿ = 0.2 dayβˆ’1, πœ€ = 0.015 dayβˆ’1, 𝑔 = 1 dayβˆ’1, 𝛿 = 5 cells/liter/day/person, 𝛾 = 0.0000000014 liters/virion/day, πœ‡2 = 2 dayβˆ’1, 𝛽 = 80 virions/cell, πœ‡3 = 0.000001 virion/day, we get 𝐸1 = (43048.7, 175.421, 877.106, 0) and 𝐸2 = (90211.9, 3.74536, 8.92857, 783856000). Consequently, the disease is endemic in nature and it is under controlled due to the coexistence of Vibrio cholerae and phage, which is clearly reflected from the second component of 𝐸1 and 𝐸2.

Case IV (A special case). Choosing the parameter values for the simulation in Figure 4, we obtain three equilibrium points 𝐸0 = (91240.9, 0, 0, 0), 𝐸4 = (43048.7, 175.421, 877.106, 0.00202946), and 𝐸5 = (91236.3, 0.0168352, 0.0396825, 800880000). From the numerical values we see that in one endemic region, there is scarcity of phage population where as in the other endemic region there is an abundance of phage population. Moreover, the number of infective is also greater in the region where there is scarcity of phage. Both equilibria 𝐸0 and 𝐸4 ultimately switch to 𝐸5 in the long run as they become asymptotically stable. Figure 4 depicts the dynamical behavior of 𝐸5. From simulations we can infer that abundance of bacteriophage in the environment reduces the number of Vibrio cholerae in the system and consequently infectives decrease in number. Hence in this case the advancement of disease may be controlled with an abundance of phage population.

fig4
Figure 4: Trajectories of the system for π‘Ž = 5 dayβˆ’1; π‘˜ = 1000000 cells/liter; 𝛼 = 1 dayβˆ’1; πœ‡1 = 0.0000548 dayβˆ’1; π‘Ÿ = 0.2 dayβˆ’1; πœ€ = 0.015 dayβˆ’1; 𝑔 = 1 dayβˆ’1; 𝛿 = 5 cells/liter/day/person; 𝛾 = 0.0000000014 liters/virion/day; πœ‡2 = 2 dayβˆ’1; 𝛽 = 90 virions/cell; πœ‡3 = 0.000000005 virion/day with 𝑆(0) = 80000; 𝐼(0) = 500; 𝑉(0) = 200000; 𝑃(0) = 100000 when 𝑅0 = 2.1213 and 𝜏 = 1.00001. 𝐸5 is stable.

The persistence result reflects that if Vibrio cholerae persists in the environment uniformly, then cholera also persists uniformly within the society. We observe that if the concentration of each infected population of Vibrio cholerae in the aquatic environment is less than a certain threshold value depending on mortality of human population (natural death and cholera-induced death), then cholera is endemic and moderate. So the contribution from human population to the aquatic environment has some impact in disease dynamics of cholera. Further from global stability results it appears that if death rate of human dominates the rate of exposure to contaminated water, then the system tends to be stable.

Positive net growth rate of Vibrio cholerae indicates that the disease cholera may be periodic in nature and persists in oscillating manner in the environment when phage burst size is less than a certain threshold value, that is, when number of phages produced per infected bacterium is less in number (see Figure 5). But when phage burst size is above that value, then the system becomes asymptotically stable (see Figure 6). In other words there is a stable nontrivial periodic solution bifurcated from the positive equilibrium; that is, increasing the average lytic time (i.e., decreasing the lysed rate) can also destabilize the interaction between Vibrio cholerae and phage population.

fig5
Figure 5: Simulated solution for π‘Ž = 5 dayβˆ’1; π‘˜ = 1000000 cells/liter; 𝛼 = 0.5 dayβˆ’1; πœ‡1 = 0.0001 dayβˆ’1; π‘Ÿ = 0.2 dayβˆ’1; πœ€ = 0.015 dayβˆ’1; 𝑔 = 1 dayβˆ’1; 𝛿 = 5 cells/liter/day/person; 𝛾 = 0.0000000014 liters/virion/day; πœ‡2 = 0.1 dayβˆ’1; 𝛽 = 172.1 virions/cell; πœ‡3 = 1 virion/day; 𝑆(0) = 1000; 𝐼(0) = 10; 𝑉(0) = 20000; 𝑃(0) = 10000 with 𝐸0= (50000, 0, 0, 0) and 𝐸2= (176.15, 329.959, 4150410, 643141000).
fig6
Figure 6: Simulated solution for a = 5 dayβˆ’1; k =1000000 cells/liter; 𝛼=0.5dayβˆ’1; πœ‡1=0.0001 dayβˆ’1; π‘Ÿ=0.2dayβˆ’1; πœ€=0.015 dayβˆ’1; 𝑔 = 1 dayβˆ’1; 𝛿 = 5 cells/liter/day/person; 𝛾=0.0000000014 liters/virion/day; πœ‡2=0.1dayβˆ’1; 𝛽 = 172.2 virions/cell; πœ‡3 = 1 virion/day; 𝑆(0) = 1000; 𝐼(0) = 10; 𝑉(0) = 20000; 𝑃(0) = 10000 with 𝐸0 (50000, 0, 0, 0) and 𝐸2= (176.169, 329.959, 4148000, 643141000).

It is interesting to note that when growth rate and death rate of Vibrio cholerae balance each other, we observe that there are two equilibria 𝐸0 and 𝐸2 appearing in the system. In this situation 𝐸0 becomes unstable and 𝐸2 is locally asymptotically stable (see Figure 7). Hence the disease remains endemic in this situation although it is rare to happen.

fig7
Figure 7: Simulated solution for π‘Ž = 5 dayβˆ’1; π‘˜ = 1000000 cells/liter; 𝛼 = 0.5 dayβˆ’1; πœ‡1 = 0.0001 dayβˆ’1; π‘Ÿ = 0.2 dayβˆ’1; πœ€ = 0.015 dayβˆ’1; 𝑔 = 1 dayβˆ’1; 𝛿 = 5 cells/liter/day/person; 𝛾 = 0.0000000014 liters/virion/day; πœ‡2 = 1 dayβˆ’1; 𝛽 = 10000 virions/cell; πœ‡3 = 1 virion/day; 𝑆(0) = 2000; 𝐼(0) = 50; 𝑉(0) = 50000; 𝑃(0) = 10000000 and 𝐸2= (2049.18, 317.555, 71428.6, 15877800) is stable.

Further stochastic analysis of the model system shows that system (3) with the association of random noise is stable about equilibrium 𝐸2 when the strengths of white noise are less than some specific quantities. For the choice of parameter values in Figure 8, we observe that trajectory graphs initially fluctuate randomly but ultimately approach its asymptotic level.

fig8
Figure 8: π‘Ž = 5 dayβˆ’1; π‘˜ = 1000000 cells/liter; 𝛼 = 1 dayβˆ’1; πœ‡1 = 0.0000548 dayβˆ’1; π‘Ÿ = 0.2 dayβˆ’1; πœ€ = 0.015 dayβˆ’1; 𝑔 = 1 dayβˆ’1; 𝛿 = 5; 𝛾 = 0.0000000014 liters/virion/day; πœ‡2 = 2 dayβˆ’1; 𝛽 = 80 virions/cell; πœ‡3 = 0.000001 dayβˆ’1; 𝛼1 = 3; 𝛼2 = 3; 𝛼3 = 3; 𝛼4 = 3 with 𝑆(0) = 80000; 𝐼(0) = 500; 𝑉(0) = 200000; 𝑃(0) = 100000.

On the basis of mathematical results and numerical simulations, we remark that for negative growth rate of Vibrio cholerae, that is, when πœ‡2>𝑔, the only equilibrium 𝐸0 exists and is stable if 0<𝑅0<1; that is, disease is incapable to invade into the system in this situation. But 𝐸0 and 𝐸1 coexist for 1<𝑅0<𝜏, and in this situation it is observed that 𝐸1 is stable whereas 𝐸0 is unstable. As a result, cholera is able to invade the system and becomes endemic. Again, if 𝑅0>𝜏, then 𝐸0, 𝐸1, and 𝐸2 all coexist. 𝐸0 and 𝐸1 are unstable while 𝐸2 is stable in this scenario. Furthermore, if concentration of each infected population of Vibrio cholerae in the aquatic environment is also lowered, then 𝐸1 and 𝐸2 are globally stable for 1<R0<𝜏 and 𝑅0>𝜏, respectively. It is interesting to note that global stability of 𝐸2 depends also on phage population, but disease is still endemic in this situation with less intensity because of the presence of phage population. Moreover, it is well observed that there is an inverse relationship among the Vibrio cholerae and Bacteriophage in the environment [27]. Human behaviour toward aquatic environment also influences the propagation of cholera. Our analysis reveals that the intensity of the disease may be kept under control by lowering the reproduction number under a certain threshold value.

The previous observations imply that cholera may spread into an epidemic when the rate of exposure to contaminated water increases, or when there are increasing number of immigrants into the region, or when each infected people contributes a lot of Vibrio cholera into the aquatic environment. The disease may be initiated in absence of bacteriophage virus. But as the phage population appears in the environment, it could slowly control the progress of Vibrio cholerae and ultimately check the endemicity of the disease. Hence with the association of phage virus, cholera is self-limiting in nature. We have also identified the parameters such as growth rate and death rate of Vibrio cholerae along with mortality rate of phage virus that play a significant role over the disease dynamics.

In brief, we conclude that, regardless whether phage exists in the environment or not, cholera persists in the environment. However, it is important to note that phage can reduce the number of Vibrio cholerae in the environment. Consequently, number of infective within the society is also decreased and severity of the disease is also checked. Hence by using phage as a biological control agent in endemic areas, we may also influence the temporal dynamics of cholera epidemics while reducing the excessive use of chemicals.

Acknowledgments

The authors are grateful to Professor Ying-Hen Hsieh for his valuable suggestions to improve this paper. P. Das is thankful to the National Science Council (NSC) and is funded by NSC postdoctoral fellowship (097-2811-M-039-003).

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