Qualitative Analysis of a Cholera Bacteriophage Model

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.

Figure 1 

Model Flow Diagram.

Our proposed mathematical model is as follows:𝑑𝑆𝑑𝑡=𝑎−𝑘𝛼+𝑉𝑆𝑉−𝜇¹𝑑𝐼𝑆+𝑟𝐼,𝑑𝑑𝑉𝑡=𝑘𝛼+𝑉𝑆𝑉−𝑟+𝜀+𝜇¹𝐼,𝑑𝑡=𝑔𝑉+𝛿𝐼−𝛾𝑉𝑃−𝜇²𝑑𝑃𝑉,𝑑𝑡=𝛽𝛾𝑉𝑃−𝜇³𝑃,(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 𝑡. Further the parameters 𝑎, 𝛼, 𝑟, and 𝜀 represent the constant immigration rate, the rate of exposure to contaminated water, the recovery rate, and the disease induced death rate, respectively. The parameter 𝑘 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, 𝜇₁, 𝜇₂ and 𝜇₃, 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 𝜇₂>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>𝜇₂, we derive the local stability conditions for the disease-free and endemic equilibrium. Hopf-bifurcation of the system is also investigated. The case 𝑔=𝜇₂ 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=(𝑎/𝜇₁,0,0,0).(ii)Phage-free equilibrium: 𝐸₁=(𝑆₁,𝐼₁,𝑉₁,0), where 𝑆₁=𝑟+𝜀+𝜇¹𝑎𝛿+𝑘𝜇₁+𝜀𝜇₂−𝑔𝐼₁𝛿𝛼𝜀+𝑟+𝛼+𝜀+𝜇₁𝜇₁,=𝑎𝛼𝛿−𝑘𝜇¹𝜇₂−𝑔𝑟+𝜀+𝜇₁𝑉₁𝛿𝛼𝜀+𝑟+𝛼+𝜀+𝜇₁𝜇₁,=𝑎𝛼𝛿−𝑘𝜇¹𝜇₂−𝑔𝑟+𝜀+𝜇₁𝜇₂−𝑔𝛼𝜀+𝑟+𝛼+𝜀+𝜇₁𝜇₁,(2) provided that 0<𝜇₂−𝑔<𝑎𝛼𝛿/𝑘𝜇₁(𝑟+𝜀+𝜇₁).(iii) Phage-cholera equilibrium: 𝐸₂=(𝑆₂,𝐼₂,𝑉₂,𝑃₂), where 𝑆₂=𝑎𝑟+𝜀+𝜇¹𝑘𝛽𝛾+𝜇₃𝐼𝛼𝜇₃𝜇₁+𝜀+𝜇₁𝑘𝛽𝛾+𝜇₃𝜇₁+𝑟+𝜀₂,=𝑎𝛼𝜇³𝑉𝛼𝜇₃𝜇₁+𝜀+𝜇₁𝑘𝛽𝛾+𝜇₃𝜇₁+𝑟+𝜀₂,=𝜇³𝑃𝛽𝛾₂,=𝑎𝛼𝛽𝛿−𝛼𝜇1₃𝜇₁+𝜀+𝜇₁𝑘𝛽𝛾+𝜇₃𝜇₁+𝑟+𝜀𝛾𝜇²−𝑔,(3) provided that 0<𝜇₂−𝑔<𝑎𝛼𝛽𝛿𝛾𝛼𝜇₃𝜇₁+𝜀+𝜇₁𝑘𝛽𝛾+𝜇₃𝜇₁+𝑟+𝜀(4) or 𝜇₂−𝑔≤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 get𝐅=⎛⎜⎜⎜⎝0𝑘𝑎𝜇𝛼¹00⎞⎟⎟⎟⎠,𝐕=⎛⎜⎜⎝𝑟+𝜀+𝜇¹−𝛿𝜇₂0−𝑔⎞⎟⎟⎠.(6) By definition, 𝑅₀=𝜌𝐅𝐕^−𝟏=𝑎𝛼𝛿𝑘𝜇₁𝑟+𝜀+𝜇₁𝜇₂−𝑔,(7) where 𝜌 is the spectral radius.

We can see that 𝑑𝑅₀/𝑑𝑎=𝛼𝛿/𝑘𝜇₁(𝑟+𝜀+𝜇₁)(𝜇₂−𝑔)>0. Therefore 𝑅₀ increases as 𝑎 increases. Again 𝑅₀ increases rapidly even for relatively small value of 𝑎, 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.

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+𝜇₃{𝑟𝜇₁+(𝜀+𝜇₁)(𝛼+𝜇₁)}/𝑘𝛽𝛾𝜇₁(𝑟+𝜀+𝜇₁). Consequently, the equilibrium points in the system, on the basis of 𝑅₀, are graphically presented in Figure 3.

Figure 3 

Scale of 𝑅0.

We now present dynamical behaviour of the system for three cases 𝜇₂>𝑔, 𝑔>𝜇₂, and 𝑔=𝜇₂ in the following section.

4. Stability Analysis

Case I (𝜇₂>𝑔). If 𝜇₂>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 𝜇₂−𝑔>0 and 𝛿<𝜀+𝜇₁. Then all solutions of system (1) which starts in ℜ⁴⁺ are bounded.

Proof. Let us define the function 𝑊=𝑆+𝐼+𝑉+𝑃/𝛽. Then the time derivative along the solution of system (1) is 𝑊̇=𝑎−𝜇₁𝑆−(𝜀+𝜇₁−𝛿)𝐼−(𝜇₂−𝑔)𝑉−(𝜇₃/𝛽)𝑃. For each 𝜎>0, the inequality 𝑊̇+𝜆𝑊≤𝑎 is satisfied, where 𝜎=min{𝜇₁,𝜀+𝜇₁−𝛿,𝜇₂−𝑔,𝜇₃}. Applying differential inequality argument, we obtain 0≤𝑊≤𝑎/𝜎. Hence system (1) is bounded and the solution of the system enter into the region 𝐵={(𝑆,𝐼,𝑉,𝑃)∈ℜ⁴⁺∶0≤𝑊≤𝑎/𝜎}.

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

Theorem 2. If 𝜇₂>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 (𝜆+𝜇₁)(𝜆+𝜇₃){(𝜆+𝑟+𝜀+𝜇₁)(𝜆+𝜇₂−𝑔)−𝑎𝛼𝛿/𝑘𝜇₁}=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 (𝛽𝛾𝑉₁−𝜇₃−𝜆)(𝜆³+𝐴𝜆²+𝐵𝜆+𝐶)=0, where 𝐴=𝜇₂−𝑔+𝑟+𝜀+2𝜇₁+𝛼𝑉¹𝑘+𝑉₁𝐵=𝜇₂,−𝑔𝑟+𝜀+2𝜇₁+𝛼𝑉¹+𝛼𝑉₁𝑘+𝑉₁𝜀+𝜇₁𝑘+𝑉₁+𝜇₁𝑟+𝜀+𝜇₁−𝛼𝛿𝑘𝑆¹𝑘+𝑉₁²𝐶=𝜇₂,−𝑔𝛼𝑉¹𝜀+𝜇₁𝑘+𝑉₁+𝜇₁𝜇²−𝑔𝑟+𝜀+𝜇₁−𝛼𝛿𝑘𝑆¹𝑘+𝑉₁².(8) Again, 𝐴, 𝐵, 𝐶, and (𝐴𝐵−𝐶) are all positive if (𝜇₂−𝑔)(𝑟+𝜀+𝜇₁)−𝛼𝛿𝑘𝑆₁/(𝑘+𝑉₁)²>0 since 𝜇₂−𝑔>0. Furthermore, the existence of 𝐸₁ implies (𝜇₂−𝑔)(𝑟+𝜀+𝜇₁)−𝛼𝛿𝑘𝑆₁/(𝑘+𝑉₁)²>0.
Hence, 𝐸₁ is locally asymptotically stable if 𝜇₃>𝛽𝛾𝑉₁ and unstable if 𝜇₃<𝛽𝛾𝑉₁, whenever it exists. Moreover, 𝜇₃>𝛽𝛾𝑉₁⇒𝑅₀<𝜏.

Theorem 4. Endemic equilibrium point 𝐸₂ is locally asymptotically stable if and only if 𝐴₄<(𝐴₁𝐴₂−𝐴₃)𝐴₃/𝐴²¹ holds.

Proof. The characteristic equation of the Jacobian matrix of system (1) at 𝐸₂=(𝑆₂,𝐼₂,𝑉₂,𝑃₂) is given by 𝜆⁴+𝐴₁𝜆³+𝐴₂𝜆²+𝐴₃𝜆+𝐴₄=0, where 𝐴₁=𝑟+𝜀+2𝜇₁+𝜇₂−𝑔+𝛾𝑃₂+𝛼𝑉²𝑘+𝑉₂𝐴₂,=𝛼𝑉²+𝑘+𝑟𝑉+₂𝜀𝜀++2𝜇𝜇₁₁+𝜇₁𝑟+𝜀+𝜇₁+𝛼𝑉²𝑘+𝑉₂𝜇²+𝛽𝛾₂−𝑔+𝛾𝑃₂𝑃₂𝑉₂−𝛼𝛿𝑘𝑆²𝑘+𝑉₂²𝐴₃,=𝛼𝑉²×𝑘𝜇+₂𝑉₂𝜀+𝜇₁+𝜇₁𝑟+𝜀+𝜇₁−𝑔+𝛾𝑃₂+𝛽𝛾²𝑃₂𝑉₂×𝑟+𝜀+2𝜇₁+𝛼𝑉²𝑘+𝑉₂−𝛼𝛿𝑘𝜇¹𝑆₂𝑘+𝑉₂²𝐴₄,=𝛼𝑉²𝑘+𝑉₂𝜀+𝜇₁+𝜇₁𝑟+𝜀+𝜇₁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(𝑡) is uniformly persistent whenever 𝑆(0)>0.

Proof. From the first equation of system (1), we get 𝑑𝑆/𝑑𝑡≥𝑎−(𝛼+𝜇₁)𝑆, since 𝑉/(𝑘+𝑉)<1⇒𝑑𝑆/𝑑𝑡+(𝛼+𝜇₁)𝑆≥𝑎. Applying differential inequality argument, we obtain lim_𝑡→∞inf𝑆(𝑡)≥𝑎/(𝛼+𝜇₁).

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

Proof. From the last equation of system (1) we get 𝑃̇<0 when 𝑉<𝜇₃/𝛽𝛾. In this case, 𝑃→0 as 𝑡→∞ which is impossible from biological point of view. Hence, 𝑉>𝜇₃/𝛽𝛾. Now from the second equation of the system, we get 𝐼(𝑡)≥𝛼𝑉𝑆≥(𝑘+𝑉)𝑟𝛼+𝜇𝜀₃+𝜇₁≥(𝑘+𝑉)𝛼𝑟𝑉+𝑎𝜀+𝜇₁𝜎𝑎𝑘𝛽𝛾+𝜇₃𝑟+𝜀+𝜇₁𝜎.(10) Moreover, if 𝐼(𝑡)<𝛼𝑉𝑆/(𝑘+𝑉)(𝑟+𝜀+𝜇₁), then, 𝑑𝐼/𝑑𝑡≥0 and hence 𝐼(𝑡) is increasing. So 𝐼(𝑡) becomes unbounded and hence we have 𝐼(𝑡)≥𝛼𝑉𝑆/(𝑘+𝑉)(𝑟+𝜀+𝜇₁).

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 ⎛⎜⎜⎝𝑑𝐼(𝑡)𝑑𝑉𝑑𝑡(𝑡)𝑑𝑡⎞⎟⎟⎠=(𝐹−𝑉)⎛⎜⎜⎝𝑉𝐼((𝑡𝑡))⎞⎟⎟⎠−⎛⎜⎜⎜⎝0𝑎𝛼𝑘𝜇₁−𝛼𝑆(𝑡)0𝛾𝑃𝑘(+𝑡)𝑉(𝑡)⎞⎟⎟⎟⎠⎛⎜⎜⎝𝑉𝐼((𝑡𝑡))⎞⎟⎟⎠.(11) Since 𝑎𝜇₁>(𝑘𝑆(𝑡))(𝑘+𝑉(𝑡))∀𝑡≥0in𝑅⁴⁺⎛⎜⎜⎝𝑑𝐼(𝑡),𝑑𝑉𝑑𝑡(𝑡)𝑑𝑡⎞⎟⎟⎠≤(𝐹−𝑉)⎛⎜⎜⎝𝑉𝐼((𝑡𝑡))⎞⎟⎟⎠.(12) Using the fact that eigenvalues of 𝐹−𝑉 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(𝑡),𝑉(𝑡))→(0,0) as 𝑡→∞. It follows by comparison theorem [19, 20] that (𝐼(𝑡),𝑉(𝑡))→(0,0). Substituting 𝐼=𝑉=0 in the first and fourth equations of model system (1), we obtain 𝑆(𝑡)→𝑎/𝜇₁ and 𝑃(𝑡)→0 as 𝑡→∞. Thus (𝑆(𝑡),𝐼(𝑡),𝑉(𝑡),𝑃(𝑡))→(𝑎/𝜇₁,0,0,0) as 𝑡→∞if 𝑅₀<1, and subsequently 𝐸₀ is globally asymptotically stable if 𝑅₀<1.

Theorem 8. Assume that 𝜇₂−𝑔>0 and 𝛿<𝜀+𝜇₁. 𝐸₁ is globally asymptotically stable, if 𝑀 is positive definite where 𝑀=[𝑎_𝑖𝑗]_4×4.

Proof. Define 𝑈(𝑆,𝐼,𝑉,𝑃)=12𝑆−𝑆¹²+12𝐼−𝐼¹²+12𝑉−𝑉¹+𝑃𝛽².(13) The time derivative of 𝑊 along the solution of system (1) is 𝑈̇=𝑆−𝑆₁𝑆̇+𝐼−𝐼₁𝐼̇+𝑉−𝑉₁+𝑃⟹𝑈̇=−𝜇₁𝛽𝑉̇+𝑃𝛽̇𝑆−𝑆₁²−𝑟+𝜀+𝜇₁𝐼−𝐼₁²−𝜇₂−𝑔𝑉−𝑉₁²−𝜇³𝛽²𝑃²+−𝑟𝜇₂𝑆−𝑆₁𝐼−𝐼₁+𝛿𝐼−𝐼₁𝑉−𝑉₁−𝑔+𝜇₃−𝛼𝛽𝑉𝑆𝑃𝑉−𝑉¹+𝛽𝛿𝑃𝐼−𝐼¹𝑘+𝑉−𝑉¹𝑆₁⟹𝑈̇=−𝜇₁𝑘+𝑉₁𝑆−𝑆¹−𝐼−𝐼₁𝑆−𝑆₁²−𝑟+𝜀+𝜇₁𝐼−𝐼₁²−𝜇₂−𝑔𝑉−𝑉₁²−𝜇³𝛽²𝑃²+−𝑟𝜇₂𝑆−𝑆₁𝐼−𝐼₁+𝛿𝐼−𝐼₁𝑉−𝑉₁−𝑔+𝜇₃−𝛼𝛽𝑉𝑆𝑃𝑉−𝑉¹+𝛽𝛿𝑃𝐼−𝐼¹𝑘+𝑉−𝑘𝑉+𝑆¹𝑉+𝑘𝑉+𝑆¹𝑉−𝑉¹𝑆₁×𝑆−𝑆₁𝑘+𝑉₁⟹𝑈̇≤−𝜇₁−𝐼−𝐼₁+𝛼𝑆−𝑆₁²−𝑟+𝜀+𝜇₁𝐼−𝐼₁²−𝜇₂−𝑔𝑉−𝑉₁²−𝜇³𝛽²𝑃²++(𝑟𝛿++𝛼)||𝛼𝑆𝑆−₁𝑆₁||||𝐼−𝐼₁||𝑘+𝑉₁||𝐼−𝐼¹+𝜇||||𝑉−𝑉₂||−𝑔+𝜇₃+𝛿𝛽𝑃||𝑉−𝑉||𝛽𝑃||𝐼−𝐼¹||+𝛼𝑆¹𝑘+𝑉₁=−𝑎₁₁||𝑆−𝑆₁||||𝑉−𝑉||𝑆−𝑆₁²−𝑎₂₂𝐼−𝐼₁²×−𝑎||𝐼₃₃−𝑉𝐼₁−𝑉₁²−𝑎₄₄𝑃²+2𝑎₁₂||𝑆−𝑆₁||×||𝑉−𝑉||||++22𝑎𝑎₁₃₂₄||𝑆−𝑆₁||||𝑉−𝑉||+2𝑎₂₃||𝐼−𝐼₁||=−𝑋_𝑇𝑃||𝐼−𝐼₁||+2𝑎₃₄𝑃||𝑉−𝑉||𝑀𝑋,(14) where 𝑋^𝑇={|𝑆−𝑆₁|,|𝐼−𝐼₁|,|𝑉−𝑉₁|,𝑃}and𝑀=[𝑎_𝑖𝑗]_4×4. Elements of the matrix 𝑀 are given by 𝑎₁₁=𝛼+𝑎𝜇₁,𝑎₂₂=𝑟+𝜀+𝜇₁,𝑎₃₃=𝜇₂−𝑔,₄₄=𝜇³𝛽²,𝑎₁₂=𝑎₂₁=−(𝑟+𝛼)𝑎₁₃2,=𝑎₃₁=−𝛼𝑆¹2𝑘+𝑉₁,𝑎¹⁴𝑎=𝑎₄₁=0,₂₃=𝑎₃₂=−1𝑎₂₄2𝛿+𝑘𝛼+𝑆𝑉¹₁,=𝑎₄₂=−𝛿2𝛽,𝑎³⁴=𝑎₄₃=−𝜇²−𝑔+𝜇₃2𝛽.(15) Here, 𝑀 is positive definite if ||||||𝑎₁₁𝑎₁₂𝑎₂₁𝑎₂₂||||||>0,|||||||||𝑎₁₁𝑎₁₂𝑎₁₃𝑎₂₁𝑎₂₂𝑎₂₃𝑎₃₁𝑎₃₂𝑎₃₃||||||||||||𝑎|||||||||>0,₁₁𝑎₂₁𝑎₁₂𝑎₁₃0𝑎₂₂𝑎₂₃𝑎₂₄𝑎₃₁𝑎₃₂𝑎₃₃𝑎₃₄0𝑎₄₂𝑎₄₃𝑎₄₄||||||||||||>0holdsimultaneously.(16) Further since 𝐵 is a global attractor, we may restrict our attention to solutions initiating in 𝐵. From the aforementioned inequalities, the right-hand side of (14), which is considered as a quadratic form in the variables |𝑆−𝑆₁|,|𝐼−𝐼₁|,and|𝑉−𝑉₁|,𝑃 is negative definite for (𝑆,𝐼,𝑉,𝑃)∈𝐵. Hence (𝑑/𝑑𝑡)𝑈(𝑆,𝐼,𝑉,𝑃) is negative definite about 𝐸₁ and consequently 𝑈(𝑆,𝐼,𝑉,𝑃) is a Lyapunov function for (𝑆,𝐼,𝑉,𝑃)∈𝐵. This completes the proof.

Theorem 9. Assume that 𝜇₂−𝑔>0 and 𝛿<𝜀+𝜇₁. 𝐸₂ is globally asymptotically stable, if 𝑁 is positive definite, where𝑁=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝛼+𝜇−(𝑟+𝛼)−(𝑟+𝛼)2−2𝑘𝛼𝑆+²𝑉₂02𝑟+𝜀+𝜇¹−1−𝛼𝑆₂2𝛿+𝑘𝛼+𝑆𝑉²₂−2𝛿𝛽2𝑘+𝑉₂−21𝛿+𝑘𝛼+𝑆𝑉²₂𝜇²−𝑔−𝜇²−𝑔+𝜇₃0−𝛿2𝛽2𝛽−𝜇²−𝑔+𝜇₃2𝛽𝜇𝛽³²⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(17)

Proof. Theorem 9 can be proved in a similar fashion as the proof of Theorem 8 by considering the positive definite function: 𝜓(𝑆,𝐼,𝑉,𝑃)=12𝑆−𝑆²+12𝐼−𝐼²²+12𝑉−𝑉²+𝑃𝛽−𝑃𝛽²².(18)

Case II (𝑔>𝜇₂). If 𝑔>𝜇₂, 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>𝜇₂ or if 𝑅₀=1 and 𝑔>𝜇₂+𝑟+𝜀+𝜇₁ 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 𝐷_𝑖>0(𝑖=1,2,3,4) for 𝜇₂<𝑔, where 𝐷₁=𝐴₁, 𝐷₂=𝐴₁𝐴₂−𝐴₃, 𝐷₃=𝐴₁(𝐴₂𝐴₃−𝐴₁𝐴₄)−𝐴²³, 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=𝛽_𝑐. For simplicity of notation, we assume that 𝐸^∗(𝑆^∗(𝛽),𝐼^∗(𝛽),𝑉^∗(𝛽),𝑃^∗(𝛽)) is the positive equilibrium of the system and let 𝜆⁴+𝐴₁(𝛽)𝜆³+𝐴₂(𝛽)𝜆²+𝐴₃(𝛽)𝜆+𝐴₄(𝛽)=0(19) be the characteristic equation of the variational system associated with system (1) about it. Let 𝜆_𝑖(𝛽),𝑖=1,2,3,4 be the roots of (19) with 𝑓(R𝛽e)𝜆=₁𝐴(𝛽₁)≤Re𝜆₂(𝛽)≤Re𝜆₃(𝛽)≤Re𝜆₄(𝛽),(𝛽)𝐴₂(𝛽)𝐴₃(𝛽)−𝐴²¹(𝛽)𝐴₄(𝛽)−𝐴²³(𝛽).(20)

Now we state the following theorem.

Theorem 12. If there exists 𝛽=𝛽_𝑐 such that (i)   𝑔>𝜇₂, (ii)𝑓(𝛽_𝑐)=0, 𝑓^(𝛽_𝑐)>0, and (iii)   𝐴₃(𝛽_𝑐)>0, then positive equilibrium 𝐸^∗(𝑆^∗(𝛽),𝐼^∗(𝛽),𝑉^∗(𝛽),𝑃^∗(𝛽)) is locally stable if 𝛽>𝛽_𝑐, 𝛽−𝛽_𝑐≪1 but unstable for 𝛽<𝛽_𝑐, 𝛽_𝑐−𝛽≪1,and a Hopf bifurcation of periodic solution occurs at 𝛽=𝛽_𝑐.

Proof. Here 𝑓′(𝛽_𝑐)>0 indicates that 𝑓(𝛽) is a monotonic increasing function in the neighborhood of 𝛽=𝛽_𝑐. This, together with 𝐴₃(𝛽_𝑐)>0, implies that 𝑓(𝛽_𝑐)<0, 𝐴₁(𝛽)𝐴₂(𝛽)−𝐴₃(𝛽)<0 for 𝛽>𝛽_𝑐, 𝛽−𝛽_𝑐≪1. Thus we have Re𝜆₃(𝛽_𝑐)<0 and Re𝜆₄(𝛽)<0, that is, the positive equilibrium 𝐸^∗ is locally stable.
Again monotonic increasing function 𝑓(𝛽) in the neighborhood of 𝛽=𝛽_𝑐 also implies that 𝑓(𝛽_𝑐)>0 for 𝛽<𝛽_𝑐, 𝛽_𝑐−𝛽≪1. Hence we have Re𝜆₄(𝛽)>0, that is, 𝐸^∗ is unstable.
It is easy to see that 𝑓(𝛽_𝑐)=𝐴₁(𝛽_𝑐)𝐴₂(𝛽_𝑐)𝐴₃(𝛽_𝑐)−𝐴²¹(𝛽_𝑐)𝐴₄(𝛽_𝑐)−𝐴²³(𝛽_𝑐)=0; 𝐴₁(𝛽_𝑐)>0 implies that 𝐴₂(𝛽_𝑐)𝐴₃(𝛽_𝑐)−𝐴₁(𝛽_𝑐)𝐴₄(𝛽_𝑐)>0.
Since𝑓′(𝛽_𝑐)>0, then by a theorem of [21] we have a simple Hopf bifurcation of periodic solution occurs at 𝛽=𝛽_𝑐.

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>𝜇₂⇒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=𝜇₂). If 𝑔=𝜇₂, 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 𝜇₂=𝑔, 𝐸₀ is neutrally stable when 𝑅₀=1 and unstable otherwise. Further 𝐸₂ coexists with 𝐸₀ and is locally asymptotically stable when 𝐴₄<(𝐴₁𝐴₂−𝐴₃)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(𝑡),𝐼(𝑡),𝑉(𝑡), and 𝑃(𝑡) 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<𝜇₂−𝑔<𝑎𝛼𝛽𝛿𝛾/(𝛼𝜇₃(𝜇₁+𝜀)+𝜇₁(𝑘𝛽𝛾+𝜇₃)(𝜇₁+𝑟+𝜀)) or 𝜇₂−𝑔≤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(𝑡),𝐼(𝑡),𝑉(𝑡), and 𝑃(𝑡) from values 𝑆₂,𝐼₂,𝑉₂, and 𝑃₂.

Subseqsuently we have𝑑𝑆=𝑎−𝛼𝑉𝑆𝑘+𝑉−𝜇¹𝑑𝐼=𝛼𝑉𝑆𝑆+𝑟𝐼𝑑𝑡+𝜎₁𝑆−𝑆₂𝑑𝜉^1𝑡,𝑘+𝑉−𝑟+𝜀+𝜇¹𝐼+𝜎₂𝐼−𝐼₂𝑑𝜉^2𝑡𝑑𝑉=𝑔𝑉+𝛿𝐼−𝛾𝑉𝑃−𝜇₂,𝑑𝑃=𝛽𝛾𝑉𝑃−𝜇₃𝑉+𝜎₃𝑉−𝑉₂𝑑𝜉^3𝑡,𝑃+𝜎₄𝑃−𝑃₄𝑑𝜉^4𝑡.(21) Here 𝜎_𝑗,𝑗=1, 2, 3, 4, are real constants, and 𝜉^𝑗𝑡=𝜉_𝑗(𝑡),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 variables𝑢𝑢₁₃=𝑆−𝑆₂,𝑢₂=𝐼−𝐼₂,=𝑉−𝑉₂,𝑢₄=𝑃−𝑃₂.(22) The linearized SDEs around 𝐸₂ take the form𝑑𝑢(𝑡)=𝑓(𝑢(𝑡))𝑑𝑡+𝑔(𝑢(𝑡))𝑑𝜉(𝑡),(23) where 𝑢(𝑡)=col𝑢₁(𝑡),𝑢₂(𝑡),𝑢₃(𝑡),𝑢₄(𝑡),(24)𝑓(𝑢(𝑡))=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−𝛼𝑉²𝑘+𝑉₂+𝜇₁𝑟−𝑘𝛼𝑆²𝑘+𝑉₂²𝛼𝑉₂0𝑘+𝑉₂−𝑟+𝜀+𝜇₁𝑘𝛼𝑆²𝑘+𝑉₂²0𝛿𝑔−𝛾𝑃₂0−𝜇₂−𝛾𝑉₂00𝛽𝛾𝑃₂𝛽𝛾𝑉₂−𝜇₃⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦𝑢(𝑡),(25)𝑔(𝑢)=⎡⎢⎢⎢⎢⎢⎢⎣𝜎¹0𝑢₁𝜎₂00000𝑢₂𝜎₃00000𝑢₃𝜎₂0𝑢₂⎤⎥⎥⎥⎥⎥⎥⎦.(26) In (23) the positive equilibrium 𝐸₂ corresponds to the trivial solution, 𝑢(𝑡)=0.

Let 𝑈=(𝑡≥𝑡₀)×𝑅^𝑛,𝑡₀∈𝑅⁺, and consider a twice continuously differentiable function, 𝑊. With reference to [26], the following theorem holds. Note that for (23),𝐿𝑊(𝑡,𝑢)=𝜕𝑊(𝑡,𝑢)𝜕𝑡+𝑓^𝑇(𝑢)𝜕𝑊(𝑡,𝑢)+1𝜕𝑢2𝑇𝑟𝑔^𝑇(𝑢)𝜕²𝑊(𝑡,𝑢)𝜕𝑢²𝑔(𝑢),(27) where𝜕𝑊𝜕𝑢=Col𝜕𝜕𝑊𝑢₁,𝜕𝑊𝜕𝑢₂,𝜕𝑊𝜕𝑢₃,𝜕𝑊𝜕₂𝜕𝑢₄,𝑊(𝑡,𝑢)𝜕𝑢²=𝜕²𝑊𝜕𝑢_𝑗𝜕𝑢_𝑖,𝑖,𝑗=1,2,3,4,(28)

and “𝑇” represents transposition.

Theorem 14. Assume that there exists a function 𝑊(𝑡,𝑢)∈𝐶⁰²(𝑈) satisfying the following inequalities: 𝐾₁𝐿|𝑢𝑊|^𝑝≤(𝑡,𝑊𝑢)(≤𝑡,𝑢−)≤𝐾₃𝐾₂|𝑢|^𝑝,(29)|𝑢|^𝑝,(30) where 𝐾_𝑖and p are positive constants. Then the trivial solution of (23) is exponentially p-stable for 𝑡≥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<𝜇₂−𝑔<𝑎𝛼𝛽𝛿𝛾/(𝛼𝜇₃(𝜇₁+𝜀)+𝜇₁(𝑘𝛽𝛾+𝜇₃)(𝜇₁+𝑟+𝜀)) or 𝜇₂−𝑔≤0. Further assume that 𝜎²¹<4(𝜇₁−𝑟),𝜎²²<4(𝜀+𝜇₁−𝛼𝑉₂/(𝑘+𝑉₂)),𝜎²³<4(−𝑔+𝛾𝑃₂+𝜇₂), and 𝜎²⁴<4(−𝛽𝛾𝑉₂+𝜇₃)   hold. Then the zero solution of (23) is asymptotically mean square stable.

Proof. Let us consider the Lyapunov function 𝑊(𝑢)=(1/2)(𝑢²¹+𝑢²²+𝑢²³+𝑤𝑢²⁴), where 𝑤>0 is a constant to be chosen later. It is easy to check that inequality (29) holds true with 𝑝=2. Moreover, 𝐿𝑊(𝑢)=−2𝛼𝑉²𝑘+𝑉₂+𝜇₁−21𝜎²¹𝑢²¹−2𝑟+𝜀+𝜇₁−12𝜎²²𝑢²²−2−𝑔+𝛾𝑃₂+𝜇₃−12𝜎²³𝑢²³−𝑤2−𝛽𝛾𝑉₂+𝜇₄−12𝜎²⁴𝑢²⁴+2𝑟+𝛼𝑉²𝑘+𝑉₂𝑢¹𝑢₂−2𝛼𝑘𝑆²𝑘+𝑉₂²𝑢₁𝑢₃+2−𝛾𝑉₂+𝑤𝛽𝛾𝑃₂𝑢₃𝑢₄.(31) Choosing 𝑤=𝑉₂/𝛽𝑃₂ and using 𝑢²¹+𝑢²²≥2𝑢₁𝑢₂ from (31), we obtain 𝐿𝑊(𝑢)≤−2𝜇₁−𝑟−12𝜎²¹𝑢²¹−2𝜀+𝜇₁−𝛼𝑉²𝑘+𝑉₂−21𝜎²²𝑢²²−2−𝑔+𝛾𝑃₂+𝜇₃−12𝜎²³𝑢²³−𝑤2−𝛽𝛾𝑉₂+𝜇₄−12𝜎²⁴𝑢²⁴≤−𝑌^𝑇𝑍𝑌,(32) where 𝑍=[𝑎_𝑖𝑗]_4×4 and 𝑌=(𝑢₁,𝑢₂,𝑢₃,𝑢₄)^𝑇. Here 𝑎₁₁=2𝜇₁−𝑟−12𝜎²¹𝑎₂₂,=2𝜀+𝜇₁−𝛼𝑉²𝑘+𝑉₂−21𝜎²²𝑎₃₃,=2−𝑔+𝛾𝑃₂+𝜇₃−12𝜎²³𝑎₄₄,=2−𝛽𝛾𝑉₂+𝜇₄−12𝜎²⁴𝑎₁₂,𝑎=𝑎₂₁=0,𝑎₁₃=𝑎₃₁=0,₁₄𝑎=𝑎₄₁=0,𝑎₂₃=𝑎₃₂=0,₂₄=𝑎₄₂=0,𝑎₃₄=𝑎₄₃=0.(33)𝑍 is positive definite if 𝜎²¹<4(𝜇₁+𝛼𝑉₂/(𝑘+𝑉₂)),𝜎²²<4(𝑟+𝜀+𝜇₁),𝜎²³<4(−𝑔+𝛾𝑃₂+𝜇₂), and 𝜎²⁴<4(−𝛽𝛾𝑉₂+𝜇₃) hold.
Hence inequality (32) can be written as 𝐿𝑊(𝑡,𝑢(𝑡))≤−𝜆|𝑢|², where 𝜆 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: 𝑎 = 5 day⁻¹, 𝑘 = 1000000 cells/liter, 𝛼 = 1 day⁻¹, 𝜇₁ = 0.0000548 day⁻¹, 𝑟 = 0.2 day⁻¹, 𝜀 = 0.015 day⁻¹, 𝑔 = 1 day⁻¹, 𝛿 = 5 cells/liter/day/person, 𝛾 = 0.0000000014 liters/virion/day, 𝜇₂ = 2 day⁻¹, 𝛽 = 80 virions/cell, 𝜇₃ = 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.

(a)

(b)

(c)

(d)

Figure 4 

Trajectories of the system for 𝑎 = 5 day−1; 𝑘 = 1000000 cells/liter; 𝛼 = 1 day−1; 𝜇1 = 0.0000548 day−1; 𝑟 = 0.2 day−1; 𝜀 = 0.015 day−1; 𝑔 = 1 day−1; 𝛿 = 5 cells/liter/day/person; 𝛾 = 0.0000000014 liters/virion/day; 𝜇2 = 2 day−1; 𝛽 = 90 virions/cell; 𝜇3 = 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.

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Figure 5 

Simulated solution for 𝑎 = 5 day−1; 𝑘 = 1000000 cells/liter; 𝛼 = 0.5 day−1; 𝜇1 = 0.0001 day−1; 𝑟 = 0.2 day−1; 𝜀 = 0.015 day−1; 𝑔 = 1 day−1; 𝛿 = 5 cells/liter/day/person; 𝛾 = 0.0000000014 liters/virion/day; 𝜇2 = 0.1 day−1; 𝛽 = 172.1 virions/cell; 𝜇3 = 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).

(a)

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Figure 6 

Simulated solution for a = 5 day−1; k =1000000 cells/liter; 𝛼 =0.5day−1; 𝜇1 =0.0001 day−1; 𝑟=0.2day−1; 𝜀 =0.015 day−1; 𝑔 = 1 day−1; 𝛿 = 5 cells/liter/day/person; 𝛾 =0.0000000014 liters/virion/day; 𝜇2 =0.1day−1; 𝛽 = 172.2 virions/cell; 𝜇3 = 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.

(a)

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Figure 7 

Simulated solution for 𝑎 = 5 day−1; 𝑘 = 1000000 cells/liter; 𝛼 = 0.5 day−1; 𝜇1 = 0.0001 day−1; 𝑟 = 0.2 day−1; 𝜀 = 0.015 day−1; 𝑔 = 1 day−1; 𝛿 = 5 cells/liter/day/person; 𝛾 = 0.0000000014 liters/virion/day; 𝜇2 = 1 day−1; 𝛽 = 10000 virions/cell; 𝜇3 = 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.

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Figure 8 

𝑎 = 5 day−1; 𝑘 = 1000000 cells/liter; 𝛼 = 1 day−1; 𝜇1 = 0.0000548 day−1; 𝑟 = 0.2 day−1; 𝜀 = 0.015 day−1; 𝑔 = 1 day−1; 𝛿 = 5; 𝛾 = 0.0000000014 liters/virion/day; 𝜇2 = 2 day−1; 𝛽 = 80 virions/cell; 𝜇3 = 0.000001 day−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 𝜇₂>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).