# Analysis of Cholera Epidemic Controlling Using Mathematical Modeling

Authors: Kinfe Hailemariam Hntsa; Berhe Nerea Kahsay
Journal: International Journal of Mathematics and Mathematical Sciences
 (2020)

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## Abstract

The purpose of this study is to see whether it is possible to eradicate the disease theoretically using mathematical modeling with the aid of numerical simulation when disease occurs in a population by implementing adequate preventive measures. For this, we consider a mathematical model for the transmission dynamics of cholera and its preventive measure as one cohort of individuals, namely, a protected cohort in addition to susceptible, infected, and recovered cohorts of individuals including the concentration ofVibrio cholerae in the contaminated aquatic reservoir with small modifications. We calculate the basic reproduction number, ℛ0, and investigate the existence and stability of equilibria. The model possessed forward bifurcation. Moreover, we compute the sensitivity indices of each parameter in relating to ℛ0 of the model. Numerical simulations are carried out to validate our theoretical results. The result indicates that the disease dies out in areas with adequate preventive measures and widespread and kills more people in areas with the inadequate preventive measures.


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## Body

## 1. Introduction

The cholera epidemic is a fatal waterborne disease causing diarrhea, dehydration, and vomiting in an individual [1, 2]. It is caused by a bacterium called Vibrio cholerae. Cholera is transmitted through ingesting contaminated drinks and food, contact with cholera patient’s feces, and touching vomit and corpse killed by the bacterium without using protective agents [1–3]. The incubation period of cholera is less than 24 hours to 5 days. The infection is frequently asymptomatic. Not more than 25% of the infected persons become symptomatic; of these, 10–20% experience severe disease [1]. A continuous drastic loss of body fluids leads to dehydration, thus neglecting treatment as soon as the case occurs, and accelerates the death of the infected person within hours [4].There are environmental factors that play a great role in the propagation of cholera infection. AsVibrio cholerae can move in the aquatic environment, every alteration in the hydrological cycle has a probability to affect the pathogenic concentration in water. The rain and its seasonal behavior, droughts and floods, can increase or decrease the transmission process [5].Nevertheless, relevance has to be given to the environmental matrix in which the disease spreads into disease-free regions [6, 7] together with consideration on individual mobility and travelers carrying the disease in long-distance journeys. Susceptible people traveling on a daily basis may contact the disease in destination sites and take the disease back to the possibly uninfected communities where they regularly live. At the same time, infected individuals not showing severe symptoms can carry the illness releasing bacteria via their feces [8]. Finally, symptomatic infected individuals locally increase the bacteria concentration, which is, then, spread along the hydrological network.Experiments for spreading of infectious disease in individual are unethical. Epidemiologists and other researchers use mathematical modeling and numerical simulation for scientific understanding about the dynamics and preventive method of an infectious disease, for determining sensitivities, changes of parameter values, and to forecasting [5, 9–19]. The models are based on cohorts, namely, susceptible, infected, and recovered, for individual population incorporated with some preventive measures such as treatment, vaccination, chlorination, hygienic, and sanitation through education. Treatment is more recommended when cases occur, that is, a short-time plan of controlling and eradicating the disease. Education is the most recommended option in the disease controlling and eradicating from the community whenever. Education is a long-term plan of controlling and eradicating the disease, especially in creating awareness of water treatment, sewage removal, food safety, personal hygiene, and environment sanitation [3, 20–22]. It is an undeniable truth that there are individual populations who have awareness on the adequate preventive measures of cholera diseases and compliance them. In our best knowledge, those populations are not considered as one cohort of individuals in addition to susceptible, infected, and recovered individual cohorts in order to analyse the dynamics, control, and eliminate cholera.The objective of this study is to analyse qualitatively the controlling mechanisms of cholera epidemics in a population by implementation of adequate preventive measures.


## 2. Formulation of Mathematical Modeling

We divided the population denoted byNt according to the infection status into St- susceptible, It- infected, Rt- recovered, and Ut- prevented individuals at given time t. Moreover, Bt is the amount of concentration of Vibro cholerae in an aquatic environment at time t. New susceptible individuals are recruited into the community at a rate of b and 1−pρ, where b is birth or immigration rate of individuals, p is the fraction of the compliance of hygienic, ingestion of cholera bacterium and contact with cholera patients of susceptible and recovered cohort, and ρ is the loss rate of immunity of recovered individuals. They either die from suffering at a rate μ, where μ is the sum of the natural death rate and population-dependent death rate, or move to an infectious cohort by acquiring cholera through contact with the aquatic reservoir at a rate of 1−pβB/B+K or move to the protected cohort at a rate of pα, where α is progression rate of susceptible individuals to the protected cohort due to the preventive method.We note that an individual must consume at least the concentration,K, of Vibrios cholerae equivalent to an amount that increases the possibility of being infected to about 50% if they are to contact the infection. Infected individuals are assumed to recover at rate τ. Infectious individuals who do not recover die at rate μ+d, where d is the cholera-induced death rate. Recovered individuals move to protected cohorts at a rate of pρ. Infected individuals in the community shed the pathogen into the aquatic environment at rate 1−θϕ, where θ is the fraction of the compliance of sanitation of an infected cohort and ϕ is the average contribution of each infected individual to the pathogen population of Vibrio cholerae. The individual population follows logistic growth due to intraspecific competition for limited environmental resources in a disease-free environment. The complete description of parameters is in Table 1.Table 1
Parameters for system of (1), where c is capita, l is litter, and T is time.
ParameterParameter definitionDimensionbConstant recruitment (birth or immigration) rate of an individualb=T−1μiDensity-independent death rate of an individualμi=T−1μdDensity-dependent death rate of an individualμd=c−1T−1dDisease-induced death rate of an individuald=T−1βRates of ingestingVibrio cholerae from the contaminated aquatic reservoirβ=T−1τProgression rate of infected individuals to the recovered class due to treatmentτ=T−1αProgression rate of susceptible individuals to the protected class due to the preventive methodα=T−1ρThe rate of loss of immunity of recovered individualsρ=T−1ϕThe average contribution of each infected individual to the pathogen population ofVibrio choleraeϕ=l−1T−1KThe concentration ofVibrio cholerae in food and water that yield 50% chance of acquiring cholera diseaseK=cl−1pFraction of the compliance of hygienic, ingestion of cholera bacterium and contact with cholera patients of the susceptible and recovered cohortp=1θFraction of the compliance of sanitation of the infected cohortθ=1bBBirth rate ofVibrio choleraebB=T−1μBDeath rate ofVibrio choleraeμB=T−1We assumed that the adequate preventing measures are using of safe water for drinking, washing, and food preparation; disposing feces in a sanitary manner; removing or washing any bedding or clothing that might have contact with cholera-infected people; restraining a cholera-infected person from swimming even until two weeks after individuals recover; washing hands always with soap and clean water before preparing food, eating, feeding children, after using a latrine, and taking care of patients with diarrhea; cleaning food preparation areas; and storing treated water and food in clean and covered containers.Based on these assumptions and the flow diagram in Figure1, we formulate the following SIRUB model, which is a system of nonlinear autonomous ordinary differential equations:(1)dSdt=bN+1−pρR−1−pβBB+K+pα+μS,dIdt=1−pβBSB+K−τ+d+μI,dRdt=τI−ρ+μR,dUdt=pαS+pρR−μU,dBdt=1−θϕI+bB−μBB,where μ=μi+μdN and Nt=St+It+Rt+Ut, and with initial conditions given, S0=S0≥0,I0=I0≥0,R0=R0≥0,U0=U0≥0,B0=B0≥0.Figure 1
The schematic depiction of the dynamics and prevention measures of cholera.Theorem 1.
(mathematically and epidemiologically well posed). Assuming that the initial conditions lie inΩ, the dynamical system of equation (1) has a unique bounded positive solution in Ω for all time t≥0, where(2)Ω=S,I,R,U,B∈ℝ+5:S+I+R+U=N,0≤N≤L,0≤B≤1−θϕLμB−bB,provided that μB>bB.Proof.
We show first boundness. The total individual population becomes(3)dNdt=dSdt+dIdt+dRdt+dUdt=bN−μN−dI=Nb−μi−μdN−dI,=b−μiN1−Nb−μi/μd−dI=rhN1−NL−dI,where rh=b−μi and L=b−μi/μd which are known as the intrinsic growth rate and carrying capacity of an individual population, respectively. In the absence of the disease (I=0), equation (3) can be written as(4)dNdt≤rhN1−NL⟹Nt≤LL/N0−1e−rht+1.
Now, it is observed that(5)0≤Nt≤limt⟶∞LL/N0−1e−rht+1=L.
Moreover, the last equation of the dynamical system can be expressed as(6)dBdt+μB−bBB=1−θϕI,and its solution is given by(7)Bt=ce−μB−bBt+1−θϕIμB−bB.
Thus, we get(8)0≤Bt≤limt⟶∞ce−μB−bBt+1−θϕIμB−bB=1−θϕIμB−bB≤1−θϕLμB−bB.
Therefore,Nt and Bt are bounded.
The dynamical system of equation (1) has a unique solution because the right-hand side of equation (1) is continuous with continuous partial derivatives in Ω.
To show the positivity invariant, let us consider the first equation of system (1), and it becomes(9)dSdt≥−1−pβBB+K+pα+μS.
This gives(10)St≥S0e−1−pβB/B+K+pα+μt≥0.
Similarly, we get(11)It≥I0e−τ+d+μt≥0,Rt≥R0e−ρ+μt≥0,Ut≥U0e−μt≥0,Bt≥B0e−μB−bBt≥0.
The solutions inΩ are all nonnegative. Hence, the domain of biological significance is positively invariant and attracting. Therefore, the dynamical system in equation (1) has a unique bounded positive solution in Ω.


## 3. Stability Analysis of Equilibrium Points

### 3.1. Disease-Free Equilibrium Point

The disease-free equilibrium point is the steady-state solutions determined when there is no diseaseI=0, and it is unique given by(12)E0=S∗,0,0,U∗,0,where S∗=b/pα+bN∗ and U∗=pα/pα+bN∗, N∗=L. It is observed that if pα≫b, then b/pα+b≈0 and pα/pα+b≈1 which indicate that St≈0 and Ut≈Nt. From the biological point of view, the disease can be controlled and eliminated from the community when the rate of susceptible individuals moving to a protected cohort due to preventive methods is greater than the recruitment rate of individuals.


### 3.2. Basic Reproduction Number

The basic reproduction number represents the average number of new infected individuals that result from one infected individual introduced into a fully susceptible population. The basic reproduction numberℛ0 can be determined using the method of next-generation matrix, as presented in [23, 24]. Thus, to determine ℛ0, let us find the spectral radius of the matrix FV−1, where(13)F=01−pβS∗K00,V=τ+d+b0−1−pϕμB−bB.Letk1=τ+d+b and k2=μB−bB, and we get the inverse of V and FV−1 as(14)V−1=1k101−θϕk1k21k2,FV−1=1−p1−θβϕS∗Kk1k21−pβS∗Kk100.Hence, the spectral radius of the matrixFV−1 is found to be(15)ℛ0=1−p1−θβϕS∗Kk1k2,=1−pβbb−μiμdτ+d+bpα+b1−θϕKμB−bB.The quantity1/τ+d+b indicates the maximum time an individual is expected to stay in an infected cohort. The basic reproduction number consists of two terms which characterize the contribution from the different pathways to new infections with cholera. The quantity ℛ0h=1−pβbb−μi/μdτ+d+bpα+b shows new infection of individuals due to person-to-person contact, and ℛ0B=1−θϕ/KμB−bB represents the new infection of individuals that results from consumption of the pathogen from the aquatic environment.


### 3.3. Stability of the Disease-Free Equilibrium Point

The dynamical system (1) is nonlinear. Its local stability is determined from the sign of the eigenvalues of the corresponding Jacobian matrix at the disease-free equilibrium point. The Jacobian matrix corresponding to the system in equation (1) at any point E=S,I,R,U,B is(16)JE=−1−pβBK+B+pα+μ01−pρ0−1−pβKSK+B21−pβBK+B−τ+d+μ001−pβKSK+B20τ−ρ+μ00pα0pρ−μ001−θϕ00−μB−bB.Therefore, the Jacobian matrix at the disease-free equilibrium point is(17)JE0=−pα+b01−pρ0−1−pβS∗K0−τ+d+b001−pβS∗K0τ−ρ+b00pα0pρ−b001−θϕ00−μB−bB.Theorem 2.
(local stability of the disease-free equilibrium point). Ifℛ0<1, then the disease-free equilibrium point E0 of the dynamical system (1) is locally asymptotically stable, and if ℛ0>1, then E0 is unstable.Proof.
Consider that the characteristics equation of the Jacobian matrix at the disease-free equilibrium point is given byJE0−λI5=0, where I5 is the identity matrix and can be written as(18)λ+pα+bλ+bλ+ρ+bλ2+a1λ+a2=0,where a1=k1+k2 and a2=k1k21−ℛ0.
The roots of the polynomialλ+pα+bλ+bλ+ρ+b=0 are a negative real value because pα+b>0,b>0 and ρ+b>0. We use the Routh–Hurwitz criterion to show that the roots of the polynomial,(19)λ2+a1λ+a2=0,have negative real values. To have negative real values for roots of the polynomial, in equation (19), both a1=k1+k2 and a2=k1k21−ℛ0 must be positive. k1=τ+d+b>0 and k2=μB−bB>0 because all parameters are positive constants, and μB>bB . Thus, a1=k1+k2>0 because it is the sum of the positive number and a2=k1k21−ℛ0>0 if and only if ℛ0<1. Hence, the roots of a polynomial in equation (19) have a negative real value. Therefore, the roots of the characteristic equation (18) have a negative real value which indicates the disease-free equilibrium point is locally asymptotically stable. This means, the disease cannot sustain in the population, and thus, it is eventually disappearing. On the other hand, if a2<0, then ℛ0>1. Thus, there exists at least one positive eigenvalue, and thus, the Routh–Hurwitz criterion tells us the disease-free equilibrium point is unstable. Biologically, the disease becomes endemic or the disease persists in the population.Theorem 3.
(global stability of the disease-free equilibrium point). Ifℛ0<1, then the cholera-free equilibrium point E0 of the dynamical system in equation (1) is globally asymptotically stable, and if ℛ0>1, then E0 is unstable.Proof.
Consider a Lyapunov functionVt=B. From the dynamical system in equation (1) by setting dI/dt=0, we get(20)I=1−pβBSτ+d+μB+K.
Moreover,S=S∗, μ=b at the disease-free equilibrium point, and B/B+K≤B/K and takes the equality. Then,(21)dVdt=1−θϕI+bB−μBB=1−p1−θϕβSBτ+d+μB+K−μB−bBB≤μB−bB1−p1−θϕβS∗Kτ+d+bμB−bB−1B,atE0,=μB−bBℛ0−1B.
Whenℛ0≤1, dV/dt is negative semidefinite, with equality at ℛ0=1. Therefore, the largest compact invariant set in Ω such that dV/dt=0 when R0≤1 is the singleton E0. Hence, by the LaSalle invariance principle [25], the disease-free equilibrium E0 is globally asymptotically stable in Ω and unstable otherwise.
Epidemiologically, the implication of Theorem3 is that when ℛ0 is less than unity, a small inflow of cholera-infected individuals into the community does not yield an outbreak. The subsequent numbers of those infected will be less than that of their predecessors, and eventually, the disease will be eradicated. We note also that a number of parameters are necessary in reducing the disease threshold to a value ℛ0<1.


### 3.4. Endemic Equilibrium Point

The endemic equilibrium points are steady-state solutions determined whenI>0, and there exist at least one endemic equilibrium point, E=S,I,R,U,B.Theorem 4.
(existence of at least one endemic equilibrium point). IfI>0, then ℛ0>1 and there exists at least one endemic equilibrium point for the model in equation (1).Proof.
Suppose thatI>0 and d=0 (there is no disease-induced death). Thus, we get N=N∗=L and μ=b. Setting dB/dt=0 and dR/dt=0, we obtain(22)B=1−θϕIk2,R=τIρ+b.
Similarly, by settingdS/dt+dI/dt+dR/dt=0, we yield(23)S=bN−pρ+bR+bIpα+b=S∗−pρ+bτ+b+ρbb+ρpα+bI≤S∗=bN∗pα+b.
Moreover, by settingdU/dt=0, we get(24)U=pαS+pρRb=pαbS∗−pρ+bτ+b+ρbb+ρpα+bI+pρτIbρ+b.
Finally, settingdI/dt=0 results in(25)I=1−pβBSB+Kk1=1−p1−θβϕIS∗−pρ+bτ+b+ρb/b+ρpα+bI1−θϕI+Kk2k1,and by solving I explicitly, we obtain(26)I=Kk1k21−p1−θβϕS∗/Kk1k2−1b+ρpα+b1−θϕk1b+ρpα+b+π,=Kk1k2ℛ0−1b+ρpα+b1−θϕk1b+ρpα+b+π,where π=1−p1−θβϕpρ+bτ+b+ρb. Here, we observe that I>0 if and only if ℛ0>1. Therefore, if I>0, then ℛ0>1 and there exists at least one endemic equilibrium point for the dynamical equation (1) given by(27)E=S∗,I∗,R∗,U∗,B∗,whereS∗=S∗−pρ+bτ+b+ρbb+ρpα+bI∗,I∗=Kk1k2ℛ0−1b+ρpα+b1−θϕk1b+ρpα+b+π,R∗=τI∗ρ+b,U∗=pαbS∗−pρ+bτ+b+ρbb+ρpα+bI∗+pρτI∗bρ+b,B∗=1−θϕI∗k2.Theorem 5.
(global stability of the endemic equilibrium point). Ifℛ0>1, then the endemic equilibrium point E of the dynamical system in equation (1) is globally asymptotically stable, and if ℛ0<1, then E is unstable.Proof.
Consider a Lyapunov functionVt=B−B∗−B∗lnB/B∗. From the dynamical system in equation (1), by setting dI/dt=0, we get(28)I=1−pβBSk1B+K.
Then,(29)dVdt=dBdt−B∗2BdBdt=1−θϕI−k2B−B∗2B1−θϕI−k2B,=1−p1−θϕβSBB+Kk1−k2B+B∗2Bk2B−1−p1−θϕβSBB+Kk1,letμ=bandB=2K=1−p1−θϕβ2KS3Kk1−2k2K+B∗22K2k2K−1−p1−θϕβ2KS3Kk1≤1−p1−θϕβ2KS∗3Kk1−2k2K3+B∗22K2k2K−1−p1−θϕβ2KS∗Kk1,=23k2Kℛ0−1+k2B∗21−ℛ0,=k21−ℛ0−23K+B∗2≤k21−ℛ0B∗2.
Whenℛ0≥1, dV/dt is negative semidefinite, with equality at ℛ0=1. Therefore, the largest invariant set in dV/dt=0 is the singleton E. By LaSalle’s invariance principle [25], we therefore, conclude that the endemic equilibrium E is globally asymptotically stable if ℛ0>1.


## 3.1. Disease-Free Equilibrium Point

The disease-free equilibrium point is the steady-state solutions determined when there is no diseaseI=0, and it is unique given by(12)E0=S∗,0,0,U∗,0,where S∗=b/pα+bN∗ and U∗=pα/pα+bN∗, N∗=L. It is observed that if pα≫b, then b/pα+b≈0 and pα/pα+b≈1 which indicate that St≈0 and Ut≈Nt. From the biological point of view, the disease can be controlled and eliminated from the community when the rate of susceptible individuals moving to a protected cohort due to preventive methods is greater than the recruitment rate of individuals.


## 3.2. Basic Reproduction Number

The basic reproduction number represents the average number of new infected individuals that result from one infected individual introduced into a fully susceptible population. The basic reproduction numberℛ0 can be determined using the method of next-generation matrix, as presented in [23, 24]. Thus, to determine ℛ0, let us find the spectral radius of the matrix FV−1, where(13)F=01−pβS∗K00,V=τ+d+b0−1−pϕμB−bB.Letk1=τ+d+b and k2=μB−bB, and we get the inverse of V and FV−1 as(14)V−1=1k101−θϕk1k21k2,FV−1=1−p1−θβϕS∗Kk1k21−pβS∗Kk100.Hence, the spectral radius of the matrixFV−1 is found to be(15)ℛ0=1−p1−θβϕS∗Kk1k2,=1−pβbb−μiμdτ+d+bpα+b1−θϕKμB−bB.The quantity1/τ+d+b indicates the maximum time an individual is expected to stay in an infected cohort. The basic reproduction number consists of two terms which characterize the contribution from the different pathways to new infections with cholera. The quantity ℛ0h=1−pβbb−μi/μdτ+d+bpα+b shows new infection of individuals due to person-to-person contact, and ℛ0B=1−θϕ/KμB−bB represents the new infection of individuals that results from consumption of the pathogen from the aquatic environment.


## 3.3. Stability of the Disease-Free Equilibrium Point

The dynamical system (1) is nonlinear. Its local stability is determined from the sign of the eigenvalues of the corresponding Jacobian matrix at the disease-free equilibrium point. The Jacobian matrix corresponding to the system in equation (1) at any point E=S,I,R,U,B is(16)JE=−1−pβBK+B+pα+μ01−pρ0−1−pβKSK+B21−pβBK+B−τ+d+μ001−pβKSK+B20τ−ρ+μ00pα0pρ−μ001−θϕ00−μB−bB.Therefore, the Jacobian matrix at the disease-free equilibrium point is(17)JE0=−pα+b01−pρ0−1−pβS∗K0−τ+d+b001−pβS∗K0τ−ρ+b00pα0pρ−b001−θϕ00−μB−bB.Theorem 2.
(local stability of the disease-free equilibrium point). Ifℛ0<1, then the disease-free equilibrium point E0 of the dynamical system (1) is locally asymptotically stable, and if ℛ0>1, then E0 is unstable.Proof.
Consider that the characteristics equation of the Jacobian matrix at the disease-free equilibrium point is given byJE0−λI5=0, where I5 is the identity matrix and can be written as(18)λ+pα+bλ+bλ+ρ+bλ2+a1λ+a2=0,where a1=k1+k2 and a2=k1k21−ℛ0.
The roots of the polynomialλ+pα+bλ+bλ+ρ+b=0 are a negative real value because pα+b>0,b>0 and ρ+b>0. We use the Routh–Hurwitz criterion to show that the roots of the polynomial,(19)λ2+a1λ+a2=0,have negative real values. To have negative real values for roots of the polynomial, in equation (19), both a1=k1+k2 and a2=k1k21−ℛ0 must be positive. k1=τ+d+b>0 and k2=μB−bB>0 because all parameters are positive constants, and μB>bB . Thus, a1=k1+k2>0 because it is the sum of the positive number and a2=k1k21−ℛ0>0 if and only if ℛ0<1. Hence, the roots of a polynomial in equation (19) have a negative real value. Therefore, the roots of the characteristic equation (18) have a negative real value which indicates the disease-free equilibrium point is locally asymptotically stable. This means, the disease cannot sustain in the population, and thus, it is eventually disappearing. On the other hand, if a2<0, then ℛ0>1. Thus, there exists at least one positive eigenvalue, and thus, the Routh–Hurwitz criterion tells us the disease-free equilibrium point is unstable. Biologically, the disease becomes endemic or the disease persists in the population.Theorem 3.
(global stability of the disease-free equilibrium point). Ifℛ0<1, then the cholera-free equilibrium point E0 of the dynamical system in equation (1) is globally asymptotically stable, and if ℛ0>1, then E0 is unstable.Proof.
Consider a Lyapunov functionVt=B. From the dynamical system in equation (1) by setting dI/dt=0, we get(20)I=1−pβBSτ+d+μB+K.
Moreover,S=S∗, μ=b at the disease-free equilibrium point, and B/B+K≤B/K and takes the equality. Then,(21)dVdt=1−θϕI+bB−μBB=1−p1−θϕβSBτ+d+μB+K−μB−bBB≤μB−bB1−p1−θϕβS∗Kτ+d+bμB−bB−1B,atE0,=μB−bBℛ0−1B.
Whenℛ0≤1, dV/dt is negative semidefinite, with equality at ℛ0=1. Therefore, the largest compact invariant set in Ω such that dV/dt=0 when R0≤1 is the singleton E0. Hence, by the LaSalle invariance principle [25], the disease-free equilibrium E0 is globally asymptotically stable in Ω and unstable otherwise.
Epidemiologically, the implication of Theorem3 is that when ℛ0 is less than unity, a small inflow of cholera-infected individuals into the community does not yield an outbreak. The subsequent numbers of those infected will be less than that of their predecessors, and eventually, the disease will be eradicated. We note also that a number of parameters are necessary in reducing the disease threshold to a value ℛ0<1.


## 3.4. Endemic Equilibrium Point

The endemic equilibrium points are steady-state solutions determined whenI>0, and there exist at least one endemic equilibrium point, E=S,I,R,U,B.Theorem 4.
(existence of at least one endemic equilibrium point). IfI>0, then ℛ0>1 and there exists at least one endemic equilibrium point for the model in equation (1).Proof.
Suppose thatI>0 and d=0 (there is no disease-induced death). Thus, we get N=N∗=L and μ=b. Setting dB/dt=0 and dR/dt=0, we obtain(22)B=1−θϕIk2,R=τIρ+b.
Similarly, by settingdS/dt+dI/dt+dR/dt=0, we yield(23)S=bN−pρ+bR+bIpα+b=S∗−pρ+bτ+b+ρbb+ρpα+bI≤S∗=bN∗pα+b.
Moreover, by settingdU/dt=0, we get(24)U=pαS+pρRb=pαbS∗−pρ+bτ+b+ρbb+ρpα+bI+pρτIbρ+b.
Finally, settingdI/dt=0 results in(25)I=1−pβBSB+Kk1=1−p1−θβϕIS∗−pρ+bτ+b+ρb/b+ρpα+bI1−θϕI+Kk2k1,and by solving I explicitly, we obtain(26)I=Kk1k21−p1−θβϕS∗/Kk1k2−1b+ρpα+b1−θϕk1b+ρpα+b+π,=Kk1k2ℛ0−1b+ρpα+b1−θϕk1b+ρpα+b+π,where π=1−p1−θβϕpρ+bτ+b+ρb. Here, we observe that I>0 if and only if ℛ0>1. Therefore, if I>0, then ℛ0>1 and there exists at least one endemic equilibrium point for the dynamical equation (1) given by(27)E=S∗,I∗,R∗,U∗,B∗,whereS∗=S∗−pρ+bτ+b+ρbb+ρpα+bI∗,I∗=Kk1k2ℛ0−1b+ρpα+b1−θϕk1b+ρpα+b+π,R∗=τI∗ρ+b,U∗=pαbS∗−pρ+bτ+b+ρbb+ρpα+bI∗+pρτI∗bρ+b,B∗=1−θϕI∗k2.Theorem 5.
(global stability of the endemic equilibrium point). Ifℛ0>1, then the endemic equilibrium point E of the dynamical system in equation (1) is globally asymptotically stable, and if ℛ0<1, then E is unstable.Proof.
Consider a Lyapunov functionVt=B−B∗−B∗lnB/B∗. From the dynamical system in equation (1), by setting dI/dt=0, we get(28)I=1−pβBSk1B+K.
Then,(29)dVdt=dBdt−B∗2BdBdt=1−θϕI−k2B−B∗2B1−θϕI−k2B,=1−p1−θϕβSBB+Kk1−k2B+B∗2Bk2B−1−p1−θϕβSBB+Kk1,letμ=bandB=2K=1−p1−θϕβ2KS3Kk1−2k2K+B∗22K2k2K−1−p1−θϕβ2KS3Kk1≤1−p1−θϕβ2KS∗3Kk1−2k2K3+B∗22K2k2K−1−p1−θϕβ2KS∗Kk1,=23k2Kℛ0−1+k2B∗21−ℛ0,=k21−ℛ0−23K+B∗2≤k21−ℛ0B∗2.
Whenℛ0≥1, dV/dt is negative semidefinite, with equality at ℛ0=1. Therefore, the largest invariant set in dV/dt=0 is the singleton E. By LaSalle’s invariance principle [25], we therefore, conclude that the endemic equilibrium E is globally asymptotically stable if ℛ0>1.


## 4. Bifurcation Analysis

A bifurcation is a qualitative change of the phase portrait due to the variation of parameter values. The change in the qualitative character of a solution as a control parameter is varied is known as a bifurcation point. At the bifurcation point, the equilibrium points, or stability properties of the equilibrium points, or both, change. We investigate bifurcation analysis of the dynamical system (1) by using center manifold theory [26].Theorem 6.
Ifa∗>a°, then the dynamical system equation (1) exhibits forward bifurcation at R0=1, where a∗=k1/pα+bS∗+k2/1−θϕK and a°=1−pρτ/ρ+bpα+bS∗.Proof.
We prove the possibility of bifurcation atℛ0=1 using the center manifold theorem [26]. For more convenience, let(30)S=x1,I=x2,R=x3,U=x4,B=x5,dSdt=f1,dIdt=f2,dRdt=f3,dUdt=f4,dBdt=f5.
We considerβ as the bifurcation parameter at ℛ0=1, and we obtain β=Kk1k2/1−p1−θϕS∗. The Jacobian matrix at disease-free equilibrium point E0 given in equation (17) has a simple zero eigenvalue when β=Kk1k2/1−p1−θϕS∗ or ℛ0=1.
The right eigenvector,w=w1,w2,w3,w4,w5t, associated with the simple zero eigenvalue of JE0 when β=Kk1k2/1−p1−θϕS∗ can be obtained from JE0w=0. The system of equation becomes(31)pα+bw1+1−pρw3−1−pβS∗Kw5=0,−k1w2+1−pβS∗Kw5=0,τw2−ρ+bw3=0,pαw1+pρw3−bw4=0,1−θϕw2−k2w5=0.
The solution of the system in (31) is obtained as(32)w1=1−pρτρ+bpα+b−k1pα+bw2,w3=τρ+bw2,w4=pαbw1+pρbw3,w5=1−θϕk2w2.
In a similar manner, we compute the left eigenvector,v=v1,v2,v3,v4,v5 associated with the simple zero eigenvalue of JE0 when β=Kk1k2/1−p1−θϕS∗ obtained from vJE0=0, and its solution becomes(33)v1=v3=v4=0,v5=k11−θϕv2,where v2 is calculated to ensure that the eigenvectors satisfy the condition vw=1. Since v1=v3=v4=0, we do not require the derivatives of f1,f3, and f4. The only nonzero from the derivative of f2 and f5 is the following:(34)∂2f2E0∂x1∂x5=∂2f2E0∂x5∂x1=1−pβK,∂2f2E0∂x22=−21−pβS∗K2,∂2f2E0∂x5∂β=1−pS∗K,and otherwise, all the other partial derivatives of f2 and f5 are zero. The direction of the bifurcation at ℛ0=1 is determined from the signs of the bifurcation coefficients a and b given by(35)a=∑k,i,j=15vkwiwj∂2fkE0∂xi∂xj,=v2w1w51−pβK+v2w5w11−pβK−2v2w2w21−pβS∗K2,=2v2w1w51−pβK−2v2w221−pβS∗K2,=−2v2w22k1k1pα+bS∗+k21−θϕK−1−pρτρ+bpα+bS∗,=−2v2w22k2a∗−a°,b=∑k,i=15vkwi∂2fk0,0∂xi∂β=v2w51−pS∗K=v2w21−p1−θϕS∗k2K>0.
Ifa∗>a°, then a<0. We observe that the coefficient b is always positive and the sign of the coefficient a is negative provided that a∗>a°. Therefore, the dynamical system in equation (1) exhibits a forward bifurcation using the center manifold theorem [26]. This indicates that an exchange of stability occurs at ℛ0=1. Using expression for I∗ in the endemic equilibrium and parameter values in Table 2, we plotted a forward bifurcation diagram as shown in Figure 2.Table 2
Parameter values used for sensitivity analysis and simulation.
ParameterValueSourcesSensitivity indicesb0.00082Assumed2.03μd3.245×10−8Assumed−1β1[5]1τ0.2[16]−0.95ρ0.01Assumed0K106[5]−1θ0.8Assumed−4μi4.21×10−5Assumed−0.05d0.01[15]−0.05bB−μB−0.33[5]−1α0.1Assumed−0.98ϕ10[16]1p0.7Assumed−3.32Figure 2
Forward bifurcation of the dynamical system in equation (1) at ℛ0=1.


## 5. Sensitivity Analysis of Parameters

In epidemiology, it is crucial to understand the relative importance of the different factors responsible for its transmission and prevalence in determining how to best reduce mortality and morbidity due to the disease. Sensitivity analysis is used to study the sensitivity changes of parameters on the transmission and prevalence of the disease [27, 28]. Sensitivity indices are used to measure the relative change in a state variable as a parameter value changes. The normalized forward sensitivity index of a variable to a parameter is determined by the ratio of the relative change in the variable to the relative change in the parameter. If the variable is a differentiable function of the parameter, then the sensitivity index can be defined using partial derivatives as follows: the normalized forward sensitivity index of a variable, η, that depends differentiably on a parameter, ξ, is defined as(36)Λξη=∂η∂ξ×ξη,where ξ represents all the basic parameters [27, 28].For example,ℛ0=1−p1−θβϕbb−μi/Kμdτ+d+bμB−bBpα+b,(37)Λpℛ0=∂ℛ0∂p×pℛ0=−pα+b1−ppα+b.The sensitivity indices of the basic reproductive number with respect to parameters are found in Table2. Thus, the negative indices are (p,θ,K,μd,μB−bB,τ,α) indicating that they had a negative impact on the transmission of the disease as the values of the parameter are increasing. The positive indices are (b,β,ϕ) indicating that they had a great impact on the transmission of the disease as the values of the parameter are increasing. This indicates that, to eradicate cholera, stakeholders should devise strategies and implement it to reduce these parameters.


## 6. Numerical Simulation

In this section, numerical simulations for model (1) is carried out. The parameter values that are used for numerical simulations are listed in Table 2 in which the parameter values of p and θ are varied. It is verified that the disease-free equilibrium point is stable when ℛ0<1 and unstable whenever ℛ0>1.Figures3–6 are the time series solution of individual cohorts and the concentration of the cholera bacterium in an aquatic reservoir when ℛ0=0.0024<1, ℛ0=0.0120<1, ℛ0=0.6891<1, and ℛ0=3.4457>1. The total individuals and protected individuals are coincided and increased towards the carrying capacity, whereas the susceptible, infected, and recovered individuals and the concentration of cholera bacterium decreased rapidly and became zero, see Figures 3 and 4. The parameter values for the plot of Figure 3 are in Table 2, and the parameter values for the plot of Figure 4 are in Table 2 by replacing θ=0. This indicates that the disease dies out when the progression rate of susceptible and recovered individuals to the protected class due to the compliance of hygienic, ingestion of cholera bacterium and contact with cholera patients, is increased more than the recruitment rate (i.e., pα>b). Moreover, the total individuals and susceptible individuals attract towards the carrying capacity, whereas the protected, infected, recovered individuals and the concentration of cholera bacterium decreased and attracted towards zero for the parameter value in Table 2 and p=0 or for ℛ0=0.6891<1, see Figure 5. This indicates that the disease also dies out when infected individuals comply to their sanitation. In other words, the disease is not transmitted if the infected individuals do not contribute the bacterium to the aquatic reservoir due to keeping sanitation of an infected individual. This indicates that reducing the contribution of the concentrated bacteria by infected individuals reduces the disease.Figure 3
Time series solution of individual (a) and bacteria (b) cohorts whenp=0.7 and θ=0.8. The diseases die out (ℛ0=0.0024<1).
(a)(b)Figure 4
Time series solution of individual (a) and bacteria (b) cohorts whenp=0.7 and θ=0. The diseases die out (ℛ0=0.0120<1).
(a)(b)Figure 5
Time series solution of individual (a) and bacteria (b) cohorts whenp=0 and θ=0.8. The diseases die out (ℛ0=0.6891<1).
(a)(b)Figure 6
Time series solution of individual (a) and bacteria (b) cohorts whenp=θ=0. The diseases do not die out (ℛ0=3.4457>1).
(a)(b)The solution curves do not settle down to their corresponding disease-free equilibrium point in the absence of implementation of adequate prevention measures, see Figure6. That is, for parameter values in Table 2 with p=θ=0 or ℛ0=3.4457>1, the susceptible, protected, and total individuals decreased. The infected, recovered, and the concentration of cholera bacterium are nonzero after initially increasing and decreasing. This indicates that the disease infects and kills many people in the absence of knowledge and compliance of the prevention measure properly.In summary, having knowledge and compliance of the prevention measure properly reduces the basic reproductive number less than one. This means that each infected individual over infectious time transmits the infection on average to less than one new individual so that the disease dies out. When an individual has no knowledge and compliance of the prevention measures properly, the basic reproductive number increases greater than one. This means that each infected individual over an infection period transmits the infection on average to more than one new individual so that the disease spreads in the population.


## 7. Conclusions

In this study, effects of implementation of adequate preventive measures on the dynamics of cholera are examined using a mathematical model. To analyze the effect of implementation of preventive measures, a deterministic mathematical model system of nonlinear ordinary differential equations with the inclusion of the protected individuals cohort as one cohort is formulated. The disease-free equilibrium point and basic reproductive number are determined. The disease-free equilibrium point is locally and globally asymptotically stable whenℛ0<1 and otherwise, unstable. The model is found to show the phenomenon of forward bifurcation at ℛ0=1 which indicates that a stable disease-free equilibrium does not coexist with a stable endemic equilibrium. The numerical result shows ℛ0<1 when susceptible and recovered individual compliance of hygienic, ingestion of cholera bacterium and contact with cholera patients, is increased or the compliance of sanitation of infected people is increased. Thus, having knowledge and compliance of the prevention measures properly reduces the basic reproductive number below one. On the contrary, when an individual has no knowledge and compliance of the prevention measure properly, the basic reproductive number increases greater than one. The fraction of the compliance of hygiene, ingestion of cholera bacterium and contact with cholera patients of the susceptible and recovered cohorts (p), and the fraction of the compliance of sanitation of the infected cohort (θ) are the most sensitive parameters that can be used to control the propagation of the disease. Hence, these parameters are those that should be targeted the most by stakeholders to combat the disease. Therefore, from the qualitative study, it is concluded that the disease can be controlled and eliminated from the community if susceptible and recovered individuals comply with hygiene, ingestion of cholera bacterium and contact with cholera patients, or infected individuals comply with sanitation.



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