# A Mathematical Model on the Dynamics of In-Host Infection Cholera Disease with Vaccination

Authors: Owade Kennedy Jackob; Okaka Akinyi; Frankline Tireito
Journal: Discrete Dynamics in Nature and Society
 (2023)

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## Abstract

In this paper, a within-host cholera mathematical model has been developed using a system of ordinary differential equations incorporating vaccine efficacy. The formulated model considers cells in an already vaccinated individual with a vaccine whose efficacy isγ. The solutions of the model have been shown to be both positive and bounded hence well-posed. The vaccine basic reproduction number has been carried out using the next generation matrix approach and is given by R0V=γ/d+μ2 and R0V<1 if γ<d+μ2. Analysis of the model shows that infectionfreeequilibriumIFE point is both locally and globally asymptotically stable when R0V<1 and infectionequilibriumIE point is locally asymptotically stable when R0V>1. Furthermore, analysis of the model shows that R0V<1 is not sufficient enough to eradicate in-host cholera disease, hence the existence of backward bifurcation which is an indication as to why cholera disease is persistent. To highlight the relevance of vaccine efficacy, a numerical simulation of the model with respect to vaccination is carried out and shows that when the vaccine efficacy γ is high, there will be a lower infection rate of cells, hence the need to improve cholera vaccine efficacy.


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## Body

## 1. Introduction

Cholera is an acute illness caused by an enterotoxin elaborated byVibrio cholerae that when ingested colonises the small intestine. After 24 hrs to 48 hrs incubation period, cholera begins with sudden onset of painless water diarrhea that mainly quickly become voluminous and is often followed shortly by vomiting. In severe cases, stool volume can exceed 250 ml/kg in the first 24 hrs. Fever is usually absent, and muscle cramps due to electrolyte disturbances are common. The stool has a characteristic appearance; a nonbilious, grey, slightly cloudy fluid with flecks of mucus, no blood, and somewhat sweet and in-offensive odor. It has been called ‘rice-water” stool because of its resemblance to the water in which rice has been washed. Clinical symptom parallel volume contraction shows losses of 3−5 percent of normal body weight. If fluids and electrolytes are not replaced, hypovolemic shock and death will occur [1].According to the studies of natural infection, the human body has T-lymphocytes cell and B-lymphocytes cell which produce antibodies that attack the pathogen, thereby providing protection. However, in a situation of excess consumption of bacteria, the body may not defend itself; as a result, there is a need for vaccination [2].The World Health Organization (WHO) recommends the use of oral cholera vaccines, Dukoral and Shanchol for those at high risk [3]. When these vaccines are administered, the cholera strain within the vaccine produces an incomplete, nontoxic version of the toxin; the body then responds to the safe version of cholera and creates immunity to the infection through the production of antibodies (T-lymphocytes cells and B-lymphocytes). Despite the availability and access to cholera vaccines, there is still a high burden of cholera in endemic areas [3].Wang and Wang [4] developed a within-host dynamics of cholera with bacterial-viral interaction given as follows:(1)dBdt=Λ−αBK+BV−σ1B,dZdt=gZ+θ1αBK+BV−σ2Z,dVdt=hV+θ2αBK+BV−σ3V,where B represents environmental vibrios, Z human vibrios, V concentration of the virus, Λ is the influx rate of the ingested environmental vibrios, α is the contact rate between the environmental vibrios and the virus, and gZ and hV intrinsic growth rate of Z and V, respectively. Their main focus is the interaction of environmental vibrios, human vibrios, the virus, and the vibrios within the human host since such interaction is critical in shaping the evolution of the pathogen within the human body and directly contributes to the epidemiology of cholera at the population level since the human vibrios shed out of the human body will remain highly infectious for a certain period of time and can be transmitted among human hosts. They established the basic reproduction number as a sharp threshold for disease dynamics such that when R0<1, the highly infectious vibrios will not grow within the human host, and the environmental vibrios ingested into the human body will not cause cholera infection and when R0>1, the human vibrios will grow and persist, leading to human cholera.Ratchford and Wang [5] developed a cholera within-host model for an average infected individual given as follows:(2)dZdt=C1BV−d1MZ−ϵZ,dVdt=C2BV−d2MV−τV,dMdt=e1MZ+e2MV−pM,which is the fast scale system where Z, V, and M represent the concentrations of human vibrios, pathogen, and host immune cells, respectively. The dynamics of the environmental evolution of the vibrios is governed by the equation given as follows:(3)dBdt=ϵZI−δB,which is the low scale system, where ϵZ is the host shedding rate that depends on the human vibrios. In their analysis, the within-host dynamical system proposed is expanded to include the influence of human virus and immune cell interaction with the infectious vibrios, and the findings show that the slow-scale and intermediate scale systems act predictably and the dynamics of two smaller combined systems depends mostly on R0. However, the study recommends that a complete stability analysis of the equilibrium of the full system is of particular interest and it suggests that numerical simulation using real world data should be carried out to shed more light onto the likelihood of various assumptions. In this paper, the focus is to develop a within-host cholera model with vaccination investigating the impact of vaccination on the dynamics of in-host infection of cholera disease.


## 2. Model Formulation and Analysis

Cholera infection involves the interaction between the vibrioV and the target cells B. The target cells B are recruited at a constant rate Λ and they die naturally at a rate μ1. Multiplication of the vibrio when infected cells are shed out is given by βVV within the human small intestine, where hV is the rate of recruitment of the vibrio. The interaction of vibrio within the small intestine with the target cells B leads to the infection of target cells at the rate αBV, where α is the contact rate between the target cells and the vibrios in the small intestine that leads to the production of infected cells Z. The infected cells die naturally at a rate μ2, and they are shed out at a rate d due to the action of the vaccine. This shedding rate increases with an increase in vaccine efficacy. The saturation incidence rate is given by αBV/1+δV such that δ is the measure of saturation level of the vibrio. The natural clearance rate of the vibrio and shedding rate is given by σ and s, respectively. The vaccine whose efficacy is γ with the proportion of noneffectiveness of the vaccine given by 1−γ exposes an individual to a dose of live cholera bacteria, which causes the body to produce antibodies against the disease.The system of ordinary differential equation governing the description previously is as follows:(4)dBdt=Λ−1−γαBV1+δV−μ1B+γZ,dZdt=−γZ+1−γαBV1+δV−d+μ2Z,dVdt=hV+βVV−σ+sV.


### 2.1. Assumptions of the Model

(i)
There is the recovery of cells due to immunity gained as a result of vaccination(ii)
Target cells are recruited at a constant rate


## 2.1. Assumptions of the Model

(i)
There is the recovery of cells due to immunity gained as a result of vaccination(ii)
Target cells are recruited at a constant rate


## 3. Positivity and Boundedness of the Model

Since model [4] describes human cells, it should be well-posed. Thus, in this section, positivity and boundedness solutions are discussed. The positivity of solutions of the model [4] is determined using the following initial conditions [5]:(5)B0=B0≥0,Z0=Z0≥0,V0=V0≥0.


### 3.1. Positivity of the Model

Proposition 1.
Solutions of model (4) with the initial conditions (5) are positive in the region defined as R+=B,Z,VB≥0,Z≥0,V≥0.Proof.
LetN=supt>0B≥0,Z>0,V>0. The first equation of model (4) is given as follows:(6)dBdt=Λ−1−γαBV1+δV−μ1B+γZ=Λ+γZ−1−γαV1+δV+μ1B.
From this, the integrating factor ise∫0t1−γαVτ/1+δVτdτ+μ1t
Multiplying (6) by the integrating factor yields dBte∫0t1−γαVτ/1+δVτdτ+μ1t/dt≥Λ+γZe∫0t1−γαVτ/1+δVτdτ+μ1t
Solving the inequality yieldsBte∫0t1−γαVτ/1+δVτdτ+μ1t−B0≥∫0tΛ+γZe∫0k1−γαVτ/1+δVτdτ+μ1kdk
Therefore,Bt becomes Bt≥B0e−∫0t1−γαVτ/1+δVτdτ+μ1t+e−∫0t1−γαVτ/1+δVτdτ+μ1t×∫0tΛ+γZe∫0k1−γαVτ/1+δVτdτ+μ1kdk>0
Thus,Bt>0.
The second equation of the model (4) is given as follows:(7)dZdt=−γZ+1−γαBV1+δV−d+μ2Z=1−γαBV1+δV−γ+d+μ2Z.
The integrating factor ise∫0tγZτdτ+d+μ2t
Multiplying (7) by the integrating factor yields dZte∫0tγZτdτ+d+μ2t/dt≥1−γαBV/1+δVe∫0tγZτdτ+d+μ2t
Solving the inequality yieldsZte∫0tγZτdτ+d+μ2t−Z0≥∫0t1−γαBV/1+δVe∫0kγZτdτ+d+μ2tdk
Therefore,Zt becomes Zt≥Z0e−∫0tγZτdτ+d+μ2t+e−∫0tγZτdτ+d+μ2t×∫0t1−γαBV/1+δVe∫0kγZτdτ+d+μ2tdk>0
Thus,Zt>0.
In a similar manner to the third equation of the model (4) as follows:(8)dVdt=hV+βVV−σ+sV.
From this, the integrating factor ise∫0tσ+sdt
Multiplying (8) by the integrating factor yields dVte∫0tσ+sdt/dt≥hV+βVVe∫0tσ+sdt
Solving the inequality yields as follows:(9)Vte∫0tσ+sdt−V0≥∫0thV+βVVe∫0kσ+sdk.
Therefore,Vt becomes Vt≥V0e−∫0tσ+sdt×∫0thV+βVVe∫0kσ+sdk>0.
Thus,Vt>0.
Hence, all the solution of model (4) given conditions (5) at any time t are positive. Therefore, the population of target cells and infected cells in the human will continue to grow positively.


### 3.2. Boundedness of the Model

Since the model formulated describes cells in a human being, the population of the target cells and the infected cells will always remain bounded.Proposition 2.
Solutions of model (4) with the initial initial condtions (5) are bounded for t≥0 in the region Φ given by Bt+Zt∈R+3:N≤Λ/μ.Proof.
LetNt:=Bt+Zt, where N is the total number of cells, and let μ=μ1=μ2. From the system of equation (4), we have(10)N′t≤Λ−μ2Z≤Λ−μNt,N′t≤Λ−μNt,∫N′tdt≤∫Λ−μNtdt,Nteμt≤∫Λμeμtdt,Nteμt≤Λμeμt+C,Nt≤Λμ+Ce−μt,limt⟶∞Nt≤Λμ.
Hence,Nt is bounded, since solutions of model (4) are positive and bounded for t≥0; therefore, model (4) is mathematically and epidemiologically meaningful and is sufficient to consider its solution in Φ.


## 3.1. Positivity of the Model

Proposition 1.
Solutions of model (4) with the initial conditions (5) are positive in the region defined as R+=B,Z,VB≥0,Z≥0,V≥0.Proof.
LetN=supt>0B≥0,Z>0,V>0. The first equation of model (4) is given as follows:(6)dBdt=Λ−1−γαBV1+δV−μ1B+γZ=Λ+γZ−1−γαV1+δV+μ1B.
From this, the integrating factor ise∫0t1−γαVτ/1+δVτdτ+μ1t
Multiplying (6) by the integrating factor yields dBte∫0t1−γαVτ/1+δVτdτ+μ1t/dt≥Λ+γZe∫0t1−γαVτ/1+δVτdτ+μ1t
Solving the inequality yieldsBte∫0t1−γαVτ/1+δVτdτ+μ1t−B0≥∫0tΛ+γZe∫0k1−γαVτ/1+δVτdτ+μ1kdk
Therefore,Bt becomes Bt≥B0e−∫0t1−γαVτ/1+δVτdτ+μ1t+e−∫0t1−γαVτ/1+δVτdτ+μ1t×∫0tΛ+γZe∫0k1−γαVτ/1+δVτdτ+μ1kdk>0
Thus,Bt>0.
The second equation of the model (4) is given as follows:(7)dZdt=−γZ+1−γαBV1+δV−d+μ2Z=1−γαBV1+δV−γ+d+μ2Z.
The integrating factor ise∫0tγZτdτ+d+μ2t
Multiplying (7) by the integrating factor yields dZte∫0tγZτdτ+d+μ2t/dt≥1−γαBV/1+δVe∫0tγZτdτ+d+μ2t
Solving the inequality yieldsZte∫0tγZτdτ+d+μ2t−Z0≥∫0t1−γαBV/1+δVe∫0kγZτdτ+d+μ2tdk
Therefore,Zt becomes Zt≥Z0e−∫0tγZτdτ+d+μ2t+e−∫0tγZτdτ+d+μ2t×∫0t1−γαBV/1+δVe∫0kγZτdτ+d+μ2tdk>0
Thus,Zt>0.
In a similar manner to the third equation of the model (4) as follows:(8)dVdt=hV+βVV−σ+sV.
From this, the integrating factor ise∫0tσ+sdt
Multiplying (8) by the integrating factor yields dVte∫0tσ+sdt/dt≥hV+βVVe∫0tσ+sdt
Solving the inequality yields as follows:(9)Vte∫0tσ+sdt−V0≥∫0thV+βVVe∫0kσ+sdk.
Therefore,Vt becomes Vt≥V0e−∫0tσ+sdt×∫0thV+βVVe∫0kσ+sdk>0.
Thus,Vt>0.
Hence, all the solution of model (4) given conditions (5) at any time t are positive. Therefore, the population of target cells and infected cells in the human will continue to grow positively.


## 3.2. Boundedness of the Model

Since the model formulated describes cells in a human being, the population of the target cells and the infected cells will always remain bounded.Proposition 2.
Solutions of model (4) with the initial initial condtions (5) are bounded for t≥0 in the region Φ given by Bt+Zt∈R+3:N≤Λ/μ.Proof.
LetNt:=Bt+Zt, where N is the total number of cells, and let μ=μ1=μ2. From the system of equation (4), we have(10)N′t≤Λ−μ2Z≤Λ−μNt,N′t≤Λ−μNt,∫N′tdt≤∫Λ−μNtdt,Nteμt≤∫Λμeμtdt,Nteμt≤Λμeμt+C,Nt≤Λμ+Ce−μt,limt⟶∞Nt≤Λμ.
Hence,Nt is bounded, since solutions of model (4) are positive and bounded for t≥0; therefore, model (4) is mathematically and epidemiologically meaningful and is sufficient to consider its solution in Φ.


## 4. Basic Reproduction Number,R0

The intervention strategy in this study is the vaccine efficacy; hence, the associated reproduction number is called vaccine reproduction number denoted asR0V; it is the threshold quantity that predicts the spread of cholera disease in a given population of the target cells in the presence of vaccination. Therefore, the vaccine reproduction number R0V=ρFK−1 is the spectral radius of the matrix FK−1,(11)F=γ−1−γαBV1+δV200,K=d+μ200β−σ+s,Therefore,(12)R0V=FK−1=γd+μ2.The calculatedR0V depends on vaccine efficacy and when d+μ2 is high due to the action of vaccine then R0V<1 which shows the elimination of infected cells due to the action of vaccine. The basic reproduction number, R0V being the measure of severity of an epidemic, determines whether or not cholera disease will invade the target cell population.Epidemiologically, this implies that ifR0V<1, cholera infection will die out in the cells and if R0V>1, cholera disease will persist in a cell population.


## 5. Stability Analysis

The stability at theIFE determines the short-term epidemics of the infection of cells, while its dynamics over a longer period of time is characterized by the global stability at the IFE. In this section, the analysis of local and global stability of IFE and local stability of IE is carried out.


### 5.1. Existence of Infection-Free Equilibrium (IFE) Points

These are steady state solutions in absence of cholera disease in the body; therefore, target cells are not infected. Stability analysis is carried out to predict the long-term behaviour of the solutions of the model (4).Proposition 3.
For model (4), the IFE point is given by IFE=B0,Z0,V0 = Λ/μ1,0,0.Proof.
At IFE,Z=0,V=0. Therefore, considering the first equation in the system (4) and replacing Z=0,V=0 and equating its right hand side to 0 yields(13)Λ−μ1B=0,Λ=μ1B.
MakingB the subject yields(14)B=Λμ1.
Therefore, the infection-free equilibrium point of the system (4) is Λ/μ1,0,0. This shows that the infected class is zero since there are no pathogens and the entire population of cells consists of target cells only and their growth is bounded by Λ/μ1.


### 5.2. Local Asymptotic Stability of the Infection-Free Equilibrium (IFE)

Theorem 1.
For any timet≥0, the infection-free equilibrium IFE=Λ/μ1,0,0 of model (4) is locally asymptotically stable when R0V<1 and unstable when R0V>1.Proof.
The jacobian matrix of (4) is given as follows:(15)J=−1−γαV1+δV−μ1γ−1−γαΛμ11+δV21−γαV1+δV−γ−d+μ21−γαΛμ11+δV200βV−σ+s.
The matrix (15) in terms of R0V yields(16)JIFE=−μ1d+μ2R0V−1−γαΛμ10−d+μ2R0V−11−γαΛμ100βV−σ+s.
To find the eigenvalues, we consider the characteristic equationJIFE−λI=0. For asymptotic stability, all the eigenvalues should be strictly negative. It can be seen clearly that λ1=−μ1 is one of the eigenvalues of (16).
UsingRouth−Hurwitz Criterion for stability as used in [6] to determine the trace and determinant, let(17)−d+μ2R0V−11−γαΛμ10βV−σ+s=0.
Equation (17) has(18)trace17=−d+μ2R0V−1+βV−σ+s<0,and a determinantdet17=−d+μ2R0V−1βV−σ+s>0 iff βV<σ+s. This implies that IFE is locally asymptotically stable when R0<1 and unstable if R0V>1.
In Theorem 1, when cholera vaccine is administered, it implies that with a small perturbation ofIFE, the solution of model (4) will eventually converge to IFE whenever R0V<1.
Epidemiologically, it implies that if a few infectious cholera pathogens are introduced into a population of target cells, the disease would die out whenR0V<1; otherwise, the disease would spread into the population of target cells.


### 5.3. Global Stability of the Infection-Free Equilibrium (IFE)

The global stability of theIFE of model (4) is explored using Castilo–Chavez et al. [7]. Model (4) is rewritten in the form as follows:(19)dHdt=LX,Z,dZdt=GX,Z,GX,0=0,where X∈R2 denotes the uninfected compartments. At IFE,(20)IFE=X∗,Z∗,X∗=Λμ1,0.The conditions(21)dXdt=HX,0,X∗,GX,Z=MZ−G^X,Z,G^X,Z≥0.X∗ is globally asymptotically stable,where M=DZGX∗,0 is a M-matrix (the off diagonal elements of M and are non-negative). If model (4) satisfies the conditions previously (21), then Theorem 2 holds.Theorem 2.
The fixed pointIFE=X∗,0 is globally asymptotically a stable equilibrium point of model (4) whenever the R0V<1 whenever the conditions (21) are satisfied, otherwise unstable.Proof.
From model (4), we obtain(22)HX,0=Λ−μ1B00,GX,Z=MZ−G^X,Z,where(23)M=−d+μ2R0V−11−γαΛμ10βV−σ+s,G^X,0=G^1X,ZG^2X,Z=00.
SinceG^X,Z>0 and the conditions (21) are satisfied; therefore, IFE is globally asymptotically stable whenever R0V<1, otherwise unstable.
This implies that given a large perturbation of theIFE, the solutions of the model (4) will eventually converge to IFE whenever R0V<1. Epidemiologically, this means that if a large number of vibrios are introduced to a population of target cells, the disease would die off, hence the elimination of infection in cells since there are no secondary infection produced whenever R0V<1, otherwise the disease would persist.


### 5.4. Local Stability of the Infection Equilibrium (IE) Points

Infection Equilibrium (IE) point is the state at which the disease persists in the cells. We investigate the local stability of infection equilibrium points atE∗B∗,Z∗,V∗.Theorem 3.
The infection equilibrium pointE∗B∗,Z∗,V∗ of system (4) is locally asymptotically stable whenever R0V>1, otherwise unstable.Proof.
The infection equilibriaE∗B∗,Z∗,V∗ of model (4) is obtained by equating model equations in (4) to zero then solving(24)Λ−1−γαBV1+δV−μ1B+γZ=0,−γZ+1−γαBV1+δV−d+μ2Z=0,hV+βVV−σ+sV=0.
The following equilibrium points are obtained by solving the previous equation simultaneously:(25)B∗=d+μ2+γ1+δV∗Z∗1−γαV∗,V∗=hVδ+s−βV.
SubstitutingB∗ into the second equation of model (4) and equating to zero, we obtain the following quadratic equation:(26)γd+μ2Z∗2+γd+μ2αhVβV−δ+sZ∗+d+μ2=0.
Let(27)a=γd+μ2=R0d+μ22,b=γd+μ2αhVβV−σ+s=R0d+μ22αhVβV−σ+s,c=d+μ2=γR0.
The quadratic equation (26) can be written as follows (28):(28)aZ∗2+bZ∗+c=0,such that Z∗ is the positive solution of the equation (28). From preceding relations, a is always positive.
Sincea>0, then there exist positive solutions of (28). Then, aZ∗2+bZ∗+c=0 at Z∗=0=c implying c=0, then (28) has a unique endemic equilibrium point. Since c=0, a unique solution of (28) is given by Z∗=−b/a, b<0.
Consider the case whenR0V<1 and c>0. If b≥0, and using Descartes’ rule of signs [8], aZ∗2+bZ∗+c has no positive root.
Ifb<0, then (28) is given as follows:(29)Z∗=b±b2−4ac2a.
Hence, the solutions are positive and distinct ifb2−4ac>0, in which case there are two endemic equilibria. The solutions of (28) coalesce into two roots when b2−4ac=0 to form one endemic equilibrium point.
Now, ifb=γd+μ2αhV/βV−σ+s>0, b will always be positive. To prove that infection equilibrium point E∗ is locally asymptotically stable when R0V>1, consider the Jacobian matrix of (4) evaluated at E∗ given as follows:(30)JE∗=−1−γαV∗1+δV∗−μ1γ−1−γαB∗1+δV∗21−γαV∗1+δV∗−d+μ2R0V−11−γαB∗1+δV∗200βV−σ+s.
For the linearized system (4), the characteristic polynomial is expressed as follows:(31)J−λI=0.
When (30) is substituted into (31), it yields(32)−1−γαV∗1+δV∗−μ1−λd+μ2R0V−1−γαB∗1+δV∗21−γαV∗1+δV∗−d+μ2R0V−1−λ1−γαB∗1+δV∗200βV−σ+s−λ=0.
Solving (32) yields(33)1−γαV∗1+δV∗−μ1−λd+μ2R0V−1−λβV−σ+s−λ+1−γαV∗1+δV∗βV−σ+s−λd+μ2R0V−1=0.
SinceR0=γ/d+μ2, then(34)−1−γ1+δV∗αV∗−μ1−λd+μ2R0V−1−λβV−δ+s−λ+d+μ2R0V1−γ1+δV∗αV∗βV−δ+s−λ=0.
Let(35)m=1−γ1+δV∗αV∗,n=βV−σ+s,V∗=hVβV−δ+s.
Then,(36)m−μ1−λd+μ2R0V−1−λn−λ+md+μ2R0Vn−λ=0.
Expanding equation (36) yields(37)λ2+λm−μ1+d+μ2R0V−1+m−μ1d+μ2R0V−1n−λ=0,and gives(38)n−λ1=0,λ1=n,λ1=βV−δ+s,k=m−μ1+d+μ2R0V−1,λ2,3=−k±k2−4m−μ1d+μ2R0V−12.
Now, the eigenvalueλ1=βV−σ+s has negative real parts, and the second eigenvalue is as follows:(39)λ2=−k−k2−4m−μ1d+μ2R0V−12,and the third eigenvalue given is as follows:(40)λ3=−k+k2−4m−μ1d+μ2R0V−12.
If4m−μ1d+μ2R0V−1>m−μ1+d+μ2R0V−12, then the square root part of λ2,3 yields an imaginary root; hence, a possibility of backward bifurcation [9] which is an indication of the oscillatory behaviour in cholera outbreaks; also, Theorem 3 implies that a small perturbation of IE, the solution of model [3], will always converge to IE whenever R0V>1. Since λ1=βV−δ+s and λ2,3=−k±k2−4m−μ1d+μ2R0V−1/2 have negative real parts at infection equilibrium point, then infection equilibrium point is locally asymptotically stable whenever R0V>1.
Epidemiologically implying that if vibrios are introduced in a population of target cells and there are new secondary infections produced wheneverR0V>1, then cholera disease would persist in the population of cells. The study of backward bifurcation in a disease dynamics can also occur when the reproduction number is less than unity, R0V<1; this is explored in the next section.


### 5.5. Existence of Backward Bifurcation

Based on the analysis in the previous section, there exists a range of values forR0 in which model (4) have two positive endemic equilibria.Let the discriminantb2−4ac be positive and be given as follows:(41)γd+μ2αhVβV−σ+s2−4γd+μ22>0.To find the value where the two endemic equilibria converge, setb2−4ac=0 and solve for the value of R0V denoted R0∗.Let(42)b2−4ac=0.Then,(43)b24ac=1,R0∗=1−b24ac,b=1−γd+μ2αhVβV−σ+s.Let(44)R0∗=1−b241−γd+μ22=1−α2hV21−γβV−σ+s.IfR0∗<R0V is equal to b2−4ac>0, then backward bifurcation will occur for values of R0V such that R0∗<R0V<1. Therefore, we show the existence of backward bifurcation due to the existence of hyperbolic fixed points as in (39) and (40). Backward bifurcation in disease transmission models is where a stable endemic equilibrium coexists with a stable disease-free equilibrium when the associated reproduction number is less than unity.Investigating the nature of bifurcation involvingIFEΛ/μ1,0,0 using Castilo and Song [10], we consider a general system of ODEs given by (45):(45)dxdt=fx,ψ;f:Rn×Rn,f∈C2Rn×Rn.Theorem 4.
Castilo–Chavez and Song [10]. We consider the following:(i)
LetI=Dxf0,0 be the linearization matrix of system (48) around the equilibrium point x=0 with ψ evaluated at 0. Zero is a simple eigenvalue of the matrix I and all other eigenvalues of I have negative real parts.(ii)
MatrixI has a non-negative right eigenvector w and a left eigenvector v corresponding to the zero eigenvalue(iii)
Letfk denote the kth component of f and(a)
a1=∑i,j,k=1nvkwiwj∂2fk/∂xi∂xj0,0(b)
a2=∑i,k=1nvkwi∂2fk/∂xi∂ϕ0,0
Then, the local dynamic of system (48) around x=0 is determined by a1 and a2.(i)
Ifa1>0, a2>0, when ψ<0, with ψ≪1, then x=0 is locally asymptotically stable and there exists a positive unstable equilibrium and when 0<ψ≪1, x=0 is unstable, and there exists a negative and locally asymptotically stable equilibrium.(ii)
Ifa1<0, a2<0 when ψ<0, with ψ≪1, then x=0 is unstable; when 0<ψ≪1, x=0 is locally asymptotically stable and there exists a positive unstable equilibrium.(iii)
Ifa1>0 and a2<0 when ψ<0, with ψ≪1, then x=0 is unstable, and there exists a locally asymptotically stable negative equilibrium, when 0<ψ≪1, x=0 is stable and positive unstable equilibrium appears.(iv)
Whena1<0 and a2>0, then there exists a forward bifurcation.(v)
Whena1>0 and a2>0, then the bifurcation at ψ=0 is backward.Proof.
Using Castilo–Chavez and Song [10], let ψ be the bifurcation parameter such that R0V<1 such that x0 is an IFE equilibrium for all values of ψ.
We consider(46)dxdt=fx,ψ,where f is a continuous differentiable function at least twice in both x and ψ. The IFE is the line x0,ψ and local stability of the IFE changes at the point x0,ψ.
Let(47)B=x1,Z=x2,V=x3.
System (4) becomes(48)dx1tdt=Λ−1−γαx1tx3t1+δx3t−μ1x1t+γx2t≕f1,dx2tdt=−γx2t+1−γαx1tx3t1+δx3t−d+μ2x2t≕f2,dx3tdt=hV+βVV−σ+sx3t≕f3.
Applying Theorem4 to investigate if system (48) exhibits a backward bifurcation when R0V=1. Therefore, we first linearize the system and then compute its eigenvalues and eigenvectors corresponding to eigenvalues with negative real part form a basis for the stable eigenspace.
Linearizing (48) around the IFE yields(49)−μ1d+μ2R0V−1−γαΛμ10−d+μ2R0V−11−γαΛμ100βV−σ+s.
The eigenvalues of (49) are given by λ1=−μ1, λ2=−d+μ2R0V−1, and λ3=βV−σ+s.
Now, letw=w1,w2,w3T be the right eigenvector associated with zero eigenvalue. Hence,(50)−μ1d+μ2R0V−1−γαΛμ10−d+μ2R0V−11−γαΛμ100βV−σ+sw1w2w3=0.
which yields(51)−μ1w1+d+μ2R0Vw2−1−γαΛμ1w3=0,−d+μ2R0V−1w2+1−γαΛμ1w3=0,βV−σ+sw3=0.
Solving forw1, w2, and w3, we obtain w1=21−γαΛ, w2=1−γαΛ/μ1d+μ2R0V−1, and w3=1.
Letv=v1,v2,v3T be the left eigenvector associated with zero eigenvalue such that(52)−μ100d+μ2R0V−d+μ2R0V−10−1−γαΛμ11−γαΛμ1βV−σ+sv1v2v3=0,−μ1v1=0,d+μ2R0Vv1−d+μ2R0V−1v2=0,−1−γαΛμ1v1+1−γαΛμ1v2+βV−σ+sv3=0.
Solving forv1, v2, and v3, we obtain v1=0, v2=0, and v3=1.
It is observed thatv.w=1. Now, evaluating the partial derivatives of system (48) at IFE, we obtain(53)∂2f1∂x1∂x3=∂2f1∂x3∂x1=−1−γαδ,∂2f2∂x1∂x3=1−γαδ.
Since all the other remaining second order partial derivatives are equal to zero. Thus, computing the coefficientsa1 and a2 is defined in Theorem 4 (iii).(54)a1=∑i,j,k=1nvkwiwj∂2fk∂xi∂xj,a2=∑i,k=1nvkwi∂2fk∂xi∂xj.
We consider system (48) and considering a1 and a2 only as the nonzero derivatives for the terms ∂2fk/∂xi∂xj and ∂2fk/∂xi∂xj, it follows that(55)a1=2v3w1w3∂2f2∂x1∂x3+v3w2w1∂2f1∂x3∂x1,a2=v3w1∂2f2∂x1∂x3+v3w2∂2f2∂x1∂x3+v3w3∂2f2∂x1∂x3.
Substitutingvs and ws in (55) yields(56)a1=21−γ2α2Λδ,a2=1−γαδ1+1−γαΛμ1d+μ2R0V−1+21−γαΛ.
The signs of coefficientsa1 and a2 determine the nature of bifurcation exhibited by system (4) around the infection-free equilibrium for R0V=1.
For model (4) to undergo backward bifurcation, then a1>0 and a2>0. We consider the parameter values used in Table 1 to verify the conditions a1>0 and a2>0 where(57)a1=21−γ2α2Λδ,a2=1−γαδ1+1−γαΛμ1d+μ2R0V−1+21−γαΛ.
Letα=0.9, 0<γ<1, δ=0.05, d=0.4, and μ1=μ2=0.27 substituting these numerical values in a1 and a2 yields a1>0=3.24∗103 and a2>0=2.68304∗105. Hence, model (4) exhibits backward bifurcation. This implies that the epidemiological implication in relation to cholera disease of backward bifurcation is that R0V<1 is necessary and not sufficient for effective control of cholera disease; hence in a backward bifurcation setting, the cholera disease will invade to a relatively high endemic level in cells.Table 1
Parameter values for the within-host cholera model.
DescriptionParametersInitial valueSourceRecruitment rate of BΛ1.0∗106cells[11]Natural death rate of Bμ10.27day−1[5]Natural death rate of Zμ20.27day−1[5]Contact rate of B and Vα0.9[5]Clearance rate of Vσ0.02[5]Vaccine efficacyγ0.66[3]Recruitment rate of VhV0.7[11]Shedding rate of Vs0.03[11]Multiplication rate of VβV0.3[11]


## 5.1. Existence of Infection-Free Equilibrium (IFE) Points

These are steady state solutions in absence of cholera disease in the body; therefore, target cells are not infected. Stability analysis is carried out to predict the long-term behaviour of the solutions of the model (4).Proposition 3.
For model (4), the IFE point is given by IFE=B0,Z0,V0 = Λ/μ1,0,0.Proof.
At IFE,Z=0,V=0. Therefore, considering the first equation in the system (4) and replacing Z=0,V=0 and equating its right hand side to 0 yields(13)Λ−μ1B=0,Λ=μ1B.
MakingB the subject yields(14)B=Λμ1.
Therefore, the infection-free equilibrium point of the system (4) is Λ/μ1,0,0. This shows that the infected class is zero since there are no pathogens and the entire population of cells consists of target cells only and their growth is bounded by Λ/μ1.


## 5.2. Local Asymptotic Stability of the Infection-Free Equilibrium (IFE)

Theorem 1.
For any timet≥0, the infection-free equilibrium IFE=Λ/μ1,0,0 of model (4) is locally asymptotically stable when R0V<1 and unstable when R0V>1.Proof.
The jacobian matrix of (4) is given as follows:(15)J=−1−γαV1+δV−μ1γ−1−γαΛμ11+δV21−γαV1+δV−γ−d+μ21−γαΛμ11+δV200βV−σ+s.
The matrix (15) in terms of R0V yields(16)JIFE=−μ1d+μ2R0V−1−γαΛμ10−d+μ2R0V−11−γαΛμ100βV−σ+s.
To find the eigenvalues, we consider the characteristic equationJIFE−λI=0. For asymptotic stability, all the eigenvalues should be strictly negative. It can be seen clearly that λ1=−μ1 is one of the eigenvalues of (16).
UsingRouth−Hurwitz Criterion for stability as used in [6] to determine the trace and determinant, let(17)−d+μ2R0V−11−γαΛμ10βV−σ+s=0.
Equation (17) has(18)trace17=−d+μ2R0V−1+βV−σ+s<0,and a determinantdet17=−d+μ2R0V−1βV−σ+s>0 iff βV<σ+s. This implies that IFE is locally asymptotically stable when R0<1 and unstable if R0V>1.
In Theorem 1, when cholera vaccine is administered, it implies that with a small perturbation ofIFE, the solution of model (4) will eventually converge to IFE whenever R0V<1.
Epidemiologically, it implies that if a few infectious cholera pathogens are introduced into a population of target cells, the disease would die out whenR0V<1; otherwise, the disease would spread into the population of target cells.


## 5.3. Global Stability of the Infection-Free Equilibrium (IFE)

The global stability of theIFE of model (4) is explored using Castilo–Chavez et al. [7]. Model (4) is rewritten in the form as follows:(19)dHdt=LX,Z,dZdt=GX,Z,GX,0=0,where X∈R2 denotes the uninfected compartments. At IFE,(20)IFE=X∗,Z∗,X∗=Λμ1,0.The conditions(21)dXdt=HX,0,X∗,GX,Z=MZ−G^X,Z,G^X,Z≥0.X∗ is globally asymptotically stable,where M=DZGX∗,0 is a M-matrix (the off diagonal elements of M and are non-negative). If model (4) satisfies the conditions previously (21), then Theorem 2 holds.Theorem 2.
The fixed pointIFE=X∗,0 is globally asymptotically a stable equilibrium point of model (4) whenever the R0V<1 whenever the conditions (21) are satisfied, otherwise unstable.Proof.
From model (4), we obtain(22)HX,0=Λ−μ1B00,GX,Z=MZ−G^X,Z,where(23)M=−d+μ2R0V−11−γαΛμ10βV−σ+s,G^X,0=G^1X,ZG^2X,Z=00.
SinceG^X,Z>0 and the conditions (21) are satisfied; therefore, IFE is globally asymptotically stable whenever R0V<1, otherwise unstable.
This implies that given a large perturbation of theIFE, the solutions of the model (4) will eventually converge to IFE whenever R0V<1. Epidemiologically, this means that if a large number of vibrios are introduced to a population of target cells, the disease would die off, hence the elimination of infection in cells since there are no secondary infection produced whenever R0V<1, otherwise the disease would persist.


## 5.4. Local Stability of the Infection Equilibrium (IE) Points

Infection Equilibrium (IE) point is the state at which the disease persists in the cells. We investigate the local stability of infection equilibrium points atE∗B∗,Z∗,V∗.Theorem 3.
The infection equilibrium pointE∗B∗,Z∗,V∗ of system (4) is locally asymptotically stable whenever R0V>1, otherwise unstable.Proof.
The infection equilibriaE∗B∗,Z∗,V∗ of model (4) is obtained by equating model equations in (4) to zero then solving(24)Λ−1−γαBV1+δV−μ1B+γZ=0,−γZ+1−γαBV1+δV−d+μ2Z=0,hV+βVV−σ+sV=0.
The following equilibrium points are obtained by solving the previous equation simultaneously:(25)B∗=d+μ2+γ1+δV∗Z∗1−γαV∗,V∗=hVδ+s−βV.
SubstitutingB∗ into the second equation of model (4) and equating to zero, we obtain the following quadratic equation:(26)γd+μ2Z∗2+γd+μ2αhVβV−δ+sZ∗+d+μ2=0.
Let(27)a=γd+μ2=R0d+μ22,b=γd+μ2αhVβV−σ+s=R0d+μ22αhVβV−σ+s,c=d+μ2=γR0.
The quadratic equation (26) can be written as follows (28):(28)aZ∗2+bZ∗+c=0,such that Z∗ is the positive solution of the equation (28). From preceding relations, a is always positive.
Sincea>0, then there exist positive solutions of (28). Then, aZ∗2+bZ∗+c=0 at Z∗=0=c implying c=0, then (28) has a unique endemic equilibrium point. Since c=0, a unique solution of (28) is given by Z∗=−b/a, b<0.
Consider the case whenR0V<1 and c>0. If b≥0, and using Descartes’ rule of signs [8], aZ∗2+bZ∗+c has no positive root.
Ifb<0, then (28) is given as follows:(29)Z∗=b±b2−4ac2a.
Hence, the solutions are positive and distinct ifb2−4ac>0, in which case there are two endemic equilibria. The solutions of (28) coalesce into two roots when b2−4ac=0 to form one endemic equilibrium point.
Now, ifb=γd+μ2αhV/βV−σ+s>0, b will always be positive. To prove that infection equilibrium point E∗ is locally asymptotically stable when R0V>1, consider the Jacobian matrix of (4) evaluated at E∗ given as follows:(30)JE∗=−1−γαV∗1+δV∗−μ1γ−1−γαB∗1+δV∗21−γαV∗1+δV∗−d+μ2R0V−11−γαB∗1+δV∗200βV−σ+s.
For the linearized system (4), the characteristic polynomial is expressed as follows:(31)J−λI=0.
When (30) is substituted into (31), it yields(32)−1−γαV∗1+δV∗−μ1−λd+μ2R0V−1−γαB∗1+δV∗21−γαV∗1+δV∗−d+μ2R0V−1−λ1−γαB∗1+δV∗200βV−σ+s−λ=0.
Solving (32) yields(33)1−γαV∗1+δV∗−μ1−λd+μ2R0V−1−λβV−σ+s−λ+1−γαV∗1+δV∗βV−σ+s−λd+μ2R0V−1=0.
SinceR0=γ/d+μ2, then(34)−1−γ1+δV∗αV∗−μ1−λd+μ2R0V−1−λβV−δ+s−λ+d+μ2R0V1−γ1+δV∗αV∗βV−δ+s−λ=0.
Let(35)m=1−γ1+δV∗αV∗,n=βV−σ+s,V∗=hVβV−δ+s.
Then,(36)m−μ1−λd+μ2R0V−1−λn−λ+md+μ2R0Vn−λ=0.
Expanding equation (36) yields(37)λ2+λm−μ1+d+μ2R0V−1+m−μ1d+μ2R0V−1n−λ=0,and gives(38)n−λ1=0,λ1=n,λ1=βV−δ+s,k=m−μ1+d+μ2R0V−1,λ2,3=−k±k2−4m−μ1d+μ2R0V−12.
Now, the eigenvalueλ1=βV−σ+s has negative real parts, and the second eigenvalue is as follows:(39)λ2=−k−k2−4m−μ1d+μ2R0V−12,and the third eigenvalue given is as follows:(40)λ3=−k+k2−4m−μ1d+μ2R0V−12.
If4m−μ1d+μ2R0V−1>m−μ1+d+μ2R0V−12, then the square root part of λ2,3 yields an imaginary root; hence, a possibility of backward bifurcation [9] which is an indication of the oscillatory behaviour in cholera outbreaks; also, Theorem 3 implies that a small perturbation of IE, the solution of model [3], will always converge to IE whenever R0V>1. Since λ1=βV−δ+s and λ2,3=−k±k2−4m−μ1d+μ2R0V−1/2 have negative real parts at infection equilibrium point, then infection equilibrium point is locally asymptotically stable whenever R0V>1.
Epidemiologically implying that if vibrios are introduced in a population of target cells and there are new secondary infections produced wheneverR0V>1, then cholera disease would persist in the population of cells. The study of backward bifurcation in a disease dynamics can also occur when the reproduction number is less than unity, R0V<1; this is explored in the next section.


## 5.5. Existence of Backward Bifurcation

Based on the analysis in the previous section, there exists a range of values forR0 in which model (4) have two positive endemic equilibria.Let the discriminantb2−4ac be positive and be given as follows:(41)γd+μ2αhVβV−σ+s2−4γd+μ22>0.To find the value where the two endemic equilibria converge, setb2−4ac=0 and solve for the value of R0V denoted R0∗.Let(42)b2−4ac=0.Then,(43)b24ac=1,R0∗=1−b24ac,b=1−γd+μ2αhVβV−σ+s.Let(44)R0∗=1−b241−γd+μ22=1−α2hV21−γβV−σ+s.IfR0∗<R0V is equal to b2−4ac>0, then backward bifurcation will occur for values of R0V such that R0∗<R0V<1. Therefore, we show the existence of backward bifurcation due to the existence of hyperbolic fixed points as in (39) and (40). Backward bifurcation in disease transmission models is where a stable endemic equilibrium coexists with a stable disease-free equilibrium when the associated reproduction number is less than unity.Investigating the nature of bifurcation involvingIFEΛ/μ1,0,0 using Castilo and Song [10], we consider a general system of ODEs given by (45):(45)dxdt=fx,ψ;f:Rn×Rn,f∈C2Rn×Rn.Theorem 4.
Castilo–Chavez and Song [10]. We consider the following:(i)
LetI=Dxf0,0 be the linearization matrix of system (48) around the equilibrium point x=0 with ψ evaluated at 0. Zero is a simple eigenvalue of the matrix I and all other eigenvalues of I have negative real parts.(ii)
MatrixI has a non-negative right eigenvector w and a left eigenvector v corresponding to the zero eigenvalue(iii)
Letfk denote the kth component of f and(a)
a1=∑i,j,k=1nvkwiwj∂2fk/∂xi∂xj0,0(b)
a2=∑i,k=1nvkwi∂2fk/∂xi∂ϕ0,0
Then, the local dynamic of system (48) around x=0 is determined by a1 and a2.(i)
Ifa1>0, a2>0, when ψ<0, with ψ≪1, then x=0 is locally asymptotically stable and there exists a positive unstable equilibrium and when 0<ψ≪1, x=0 is unstable, and there exists a negative and locally asymptotically stable equilibrium.(ii)
Ifa1<0, a2<0 when ψ<0, with ψ≪1, then x=0 is unstable; when 0<ψ≪1, x=0 is locally asymptotically stable and there exists a positive unstable equilibrium.(iii)
Ifa1>0 and a2<0 when ψ<0, with ψ≪1, then x=0 is unstable, and there exists a locally asymptotically stable negative equilibrium, when 0<ψ≪1, x=0 is stable and positive unstable equilibrium appears.(iv)
Whena1<0 and a2>0, then there exists a forward bifurcation.(v)
Whena1>0 and a2>0, then the bifurcation at ψ=0 is backward.Proof.
Using Castilo–Chavez and Song [10], let ψ be the bifurcation parameter such that R0V<1 such that x0 is an IFE equilibrium for all values of ψ.
We consider(46)dxdt=fx,ψ,where f is a continuous differentiable function at least twice in both x and ψ. The IFE is the line x0,ψ and local stability of the IFE changes at the point x0,ψ.
Let(47)B=x1,Z=x2,V=x3.
System (4) becomes(48)dx1tdt=Λ−1−γαx1tx3t1+δx3t−μ1x1t+γx2t≕f1,dx2tdt=−γx2t+1−γαx1tx3t1+δx3t−d+μ2x2t≕f2,dx3tdt=hV+βVV−σ+sx3t≕f3.
Applying Theorem4 to investigate if system (48) exhibits a backward bifurcation when R0V=1. Therefore, we first linearize the system and then compute its eigenvalues and eigenvectors corresponding to eigenvalues with negative real part form a basis for the stable eigenspace.
Linearizing (48) around the IFE yields(49)−μ1d+μ2R0V−1−γαΛμ10−d+μ2R0V−11−γαΛμ100βV−σ+s.
The eigenvalues of (49) are given by λ1=−μ1, λ2=−d+μ2R0V−1, and λ3=βV−σ+s.
Now, letw=w1,w2,w3T be the right eigenvector associated with zero eigenvalue. Hence,(50)−μ1d+μ2R0V−1−γαΛμ10−d+μ2R0V−11−γαΛμ100βV−σ+sw1w2w3=0.
which yields(51)−μ1w1+d+μ2R0Vw2−1−γαΛμ1w3=0,−d+μ2R0V−1w2+1−γαΛμ1w3=0,βV−σ+sw3=0.
Solving forw1, w2, and w3, we obtain w1=21−γαΛ, w2=1−γαΛ/μ1d+μ2R0V−1, and w3=1.
Letv=v1,v2,v3T be the left eigenvector associated with zero eigenvalue such that(52)−μ100d+μ2R0V−d+μ2R0V−10−1−γαΛμ11−γαΛμ1βV−σ+sv1v2v3=0,−μ1v1=0,d+μ2R0Vv1−d+μ2R0V−1v2=0,−1−γαΛμ1v1+1−γαΛμ1v2+βV−σ+sv3=0.
Solving forv1, v2, and v3, we obtain v1=0, v2=0, and v3=1.
It is observed thatv.w=1. Now, evaluating the partial derivatives of system (48) at IFE, we obtain(53)∂2f1∂x1∂x3=∂2f1∂x3∂x1=−1−γαδ,∂2f2∂x1∂x3=1−γαδ.
Since all the other remaining second order partial derivatives are equal to zero. Thus, computing the coefficientsa1 and a2 is defined in Theorem 4 (iii).(54)a1=∑i,j,k=1nvkwiwj∂2fk∂xi∂xj,a2=∑i,k=1nvkwi∂2fk∂xi∂xj.
We consider system (48) and considering a1 and a2 only as the nonzero derivatives for the terms ∂2fk/∂xi∂xj and ∂2fk/∂xi∂xj, it follows that(55)a1=2v3w1w3∂2f2∂x1∂x3+v3w2w1∂2f1∂x3∂x1,a2=v3w1∂2f2∂x1∂x3+v3w2∂2f2∂x1∂x3+v3w3∂2f2∂x1∂x3.
Substitutingvs and ws in (55) yields(56)a1=21−γ2α2Λδ,a2=1−γαδ1+1−γαΛμ1d+μ2R0V−1+21−γαΛ.
The signs of coefficientsa1 and a2 determine the nature of bifurcation exhibited by system (4) around the infection-free equilibrium for R0V=1.
For model (4) to undergo backward bifurcation, then a1>0 and a2>0. We consider the parameter values used in Table 1 to verify the conditions a1>0 and a2>0 where(57)a1=21−γ2α2Λδ,a2=1−γαδ1+1−γαΛμ1d+μ2R0V−1+21−γαΛ.
Letα=0.9, 0<γ<1, δ=0.05, d=0.4, and μ1=μ2=0.27 substituting these numerical values in a1 and a2 yields a1>0=3.24∗103 and a2>0=2.68304∗105. Hence, model (4) exhibits backward bifurcation. This implies that the epidemiological implication in relation to cholera disease of backward bifurcation is that R0V<1 is necessary and not sufficient for effective control of cholera disease; hence in a backward bifurcation setting, the cholera disease will invade to a relatively high endemic level in cells.Table 1
Parameter values for the within-host cholera model.
DescriptionParametersInitial valueSourceRecruitment rate of BΛ1.0∗106cells[11]Natural death rate of Bμ10.27day−1[5]Natural death rate of Zμ20.27day−1[5]Contact rate of B and Vα0.9[5]Clearance rate of Vσ0.02[5]Vaccine efficacyγ0.66[3]Recruitment rate of VhV0.7[11]Shedding rate of Vs0.03[11]Multiplication rate of VβV0.3[11]


## 6. Numerical Simulation

In this section, we give some numerical simulation to illustrate our theoretical results by numerically solving the model system (4). This is carried out by first taking the initial values from the existing literature as shown in Table 1. Using the parameter values in Table 1 and fitting the model system (4) in MATLAB software, the ode45 command in MATLAB is used to solve it and the results in Section 6.2 are obtained.


### 6.1. Parameter Values

The parameters values are described in Table1.


### 6.2. Simulation Results

The graphs and the results obtained are described as in Figures1–4.Figure 1
Graph of (B, Z, V) against time when γ=0.1.Figure 2
Graph of (B, Z, V) against time when γ=0.6.Figure 3
Graph of (B, Z, V) against time when γ=0.9.Figure1–3 shows a graph of target cells B, infected cells Z, and the vibrio V against time in hours when γ=0.1, when γ=0.6 and when γ=0.9, respectively.Figure 4
Bifurcation curve.In Figure1, it is observed that if the vaccine efficacy is low i.e., γ=0.1, there are more cells infected which is an indication that there are pathogens invading the target cells in the small intestine as a result decrease in number of target cells as they become infected due to their interaction with the pathogen. This implies that the lower the vaccine efficacy, the higher the rate of infection of target cells.In Figure2, the vaccine efficacy is γ=0.6, the number of cells infected decreases compared to when γ=0.1. This shows that with improved vaccine efficacy, there will be low infection of cells without which the vibrios will continue to invade the cells to cause infection and this explains why cholera disease is persistence.In Figure3, the vaccine efficacy is γ=0.9; this leads to decrease in the number of cells infected which implies that with vaccine efficacy being high, the chances of cells getting infected will be low compared to when the vaccine efficacy is low. It also implies that when the vaccine efficacy is high, it takes longer time for healthy cells to be infected. Despite this, vibrios still find their way to infect the cells.Figure4 shows backward bifurcation diagram for the force of infection against reproduction number of model (4) using parameter values α=0.9, γ=0.66, δ=0.05, d=0.4, and μ1=μ2=0.27. This shows that backward bifurcation occurs for values of R0V such that R0∗<R0V<1, hence reoccurrence of in-host infection cholera disease when R0V<1.


## 6.1. Parameter Values

The parameters values are described in Table1.


## 6.2. Simulation Results

The graphs and the results obtained are described as in Figures1–4.Figure 1
Graph of (B, Z, V) against time when γ=0.1.Figure 2
Graph of (B, Z, V) against time when γ=0.6.Figure 3
Graph of (B, Z, V) against time when γ=0.9.Figure1–3 shows a graph of target cells B, infected cells Z, and the vibrio V against time in hours when γ=0.1, when γ=0.6 and when γ=0.9, respectively.Figure 4
Bifurcation curve.In Figure1, it is observed that if the vaccine efficacy is low i.e., γ=0.1, there are more cells infected which is an indication that there are pathogens invading the target cells in the small intestine as a result decrease in number of target cells as they become infected due to their interaction with the pathogen. This implies that the lower the vaccine efficacy, the higher the rate of infection of target cells.In Figure2, the vaccine efficacy is γ=0.6, the number of cells infected decreases compared to when γ=0.1. This shows that with improved vaccine efficacy, there will be low infection of cells without which the vibrios will continue to invade the cells to cause infection and this explains why cholera disease is persistence.In Figure3, the vaccine efficacy is γ=0.9; this leads to decrease in the number of cells infected which implies that with vaccine efficacy being high, the chances of cells getting infected will be low compared to when the vaccine efficacy is low. It also implies that when the vaccine efficacy is high, it takes longer time for healthy cells to be infected. Despite this, vibrios still find their way to infect the cells.Figure4 shows backward bifurcation diagram for the force of infection against reproduction number of model (4) using parameter values α=0.9, γ=0.66, δ=0.05, d=0.4, and μ1=μ2=0.27. This shows that backward bifurcation occurs for values of R0V such that R0∗<R0V<1, hence reoccurrence of in-host infection cholera disease when R0V<1.


## 7. Conclusion

In this paper, a within-host cholera model with vaccination is developed and analysed. The existence and stability of the steady states of model have been determined and show that it is locally and globally asymptotically stable whenR0V<1 and unstable when R0V>1. The analysis of the model also shows that infection equilibrium point is locally asymptotically stable when R0V>1; this means that there is persistent infection. Furthermore, analysis shows that when the vaccine efficacy γ is high, there is high clearance rate of Z and V. In this study, R0V<1 is not sufficient enough to eradicate in-host cholera disease; as a result, model (4) undergoes backward bifurcation indicating coexistence of infection-free equilibrium point and infection equilibrium point when R0V<1 and this explains why cholera disease has remained persistently endemic. Since in the model analysis, it is observed that when vaccine efficacy is high, the number of cells getting infected reduces. The study, therefore, recommends that vaccines with high efficacy than the currently existing vaccines should be manufactured. This could help reduce the burden of cholera disease when there is an outbreak.



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*Source: 1465228-2023-03-01.xml*