# Assessing the Impact of Increasing Antimicrobial Resistance ofVibrio cholerae on the Future Trends of Cholera Epidemic

Authors: Steady Mushayabasa; Claver P. Bhunu
Journal: ISRN Biomathematics
 (2012)

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## Abstract

Cholera, an acute intestinal infection caused by the bacteriumVibrio cholerae, remains a major public health problem in many parts of Africa, Asia, and Latin America. A mathematical model is developed, to assess the impact of increasing antimicrobial resistance of Vibrio cholerae on the future trends of the cholera epidemic. Equilibrium states of the model are determined and their stabilities have been examined. The impacts of increasing antimicrobial resistance of Vibrio cholerae on the future trends of cholera epidemic have been investigated through the reproductive number. Numerical results are provided to support analytical findings.


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## Body

## 1. Introduction

The global burden of cholera remains substantial. In 2005, 131,943 cases and 2,272 deaths were reported to the WHO (World Health Organization), and recently major, sustained epidemics have been reported in many parts of Africa, such as Zimbabwe and Nigeria to mention a few [1]. These statistics are gross underestimates, as many cholera-endemic countries do not report cholera to the WHO, including Zimbabwe, which has one of the highest rates of cholera in the world. More realistic estimates of the global burden of cholera mortality place the figure at 100,000–150,000 deaths per year [2]. Cholera is contracted by ingestion of food or water contaminated with the Gram-negative bacterium Vibrio cholerae. The bacteria pass through the human gastric acid barrier into the small intestine where they colonize, multiply, and begin to secrete cholera toxin. The primary treatment for cholera is rehydration with oral or intravenous fluids [3]. For severe cases, antimicrobial agents may reduce the volume and duration of diarrhea [3]. Tetracyclines (e.g., doxycycline), fluoroquinolones (e.g., ciprofloxacin), macrolides (e.g., erythromycin), and trimethoprim/sulfamethoxazole have commonly been used to treat cholera [3].Antimicrobial drug resistance can undermine the success of antimicrobial therapy.V. cholerae becomes drug resistant by exporting drugs through efflux pumps, chromosomal mutations, or developing genetic resistance via the exchange of conjugative plasmids, conjugative transposons, and integrons [4]. Several reports have documented tetracycline- and fluoroquinolone-resistant V. cholerae, and multidrug resistance is increasing [4]. The aim of this study is to assess the impact of an increase on antimicrobial resistance of Vibrio cholerae on the future trends of the cholera epidemic, with the aid of a simple mathematical model. Mathematical models have become an important tool in describing the dynamics of the spread of an infectious disease and the effects intervention strategies have on its dynamics. Modeling of epidemiological phenomena has a very long history with the first model for small pox formulated by Daniel Bernoulli in 1760 [5].A brief survey on previous studies provides the context of this paper. Various theoretical studies have been carried out on mathematical modeling of cholera transmission dynamics, focusing on a number of different issues, see [6–11], to mention a few. King et al. (2008) [11] proposed a two-path cholera model including a class for severe infections as well as a class for mild or inapparent infections. Log-likelihood estimates using 1900’s mortality data in 26 Bengal districts were generated by King et al. [11] and indicate that the districts of Bogra and Calcutta are well described by the two-path model, yet differ greatly in several parameter estimates, most notably the estimated proportion of infections resulting in severe symptoms (approximately 0.24 and 0.02, resp.). In addition to emphasizing the need for two infectious classes, the estimates show that natural immunity to cholera may wane within a year, as opposed to the previously suspected period of 3–10 years. Neilan et al. (2010) [8] proposed a mathematical model for investigating the optimal intervention strategies for cholera. Their work managed to provide a framework for designing cost-effective strategies for diseases with multiple intervention methods. More recently, Das and Mukherjee (2012) [6] proposed a mathematical model to investigate the role of lytic bacteriophage in the cyclic behaviour of cholera outbreaks. Results from their study suggest that, regardless whether phage exists in the environment or not, cholera persists in the environment.The paper is structured as follows. The cholera transmission model is formulated in Section2. Analytical results for the model are presented in Section 3. The future trend and projection profiles of the proposed model are presented in Section 4. A short discussion rounds up the paper.


## 2. Model Formulation

Our objective is to formulate a model for cholera that includes drug resistant and drug sensitive bacteria. Thus, based on the individual’s epidemiological status the total human populationN is made up of the following subgroups: susceptible individuals S, individuals infected with cholera drug sensitive strain I1, and individuals infected with drug resistance bacteria I2. The total bacteria population B constitutes the drug sensitive bacteria B1 and drug resistant bacteria B2. The susceptible human population (S) becomes infected with cholera through ingestion of bacteria contaminated water or food which contains either drug sensitive strain bacteria or drug resistant strain at rate λ1=ϑB1/(K1+B1) and λ2=ϑB2/(K2+B2), for drug sensitive and drug resistant, respectively. The parameter ϑ denotes the ingestion rate, K1,K2 captures the bacteria concentrations measured with respect to their infection doses, for drug sensitive strain and drug resistant strain, respectively. Assuming homogeneous mixing and constant recruitment of the human population into the susceptible class at rate Λ, the model takes the form
(1)dSdt=Λ-ϑB1SK1+B1-ϑB2SK2+B2-μS,dI1dt=ϑB1SK1+B1-ϑB2I1K2+B2-(μ+(1-p)δ1+r1)I1,dI2dt=ϑB2(S+I1)K2+B2+pδ1I1-(μ+δ2+r2)I2,dB1dt=ϕ1I1-ωB1,dB2dt=ϕ2I2-ωB2.
Natural mortality μ is assumed to occur in all human population classes at a constant rate; ω represents the net death rate of vibros; ϕ1,ϕ2 defines the average contribution of each cholera infected individual to the aquatic population of V. cholerae, for those infected with drug sensitive strain and drug resistant strain, respectively. Treated individuals recover at rate r1 (for those infected with drug sensitive strain) and r2 (for those infected with drug resistant strain). We assume that a proportion 1-p of individuals infected with drug sensitive strain on treatment will successfully recover from the disease, and the remainder p develop resistance due to chromosomal mutations or genetic resistance. Individuals infected with drug sensitive bacteria are assumed to be susceptible to infection by the drug resistance bacteria at rate λ2. Further, untreated infected individuals may succumb to cholera related mortality at rate δ1 and δ2, for individuals infected with drug sensitive strain and drug resistant strain, respectively. The model flow diagram is depicted in Figure 1.Figure 1
Model flow diagram.For system (1), the first octant in the state space is positively invariant and attracting; that is, solutions that start where all the variables are nonnegative remain there. Thus, system (1) will be analyzed in a suitable region
(2)Φ={(S,I1,I2)∈ℝ+3:N≤Λμ,(B1,B2)∈ℝ+2:B≤Λ(ϕ1+ϕ2)μω,
which is positively invariant and attracting. Existence, uniqueness, and continuation results for system (1) holds in this region.


## 3. Analytical Results

### 3.1. Disease-Free Equilibrium and Its Stability

System (1) has a disease-free ℰ0, given by
(3)ℰ0=(S0,I10,I20,B10,B20)=(Λμ,0,0,0,0).
Following Driessche and Watmough [12], the reproductive number for system (1), defined as the spectral radius of an irreducible or primitive nonnegative matrix is given by
(4)ℛ0=max(ℛ1,ℛ2)=(ϑϕ1ΛK1ωμ(μ+δ1+(1-p)r1),ϑϕ2ΛK2ωμ(μ+r2+δ2)).
Biologically ℛ1 and ℛ2 measure the average number of secondary infections generated by a cholera sensitive strain and resistant strain bacteria, respectively, when introduced in a disease-free population. The reproductive number ℛ0 controls the number of equilibria of system (1). If ℛ0≤1, then the disease-free equilibrium ℰ0 is the only equilibrium in Φ. If ℛ0>1, then there exists a unique endemic equilibrium ℰ*. Using Theorem 2 in van den Driessche and Watmough (2002) [12], the following result is established.Lemma 1.
The disease-free equilibriumℰ0 of system (1) is locally-asymptotically stable (LAS) if ℛ0<1 and unstable if ℛ0>1.Following Kamgang and Sallet (2008) [13], we write system (1) in the form
(5)x1′=A1(x)·(x1-x1*)+A12·(x2),x2′=A2(x)x2,
on the positively invariant set Φ⊂ℝ+n1+n2. Here x1=S and x2=(I1,I2,B1,B2). Here x1∈ℝ+1 denotes the uninfected components and x2∈ℝ+4 denoting the number of infected components, and x1*=ℰ0. We have to prove that the following conditions hold. (H1)
The system is defined on a positively invariant setΦ of the nonnegative orthant. The system is dissipative on Φ.(H2)
The subsystemx1′=A1·(x1,0)·(x1-x1*) is globally asymptotically stable at the equilibrium x1* on the canonical projection of Φ on ℝ+n1.(H3)
The matrixA2(x) is Metzler (a Metzler matrix is a matrix with off-diagonal entries nonnegative [14]) and irreducible for any given x∈Φ.(H4)
There exists an upper-bound matrixA-2 for 𝕄={A2(x)/x∈Φ} with the property that either A2∉𝕄 or if A2∉𝕄 (i.e., A2=maxΦ𝕄) then for any x-∈Φ such that A-2=A2(x-),x-∈ℝ+n1×{0} (i.e., the points where the maximum is realized are contained in the disease-free submanifold).(H5)
α(A-2)≤0 are satisfied. If conditions (H1–H5) are satisfied, then ℰ0 is globally asymptotically stable for in Φ.We express the subsystemx1′=A1·(x1,0)·(x1-x1*):
(6)S′=Λ-μS.
This is a linear system which is globally asymptotically stable at the equilibrium Λ/μ corresponding to ℰ0, which satisfies conditions H1 and H2. The matrix A2(x) is given by(7)A2(x)=[-(μ+δ1+(1-p)r1+ϑB2K2+B2)0ϑK1S(K1+B1)20ϑB2K2+B2+pr1-(μ+r2+δ2)0ϑK2S(K2+B2)2ϕ10-ω00ϕ20-ω],which is an irreducible matrix. The upper bound forx∈Φ is given by
(8)A-2=[-(μ+(1-p)r1+δ1)0ϑΛμK10pr1-(μ+r2+δ2)0ϑΛμK2ϕ10-ω00ϕ20-ω],
which is not attained on Φ and which is not the corresponding block in the Jacobian matrix of the system at ℰ0. Thus we obtain only a sufficient condition. Condition H5 requires that α(A)≤0; thus
(9)ϑϕ1ΛK1ωμ(μ+δ1+(1-p)r1)≤1,ϑϕ1ΛK2ωμ(μ+r2+δ2)≤1.
The maximum of two quantities in (9) is the reproductive number for system (1). Thus, ℰ0 is globally asymptotically stable whenever ℛ0≤1. We summarize the result in Theorem 2.Theorem 2.
The disease-free equilibrium(ℰ0) of model system (1) is globally-asymptotically stable (GAS) if ℛ0≤1 and unstable if ℛ0>1.


### 3.2. Endemic Equilibrium and Its Stability Analysis

There are three possible endemic equilibrium states for system (1), namely, (a)
ifℛ1>1 and ℛ2<1, then there exists a drug sensitive strain induced endemic equilibrium point;(b)
ifℛ1<1 and ℛ2>1, then there exists a drug resistant strain induced endemic equilibrium point;(c)
ifℛ1>1 and ℛ2>1, then there coexists a drug sensitive and drug resistant strain induced endemic equilibrium point.If either case (a) or case (b) exists, then the global stability of resultant system follows from [15]. Now we investigated the local stability for case (c), with the aid of the centre manifold theory. When case (c) exists, then the endemic equilibrium for system (1) in terms of the forces of infection is given by (10)ℰ*{S*=Λλ1*+λ2*+μ,I1*=Λλ1*(μ+δ1+(1-p)r1+λ2*)(μ+λ1*+λ2*),I2*=Λ(λ2*(μ+δ1+λ1*+λ2*)+r1(pλ1+(1-p)λ2))(μ+δ1+δ2)(μ+δ1+(1-p)r1+λ2)(μ+λ1*+λ2*),B1*=Λϕλ1*ω(μ+δ1+(1-p)r1+λ2*)(μ+λ1*+λ2*),B2*=Λϕ2(λ2*(μ+δ1+λ1*+λ2*)+r1(pλ1+(1-p)λ2))ω(μ+r2+δ2)(μ+(1-p)r1+δ1+λ2*)(μ+λ1*+λ2*),with,λ1*=ϑB1*K1+B1*,λ2*=ϑB2*K1+B2*.Before starting our main results, we give the following lemma which will be useful in the subsequent section.Theorem 3 (see [16]).
Consider the following general system of ordinary differential equations with a parameterϕ,
(11)dxdt=f(x,ϕ),f:ℝn×ℝ→ℝn,f∈ℂ2(ℝn×ℝ).
Without loss of generality, it is assumed that 0 is an equilibrium for System (11) for all values of the parameter ϕ, that is, f(0,ϕ)=0 for all ϕ and assumed (A1)
A=Dxf(0,0)=((∂fi/∂xj)(0,0)) is the linearization of system (11) around the equilibrium 0 with ϕ evaluated at 0. Zero is a simple eigenvalue of A and other eigenvalues of A have negative real parts;(A2)
matrixA has a right eigenvector w and a left eigenvector v corresponding to the zero eigenvalue.
Letfk be the Kth component of f and
(12)a=∑k,i,j=1n‍vkwiwj∂2fk∂xi∂xj(0,0),b=∑k,i=1n‍vkwi∂2fk∂xi∂ϕ(0,0).
The local dynamics of (11) around 0 are totally governed by a and b. (i)
a>0, b>0. When ϕ<0 with |ϕ|≪1,0 is locally asymptotically stable, and there exists a positive unstable equilibrium; when 0<ϕ≪1,0 is unstable and there exists a negative and locally asymptotically stable equilibrium;(ii)
a<0, b<0. When ϕ<0 with |ϕ|≪1,0 is unstable; when 0<ϕ≪1, asymptotically stable, and there exists a positive unstable equilibrium;(iii)
a>0, b<0. When ϕ<0 with |ϕ|≪1,0 is unstable, and there exists a locally asymptotically stable negative equilibrium; when 0<ϕ≪1,0 is stable, and a positive unstable equilibrium appears;(iv)
a<0, b>0. When ϕ changes from negative to positive, 0 changes its stability from stable to unstable. Correspondingly, a negative equilibrium becomes positive and locally asymptotically stable.To apply this method, the following change of variables is required. LetS=x1, I1=x2, I2=x3, B1=x4, and B2=x5, further, by using vector notation x=(x1,x2,x3,x4,x5)T, model system (1) can be written in the form dx/dt=F(x), with F=(f1,f2,f3,f4,f5)T:
(13)dx1dt=f1=Λ-ϑx4x1K1+x4-ϑx5x1K2+x5-μx1,dx2dt=f2=ϑx4x1K1+x4-ϑx5x2K2+x5-(μ+δ1+(1-p)r1)x2,dx3dt=f3=ϑB2(x1+x2)K2+x5+pr1x2-(μ+r2+δ2)x3,dx4dt=f4=ϕ1x2-ωx4,dx5dt=f5=ϕ2x3-ωx5.
Linearizing system (13) about ℰ0 gives(14)J=[-μ00-ϑΛμK1-ϑΛμK20-(μ+δ1+(1-p)r1)0ϑΛμK100pr1-(μ+r2+δ2)0ϑΛμK20ϕ10-ω000ϕ20-ω],from (8) it can be shown that
(15)ℛ0=max(ℛ1,ℛ2)=(ϑϕ1ΛK1ωμ(μ+δ1+(1-p)r1),ϑϕ1ΛK2ωμ(μ+r2+δ2)).Consider a caseℛ2=1 (i.e., ℛ1<ℛ2=1). Suppose, further, that ϑ=ϑ* is chosen as a bifurcation parameter. Solving for ϑ=ϑ*, we have
(16)ϑ*=K2ωμ(μ+r2+δ2)ϕ1Λ.Thus, the linearized system of the transformed equation (13) with ϑ=ϑ* chosen as the bifurcation parameter has a simple zero eigenvalue. Hence, it can be shown that the Jacobian of (13) at ϑ=ϑ* has right and left eigenvectors (associated with the zero eigenvalue) given below.


### 3.3. Eigenvectors ofJ

It can be shown that the JacobianJ of system (13) at ϑ=ϑ* has a right eigenvector (corresponding to the zero eigenvalue) given by w=(w1,w2,w3,w4,w5)T, where
(17)w1=-ϑ*Λμ2(w4K1+w5K2),w2=ϑ*Λw4μ(μ+δ1+(1-pr1),w3=ϑ*Λμ(μ+r2+δ2)(w5K2+pr1w4(μ+(1-p)δ1+r1)),w4>0,w5>0.
Further, J has a left eigenvector (associated with the zero eigenvalue) given byv
=
(
v
1
,
v
2
,
v
3
,
v
4
,
v
5
)
T, where
(18)v1=0,v4>0,v5>0,v2=1(μ+(1-p)r1+δ1)(ϕ1v4+pr1ϕ2v5μ+r2+δ2),v3=ϕ2v5μ+r2+δ2.


### 3.4. Computations of the Bifurcation Coefficientsa and b

For the sign ofa, it can be shown that the associated nonvanishing partial derivatives of F are
(19)∂2f2∂x1∂x4=2ϑ*K1,∂2f2∂x2∂x5=-2ϑ*K2,∂2f2∂x4∂x4=-2ϑ*ΛμK12,∂2f3∂x1∂x5=2ϑ*K1,∂2f3∂x2∂x5=2ϑ*K2,(20)∂2f3∂x5∂x5=-2ϑ*ΛμK22.It follows from (19) that
(21)a=-2ϑ*2Λμ(w4K1+w5K2)×(ϕ1v4w4(μ+δ1+(1-p)r1)+pr1ϕ2w5v5(μ+r2+δ2))-2ϑ*2Λw4w5μ(μ+r1+(1-p)δ1)×(ϕ1v4K1(μ+δ1+(1-p)r1)-ϕ2v5K2(μ+r2+δ2))-2ϑ*Λμ(ϕ1v4w42K12(μ+δ1+(1-p)r1)+pr1ϕ2v5w52K22(μ+r2+δ2)).
For the sign of b, it can be shown that the associated nonvanishing partial derivatives of F are
(22)∂2f2∂x4∂ϑ*=ΛμK1,∂2f3∂x5∂ϑ*=ΛμK2.
It follows from (22) that
(23)b=Λμ(v2w4K1+v3w5K2).
Based on the results in (21), (23), and Theorem 3, we establish the following results.Theorem 4.
The endemic equilibriumℰ* is locally asymptotically stable for ℛ0>1, but close to 1.


### 3.5. Impact of Drug Resistance on the Dynamics of Cholera

The reproduction number of an epidemic (the mean number of secondary cases infected by a single infectious case) is a key parameter for the analysis of infectious diseases because it summarizes the potential transmissibility of the disease and indicates whether an epidemic is under control. Now, we investigate the impact of increasingℛ2 on ℛ1, through partial differentiation, as follows:
(24)∂ℛ1∂ℛ2=∂ℛ1/∂Λ∂ℛ2/∂Λ+∂ℛ1/∂μ∂ℛ2/∂μ+∂ℛ1/∂ω∂ℛ2/∂ω+∂ℛ1/∂ϑ∂ℛ2/∂ϑ=ϕ1ϕ2(μ+δ1+(1-p)r1)(μ+r2+δ2)×(3+(μ+r2+δ2)(2μ+δ1+(1-p)r1)(μ+δ1+(1-p)r1)(2μ+r2+δ2))>0.Results in (24) suggest that an increase on antimicrobial resistance of Vibrio cholerae may lead to an increase in cholera prevalence in the community.


## 3.1. Disease-Free Equilibrium and Its Stability

System (1) has a disease-free ℰ0, given by
(3)ℰ0=(S0,I10,I20,B10,B20)=(Λμ,0,0,0,0).
Following Driessche and Watmough [12], the reproductive number for system (1), defined as the spectral radius of an irreducible or primitive nonnegative matrix is given by
(4)ℛ0=max(ℛ1,ℛ2)=(ϑϕ1ΛK1ωμ(μ+δ1+(1-p)r1),ϑϕ2ΛK2ωμ(μ+r2+δ2)).
Biologically ℛ1 and ℛ2 measure the average number of secondary infections generated by a cholera sensitive strain and resistant strain bacteria, respectively, when introduced in a disease-free population. The reproductive number ℛ0 controls the number of equilibria of system (1). If ℛ0≤1, then the disease-free equilibrium ℰ0 is the only equilibrium in Φ. If ℛ0>1, then there exists a unique endemic equilibrium ℰ*. Using Theorem 2 in van den Driessche and Watmough (2002) [12], the following result is established.Lemma 1.
The disease-free equilibriumℰ0 of system (1) is locally-asymptotically stable (LAS) if ℛ0<1 and unstable if ℛ0>1.Following Kamgang and Sallet (2008) [13], we write system (1) in the form
(5)x1′=A1(x)·(x1-x1*)+A12·(x2),x2′=A2(x)x2,
on the positively invariant set Φ⊂ℝ+n1+n2. Here x1=S and x2=(I1,I2,B1,B2). Here x1∈ℝ+1 denotes the uninfected components and x2∈ℝ+4 denoting the number of infected components, and x1*=ℰ0. We have to prove that the following conditions hold. (H1)
The system is defined on a positively invariant setΦ of the nonnegative orthant. The system is dissipative on Φ.(H2)
The subsystemx1′=A1·(x1,0)·(x1-x1*) is globally asymptotically stable at the equilibrium x1* on the canonical projection of Φ on ℝ+n1.(H3)
The matrixA2(x) is Metzler (a Metzler matrix is a matrix with off-diagonal entries nonnegative [14]) and irreducible for any given x∈Φ.(H4)
There exists an upper-bound matrixA-2 for 𝕄={A2(x)/x∈Φ} with the property that either A2∉𝕄 or if A2∉𝕄 (i.e., A2=maxΦ𝕄) then for any x-∈Φ such that A-2=A2(x-),x-∈ℝ+n1×{0} (i.e., the points where the maximum is realized are contained in the disease-free submanifold).(H5)
α(A-2)≤0 are satisfied. If conditions (H1–H5) are satisfied, then ℰ0 is globally asymptotically stable for in Φ.We express the subsystemx1′=A1·(x1,0)·(x1-x1*):
(6)S′=Λ-μS.
This is a linear system which is globally asymptotically stable at the equilibrium Λ/μ corresponding to ℰ0, which satisfies conditions H1 and H2. The matrix A2(x) is given by(7)A2(x)=[-(μ+δ1+(1-p)r1+ϑB2K2+B2)0ϑK1S(K1+B1)20ϑB2K2+B2+pr1-(μ+r2+δ2)0ϑK2S(K2+B2)2ϕ10-ω00ϕ20-ω],which is an irreducible matrix. The upper bound forx∈Φ is given by
(8)A-2=[-(μ+(1-p)r1+δ1)0ϑΛμK10pr1-(μ+r2+δ2)0ϑΛμK2ϕ10-ω00ϕ20-ω],
which is not attained on Φ and which is not the corresponding block in the Jacobian matrix of the system at ℰ0. Thus we obtain only a sufficient condition. Condition H5 requires that α(A)≤0; thus
(9)ϑϕ1ΛK1ωμ(μ+δ1+(1-p)r1)≤1,ϑϕ1ΛK2ωμ(μ+r2+δ2)≤1.
The maximum of two quantities in (9) is the reproductive number for system (1). Thus, ℰ0 is globally asymptotically stable whenever ℛ0≤1. We summarize the result in Theorem 2.Theorem 2.
The disease-free equilibrium(ℰ0) of model system (1) is globally-asymptotically stable (GAS) if ℛ0≤1 and unstable if ℛ0>1.


## 3.2. Endemic Equilibrium and Its Stability Analysis

There are three possible endemic equilibrium states for system (1), namely, (a)
ifℛ1>1 and ℛ2<1, then there exists a drug sensitive strain induced endemic equilibrium point;(b)
ifℛ1<1 and ℛ2>1, then there exists a drug resistant strain induced endemic equilibrium point;(c)
ifℛ1>1 and ℛ2>1, then there coexists a drug sensitive and drug resistant strain induced endemic equilibrium point.If either case (a) or case (b) exists, then the global stability of resultant system follows from [15]. Now we investigated the local stability for case (c), with the aid of the centre manifold theory. When case (c) exists, then the endemic equilibrium for system (1) in terms of the forces of infection is given by (10)ℰ*{S*=Λλ1*+λ2*+μ,I1*=Λλ1*(μ+δ1+(1-p)r1+λ2*)(μ+λ1*+λ2*),I2*=Λ(λ2*(μ+δ1+λ1*+λ2*)+r1(pλ1+(1-p)λ2))(μ+δ1+δ2)(μ+δ1+(1-p)r1+λ2)(μ+λ1*+λ2*),B1*=Λϕλ1*ω(μ+δ1+(1-p)r1+λ2*)(μ+λ1*+λ2*),B2*=Λϕ2(λ2*(μ+δ1+λ1*+λ2*)+r1(pλ1+(1-p)λ2))ω(μ+r2+δ2)(μ+(1-p)r1+δ1+λ2*)(μ+λ1*+λ2*),with,λ1*=ϑB1*K1+B1*,λ2*=ϑB2*K1+B2*.Before starting our main results, we give the following lemma which will be useful in the subsequent section.Theorem 3 (see [16]).
Consider the following general system of ordinary differential equations with a parameterϕ,
(11)dxdt=f(x,ϕ),f:ℝn×ℝ→ℝn,f∈ℂ2(ℝn×ℝ).
Without loss of generality, it is assumed that 0 is an equilibrium for System (11) for all values of the parameter ϕ, that is, f(0,ϕ)=0 for all ϕ and assumed (A1)
A=Dxf(0,0)=((∂fi/∂xj)(0,0)) is the linearization of system (11) around the equilibrium 0 with ϕ evaluated at 0. Zero is a simple eigenvalue of A and other eigenvalues of A have negative real parts;(A2)
matrixA has a right eigenvector w and a left eigenvector v corresponding to the zero eigenvalue.
Letfk be the Kth component of f and
(12)a=∑k,i,j=1n‍vkwiwj∂2fk∂xi∂xj(0,0),b=∑k,i=1n‍vkwi∂2fk∂xi∂ϕ(0,0).
The local dynamics of (11) around 0 are totally governed by a and b. (i)
a>0, b>0. When ϕ<0 with |ϕ|≪1,0 is locally asymptotically stable, and there exists a positive unstable equilibrium; when 0<ϕ≪1,0 is unstable and there exists a negative and locally asymptotically stable equilibrium;(ii)
a<0, b<0. When ϕ<0 with |ϕ|≪1,0 is unstable; when 0<ϕ≪1, asymptotically stable, and there exists a positive unstable equilibrium;(iii)
a>0, b<0. When ϕ<0 with |ϕ|≪1,0 is unstable, and there exists a locally asymptotically stable negative equilibrium; when 0<ϕ≪1,0 is stable, and a positive unstable equilibrium appears;(iv)
a<0, b>0. When ϕ changes from negative to positive, 0 changes its stability from stable to unstable. Correspondingly, a negative equilibrium becomes positive and locally asymptotically stable.To apply this method, the following change of variables is required. LetS=x1, I1=x2, I2=x3, B1=x4, and B2=x5, further, by using vector notation x=(x1,x2,x3,x4,x5)T, model system (1) can be written in the form dx/dt=F(x), with F=(f1,f2,f3,f4,f5)T:
(13)dx1dt=f1=Λ-ϑx4x1K1+x4-ϑx5x1K2+x5-μx1,dx2dt=f2=ϑx4x1K1+x4-ϑx5x2K2+x5-(μ+δ1+(1-p)r1)x2,dx3dt=f3=ϑB2(x1+x2)K2+x5+pr1x2-(μ+r2+δ2)x3,dx4dt=f4=ϕ1x2-ωx4,dx5dt=f5=ϕ2x3-ωx5.
Linearizing system (13) about ℰ0 gives(14)J=[-μ00-ϑΛμK1-ϑΛμK20-(μ+δ1+(1-p)r1)0ϑΛμK100pr1-(μ+r2+δ2)0ϑΛμK20ϕ10-ω000ϕ20-ω],from (8) it can be shown that
(15)ℛ0=max(ℛ1,ℛ2)=(ϑϕ1ΛK1ωμ(μ+δ1+(1-p)r1),ϑϕ1ΛK2ωμ(μ+r2+δ2)).Consider a caseℛ2=1 (i.e., ℛ1<ℛ2=1). Suppose, further, that ϑ=ϑ* is chosen as a bifurcation parameter. Solving for ϑ=ϑ*, we have
(16)ϑ*=K2ωμ(μ+r2+δ2)ϕ1Λ.Thus, the linearized system of the transformed equation (13) with ϑ=ϑ* chosen as the bifurcation parameter has a simple zero eigenvalue. Hence, it can be shown that the Jacobian of (13) at ϑ=ϑ* has right and left eigenvectors (associated with the zero eigenvalue) given below.


## 3.3. Eigenvectors ofJ

It can be shown that the JacobianJ of system (13) at ϑ=ϑ* has a right eigenvector (corresponding to the zero eigenvalue) given by w=(w1,w2,w3,w4,w5)T, where
(17)w1=-ϑ*Λμ2(w4K1+w5K2),w2=ϑ*Λw4μ(μ+δ1+(1-pr1),w3=ϑ*Λμ(μ+r2+δ2)(w5K2+pr1w4(μ+(1-p)δ1+r1)),w4>0,w5>0.
Further, J has a left eigenvector (associated with the zero eigenvalue) given byv
=
(
v
1
,
v
2
,
v
3
,
v
4
,
v
5
)
T, where
(18)v1=0,v4>0,v5>0,v2=1(μ+(1-p)r1+δ1)(ϕ1v4+pr1ϕ2v5μ+r2+δ2),v3=ϕ2v5μ+r2+δ2.


## 3.4. Computations of the Bifurcation Coefficientsa and b

For the sign ofa, it can be shown that the associated nonvanishing partial derivatives of F are
(19)∂2f2∂x1∂x4=2ϑ*K1,∂2f2∂x2∂x5=-2ϑ*K2,∂2f2∂x4∂x4=-2ϑ*ΛμK12,∂2f3∂x1∂x5=2ϑ*K1,∂2f3∂x2∂x5=2ϑ*K2,(20)∂2f3∂x5∂x5=-2ϑ*ΛμK22.It follows from (19) that
(21)a=-2ϑ*2Λμ(w4K1+w5K2)×(ϕ1v4w4(μ+δ1+(1-p)r1)+pr1ϕ2w5v5(μ+r2+δ2))-2ϑ*2Λw4w5μ(μ+r1+(1-p)δ1)×(ϕ1v4K1(μ+δ1+(1-p)r1)-ϕ2v5K2(μ+r2+δ2))-2ϑ*Λμ(ϕ1v4w42K12(μ+δ1+(1-p)r1)+pr1ϕ2v5w52K22(μ+r2+δ2)).
For the sign of b, it can be shown that the associated nonvanishing partial derivatives of F are
(22)∂2f2∂x4∂ϑ*=ΛμK1,∂2f3∂x5∂ϑ*=ΛμK2.
It follows from (22) that
(23)b=Λμ(v2w4K1+v3w5K2).
Based on the results in (21), (23), and Theorem 3, we establish the following results.Theorem 4.
The endemic equilibriumℰ* is locally asymptotically stable for ℛ0>1, but close to 1.


## 3.5. Impact of Drug Resistance on the Dynamics of Cholera

The reproduction number of an epidemic (the mean number of secondary cases infected by a single infectious case) is a key parameter for the analysis of infectious diseases because it summarizes the potential transmissibility of the disease and indicates whether an epidemic is under control. Now, we investigate the impact of increasingℛ2 on ℛ1, through partial differentiation, as follows:
(24)∂ℛ1∂ℛ2=∂ℛ1/∂Λ∂ℛ2/∂Λ+∂ℛ1/∂μ∂ℛ2/∂μ+∂ℛ1/∂ω∂ℛ2/∂ω+∂ℛ1/∂ϑ∂ℛ2/∂ϑ=ϕ1ϕ2(μ+δ1+(1-p)r1)(μ+r2+δ2)×(3+(μ+r2+δ2)(2μ+δ1+(1-p)r1)(μ+δ1+(1-p)r1)(2μ+r2+δ2))>0.Results in (24) suggest that an increase on antimicrobial resistance of Vibrio cholerae may lead to an increase in cholera prevalence in the community.


## 4. Population Level Effects

In order to support analytical findings in this study, we now simulate system (1) using the parameters in Table 1. The average life span 1/μ was set to be 50 years.Table 1
Model parameters and their interpretations.
Parameter definition
Symbol
Units
Point estimate
Range
Source
Recruitment rate
Λ
People/year
100 000
—
[15]
Bacteria death rate
ω
/day
0.033
0.03–0.033
[7, 8, 15]
Bacteria shedding rate
ϕ
1
cells/mL day
30.0
10.0–50.0
[8, 15]
Bacteria shedding rate
ϕ
2
cells/mL day
50.0
10.0–50.0
[8, 15]
Natural mortality rate
μ
/year
0.0142
0.0142–0.02
[15]
Bacteria ingestion rate
ϑ
/day
0.5
0.15–1.5
[8, 15]
Half saturation constant
K
1
cells/mL
1
0
7
1
0
6
–
1
0
9
[8, 15]
Half saturation constant
K
2
cells/mL
1
0
6
1
0
6
–
1
0
9
[8, 15]
Cholera related mortality rate
δ
1
/year
0.0240
0.0240–0.4622
[8]
Cholera related mortality rate
δ
2
/year
0.4622
0.0240–0.4622
[8]
Recovery rate for sensitive strain
r
1
/year
22.6
11.3–43.2
[8]
Recovery rate for resistant strain
r
2
/year
21.6
11.3–43.2
[8]
Proportion of infected individuals who develop drug resistance
p
—
0.5
0.001–0.5
AssumedNumerical results in Figure2 suggests that whenever cholera sensitive strain and resistant strain coexist in the community, then in the long run (averagely 25 days or less according to Figures 2(a) and 2(c)) cumulative resistant strains will outnumber cumulative sensitive strains. In Figure 2(b) we note that if we have more drug resistant cases than drug sensitive cases on the outbreak, then they will be a pronounced increase of cumulative drug resistant strain cases on a period, averagely of 10 days or less.Simulations of model system (1) showing the expected long-term cumulative cholera cases, for different initial conditions, and parameter values used are as in Table 1. In all cases S=20000,B1=B2=1000,0000,000, and for (a) I1=I2=500, (b) I1=100,I2=500, (c) I1=500,andI2=100.
(a)
(b)
(c)Here (Figure3) we note that an increase of p will lead to a decrease on the susceptible population and cumulative infectious drug sensitive cases, while cumulative infectious drug resistant cases will rise. These results further highlight the impact drug resistance has on the long-term dynamics of cholera.Small simulations of model system (1) showing the effects of increasing the proportion of individuals who develop drug resistance through high mutation rate of the bacteria within their body (p) on the dynamics of (a) individuals infected with drug sensitive strain (I1), (b) individuals infected with drug resistant strain (I2), (c) susceptible individuals (S) over a period of 50 days. The rest of the parameters are fixed on their baseline values from Table 1, and the following assumed initial conditions S=20000, I1=500,I2=500, and B1=B2=1000,0000,000.
(a)
(b)
(c)Simulations in Figure4 illustrate the effect of varying bacteria ingestion rate ϑ, on cumulative cholera cases. Figures 4(a) and 4(b) depict the associated infection levels when cholera drug sensitive and drug resistant coexist. Figure 4(c) illustrates cumulative infections in the absence of drug resistant strain. For ϑ>0.5 we note that infection levels have a sharp increase and they attain their maximum in a period of 20 days or less, despite whether there is coexistence of drug sensitive and drug resistant or not, then decreases sharply, until they attain stability. Further analysis of Figure 4 reveals that for low bacteria ingestion rate (ϑ<0.5) the cumulative infections increase gradually and attain the maximum after a period of 20 days, and then decrease gradually until stability is attained.Simulation of system (1) showing the effects of varying bacteria ingestion rate, ϑ.
(a)
(b)
(c)


### 4.1. Sensitivity Analysis ofℛ0

Sensitivity analysis of model parameters is very important to design and control strategies as well as a direction to future research. There are many methods [17] available for conducting sensitivity analysis such as differential analysis, response surface methodology, the Fourier amplitude sensitivity test (FAST) and other variance decomposition procedures, fast probability integration, and sampling-based procedures. In this section, sensitivity analysis based on Latin hypercube sampling (LHS) [17, 18] has been performed using relevant parameters’ values in Table 1. Sensitivity analysis assesses the amount and type of change inherent in the model as captured by the terms that define the reproductive number (ℛ0). If ℛ0 is very sensitive to a particular parameter, then a perturbation of the conditions that connect the dynamics to such a parameter may prove useful in identifying policies or intervention strategies that reduce epidemic prevalence. In this section the partial rank correlation coefficients (PRCCs) were calculated to estimate the correlation between values of ℛ0 and the model parameters across 1000 random draws from the empirical distribution of ℛ0 and its associated parameters.Results in Figure5 suggest that reduction of the cholera bacteria through death has a significant impact on reducing the magnitude of the reproductive number. Although, there is a need to treat infected individuals, results on Figure 5 suggest that implementing intervention strategies targeting at reducing the bacteria population may have more impact on controlling cumulative cholera cases and related mortality since half saturation constant and human recruitment rate are the two parameters, which have the greatest impact on decreasing and increasing the magnitude of ℛ2, respectively. We now examine the dependence of the ℛ2 using the Latin hypercube sampling technique. Results in Figure 6 support the earlier findings in Figure 5, that high values of K2 and Λ lead to low and high values, respectively, of ℛ2.Figure 5
Partial rank correlation coefficients showing the effects of parameter variation on,ℛ2 using ranges in Table 1. Parameters with positive PRCCs will increase ℛ2 when they are increased, whereas parameters with negative PRCCs will decrease ℛ2 when they are increased.Scatter plots for the basic reproductive numberℛ2 and the two model parameter values (half saturation constant and human recruitment rate). These results were obtained from Latin hypercube sampling using a sample size of 1000.
(a)
(b)Overall, results in Figure7 are in agreement to the results in Figure 5; however it is worth noting that the inclusion of a proportion of individuals who develop drug resistance has an impact on increasing the magnitude of the reproductive number, though not highly significant compared to the other model parameters.Figure 7
Partial rank correlation coefficients showing the effects of parameter variation onℛ1 using ranges in Table 1. Parameters with positive PRCCs will increase ℛ1 when they are increased, whereas parameters with negative PRCCs will decrease ℛ1 when they are increased.


## 4.1. Sensitivity Analysis ofℛ0

Sensitivity analysis of model parameters is very important to design and control strategies as well as a direction to future research. There are many methods [17] available for conducting sensitivity analysis such as differential analysis, response surface methodology, the Fourier amplitude sensitivity test (FAST) and other variance decomposition procedures, fast probability integration, and sampling-based procedures. In this section, sensitivity analysis based on Latin hypercube sampling (LHS) [17, 18] has been performed using relevant parameters’ values in Table 1. Sensitivity analysis assesses the amount and type of change inherent in the model as captured by the terms that define the reproductive number (ℛ0). If ℛ0 is very sensitive to a particular parameter, then a perturbation of the conditions that connect the dynamics to such a parameter may prove useful in identifying policies or intervention strategies that reduce epidemic prevalence. In this section the partial rank correlation coefficients (PRCCs) were calculated to estimate the correlation between values of ℛ0 and the model parameters across 1000 random draws from the empirical distribution of ℛ0 and its associated parameters.Results in Figure5 suggest that reduction of the cholera bacteria through death has a significant impact on reducing the magnitude of the reproductive number. Although, there is a need to treat infected individuals, results on Figure 5 suggest that implementing intervention strategies targeting at reducing the bacteria population may have more impact on controlling cumulative cholera cases and related mortality since half saturation constant and human recruitment rate are the two parameters, which have the greatest impact on decreasing and increasing the magnitude of ℛ2, respectively. We now examine the dependence of the ℛ2 using the Latin hypercube sampling technique. Results in Figure 6 support the earlier findings in Figure 5, that high values of K2 and Λ lead to low and high values, respectively, of ℛ2.Figure 5
Partial rank correlation coefficients showing the effects of parameter variation on,ℛ2 using ranges in Table 1. Parameters with positive PRCCs will increase ℛ2 when they are increased, whereas parameters with negative PRCCs will decrease ℛ2 when they are increased.Scatter plots for the basic reproductive numberℛ2 and the two model parameter values (half saturation constant and human recruitment rate). These results were obtained from Latin hypercube sampling using a sample size of 1000.
(a)
(b)Overall, results in Figure7 are in agreement to the results in Figure 5; however it is worth noting that the inclusion of a proportion of individuals who develop drug resistance has an impact on increasing the magnitude of the reproductive number, though not highly significant compared to the other model parameters.Figure 7
Partial rank correlation coefficients showing the effects of parameter variation onℛ1 using ranges in Table 1. Parameters with positive PRCCs will increase ℛ1 when they are increased, whereas parameters with negative PRCCs will decrease ℛ1 when they are increased.


## 5. Discussion

As the causative agent of cholera, the bacteriumVibrio cholerae represents an enormous public health burden, especially in the developing countries around the world. A deterministic model for assessing the impact of increasing antimicrobial resistance of Vibrio cholerae on the long-term transmission dynamics of cholera has been proposed and analyzed. The reproductive number for the model has been computed. Qualitative analysis of the model reveals that the model has a globally asymptotically stable infection free equilibrium state whenever the associated reproductive number is less than unity. With the aid of centre manifold theory it has been established that for the reproductive number greater but close to unity the model has a locally asymptotically stable endemic equilibrium point. Comprehensive analytical and numerical results from the study suggest that an increase on cumulative antimicrobial resistance of Vibrio cholerae has a negative impact on controlling cholera prevalence within the community, more specifically, if the outbreak is constituted of higher drug resistant cases than drug sensitive cases.


## 6. Concluding Remarks

In conclusion, our model suggests that the level of cholera infection depends highly on the rate of human recruitment and bacteria ingestion rate. We note that for bacteria ingestion rate above 0.5 cumulative cholera cases have a sharp increase within the period of 20 days, while for bacteria have ingestion less than 0.5 cumulative cholera cases a gradual increase and attain maximum after 20 days, and they also decrease steadily until stability is attained. We also observed that if a cholera outbreak have higher drug resistant cases than drug sensitive cases on the outbreak, then there will be a pronounced increase of cumulative drug resistant strain cases on a period, averagely of 10 days or less.



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*Source: 127492-2012-12-04.xml*