Applied Mathematical Sciences, Vol. 9, 2015, no. 20, 989 - 1002HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ams.2015.41166

Dynamical Behavior of Cholera Epidemic Model

with Non-linear Incidence Rate

Muhammad Altaf Khan1, Ahmad Ali2, L. C. C. Dennis4∗, Saeed Islam1,Ilyas Khan3, Murad Ullah5 and Taza Gul1

1Department of Mathematics, Abdul Wali Khan University MardanKhyber Pakhtunkhwa, Pakistan

2Department of Mathematics, ISPaR-Bacha Khan, University CharsaddaKhyber Pakhtunkhwa, Pakistan

3College of Engineering Majmaah University, MajmaahKingdom of Saudi Arabia

4Department of Fundamental and Applied Sciences, Universiti TeknologiPETRONAS, 31750 Perak, Malaysia

4 Department of Mathematics, Islamia College University, Peshawar, Pakistan∗Corresponding author

Copyright c© 2014 Muhammad Altaf Khan et al. This is an open access article dis-

tributed under the Creative Commons Attribution License, which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Cholera is an infection of the small intestine caused by the bacteriumVibrio cholera. In this work, we consider a mathematical model SIRBwhich represents the dynamics of cholera. To understand the dynamicsof the model, the disease free and endemic stability are discussed. Thedisease free equilibrium is stable locally as well as globally when thebasic reproduction less than 1, otherwise an unstable equilibrium exist.If the basic reproduction number exceeds than unity, then the endemicequilibrium is stable locally as well as globally with some sufficient con-ditions. The numerical solution of the model illustrates the analyticalresults.

990 Muhammad Altaf Khan et al.

Keywords: Cholera, Reproduction Number, Local Stability, Global Sta-bility, Numerical Solution

1 Introduction

Cholera is a small intestine infection caused by the bacterium Vibrio cholera.The cause also contains the contaminated water and ingestion of food withbacterium Vibrio cholera. Among the 200 serogroups of Vibrio cholera, it isonly Vibrio cholera o1and o139 that are known to be the cause of the choleradisease[1]. The etiological agent,Vibrio cholera o1 (and more recently vibriocholera o139), passes through and survives the gastric acid barrier of the stom-ach and then penetrates the mucus lining that coats the intestinal epithelial[2].The main symptoms are watery diarrhea and vomiting. This may result indehydration and in severe cases grayish-bluish skin. Transmission occurs pri-marily by drinking water or eating food that has been contaminated by thefeces (waste product) of an infected person, including one with no apparentsymptoms [3].

The severity of the diarrhea and vomiting can lead to rapid dehydrationand electrolyte imbalance, and death in some cases. The primary treatment isoral rehydration therapy, typically with oral rehydration solution, to replacewater and electrolytes. If this is not tolerated or does not provide improve-ment fast enough, intravenous fluids can also be used. Antibacterial drugs arebeneficial in those with severe disease to shorten its duration and severity. Ifthe severe diarrhea is not treated, it can result in life-threatening dehydrationand electrolyte imbalances. Fever is rare and should raise suspicion for sec-ondary infection. Patients can be lethargic, and might have sunken eyes, drymouth, cold clammy skin, decreased skin turgor, or wrinkled hands and feet.Kussmaul breathing, a deep and labored breathing pattern, can occur becauseof acidosis from stool bicarbonate losses and lactic acidosis associated withpoor perfusion. Blood pressure drops due to dehydration, peripheral pulse israpid and thready, and urine output decreases with time. Muscle crampingand weakness, altered consciousness, seizures, or even coma due to electrolytelosses and ion shifts are common, especially in children [4].

Several mathematical models have been modeled on cholera as well as dif-feren infectious diseases, such as [5, 6, 7, 8, 10, 12]. In this work, we formulateda mathematical model of cholera in the form of SEIR mathematical model. Themodel consist four state variable,i.e. susceptible S(t), infected I(t) and recov-ered R(t) and pathogen population B(t). The mathematical model and theirdescriptions are presented in section 2. The fundamental properties of themodel are discussed in section 3. The stability results are discussed in section4, and 5. The brief discussion is presented in section 6.

Dynamical behavior of cholera epidemic model 991

2 Model Formulation

The human population is divided into three subgroups,S(t)-Susceptible, I(t)-Infected and R(t)-Recovered or removed, with N(t) = S(t) + I(t) +R(t). Theterm B(t) represent the pathogen population at time t. The basic governingdifferential equation consist a system of differential equation is given by

dS

dt= Λ− µS(t)− βbS(t)B(t)

1 + a1B(t)− βsS(t)I(t)

1 + a2I(t),

dI

dt=

βbS(t)B(t)

1 + a1B(t)+βsS(t)I(t)

1 + a2I(t)− (µ0 + µ+ φ)I(t),

dR

dt= φI(t)− µR(t),

dB

dt= ξI(t)− dB(t), (1)

Subject to non-negative initial conditions

S(t) = S0 ≥ 0, I(t) = I0 ≥ 0, R(t) = R0 ≥ 0, B(t) = B0 ≥ 0. (2)

Here in system (1), the human population growth rate is shown by Λ. Thenatural mortality rate of the human population is µ. The disease contactbetween S(t) and B(t) is shown by βb while between S(t) and I(t) by βs. Theterm a1 and a2 are the saturation constants. The disease death rate at I(t)is µ0 and the rate of recovery from infection is denoted by φ. The term ξrepresent the human contribution to pathogen and d is the death rate of thepathogen in the environment.Since equations S, I and B in system (1) are independent of the variable R,so it is reasonable that we reduced the system (1) to the following form:

dS

dt= Λ− µS(t)− βbS(t)B(t)

1 + a1B(t)− βsS(t)I(t)

1 + a2I(t),

dI

dt=

βbS(t)B(t)

1 + a1B(t)+βsS(t)I(t)

1 + a2I(t)− (µ0 + µ+ φ)I(t),

dB

dt= ξI(t)− dB(t), (3)

Subject to non-negative initial conditions. The total dynamics of the humanpopulation is given by

d(S + I)

dt= Λ− µS − µI − µ0I − φI,

≤ Λ− µS − µI. (4)

992 Muhammad Altaf Khan et al.

In the absence of µ0 and φ, the feasible region for the system (3), given by

Π1 ={

(S, I) ∈ R2+ : 0 ≤ S + I ≤ Λ

µ

}(5)

and the feasible region for the population of pathogen

Π2 ={B|0 ≤ B ≤ ξ

d

}, (6)

and define Θ = Π1 × Π1. The feasible region for human and pathogen popu-lation, we discuss all the solutions lies inside the region Θ.

3 Fundamental Properties

In this section, we find the fundamental properties of the system (3), which isessential in the proof of the proceeding sections.Theorem 2.1: The associated solutions of the system (3) with initial condi-tions (2) are non-negative for all t > 0.Proof: We can write the system (3) in the matrix form as follows:

Y ′ = M(Y ),

where Y = (S, I, B)T ∈ R3 and M(Y ) is given by

M(Y ) =

M1(Y )M2(Y )M3(Y )

=

Λ− µS(t)− βbS(t)B(t)

1+a1B(t)− βsS(t)I(t)

1+a2I(t)

βbS(t)B(t)1+a1B(t)

+ βsS(t)I(t)1+a2I(t)

− (µ0 + µ+ φ)I(t)

ξI(t)− dB(t)

. (7)

We have

dS

dt|S=0 = Λ− µS ≥ 0,

dI

dt|I=0 =

βbS(t)B(t)

1 + a1B(t)≥ 0,

dB

dt|B=0 = ξI ≥ 0. (8)

Therefore,

Mi|Yt=0,Yt∈C3+≥ 0, i = 1, 2, 3. (9)

Thus, the corresponding solutions of system (3) for Y (t) ∈ R3+ for all time

t ≥ 0.

Dynamical behavior of cholera epidemic model 993

3.1 Basic Reproduction Number R0

In this subsection, we determine the basic reproduction number R0 for theproposed system (3), by using the same procedure developed in [11].Suppose x = (I, B), where

F =

[βbSB1+a1B

+ βsSI1+a1I

0

]and V =

[(µ+ µ0 + φ)I−ξI + dB

],

we obtained the matrix F and V as

F =

[βsS

0 βbS0

0 0

]and V =

[(µ+ µ0 + φ) 0

−ξ d

].

The matrix FV −1 is

FV −1 =

[βsS0

(µ+µ0+φ)+ βbS

0

ξ(µ+µ0+φ)βbS

0/d

0 0

].

The required basic reproduction number R0 of the system is given by

R0 =Λβs

µ(µ+ µ0 + φ)︸ ︷︷ ︸R01

+Λβbξ

µd(µ+ µ0 + φ)︸ ︷︷ ︸R02

.

It is obvious that R01 < R0 and R02 < R0.

3.2 Equilibrium Points

In the proposed subsection, we determine the equilibria of the system (3) atE0 = (S0, 0, 0) and E1 = (S∗, I∗, B∗). Taking the right side of the system(3) equal zero, the corresponding equilibria at the point E0 = (S0, 0, 0) andE1 = (S∗, I∗, B∗) obtained as follows:

E0 = (S0, 0, 0) = (Λ

µ, 0, 0), Disease Free Equilibrium,

and at E1 = (S∗, I∗, B∗)

S∗ =βbξ − (µ+ µ0 + φ)

(d+ a1ξI∗)(1 + a2I∗),

B∗ =ξI∗

d,

called the endemic equilibria.

994 Muhammad Altaf Khan et al.

4 Disease Free stability

In this section, we find out the stability result of the disease free equilibriumat the equilibrium point E0. The local and global stability of the disease freeequilibrium is presented in the following:Theorem 4:1 The disease free equilibrium of the system at E0 is locallyasymptotically stable, if R01 < R0 < 1 and R02 < R0 < 1, otherwise unstable.Proof: At the equilibrium point E0, the Jacobian matrix J1 is given by

J1 =

−µ −βsS0 −βbS0

0 βsS0 − (µ+ µ0 + φ) βbS

0

0 ξ −d

. (10)

The characteristics equation of the Jacobian matrix J1 is

(µ+ λ)[λ2 + a1λ+ a0λ] = 0,

where

a1 =Λdβsµ

+ d(µ+ µ0 + φ)(1−R02) > 0,

a2 = d+ (µ+ µ0 + φ)(1−R01) > 0.

The first eigenvalue of the Jacobian matrix J1 is −µ, the other two eigenvaluesare negative if and only if R01 < R0 < 1 and R02 < R0 < 1. The Routh-Hurtwiz conditions are satisfied i.e., (a1 > 0, a2 > 0). All the eigenvalues ofthe Jacobian matrix J1 has negative real part, so, the disease free equilibriumE0 of the system (3) is locally asymptotically stable. �.Lemma 4.1: The global stability of the disease free equilibrium is guaranteedif it is written in the form:

dXdt

= F(X ,Z),

dZdt

= G(X ,Z), G(X ,Z) = 0, (11)

where X = S ∈ R1 and Z = (I, B) ∈ R2, repetitively represent the numberof uninfected and infected individuals. The disease free equilibrium is nowdenoted by U0 = (X 0, 0). The conditions P1 and P2 given below must basatisfied:

P1 : FordXdt

= F(X , 0) = 0,

P2 : G(X ,Z) = AZ − G(X ,Z), where G(X ,Z) ≥ 0, for (X ,Z) ∈ Θ,(12)

Dynamical behavior of cholera epidemic model 995

the matrix A = DzG(X ,Z) is M-matrix, and Θ is the biological feasible region.Then, the equilibrium point U0 = (X 0, 0) is globally asymptotically stable. �Now, we prove the global stability of the disease free equilibrium.Theorem 4.2: If R0 < 1, the disease free equilibrium of the system (3) at E0is globally asymptotically stable.Proof: According to Lemma 4.1, the conditions P1 and P2 for system (3) asfollows:

P1 : F(X , ′) = [Λ− µS],

P2 : G(X ,Z) = AZ − G(X ,Z), where G(X ,Z) ≥ 0, for (X ,Z) ∈ Θ,(13)

where

A =

[βsS

0 − (µ+ µ0 + φ) βbS0

ξ −d

],Z =

[IB

]and G(X ,Z) =

[βsI[S0 − S

1+a2I] + βbB[S0 − S

1+a1B]

0

]Since A represents a Metzler matrix and G(X ,Z) ≥ 0 at the point E0. Thus,the system (3) about the disease free equilibrium E0 is globally asymptoticallystable.

5 Stability of Endemic Equilibrium

This section investigates the local and global stability of endemic equilibriumof the system (3).Theorem 5.1: If the basic reproduction number R0 > 1 and βbξ < (µ+µ0+φ),then the endemic equilibrium E1 of the system (3)is locally asymptoticallystable.Proof: The Jacobian matrix J2 of the system (3) at the endemic equilibriumE1, is given by

J2 =

−µ− βbB

∗

1+a1B∗ − βsI∗

1+a2I∗− a2βsS∗I∗

(1+a2I∗)2− a1βbS

∗B∗

(1+a1B∗)2

βbB∗

1+a1B∗ + βsI∗

1+a2I∗a2βsS∗I∗

(1+a2I∗)2− (µ+ µ0 + φ) a1βbS

∗B∗

(1+a1B∗)2

0 ξ −d

One of the eigenvalue of the Jacobian matrix J1 is −µ− βbB

∗

1+a1B∗ − βsI∗

1+a2I∗< 0,

has negative real part. To find the other two eigenvalues, we follow Routh-Hurtwiz Criteria, i.e. traceJ2 < 0 and detJ2 > 0. From the Jacobian matrix

996 Muhammad Altaf Khan et al.

J2, we have

detJ2 =

(a2βs

(1 + a2I∗)2+

ξ2a1 βb(d+ a1ξ)

)(

(µ+ µ0 + φ)− βbξ(d+ a1ξI∗)(1 + a2I∗)

)dI∗ + (µ+ µ0 + φ) > 0,

and

traceJ2 = − a2βsI∗

(1 + a2I∗)3(d+ a1ξI∗)

(µ+ µ0 + φ)− (µ+ µ0 + φ)− d < 0.

T raceJ2 < 0 and detJ2 > 0 if βbξ < (µ + µ0 + φ). Thus, the Jacobian matrixJ2 has negative real part if βbξ < (µ + µ0 + φ) and R0 > 1. So the endemicequilibrium E1 of the system (3)is stable locally asymptotically.Theorem 5.2: If R0 > 1, then the system is stable globally asymptotically.Proof: Consider the following lyapunov function

L = (S − S∗ − S∗ln SS∗

)C1 + (I − I∗ − I∗ln II∗

)C2 + C3B, (14)

taking the time derivative of L, we obtain

L′ = C1(1−S

S∗)dS

dt+ C2(1−

I

I∗)dI

dt+ C3

dB

dt, (15)

using system (3), we obtain

L′ = C1(1−S

S∗)[Λ− µS(t)− βbS(t)B(t)

1 + a1B(t)− βsS(t)I(t)

1 + a2I(t)

]+ C2(1−

I

I∗)[βbS(t)B(t)

1 + a1B(t)

+βsS(t)I(t)

1 + a2I(t)− (µ0 + µ+ φ)I(t)

]+ C3

[φI(t)− µR(t)

]. (16)

After some arrangements, we obtain

L′ = −(S − S∗)2

Sµξ − ξ

( (βbξ)

(d+ a1ξI)+

βs(1 + a2I)

)[I∗S− S∗

I

]< 0. (17)

The positive constants are C1 = C2 = ξ and C3 = (µ+ µ0 + φ). L′ < 0, if andonly if I∗

S> S∗

I, with R0 > 1. Thus, the endemic equilibrium E1 of the system

(3) is globally asymptotically stable.

6 Discussion

The dynamical behavior of cholera epidemic model with non-linear incidencerate is investigated. The model construction and basic results are derived anddiscussed. The threshold quantity that examine the stability of the equilibriumis obtained. The disease free and endemic equilibrium obtained and discussed.

Dynamical behavior of cholera epidemic model 997

For the threshold quantity R0, we obtained the stability analysis of the model.The reduced model (3), is stable locally as well as globally when the thresholdquantity, less than unity, is derived. If the threshold exceeds than unity, thepersistence occur and the disease permanently exists in the community. For,this, we proved that the model (3) is stable both locally and globally. For thevalues of the parameters for which the threshold quantity are less than unity ispresented in Figure 1. Figure 2 represents the dynamical behavior of the humanpopulation, with the baseline S(0) = 200, I(0) = 03 and R(0) = 1. Figure 3,represents the dynamical behavior of pathogen population, with the baselineB(0) = 20. Figure 4 is the combined graph of the human with pathogen. Thedetail of the parameters values used in simulation is presented in Table 1.

Table 1: Parameter values used in Numerical simulations.

Parameter Description Value Dimension

Λ Population growth rate of human 0.6 Per day−1

βb The contact rate between susceptible S and Pathogen B 0.05 Per day−1

βs The contact rate between susceptible S and infected I 0.05 Per day−1

a1 and a2 The saturation constants 0.02, 0.03 —µ Natural death rate of human 0.1 Per day−1

µ0 Disease related death rate 0.01 Per day−1

φ The rate at which the individuals recovered 0.03 Per day−1

ξ The contribution of human to the pathogen 0.1 Per day−1

d The death rate at pathogen population 0.3 Per day−1

Acknowledgements. Authors would like to thank Universiti TeknologiPETRONAS for the financial support.

References

[1] A. K. T. Kirschner, et al., Rapid growth of planktonic Vibriocholerae non-O1/non-O139 strains in a large alkaline lake in Aus-tria: Dependence on temperature and dissolved organic carbon qual-ity, Applied and Environmental Microbiology, 74:2004, 2015, 2008.http://dx.doi.org/10.1128/aem.01739-07

[2] J. Reidl and K. E. Klose,Vibrio cholerae and cholera: out of the wa-ter and into the host, FEMS Microbiology Reviews, 26:125-139, 2002.http://dx.doi.org/10.1111/j.1574-6976.2002.tb00605.x

998 Muhammad Altaf Khan et al.

[3] McElroy, Ann and Patricia K. Townsend. Medical Anthropology in Eco-logical Perspective, Boulder, CO: Westview, 2009, 375.

[4] Sack D. A., Sack R. B., Nair G. B., Siddique A. K., Cholera. Lancet 363(9404): 223-233, 2004. http://dx.doi.org/10.1016/s0140-6736(03)15328-7

[5] C. T. Codec, Endemic and epidemic dynamics of cholera: the role of theaquatic reservoir,BMC Infectious Diseases,vol.1, article 1, 2001.

[6] Z. Mukandavire, et al.,, Estimating the reproductive numbers for the 2008-2009 cholera outbreaks in Zimbabwe, Proceedings of the National Academyof Sciences of the United States of America, vol. 108, no. 21, pp. 8767-8772, 2011. http://dx.doi.org/10.1073/pnas.1019712108

[7] S. Liao and J. Wang, Global stability analysis of epidemiological modelsbased on Volterra-Lyapunov stable matrices, Chaos, Solitons and Fractals,45, 966-977, 2012. http://dx.doi.org/10.1016/j.chaos.2012.03.009

[8] Y. Kang and H. Z. Zhang, The study status of oral cholera vaccine, Chi-nese Frontier Health Quarantine, 28, 91-92, 2005.

[9] Jing’an Cui, Zhanmin Wu, and Xueyong Zhou, MathematicalAnalysis of a Cholera Model with Vaccination, Journal of Ap-plied Mathematics, vol. 2014, Article ID 324767, 16 pages, 2014.http://dx.doi.org/10.1155/2014/324767

[10] Khan M. A., Islam S., Arif M., and Haq Z., Transmission Model ofHepatitis B Virus with the Migration Effect, Bio. Res. Int., (2013).http://dx.doi.org/10.1155/2013/150681

[11] P. van den Driessche, and J. Watmough, Reproduction num-bers and sub-threshold endemic equilibria for compartmental mod-els of disease transmission, Math. Biosci., 180, 29-48, 2002.http://dx.doi.org/10.1016/s0025-5564(02)00108-6

[12] Khan M. A., Islam S., Khan S. A., Zaman G., Global Stability of Vector-Host Disease with Variable Population Size, Bio. Res. Int., Research In-ternational, (2013). http://dx.doi.org/10.1155/2013/710917

[13] Muhammad Altaf Khan, Muhammad Parvez, Saeed Islam, Ilyas Khan,Sharidan Shafie, Taza Gul, Mathematical analysis of typhoid model withsaturated incidence rate, Advanced Studies in Biology, Vol. 7, 2015, no.2, 65-78, http://dx.doi.org/10.12988/asb.2015.41059.

Dynamical behavior of cholera epidemic model 999

[14] Muhammad Altaf Khan, Saeed Islam, Sher Afzal Khan, Ilyas Khan,Sharidan Shafie, and Taza Gul, Prevention of Leptospirosis InfectedVector and Human Population by Multiple Control Variables, Ab-stract and Applied Analysis Volume 2014, Article ID 619035,9 pageshttp://dx.doi.org/10.1155/2014/619035.

[15] Khan, Muhammad Altaf, Syed Farasat Saddiq, SherAfzal Khan, SaeedIslam, and Farooq Ahmad, Application of Homotopy Perturbation Methodto an SIR Epidemic Model J. Appl. Environ. Biol. Sci 4, no. 3 (2014): 49-54.

Received: December 1, 2014; Publsihed: February 1, 2015

0.0

0.5

1.00.0

0.5

1.00.0

0.5

1.0

Figure 1: The plot represents R0 < 1, when Λ = 0.6, β1 = 0.04, β2 = 0.05, µ = 0.2, µ0 =0.01, φ = 0.01, d = 0.02, ξ = 0.01.

1000 Muhammad Altaf Khan et al.

0 5 10 15 200

20

40

60

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120

140

160

180

200

Time(day)

Pop

ulat

ion

Beh

avio

r of h

uman

Dynamical Behavior of human

S

I

R

Figure 2: The plot represents the population behavior of human.

Dynamical behavior of cholera epidemic model 1001

0 5 10 15 2010

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20

25

30

35

40

Time(day)

Pop

ulat

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Beh

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atho

gen

Dynamical Behavior of Pathogen

B

Figure 3: The plot represents the population behavior of Pathogen.

1002 Muhammad Altaf Khan et al.

0 5 10 15 200

20

40

60

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100

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Time(day)

Pop

ulat

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Beh

avio

r of h

uman

and

Pat

hoge

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Dynamical Behavior of human and Pathogen

S

I

R

B

Figure 4: The plot represents the population behavior of human with pathogen.